HANDBOOK OF PI AND PID CONTROLLER TUNING RULES 3rd Edition P575.TP.indd 1 3/20/09 5:16:12 PM This page intentionally left blank HANDBOOK OF PI AND PID CONTROLLER TUNING RULES 3rd Edition Aidan O’Dwyer Dublin Institute of Technology, Ireland ICP P575.TP.indd 2 Imperial College Press 3/20/09 5:16:18 PM Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. HANDBOOK OF PI AND PID CONTROLLER TUNING RULES (3rd Edition) Copyright © 2009 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-1-84816-242-6 ISBN-10 1-84816-242-1 Typeset by Stallion Press Email: enquiries@stallionpress.com Printed in Singapore. Steven - Hdbook of PI & PID (3rd).pmd 1 9/2/2009, 7:13 PM b720_FM 17-Mar-2009 FA Dedication Once more, this book is dedicated with love to Angela, Catherine and Fiona and to my parents, Sean and Lillian. b720_FM 17-Mar-2009 This page intentionally left blank FA b720_FM 17-Mar-2009 FA Preface Proportional integral (PI) and proportional integral derivative (PID) controllers have been at the heart of control engineering practice for over seven decades. However, in spite of this, the PID controller has not received much attention from the academic research community until the past two decades, when work by K.J. Åström, T. Hägglund and F.G. Shinskey, among others, sparked a revival of interest in the use of this ‘workhorse’ of controller implementation. There is strong evidence that PI and PID controllers remain poorly understood and, in particular, poorly tuned in many applications. It is clear that the many controller tuning rules proposed in the literature are not having an impact on industrial practice. One reason is that the tuning rules are not accessible, being scattered throughout the control literature; in addition, the notation used is not unified. The purpose of this book, now in its third edition, is to bring together and summarise, using a unified notation, tuning rules for PI and PID controllers. The author restricts the work to tuning rules that may be applied to the control of processes with time delays (dead times); in practice, this is not a significant restriction, as most process models have a time delay term. In this edition, the structure of the book has been modified from the previous edition, with controller tuning rules for non-self-regulating process models being summarised in a different chapter from those for self-regulating process models. It is the author’s belief that this book will be useful to control and instrument engineering practitioners and will be a useful reference for students and educators in universities and technical colleges. I wish to thank the School of Electrical Engineering Systems, Dublin Institute of Technology, for providing the facilities needed to complete the book. Finally, I am deeply grateful to my mother, Lillian, and my father, Sean, for their inspiration and support over many years and to my family, Angela, Catherine and Fiona, for their love and understanding. Aidan O’Dwyer vii b720_FM 17-Mar-2009 This page intentionally left blank FA b720_FM 17-Mar-2009 FA Contents Preface ........................................................................................................ vii 1. Introduction ........................................................................................................ 1.1 Preliminary Remarks ...................................................................................... 1.2 Structure of the Book...................................................................................... 1 1 2 2. Controller Architecture........................................................................................... 2.1 Introduction .................................................................................................. 2.2 Comments on the PID Controller Structures . ................................................ 2.3 Process Modelling .......................................................................................... 2.3.1 Self-regulating process models . ........................................................ 2.3.2 Non-self-regulating process models................................................... 2.4 Organisation of the Tuning Rules ................................................................... 4 4 11 12 12 14 16 3. Controller Tuning Rules for Self-Regulating Process Models ............................... 3.1 Delay Model ................................................................................................. 3.1.1 Ideal PI controller – Table 2 .............................................................. 3.1.2 Ideal PID controller – Table 3 .......................................................... 3.1.3 Ideal controller in series with a first order lag – Table 4 ................ 3.1.4 Classical controller – Table 5 ........................................................... 3.1.5 Generalised classical controller – Table 6 ........................................ 3.1.6 Two degree of freedom controller 1 – Table 7 .................................. 3.2 Delay Model with a Zero ............................................................................... 3.2.1 Ideal PI controller – Table 8 . ............................................................ 3.3 FOLPD Model ............................................................................................... 3.3.1 Ideal PI controller – Table 9 .............................................................. 3.3.2 Ideal PID controller – Table 10 ........................................................ 3.3.3 Ideal controller in series with a first order lag – Table 11 ................ 3.3.4 Controller with filtered derivative – Table 12.................................... 3.3.5 Classical controller – Table 13 ........................................................ 3.3.6 Generalised classical controller – Table 14 . ..................................... 3.3.7 Two degree of freedom controller 1 – Table 15 ............................. 3.3.8 Two degree of freedom controller 2 – Table 16 ............................. 3.3.9 Two degree of freedom controller 3 – Table 17 ............................. 18 18 18 23 24 25 26 27 28 28 30 30 78 118 122 134 149 152 168 170 ix b720_FM x 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 3.4 FOLPD Model with a Zero ............................................................................ 3.4.1 Ideal PI controller – Table 18 ............................................................ 3.4.2 Ideal controller in series with a first order lag – Table 19 ................ 3.5 SOSPD Model .............................................................................................. 3.5.1 Ideal PI controller – Table 20 .......................................................... 3.5.2 Ideal PID controller – Table 21 ........................................................ 3.5.3 Ideal controller in series with a first order lag – Table 22 ................ 3.5.4 Controller with filtered derivative – Table 23.................................... 3.5.5 Classical controller – Table 24 ........................................................ 3.5.6 Generalised classical controller – Table 25 . ..................................... 3.5.7 Two degree of freedom controller 1 – Table 26 ............................. 3.5.8 Two degree of freedom controller 3 – Table 27 ............................. 3.6 SOSPD Model with a Zero ............................................................................ 3.6.1 Ideal PI controller – Table 28 ......................................................... 3.6.2 Ideal PID controller – Table 29 ........................................................ 3.6.3 Ideal controller in series with a first order lag – Table 30 ............... 3.6.4 Controller with filtered derivative – Table 31.................................... 3.6.5 Classical controller – Table 32 ......................................................... 3.6.6 Generalised classical controller – Table 33 ....................................... 3.6.7 Two degree of freedom controller 1 – Table 34 .............................. 3.6.8 Two degree of freedom controller 3 – Table 35 .............................. 3.7 TOSPD Model .............................................................................................. 3.7.1 Ideal PI controller – Table 36 . .......................................................... 3.7.2 Ideal PID controller – Table 37 ......................................................... 3.7.3 Ideal controller in series with a first order lag – Table 38 ................ 3.7.4 Controller with filtered derivative – Table 39 .................................... 3.7.5 Two degree of freedom controller 1 – Table 40 ................................. 3.7.6 Two degree of freedom controller 3 – Table 41 . ............................... 3.8 Fifth Order System Plus Delay Model ........................................................... 3.8.1 Ideal PID controller – Table 42 ......................................................... 3.8.2 Controller with filtered derivative – Table 43 .................................... 3.8.3 Two degree of freedom controller 1 – Table 44 . ............................... 3.9 General Model . .............................................................................................. 3.9.1 Ideal PI controller – Table 45 ........................................................... 3.9.2 Ideal PID controller – Table 46 ......................................................... 3.9.3 Ideal controller in series with a first order lag – Table 47 ................ 3.9.4 Controller with filtered derivative – Table 48 .................................. 3.9.5 Two degree of freedom controller 1 – Table 49 . .............................. 3.10 Non-Model Specific ...................................................................................... 3.10.1 Ideal PI controller – Table 50 ............................................................ 3.10.2 Ideal PID controller – Table 51 ......................................................... 3.10.3 Ideal controller in series with a first order lag– Table 52 . ................ 3.10.4 Controller with filtered derivative – Table 53 ................................... 3.10.5 Classical controller – Table 54 ......................................................... 3.10.6 Generalised classical controller – Table 55 ...................................... 180 180 182 183 183 206 232 236 238 251 253 264 277 277 279 282 284 286 288 289 292 293 293 296 297 298 299 302 303 303 305 308 310 310 312 315 316 317 318 318 324 332 336 341 343 b720_FM 17-Mar-2009 FA Contents xi 3.10.7 Two degree of freedom controller 1 – Table 56 . .............................. 346 3.10.8 Two degree of freedom controller 3 – Table 57 . .............................. 349 4. Controller Tuning Rules for Non-Self-Regulating Process Models ....................... 4.1 IPD Model ..................................................................................................... 4.1.1 Ideal PI controller – Table 58 . .......................................................... 4.1.2 Ideal PID controller – Table 59 ...................................................... 4.1.3 Ideal controller in series with a first order lag – Table 60 ................ 4.1.4 Controller with filtered derivative – Table 61 .................................. 4.1.5 Classical controller – Table 62 ......................................................... 4.1.6 Generalised classical controller – Table 63 ..................................... 4.1.7 Two degree of freedom controller 1 – Table 64 . .............................. 4.1.8 Two degree of freedom controller 2 – Table 65 . .............................. 4.1.9 Two degree of freedom controller 3 – Table 66 . .............................. 4.2 IPD Model with a Zero ................................................................................. 4.2.1 Ideal PI controller – Table 67 . .......................................................... 4.3 FOLIPD Model .............................................................................................. 4.3.1 Ideal PI controller – Table 68 ......................................................... 4.3.2 Ideal PID controller – Table 69 ........................................................ 4.3.3 Ideal controller in series with a first order lag – Table 70 ................ 4.3.4 Controller with filtered derivative – Table 71 ................................... 4.3.5 Classical controller – Table 72 ......................................................... 4.3.6 Generalised classical controller – Table 73 ...................................... 4.3.7 Two degree of freedom controller 1 – Table 74 .............................. 4.3.8 Two degree of freedom controller 2 – Table 75 .............................. 4.3.9 Two degree of freedom controller 3 – Table 76 .............................. 4.4 FOLIPD Model with a Zero ........................................................................ 4.4.1 Ideal PID controller – Table 77 . ....................................................... 4.4.2 Ideal controller in series with a first order lag – Table 78 ............... 4.4.3 Classical controller – Table 79 ......................................................... 4.5 I 2 PD Model..................................................................................................... 4.5.1 Ideal PID controller – Table 80 . ....................................................... 4.5.2 Classical controller – Table 81 ........................................................ 4.5.3 Two degree of freedom controller 1 – Table 82 .............................. 4.5.4 Two degree of freedom controller 2 – Table 83 ................................ 4.5.5 Two degree of freedom controller 3 – Table 84 .............................. 4.6 SOSIPD Model ............................................................................................. 4.6.1 Ideal PI controller – Table 85 . .......................................................... 4.6.2 Two degree of freedom controller 1 – Table 86 . .............................. 4.7 SOSIPD Model with a Zero............................................................................ 4.7.1 Classical controller – Table 87 ......................................................... 4.8 TOSIPD Model............................................................................................... 4.8.1 Two degree of freedom controller 1 – Table 88 .............................. 350 350 350 359 364 366 368 371 372 378 381 383 383 385 385 388 392 394 395 397 399 416 418 420 420 422 423 424 424 425 426 427 429 430 430 431 436 436 437 437 b720_FM xii 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 4.9 General Model with Integrator ....................................................................... 4.9.1 Ideal PI controller – Table 89 . .......................................................... 4.9.2 Two degree of freedom controller 1 – Table 90 ................................ 4.10 Unstable FOLPD Model ................................................................................ 4.10.1 Ideal PI controller – Table 91 ............................................................ 4.10.2 Ideal PID controller – Table 92 ........................................................ 4.10.3 Ideal controller in series with a first order lag – Table 93 ............... 4.10.4 Classical controller – Table 94 ....................................................... 4.10.5 Generalised classical controller – Table 95 ....................................... 4.10.6 Two degree of freedom controller 1 – Table 96 ................................ 4.10.7 Two degree of freedom controller 2 – Table 97 .............................. 4.10.8 Two degree of freedom controller 3 – Table 98 .............................. 4.11 Unstable FOLPD Model with a Zero ........................................................... 4.11.1 Ideal PI controller – Table 99 ............................................................ 4.11.2 Ideal controller in series with a first order lag – Table 100 ............. 4.11.3 Generalised classical controller – Table 101 .................................... 4.11.4 Two degree of freedom controller 1 – Table 102 ............................ 4.12 Unstable SOSPD Model (one unstable pole) ................................................ 4.12.1 Ideal PI controller – Table 103 ........................................................ 4.12.2 Ideal PID controller – Table 104 . ..................................................... 4.12.3 Ideal controller in series with a first order lag – Table 105 .............. 4.12.4 Classical controller – Table 106 ........................................................ 4.12.5 Two degree of freedom controller 1 – Table 107 .............................. 4.12.6 Two degree of freedom controller 3 – Table 108 .............................. 4.13 Unstable SOSPD Model (two unstable poles) ............................................... 4.13.1 Ideal PID controller – Table 109 ...................................................... 4.13.2 Generalised classical controller – Table 110 . ................................... 4.13.3 Two degree of freedom controller 2 – Table 111 .............................. 4.14 Unstable SOSPD Model with a Zero .............................................................. 4.14.1 Ideal PI controller – Table 112 . ........................................................ 4.14.2 Ideal controller in series with a first order lag – Table 113 .............. 4.14.3 Generalised classical controller – Table 114 ................................... 4.14.4 Two degree of freedom controller 1 – Table 115 .............................. 4.14.5 Two degree of freedom controller 3 – Table 116 .............................. 438 438 439 440 440 447 455 458 462 463 473 475 480 480 481 483 484 486 486 488 490 491 497 503 506 506 508 509 511 511 513 516 518 520 5. Performance and Robustness Issues in the Compensation of FOLPD Processes with PI and PID Controllers ................................................................... 5.1 Introduction ................................................................................................... 5.2 The Analytical Determination of Gain and Phase Margin ............................. 5.2.1 PI tuning formulae ............................................................................... 5.2.2 PID tuning formulae ........................................................................... 5.3 The Analytical Determination of Maximum Sensitivity ................................ 5.4 Simulation Results .......................................................................................... 521 521 522 522 525 529 529 b720_FM 17-Mar-2009 FA Contents 5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins, for All Values of Delay................................................................................... 5.5.1 PI controller design.............................................................................. 5.5.1.1 Processes modelled in FOLPD form...................................... 5.5.1.2 Processes modelled in IPD form ........................................... 5.5.2 PID controller design .......................................................................... 5.5.2.1 Processes modelled in FOLPD form – classical controller.... 5.5.2.2 Processes modelled in SOSPD form – series controller......... 5.5.2.3 Processes modelled in SOSPD form with a negative zero – classical controller ...................................................... 5.5.3 PD controller design . .......................................................................... 5.6 Conclusions . .................................................................................................. xiii 534 534 534 536 539 539 541 542 542 543 Appendix 1 Glossary of Symbols and Abbreviations................................................. 544 Appendix 2 Some Further Details on Process Modelling........................................... 551 Bibliography ........................................................................................................ 565 Index ........................................................................................................ 599 This page intentionally left blank b720_Chapter-01 17-Mar-2009 FA Chapter 1 Introduction 1.1 Preliminary Remarks The ability of proportional integral (PI) and proportional integral derivative (PID) controllers to compensate most practical industrial processes has led to their wide acceptance in industrial applications. Koivo and Tanttu (1991), for example, suggest that there are perhaps 5–10% of control loops that cannot be controlled by single input, single output (SISO) PI or PID controllers; in particular, these controllers perform well for processes with benign dynamics and modest performance requirements (Hwang, 1993; Åström and Hägglund, 1995). It has been stated that 98% of control loops in the pulp and paper industries are controlled by SISO PI controllers (Bialkowski, 1996) and that in process control applications, more than 95% of the controllers are of PID type (Åström and Hägglund, 1995). PI or PID controller implementation has been recommended for the control of processes of low to medium order, with small time delays, when parameter setting must be done using tuning rules and when controller synthesis is performed either once or more often (Isermann, 1989). However, despite decades of development work, surveys indicating the state of the art of control in industrial practice report sobering results. For example, Ender (1993) states that in his testing of thousands of control loops in hundreds of plants, it has been found that more than 30% of installed controllers are operating in manual mode and 65% of loops operating in automatic mode produce less variance in manual than in automatic (i.e. the automatic controllers are poorly tuned). The situation does not appear to have improved more recently, as Van Overschee and De Moor (2000) report that 80% of PID controllers are badly tuned; 30% of PID controllers operate in manual with another 30% of the controlled loops increasing the short term variability of the process to be controlled (typically due to too strong integral action). The authors state that 25% 1 b720_Chapter-01 2 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules of all PID controller loops use default factory settings, implying that they have not been tuned at all. These and other surveys (well summarised by Yu, 1999, pp. 1–2) show that the determination of PI and PID controller tuning parameters is a vexing problem in many applications. The most direct way to set up controller parameters is the use of tuning rules; obviously, the wealth of information on this topic available in the literature has been poorly communicated to the industrial community. One reason is that this information is scattered in a variety of media, including journal papers, conference papers, websites and books for a period of over seventy years. The purpose of this book is to bring together, in summary form, the tuning rules for PI and PID controllers that have been developed to compensate SISO processes with time delay. 1.2 Structure of the Book Tuning rules are set out in the book in tabular form. This form allows the rules to be represented compactly. The tables have four or five columns, according to whether the controller considered is of PI or PID form, respectively. The first column in all cases details the author of the rule and other pertinent information. The final column in all cases is labelled ‘Comment’; this facilitates the inclusion of information about the tuning rule that may be useful in its application. The remaining columns detail the formulae for the controller parameters. Chapter 2 explores the range of PI and PID controller structures proposed in the literature. It is often forgotten that different manufacturers implement different versions of the PID controller algorithm (in particular); therefore, controller tuning rules that work well in one PID architecture may work poorly in another. This chapter also details the process models used to define the controller tuning rules. Chapters 3 and 4 detail, in tabular form, PI and PID controller tuning rules (and their variations), as applied to a wide variety of self-regulating process models and non-self-regulating process models, respectively. One-hundred-andsixteen such tables are provided altogether. To allow the reader to access data readily, the author has arranged that each table starts on its own page; each table is preceded by the controller used, together with a block diagram showing the unity feedback closed loop arrangement of the controller and process model. In Chapter 5, analytical calculations of the gain and phase margins of a large sample of PI and PID controller tuning rules are determined, when the process is modelled in first order lag plus time delay (FOLPD) form, at a range of ratios b720_Chapter-01 17-Mar-2009 Chapter 1: Introduction FA 3 of time delay to time constant of the process model. Results are given in graphical form. An important feature of the book is the unified notation used for the tuning rules; a glossary of the symbols used is provided in Appendix 1. Appendix 2 outlines the range of methods used to determine process model parameters; this information is presented in summary form, as this topic could provide data for a book in itself. However, sufficient information, together with references, is provided for the interested reader. Finally, a comprehensive reference list is provided. In particular, the author would like to recommend the contributions by McMillan (1994), Åström and Hägglund (1995), (2006), Shinskey (1994), (1996), Tan et al. (1999a), Yu (1999), Lelic and Gajic (2000) and Ang et al. (2005) to the interested reader, which treat comprehensively the wider perspective of PID controller design and application. b720_Chapter-02 17-Mar-2009 FA Chapter 2 Controller Architecture 2.1 Introduction The ideal continuous time domain PID controller for a SISO process is expressed in the Laplace domain as follows: U (s ) = G c (s ) E (s ) (2.1) with G c (s) = K c (1 + 1 + Td s) Ti s (2.2) and with K c = proportional gain, Ti = integral time constant and Td = derivative time constant. If Ti = ∞ and Td = 0 (i.e. P control), then it is clear that the closed loop measured value y will always be less than the desired value r (for processes without an integrator term, as a positive error is necessary to keep the measured value constant, and less than the desired value). The introduction of integral action facilitates the achievement of equality between the measured value and the desired value, as a constant error produces an increasing controller output. The introduction of derivative action means that changes in the desired value may be anticipated, and thus an appropriate correction may be added prior to the actual change. Thus, in simplified terms, the PID controller allows contributions from present controller inputs, past controller inputs and future controller inputs. Many variations of the PID controller structure have been proposed (indeed, the PI controller structure is itself a subset of the PID controller structure). As Tan et al. (1999a) suggest, one important reason for the non-standard structures is due to the transition of the controllers from pneumatic implementation through electronic implementation to the present microprocessor implementation. A substantial number of the variations in the controller structures used may be 4 b720_Chapter-02 17-Mar-2009 Chapter 2: Controller Architecture FA 5 summarised by controller structure ‘supersets’. In this book, nine such supersets are specified, which allows a sensible restriction on the number of tables that need to be detailed. The controller structures specified are detailed below. 1. The ideal PI controller structure: 1 G c (s) = K c 1 + Ti s (2.3) 2. The ideal PID controller structure: 1 G c (s) = K c 1 + + Td s Ti s (2.4) This controller structure has also been labelled the ‘non-interacting’ controller (McMillan, 1994), the ‘ISA’ algorithm (Gerry and Hansen, 1987) or the ‘parallel non-interacting’ controller (Visioli, 2001). A variation of the controller is labelled the ‘parallel’ controller structure (McMillan, 1994). G c (s ) = K c + 1 + Td s Ti s (2.5) This variation has also been labelled the ‘ideal parallel’, ‘non-interacting’, ‘independent’ or ‘gain independent’ algorithm. The ideal PID controller structure is used in the following products: (a) Allen Bradley PLC5 product (McMillan, 1994) (b) Bailey FC19 PID algorithm (EZYtune, 2003) (c) Fanuc Series 90–30 and 90–70 Independent Form PID algorithm (EZYtune, 2003) (d) Intellution FIX products (McMillan, 1994) (e) Honeywell TDC3000 Process Manager Type A, non-interactive mode product (ISMC, 1999) (f) Leeds and Northrup Electromax 5 product (Åström and Hägglund, 1988) (g) Yokogawa Field Control Station (FCS) PID algorithm (EZYtune, 2003). b720_Chapter-02 6 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 3. Ideal controller in series with a first order lag: 1 1 + Td s G c (s) = K c 1 + Ti s Tf s + 1 (2.6) 4. Controller with filtered derivative: Td s 1 G c (s ) = K c 1 + + Ti s 1 + s Td N (2.7) This structure is used in the following products: (a) Bailey Net 90 PID error input product with N = 10 (McMillan, 1994) and FC156 Independent Form PID algorithm (EZYtune, 2003) (b) Concept PIDP1 and PID1 PID algorithms (EZYtune, 2003) (c) Fischer and Porter DCU 3200 CON PID algorithm with N = 8 (EZYtune, 2003) (d) Foxboro EXACT I/A series PIDA product (in which it is an option labelled ideal PID) (Foxboro, 1994) (e) Hartmann and Braun Freelance 2000 PID algorithm (EZYtune, 2003) (f) Modicon 984 product with 2 ≤ N ≤ 30 (McMillan, 1994; EZYtune, 2003) (g) Siemens Teleperm/PSC7 ContC/PCS7 CTRL PID products with N = 10 (ISMC, 1999) and the S7 FB41 CONT_C PID product (EZYtune, 2003). 5. Classical controller: This controller is also labelled the ‘cascade’ controller (Witt and Waggoner, 1990), the ‘interacting’ or ‘series’ controller (Poulin and Pomerleau, 1996), the ‘interactive’ controller (Tsang and Rad, 1995), the ‘rate-before-reset’ controller (Smith and Corripio, 1997), the ‘analog’ controller (St. Clair, 2000) or the ‘commercial’ controller (Luyben, 2001). 1 1 + sTd G c (s) = K c 1 + Ti s 1 + s Td N (2.8) b720_Chapter-02 17-Mar-2009 FA Chapter 2: Controller Architecture 7 The structure is used in the following products: (a) Honeywell TDC Basic/Extended/Multifunction Types A and B products with N = 8 (McMillan, 1994) (b) Toshiba TOSDIC 200 product with 3.33 ≤ N ≤ 10 (McMillan, 1994) (c) Foxboro EXACT Model 761 product with N = 10 (McMillan, 1994) (d) Honeywell UDC6000 product with N = 8 (Åström and Hägglund, 1995) (e) Honeywell TDC3000 Process Manager product – Type A, interactive mode with N = 10 (ISMC, 1999) (f) Honeywell TDC3000 Universal, Multifunction and Advanced Multifunction products with N = 8 (ISMC, 1999) (g) Foxboro EXACT I/A Series PIDA product (in which it is an option labelled series PID) (Foxboro, 1994). A subset of the classical PID controller is the so-called ‘series’ controller structure, also labelled the ‘interacting’ controller or the ‘analog algorithm’ (McMillan, 1994) or the ‘dependent’ controller (EZYtune, 2003). 1 (1 + sTd ) G c (s) = K c 1 + Ti s (2.9) The structure is used in the following products: (a) Turnbull TCS6000 series product (McMillan, 1994) (b) Alfa-Laval Automation ECA400 product (Åström and Hägglund, 1995) (c) Foxboro EXACT 760/761 product (Åström and Hägglund, 1995). A further related structure is one labelled the ‘interacting’ controller (Fertik, 1975). Td s 1 1 + G c (s) = K c 1 + Ti s 1 + s Td N The structure is used in the following products: (a) Bailey FC156 Classical Form PID product (EZYtune, 2003) (b) Fischer and Porter DCI 4000 PID algorithm (EZYtune, 2003). (2.10) b720_Chapter-02 8 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 6. Generalised classical controller: 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N (2.11) 7. Two degree of freedom controller 1: [1 − β]Td s R (s) − K 1 + 1 + Td s Y(s) (2.12) 1 + U (s) = K c [1 − α] + c Td T Ti s Ti s 1 1+ d s + s N N This controller is also labelled the ‘m-PID’ controller (Huang et al., 2000), the ‘ISA-PID’ controller (Leva and Colombo, 2001) and the ‘P-I-PD (only P is DOF) incomplete 2DOF algorithm’ (Mizutani and Hiroi, 1991). This structure is used in the following products: (a) Bailey Net 90 PID PV and SP product (McMillan, 1994) (b) Yokogawa SLPC products with α = −1 , β = −1 , N = 10 (McMillan, 1994) (c) Omron E5CK digital controller with β = 1 and N = 3 (ISMC, 1999). Notable subsets of this controller structure are: Td s 1 1 R (s) − K c 1 + • U (s) = K c 1 + Y(s) + Td Ti s Ti s 1+ s N (2.13) which is used in the following products: (a) Allen Bradley SLC5/02, SLC5/03, SLC5/04, PLC5 and Logix5550 products (EZYtune, 2003) (b) Modcomp product with N = 10 (McMillan, 1994). b720_Chapter-02 17-Mar-2009 FA Chapter 2: Controller Architecture 9 1 1 R (s) − K c 1 + • U (s) = K c 1 + + Td s Y (s) Ti s Ti s (2.14) Also labelled the ‘PI+D’ controller structure (Chen, 1996) or the dependent, ideal, non-interacting controller structure (Cooper, 2006a), it is used in the following products: (a) ABB 53SL6000 product (ABB, 2001) (b) Genesis product (McMillan, 1994) (c) Honeywell TDC3000 Process Manager Type B, non-interactive mode product (ISMC, 1999) (d) Square D PIDR PID product (EZYtune, 2003). 1 1 R (s) − K c 1 + • U (s) = K c + Td s Y(s) Ti s Ti s (2.15) which is used in the following products: (a) Toshiba AdTune TOSDIC 211D8 product (Shigemasa et al., 1987) (b) Honeywell TDC3000 Process Manager Type C non-interactive mode product (ISMC, 1999). 8. Two degree of freedom controller 2: T s 1 d 1 + b f 1s (F(s)R (s) − Y(s) ) − K Y(s) , + U(s) = K c 1 + 0 T 1 + a f 1s Ti s 1+ d s N 1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4 F(s) = (2.16) 1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4 b720_Chapter-02 10 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 9. Two degree of freedom controller 3: U (s) = F1 (s)R (s) − F2 (s)Y(s) , ( ) 1 − β T s 1 + b f 1s [1 − χ] + δ + d , F1 (s) = K c [1 − α ] + Td 1 + a f 1s + a f 2 s 2 Ti s 1 + Ti s 1+ s N 1+ b f 2s [ 1 − φ] [1 − ϕ]Td s [ ] + F2 (s) = K c 1 − ε + T Ti s 1 + a f 3s + a f 4 s 2 1+ d s N K (b + b f 4 s ) − 0 f3 (2.17) 1 + a f 5s Notable subsets of this controller structure are: Td s 1 R (s) − 1 + • U (s) = K c 1 + Y(s) Ti s 1 + (Td N )s (2.18) Also labelled the ‘reset-feedback’ controller structure (Huang et al., 1996), it is used in the following products: (a) Bailey Fisher and Porter 53SL6000 and 53MC5000 products (ISMC, 1999) (b) Moore Model 352 Single-Loop Controller product (Wade, 1994). 1 + Td s 1 R (s) − • U(s) = K c 1 + Y(s) 1 + (Td N )s Ti s (2.19) Also labelled the ‘industrial controller’ structure (Kaya and Scheib, 1988), it is used in the following products: (a) Fisher-Rosemount Provox product with N = 8 (ISMC, 1999; McMillan, 1994) (b) Foxboro Model 761 product with N = 10 (McMillan, 1994) (c) Fischer-Porter Micro DCI product (McMillan, 1994) b720_Chapter-02 17-Mar-2009 Chapter 2: Controller Architecture FA 11 (d) Moore Products Type 352 controller with 1 ≤ N ≤ 30 (McMillan, 1994) (e) SATT Instruments EAC400 product with N = 8.33 (McMillan, 1994) (f) Taylor Mod 30 ESPO product with N = 16.7 (McMillan, 1994) (g) Honeywell TDC3000 Process Manager Type B, interactive mode product with N = 10 (ISMC, 1999). Kc 1 1 + sTd R (s) − K c 1 + Y (s) • U (s) = Tis Tis 1 + sTd N (2.20) which is used in the following products: (a) Honeywell TDC3000 Process Manager Type C, interactive mode product with N = 10 (ISMC, 1999) (b) Honeywell TDC3000 Universal, Multifunction and Advanced Multifunction products with N = 8 (ISMC, 1999). 2.2 Comments on the PID Controller Structures In some cases, one controller structure may be transformed into another; clearly, the ideal and parallel controller structures (Equations 2.4 and 2.5) are very closely related. It is shown by McMillan (1994), among others, that the parameters of the ideal PID controller may be worked out from the parameters of the series PID controller, and vice versa. The ideal PID controller is given in Equation (2.22) and the series PID controller is given in equation (2.23). 1 G cp (s) = K cp 1 + + Tdp s Tip s (2.22) 1 (1 + sTds ) G cs (s) = K cs 1 + Tis s (2.23) Then, it may be shown that T Tis Tds . K cp = 1 + ds K cs , Tip = (Tis + Tds ) , Tdp = Tis Tis + Tds b720_Chapter-02 12 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules Similarly, it may be shown that, provided Tip > 4Tdp , Tdp T , T = 0.5T 1 + 1 − 4 dp , K cs = 0.5K cp 1 + 1 − 4 is ip Tip Tip Tdp . Tds = 0.5Tdp 1 − 1 − 4 Tip Åström and Hägglund (1996) point out that the ideal controller admits complex zeroes and is thus a more flexible controller structure than the series controller, which has real zeroes; however, in the frequency domain, the series controller has the interesting interpretation that the zeroes of the closed loop transfer function are the inverse values of Tis and Tds . O’Dwyer (2001b) developed a comprehensive set of tuning rules for the series PID controller, based on the ideal PID controller; as these are not strictly original tuning rules, only representative examples are included in the relevant tables. In a similar manner, it is straightforward to show that controller structures (2.6), (2.7) and (2.8) are subsets of a more general controller structure, and controller parameters may be transformed readily from one structure to another. 2.3 Process Modelling Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables (see Chapters 3 and 4). These modelling strategies are outlined in Appendix 2. Process models may be classified as selfregulating or non-self-regulating, and the models that fall into these categories are now detailed. 2.3.1 Self-regulating process models 1. 2. Delay model: G m (s) = K m e −sτm Delay model with a zero: G m (s) = K m (1 + Tm 3 s)e − sτ m or G m (s) = K m (1 − Tm 4 s)e − sτ m b720_Chapter-02 17-Mar-2009 FA Chapter 2: Controller Architecture 13 3. First order lag plus time delay (FOLPD) model: G m (s) = 4. FOLPD model with a zero: K m e −sτ m 1 + sTm K m (1 − sTm 4 )e − sτ m K m (1 + sTm 3 )e −sτm or G m (s) = G m (s ) = 1 + sTm1 1 + sTm1 5. Second order system plus time delay (SOSPD) model: K m e − sτ m or G m (s) = G m (s) = 2 (1 + Tm1s )(1 + Tm 2 s ) Tm1 s 2 + 2ξ m Tm1s + 1 K m e − sτ m 6. SOSPD model with a zero: G m (s ) = G m (s) = 7. K m (1 + sTm 3 )e − sτm 2 Tm1 s 2 + 2ξ m Tm1s + 1 or G m (s) = K m (1 − sTm 4 )e −sτ m K m (1 − sTm 4 )e − sτ m or G m (s) = 2 (1 + sTm1 )(1 + sTm 2 ) Tm1 s 2 + 2ξ m Tm1s + 1 Third order system plus time delay (TOSPD) model: G m (s ) = K m (1 + b1s + b 2 s 2 + b 3s 3 )e − sτ m or 1 + a 1s + a 2 s 2 + a 3s 3 ( G m (s ) = ) K m e − sτ m or (1 + sTm1 )(1 + sTm 2 )(1 + sTm3 ) G m (s) = K m (Tm 3 s + 1)e −sτ m (T s + 2ξ T s + 1)(T s + 1) m1 8. K m (1 + sTm 3 )e −sτm (1 + sTm1 )(1 + sTm 2 ) 2 2 m m1 m2 Fifth order system plus delay model: G m (s ) = K m (1 + b1s + b 2 s 2 + b 3s 3 + b 4 s 4 + b s s 5 )e − sτm 1 + a 1s + a 2 s 2 + a 3s 3 + a 4 s 4 + a 5s 5 ( ) b720_Chapter-02 14 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 9. General model. 10. Non-model specific. Corresponding tuning rules may also apply to non-selfregulating processes. 2.3.2 Non-self-regulating process models 1. Integral plus time delay (IPD) model: G m (s) = 2. IPD model with a zero: G m (s) = 3. K m e −sτ m s K m (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτ m or G m (s) = m s s First order lag plus integral plus time delay (FOLIPD) model: K m e −sτ m G m (s) = s(1 + sTm ) 4. FOLIPD model with a zero: G m (s) = 5. 6. 7. 8. K m (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτ m or G m (s) = m s(1 + sTm1 ) s(1 + sTm1 ) Integral squared plus time delay ( I 2 PD ) model: G m (s) = K m e − sτ m s2 Second order system plus integral plus time delay (SOSIPD) model: K m e −sτ m K m e −sτ m G ( s ) or = G m (s ) = m 2 2 s(1 + sTm1 ) s(Tm1 s 2 + 2ξ m Tm1s + 1) K m (1 + sTm 3 )e − sτ m s(1 + sTm1 )(1 + sTm 2 ) Third order system plus integral plus time delay (TOSIPD) model: SOSIPD model with a zero: G m (s) = G m (s) = K m e − sτ m K m e − sτ m or G m (s) = 3 s(1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 ) s(1 + sTm1 ) b720_Chapter-02 17-Mar-2009 FA Chapter 2: Controller Architecture 9. 15 General model with integrator: G m (s) = K m e − sτ m s(1 + sTm1 ) n or ∏ (T s + 1)∏ (T K G (s ) = s ∏ (T s + 1)∏ (T m m i i m1i i m 3i i m 2i m 4i ) e s + 2ξ T s + 1) 2 2 s + 2ξ m ni Tm 2i s + 1 2 2 mdi −sτ m m 4i K m e −sτm Tm s − 1 11. Unstable FOLPD model with a zero: 10. Unstable FOLPD model: G m (s) = G m (s) = K m (1 + sTm 3 )e −sτm K (1 − sTm 4 )e − sτ m or G m (s) = m Tm s − 1 Tm1s − 1 12. Unstable SOSPD model (one unstable pole): G m (s ) = K m e − sτm (Tm1s − 1)(1 + sTm 2 ) G m (s ) = K m e − sτ m (Tm1s − 1)(Tm 2 s − 1) 13. Unstable SOSPD model (two unstable poles): 14. Unstable SOSPD model with a zero: G m (s) = K m (1 + Tm 3 s )e −sτ m 2 Tm1 s 2 − 2ξ m Tm1s + 1 or K m (1 + Tm 3 s )e − sτ m 2 − Tm1 s 2 + 2ξ m Tm1s + 1 G m (s ) = or K m (1 − sTm 4 )e − sτ m 2 Tm1 s 2 − 2ξ m Tm1s + 1 b720_Chapter-02 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 16 Table 1 shows the number of tuning rules defined for each PI/PID controller structure and types of process model. The following data is key to the process model type: Model 1: Stable FOLPD model (with or without a zero) Model 2: Stable SOSPD model (with or without a zero) Model 3: Other stable models Model 4: Non-model specific Model 5: Models with an integrator Model 6: Open loop unstable models Table 1: PI/PID controller structure and tuning rules – a summary Process model Controller Equation (2.3) (2.4) (2.6) (2.7) (2.8) (2.11) (2.12) (2.16) (2.17) Total 1 2 3 4 5 6 Total 261 140 20 36 74 7 74 7 28 649 63 82 23 9 53 7 36 3 15 291 58 20 3 5 1 1 14 0 1 103 59 63 9 11 12 4 9 0 2 169 90 66 16 6 26 5 106 16 8 339 32 35 17 0 17 6 37 8 30 182 563 406 88 67 183 30 276 34 84 1731 Table 1 shows that for the tuning rules defined, 60% are based on a selfregulating (or stable) model, 10% are non-model specific, with the remaining 30% based on a non-self-regulating model. 2.4 Organisation of the Tuning Rules The tuning rules are organised in tabular form in Chapters 3 and 4. Within each table, the tuning rules are classified further; the main subdivisions made are as follows: (i) Tuning rules based on a measured step response (also called process reaction curve methods). (ii) Tuning rules based on minimising an appropriate performance criterion, either for optimum regulator or optimum servo action. b720_Chapter-02 17-Mar-2009 Chapter 2: Controller Architecture FA 17 (iii) Tuning rules that give a specified closed loop response (direct synthesis tuning rules). Such rules may be defined by specifying the desired poles of the closed loop response, for instance, though more generally, the desired closed loop transfer function may be specified. The definition may be expanded to cover techniques that allow the achievement of a specified gain margin and/or phase margin. (iv) Robust tuning rules, with an explicit robust stability and robust performance criterion built into the design process. (v) Tuning rules based on recording appropriate parameters at the ultimate frequency (also called ultimate cycling methods). (vi) Other tuning rules, such as tuning rules that depend on the proportional gain required to achieve a quarter decay ratio or to achieve magnitude and frequency information at a particular phase lag. Some tuning rules could be considered to belong to more than one subdivision, so the subdivisions cannot be considered to be mutually exclusive; nevertheless, they provide a convenient way to classify the rules. All symbols used in the tables are defined in Appendix 1. b720_Chapter-03 17-Mar-2009 FA Chapter 3 Controller Tuning Rules for Self-Regulating Process Models 3.1 Delay Model G m (s) = K m e − sτm 1 3.1.1 Ideal PI controller G c (s) = K c 1 + Ti s R(s) + U(s) 1 K c 1 + Ti s E(s) − K m e −sτ Y(s) m Table 2: PI controller tuning rules – G m (s) = K m e −sτ m Rule 1 Callender et al. (1935/6). Model: Method 1 Kc Ti 0.568 K m τ m Process reaction 3.64τ m 2 0.65 K m τ m 2.6τ m 3 0.79 K m τ m 3.95τ m 4 0.95 K m τ m 3.3τ m 1 Decay ratio = 0.015; period of decaying oscillation = 10.5τ m . 2 Decay ratio = 0.1; period of decaying oscillation = 8τ m . 3 Decay ratio = 0.12; period of decaying oscillation = 6.28τ m . 4 Decay ratio = 0.33; period of decaying oscillation = 5.56τ m . 18 Comment Representative results deduced from graphs. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Comment Two constraints method – Wolfe (1951). Model: Method 1 0.2 K mτm 0.3τ m Decay ratio is as small as possible. Minimum ISE – Haalman (1965). Model: Method 1 Minimum error: step load change – Gerry (1998). Minimum IAE – Gerry and Hansen (1987). Minimum IAE – Shinskey (1988), p. 123. Minimum IAE – Shinskey (1994). Model: Method 1 Minimum IAE – Shinskey (1988), p. 148. Minimum IAE – Shinskey (1996), p. 167. Minimum IAE – Huang and Jeng (2002). Minimum ISE – Keviczky and Csáki (1973). Model: Method 1 Fertik (1975). Model: Method 1 Åström and Hägglund (2000). Model: Method 1 Minimum error integral (regulator mode). Minimum performance index: regulator tuning undefined 1.5τ m G c (s) = 1 Tis M s = 1.9 ; A m = 2.36 ; φ m = 50 0 . 0.3 Km 0.42τ m Model: Method 1 0.345 Km 0.455τ m Model: Method 1 0.43 Km 0.5τ m Model: Method 1 0. 4 K m 0.5τ m p. 67. undefined 1.6K m τ m G c (s) = 1 Tis ; p. 63. 0.4255 Km 0.25Tu Model: Method 1 0. 4 K u 0.25Tu Model: Method 1 Minimum performance index: servo tuning Model: Method 1 0.36 K m 0.47τ m Minimum IAE = 1.377 τ m . A m = 2 ; φ m = 60 0 . undefined 1.33τ m G c (s) = 1 Tis ; Minimum ISE = 1.53τ m ; K m = 1 . 0.5 0.635τ m Km = 1 Minimum performance index: other tuning 0.35 K m 0.4 τ m K c , Ti values deduced from graph. K c Ti maximised 0.25 0.35τ m Km subject to M max = 2 . 19 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 20 Rule Kc Ti Comment Skogestad (2001). Model: Method 1 Skogestad (2003). Model: Method 1 Skogestad (2001). Model: Method 1 0.16 K m 0.340τ m M max = 1.4 0.200 K m 0.318τ m M max = 1.6 0.26 K m 0.306τ m M max = 2.0 x1 K m x 2τm Åström and Hägglund (2004). Model: Method 1 x1 Kc Ti undefined 2.70K m τ m undefined 2.50K m τ m undefined x 2τm 1 x2 OS x2 0.64 105% x1 1.00 x1 OS x1 5% 0.72 0.68 Kc = 2 Kc = ( Stochastic disturbance. Critically damped dominant pole. Damping factor (dominant pole) = 0.6. G c (s) = 1 Tis Coefficient values: OS OS x2 x2 OS 17% 2 4% Coefficient values: OS OS x1 x1 OS 50% 1.5 2τ m 0.75 Kc Ti 15% 0.78 20% Model: Method 1 0.223607 K m 0.309017 τm Model: Method 1 Kuwata (1987). 1 2 ) ( 10% ) e − ωd τ m τ 1 − 0.5e − ωd τ m , Ti = − ωmτ 1 − 0.5e − ωd τ m . Km e d m ( M max 0.400 1.1 0.211 0.342 1.6 0.389 1.2 0.227 0.334 1.7 0.376 1.3 0.241 0.326 1.8 0.363 1.4 0.254 0.320 1.9 0.352 1.5 0.264 0.314 2.0 Direct synthesis: time domain criteria Step disturbance. 0. 5 K m 0.5τ m Bryant et al. (1973). Model: Method 1; G c (s) = 1 Tis Nomura et al. (1993). x2 0.057 0.103 0.139 0.168 0.191 Van der Grinten (1963). Model: Method 1 Keviczky and Csáki (1973). Model: Method 1; representative coefficient values estimated from graphs; K m = 1 . Coefficient values: M max x1 x2 0.4τm K m 1 − 1 − 0.267 τm ) , T = 1.25τ 1 − 0.267τ − 0.25τ . i m m m b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kamimura et al. (1994), Manum (2005). Model: Method 1; G c (s) = 1 Tis . Åström and Hägglund (1995), pp. 187–189. Model: Method 1 Kc Ti Comment undefined 2K m τ m OS = 10% Also given by Schlegel (1998); M s = 1 . undefined 2.6667 K m τ m undefined 2.3529 K m τ m 0.305 K m 0.279τm ‘Quick’ servo response; ‘negligible’ overshoot. ξ = 0.3 0.273 K m 0.285τm ξ = 0.4 OS = 0% 0.218 K m 0.288τm ξ = 0.6 0.174 K m 0.276τm ξ = 0.8 0.154 K m 0.265τm ξ = 0.9 Åström and Hägglund (1995), pp. 187–189, Åström and Hägglund (2006), pp. 185–186. Model: Method 1 0.388 K m 0.258τm ξ = 0.1 0.343 K m 0.270τm ξ = 0.2 0.244 K m 0.288τm ξ = 0.5 0.195 K m 0.284τm ξ = 0.707 0.135 K m 0.250τm ξ = 1.0 Hansen (2000). 0. 2 K m 0.3τ m Model: Method 1 Buckley (1964). Model: Method 1; M max = 2dB . Schlegel (1998). Model: Method 1 Hägglund and Åström (2004). Model: Method 1 Åström and Hägglund (2006), p. 243. Direct synthesis: frequency domain criteria undefined 1.45K m τ m G c (s) = 1 Tis 0.075 K m 0.1τ m 0.53 K m τm 0.55 K m 1.59 τ m 0.318 K m 0.405τm 0.15 K m 0.35τ m 0.15 K u 0.17Tu 0.1677 K m 0.363τm Representative results. Ms = 1 M s = 1.4 Model: Method 1; M s = M t = 1. 4 . Robust Bequette (2003), p. 300. τm K m (2λ + τ m ) 0.5τ m Skogestad (2003). Model: Method 1 0.294 K m 0.5τ m Model: Method 1; λ > τm (Yu, 2006, p. 19). IMC based rule. 21 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 22 Rule Kc Ti Comment 1 0.5τm K m 0.5τm + λ 0.5τm Model: Method 1 undefined x1K m τm G c (s) = 1 Tis ; Skogestad (2003). 0.25 K m Other tuning 0.333τ m Model: Method 1 McMillan (2005), p. 35. 0.25 K m 0.25τ m Model: Method 1 Yu (2006), p. 18. 0.30K u Ultimate cycle 0.25Tu Model: Method 1 Åström and Hägglund (2006), p. 189. Urrea et al. (2006). Model: Method 1 3 17-Mar-2009 3 1.5 ≤ x1 ≤ 2 . λ = Tm (aggressive tuning). λ = 3Tm (robust tuning). b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 1 + Td s 3.1.2 Ideal PID controller G c (s) = K c 1 + Ti s R(s) E(s) − + 1 K c 1 + + Td s T s i Y(s) U(s) K me − sτ m Table 3: PID controller tuning rules – delay model G m (s) = K m e − sτ m Rule Callender et al. (1935/6). Model: Method 1 Ream (1954). Model: Method 1 Nomura et al. (1993). Model: Method 1 Kc Ti Td Comment 0.749 K m τm Process reaction 2.734 τ m 0.303τ m 0.2269 K m 0.3109τm 0.0264τm ‘Butterworth’. 0.2635 K m 0.3610τm 0.1911τm 0.3548 K m 0.4860τm 0.2735τm ‘Minimum ITAE’. ‘Binomial’. 0.511 K m 0.511τm 0.1247 τm Model: Method 1 0.40 K m 0.46τm 0.149τm Regulator. Representative results deduced 2 1.31τ m 0.303τ m 1.24 K m τm from graphs. Direct synthesis: time domain criteria Decay ratio 0.61 K m 0.491τm 0.015τm ≈ 33% . ‘Kitamori’. 0.230 K m 0.315τm 0.0086τm 1 Robust Gong et al. (1998). Wang (2003), pp. 15–17. Model: Method 1 Servo; ±10% overshoot. Deduced from graphs; robust to ±25% uncertainty in process model parameters. 0.29 K m 0.40τm 0.096τm 1 Decay ratio = 0.015; period of decaying oscillation = 10.5τ m . 2 Decay ratio = 0.12; period of decaying oscillation = 6.28τ m . 23 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 24 3.1.3 Ideal controller in series with a first order lag 1 1 G c (s) = K c 1 + + Td s Tf s + 1 Ti s R(s) + E(s) − 1 K c 1 + + Td s Ti s 1 1 + Tf s Y(s) U(s) K m e −sτ m Table 4: PID controller tuning rules – delay model G m (s) = K m e −sτ m Rule Kc Ti Td Comment Robust Bequette (2003), p. 300. Model: Method 1 τm K m (4λ + τ m ) 0.5τ m 0.167 τ m λ > 1.7 τm (Yu, 2006, p. 19). Tf = 2 2λ − 0.167τ m 4λ + τ m 2 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 1 1 + Td s 3.1.4 Classical controller G c (s) = K c 1 + Ti s 1 + Td s N E(s) R(s) − + 1 1 + Td s U(s) K c 1 + Td Ti s s 1 + N Y(s) K m e − sτ m Table 5: PID controller tuning rules – delay model G m (s) = K m e −sτ m Rule Kc Ti Td Comment Process reaction Hartree et al. (1937). Model: Method 1 1 2 0.70 K mτm 2.66τ m τm 0.78 K mτm 2.97 τ m 0.5τ m 1 N = 2. Decay ratio = 0.15; period of decaying oscillation = 4.49 τ m . 2 N = 2. Decay ratio = 0.042; period of decaying oscillation = 5.03τ m . Representative results. 25 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 26 3.1.5 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N R(s) E(s) + − 2 T s 1 d b f 0 + b f 1s + b f 2 s K c 1 + + 2 Ti s T 1 + a f 1s + a f 2 s 1+ d s N Rule Yu (2006), p. 18. Model: Method 1 U(s) K m e − sτ Y(s) m Table 6: PID controller tuning rules – delay model K me −sτ m Comment Kc Ti Td 0.30K u Ultimate cycle 0 0.23Tu b f 0 = 0 , b f 1 = 0.12Tu , b f 2 = 0 , a f 1 = 0.012Tu , a f 2 = 0 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 27 3.1.6 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N E(s) R(s) − + T s 1 d K c 1 + + Ti s Td + 1+ s N − U(s) Y(s) K me − sτ m Table 7: PID controller tuning rules – delay model G m (s) = K m e −sτ m Rule Kc Kuwata (1987). Model: Method 1 Nomura et al. (1993). Model: Method 1 1 Kc = ( 1 Ti Kc 0.2 K m Direct synthesis: time domain criteria 0.278τm 0 1.667 τm − 4.167 0.1 K m 0.278τm 1 Km 1.667 τm − 2.5 0.8τm K m 1 − 1 − 0.267 τm ). Comment Td 0 α =1 α = 1 ; also given by Kuwata (1987). b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 28 3.2 Delay Model with a Zero G m (s) = K m (1 + Tm 3 s)e −sτ m or G m (s) = K m (1 − Tm 4 s)e −sτ m 1 3.2.1 Ideal PI controller G c (s) = K c 1 + Ti s E(s) R(s) + 1 K c 1 + T is − Y(s) U(s) Model Table 8: PI controller tuning rules – delay model with a zero G m (s) = K m (1 + Tm 3 s)e −sτ m or G m (s) = K m (1 − Tm 4s)e −sτ m Rule Kc 2 2 ( T = 0.424τ 1 + 4.712(T ). τ ). Ti = 0.637 τm 1 + 3.142 Tm 4 τm i m Comment Direct synthesis: time domain criteria 1 ‘Smaller’ τm Tm 4 . Ti undefined Chidambaram (2002), p. 157. Model: Method 1; G c (s) = 1 Tis . 1 Ti m4 2 2 m Ti ‘Larger’ τm Tm 4 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Jyothi et al. (2001). Model: Method 1; G c (s) = 1 Tis . Jyothi et al. (2001). Model: Method 1; G c (s) = 1 Tis . Kc Ti Comment Direct synthesis: frequency domain criteria 3 τm Tm3 < 1 Ti Ti τm Tm3 > 1 5 Ti 0 < τm Tm 3 < 1 6 Ti 0 < τm Tm 3 < 10 4 undefined 1.5K m (τ m − Tm 3 ) 2K m Tm 4 undefined 3 Ti = 0.64K m τm 1 + 9.870(Tm3 τm ) . 4 Ti = 1.27 K m τm 1 + 2.467(Tm3 τm ) . 5 Ti = 2.0309K mTm3e0.419692 τ m Tm 3 . 6 Ti = 7 Ti = 1.27 K m τm 1 + 2.467(Tm 4 τm ) . 8 Ti = 9 Ti = τm Tm 4 < 1 Ti τm Tm 4 > 1 8 Ti 0 < τm Tm 4 < 1 9 Ti 0 < τm Tm 4 < 10 7 1.5K m (τm + Tm 4 ) 2 2 K mTm3 . exp[− 0.706636 ln (τm Tm3 ) − 0.962232] 2 K m Tm 4 0.5 + 0.001945(τm Tm 4 ) − 0.03091(τm Tm 4 )2 K m Tm 4 . 0.6178 − 0.1452(τm Tm 4 ) + 0.0141(τm Tm 4 )2 − 0.0004883(τm Tm 4 )3 . 29 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 30 3.3 FOLPD Model G m (s) = K m e −sτm 1 + sTm 1 3.3.1 Ideal PI controller G c (s) = K c 1 + Ti s R(s) E(s) + − U(s) 1 K c 1 + Ti s Y(s) K m e − sτ 1 + sTm m Table 9: PI controller tuning rules – FOLPD model G m (s) = Rule Callender et al. (1935/6). Model: Method 1 Ziegler and Nichols (1942). Model: Method 2 Hazebroek and Van der Waerden (1950). Model: Method 2 K m e − sτ m 1 + sTm Comment Kc Ti 1 0.568 K m τ m Process reaction 3.64τ m 2 0.690 K m τ m 2.45τ m τm = 0.3 Tm 0.9Tm K mτm 3.33τ m Quarter decay ratio; τm Tm ≤ 1 . x1Tm K m τm x 2τm τm Tm x1 x2 Coefficient values: x1 x2 τm Tm 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.68 0.70 0.72 0.74 0.76 0.79 0.81 0.84 0.87 7.14 4.76 3.70 3.03 2.50 2.17 1.92 1.75 1.61 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0.90 0.93 0.96 0.99 1.02 1.06 1.09 1.13 1.17 1.49 1.41 1.32 1.25 1.19 1.14 1.10 1.06 1.03 1 Decay ratio = 0.015; period of decaying oscillation = 10.5τ m . 2 Decay ratio = 0.043; period of decaying oscillation = 6.28τ m . τm Tm x1 x2 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 1.20 1.28 1.36 1.45 1.53 1.62 1.71 1.81 1.00 0.95 0.91 0.88 0.85 0.83 0.81 0.80 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Hazebroek and Van Tm τ 0.5 m + 1 der Waerden (1950) – K m τm Tm continued. Oppelt (1951). 3 Model: Method 2 K 2 τm 1.6τm − 1.2Tm τm Tm > 3.5 1.66τ m τ m >> Tm c Moros (1999). Model: Method 1 τ m << Tm 0.8Tm K m τ m 3τ m 0.91Tm K m τ m 3.3τ m Attributed to Rosenberg. 0.6Tm K m τm Regulator 4τ m 0% overshoot. 0.7Tm K m τm 2.33τ m 20% overshoot. 0.35Tm K m τm Servo 1.17Tm 0% overshoot. 0.6Tm K m τm Tm 20% overshoot. Reswick (1956). Model: Method 2; K c , Ti deduced from graphs; τm Tm = 6.25 . 0.20 K m Regulator 0.20τ m 0% overshoot. 0.30 K m 0.22τ m 20% overshoot. 0.15 K m Servo 0.16τ m 0% overshoot. 0.20 K m 0.14τ m 20% overshoot. Cohen and Coon (1953). Model: Method 2 T 1 0.9 m + 0.083 Km τm Two constraints method – Wolfe (1951). Model: Method 4 4 3.32τ m Attributed to Oppelt. Chien et al. (1952). Model: Method 2; 0.1 < τm Tm < 1.0 . 3 Comment Ti Kc ≤ 4 x1Tm K m τm Quarter decay ratio; 0 < τm Tm ≤ 1.0 . Ti x 2τm τm Tm Coefficient values: x1 x2 x1 τm Tm x2 0.2 0.5 4.4 1.8 1.28 0.53 3.23 2.27 1.0 5.0 0.78 0.30 1 1 Tm Tm + 1 ; recommended K c = − 1 . 1.5708 0.77 Km τm K τm m 3.33(τm Tm ) + 0.31(τm Tm ) 2 . Ti = Tm 1 + 2.22(τm Tm ) Decay ratio is as small as possible; minimum error integral (regulator mode). 31 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 32 Rule Two constraints criterion – Murrill (1967), p. 356. Model: Method 5 Heck (1969). Model: Method 2 Kc 0.928 Tm K m τ m Comment Ti 0.946 Tm τ m 1.078 Tm 0.583 Quarter decay ratio; minimum error integral (servo mode). 0.1 ≤ τm Tm ≤ 1.0 . 0.8Tm K m τm 3τ m Regulator. undefined x 2K mτm G c = 1 (Tis) Regulator – representative values (deduced from graph): τm Tm x2 τm Tm x2 τm Tm x2 τm Tm x2 0.11 0.13 0.14 0.17 2.94 0.20 2.50 0.40 2.17 0.71 1.96 2.86 0.25 2.38 0.50 2.13 2.78 0.29 2.35 0.56 2.08 2.63 0.33 2.30 0.63 2.00 Servo – representative values (deduced from graph): τm Tm x2 τm Tm x2 τm Tm x2 τm Tm x2 0.11 0.13 0.14 0.17 3.92 3.85 3.64 3.51 0.56 K m 0.20 0.25 0.29 0.33 3.33 0.40 3.08 0.50 2.94 0.56 2.82 0.63 0.65Tm 2.63 0.71 2.27 2.47 2.41 2.35 Model: Method 3 Fertik and Sharpe (1979). Parr (1989). 0.91Tm K m τm 3.3τm Model: Method 2 Sakai et al. (1989). 1.2408Tm K m τm 0.5τ m Model: Method 2 Borresen and Grindal (1990). Model: Method 2 Klein et al. (1992). Model: Method 1 1.0Tm K mτm 3τ m Also described by ABB (2001). McMillan (1994), p. 25. Model: Method 5 St. Clair (1997), p. 22. Model: Method 5 0.28Tm K m (τ m + 0.1Tm ) 0.53Tm Km 3 τm Time delay dominant processes. 0.333Tm K mτm Tm τm ≥ 0.33 Tm x1 K m x 2τm Hay (1998), Coefficient values – servo tuning: pp. 197–198. τ m Tm x1 x2 τ m Tm x1 Model: Method 1; 0.1 6.0 9.5 0.1 4.0 coefficients of K c , Ti 4.5 7.5 0.125 3.2 deduced from graphs. 0.125 0.167 3.0 5.4 0.167 2.2 0.25 2.0 3.4 0.25 1.3 0.5 0.8 1.7 0.5 0.5 x2 10.0 8.0 6.0 4.0 2.0 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Hay (1998), pp. 197–198 – continued. Hay (1998), p. 200. Model: Method 2; K c , Ti deduced from graphs. Shinskey (2000), (2001). Lipták (2001). Kc Ti x1 K m x 2τm τ m Tm Coefficient values – regulator tuning: x1 x2 τ m Tm x1 0.1 9.5 0.125 7.0 0.167 5.0 0.25 3.4 0.5 1.8 x1 K m x1 3.2 3.2 3.0 2.8 2.0 0.1 0.125 0.167 0.25 0.5 Minimum IAE – Frank and Lenz (1969). Model: Method 1 x2 2.8 2.8 2.6 2.3 1.6 τm Tm x2 4 1.6 1.7 1.0 1.2 0.8 1.0 0.7 0.667Tm K m τm 6.8 5.6 4.3 2.8 1.2 x 2 τm 4.9 4 3.0 2.6 2.1 1.9 1.76 1.51 0.2 0.3 0.4 0.5 3.78τ m 0.95Tm K m τm Regulator. Servo. Regulator. Servo. Regulator. Servo. Regulator. Servo. Model: Method 2; τm Tm = 6 . Model: Method 2 4τ m Chidambaram (2002), 1 Tm p. 154. 0.4 + 0.665 Km τm Model: Method 1 Faanes and Skogestad 0.71Tm (2004). K m τm Model: Method 1 PMA (2006). 0.39Tm K m τ m Minimum IAE – Murrill (1967), pp. 358–363. Comment ‘Improved’ Ziegler– Nichols method. 3.4τm Equivalent Ziegler– Nichols ultimate cycle method. Model: Method 2 3.3τm 6τ m Minimum performance index: regulator tuning 0.986 0.707 Model: Method 5; Tm τ m 0.984 Tm 0.1 ≤ τm Tm ≤ 1.0 . 0.608 Tm K m τm T 1 x 1 + x 2 m K m τm T τm x 1 + x 2 m x 2 τm Representative coefficient values (deduced from graph): τ m Tm x1 x2 τ m Tm x1 x2 0.2 0.4 0.6 0.8 0.37 0.42 0.44 0.45 0.77 0.85 0.91 0.98 1.0 1.2 1.4 0.46 0.46 0.46 1.05 1.11 1.17 33 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 34 Rule Kc Ti Comment Minimum IAE – Pemberton (1972a), Smith and Corripio (1997), p. 345. Tm K mτm Tm Model: Method 1; τ 0.1 ≤ m ≤ 0.5 . Tm Minimum IAE – Murata and Sagara (1977). Model: Method 2 x1 K m x2 τm Tm x1 x2 τm Tm 4.5 2.3 1.5 0.6 0.9 1.1 0.2 0.4 0.6 1.2 0.9 1.4 1.6 0.8 1.0 0.075 Ti Kc 0.05 ≤ τm T ≤ 1.0 ; 0.05 ≤ L ≤ 7.0 . Tm Tm Load disturbance 1 model = . 1 + sTL Values of x1 to x 10 for varying τm Tm provided. 0.75 1.0 x1 19.160 12.886 9.6994 6.5357 4.9749 3.3775 2.5985 2.1285 1.4947 1.1714 x2 4.7284 3.5703 4.2713 10.412 3.9767 2.3329 1.4059 1.1867 0.8493 0.7991 x3 9.4147 4.9974 3.7827 2.5854 1.9599 1.3112 1.0318 0.8244 0.5444 0.3968 τm Tm 0.05 x 2Tm x1 5 Minimum IAE – Kosinsani (1985), pp. 43–44. Model: Method 1 5 17-Mar-2009 0.1 0.15 0.2 0.3 0.4 0.5 x4 5.5495 6.1065 6.4339 0.1239 0.0182 0.1101 -3.0E-5 -5.0E-9 -8.0E-9 -3.0E-7 x5 -11.859 -2.1296 0.4052 200.86 375.61 13.563 2.8052 0.2503 -0.1163 0.8593 x6 1.0356 0.9598 0.8911 0.8734 0.8989 0.8537 0.8128 0.7849 0.7199 0.6711 x7 5.5638 3.9556 3.1631 2.2782 1.7580 1.3671 1.0831 0.9053 0.6215 0.5131 x8 0.2390 0.3740 0.4972 0.7729 1.1390 1.8367 2.4317 3.1877 3.2503 4.1661 x9 0.1162 0.1452 0.1667 0.1883 0.1789 0.2327 0.2775 0.2695 0.5488 0.6450 x 10 -1.6815 -1.5586 -1.4805 -1.4149 -1.5115 -1.2973 -1.4476 -1.5194 -1.4692 -1.5578 Kc = T T 1 x1 − e − x 2 TL Tm x 3 cosx 4 L + x 5 sin x 4 L , Km Tm Tm Ti = Tm x6 + 1 + x8 . x7 TL − x9 Tm x 10 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum IAE – Shinskey (1988), p. 123. Model: Method 1 6 0.685 TL Kc = K m Tm Ti 6 Kc Ti 7 Kc Ti 8 Kc Ti 9 Kc Ti τ m Tm > 0.35 1.00Tm K m τ m 3.0τ m τ m Tm = 0.2 1.04Tm K m τ m 2.25τ m τ m Tm = 0.5 1.08Tm K m τ m 1.45τ m τ m Tm = 1 1.39Tm K m τ m τm τ m Tm = 2 0.214 − 0.346 τm Tm τ m T m , 1.977 0.874 TL Kc = K m Tm − 0.099 + 0.159 τm Tm τ m T m 0.871 TL Kc = K m Tm τ − 0.015 + 0.384 m Tm τ m T m τm − 0.55 Tm τ 0.513 TL Kc = K m Tm 0.218 τm Tm τ T m m −1.451 , TL τ ≤ 2.641 m + 0.16 . Tm Tm , τ − 4.515 m + 0.067 Tm τ m T m 0.876 , 2.641 τm T + 0.16 ≤ L ≤ 1 . Tm Tm −1.055 , T T Ti = m L 0.444 Tm 9 1.123 m T m −1.041 T T Ti = m L 0.415 Tm 8 τm ≤ 0.35 Tm −1.256 T T Ti = m L 0.214 Tm 7 Comment Kc Minimum IAE – Yu (1988). Model: Method 1. Load disturbance K e −sτ L . model = L 1 + sTL 35 T T , Ti = m L 0.670 Tm − 0.003 − 0.217 τm Tm τm − 0.213 Tm τ − 0.084 m T m τm T m 0.56 . 0.867 , 1< TL ≤ 3. Tm b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 36 Rule Kc Minimum IAE – Hill and Venable (1989). Model: Method 1 x1 K m τm Tm Ti Comment x 2Tm Load disturbance 1 model = . 1 + sTL x1 values (this page) and x 2 values (next page) deduced from graphs; robust to ±30% process model parameter changes. 0.05 0.07 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0 TL Tm = 0.05 14.0 9.8 6.2 4.0 3.0 2.08 1.60 1.32 0.95 0.78 TL Tm = 0.1 16.8 10.3 7.2 4.3 3.2 2.14 1.63 1.33 0.97 0.77 TL Tm = 0.3 17.6 11.8 8.8 5.4 4.0 2.48 1.83 1.46 1.00 0.80 TL Tm = 0.5 17.0 11.4 8.6 5.6 4.2 2.67 1.96 1.58 1.06 0.84 TL Tm = 0.7 16.4 11.0 8.2 5.4 4.1 2.74 2.04 1.64 1.11 0.88 TL Tm = 1.0 15.7 10.5 7.8 5.2 4.0 2.70 2.07 1.68 1.17 0.92 TL Tm = 1.2 15.7 10.5 7.8 5.1 4.0 2.69 2.05 1.68 1.18 0.94 TL Tm = 1.5 15.8 10.5 7.8 5.1 3.9 2.68 2.01 1.68 1.18 0.95 TL Tm = 2.0 15.8 10.5 7.6 5.1 3.9 2.65 2.02 1.65 1.18 0.95 TL Tm = 2.5 15.9 10.5 7.8 5.1 3.9 2.64 1.95 1.63 1.17 0.95 TL Tm = 3.0 15.8 10.5 7.8 5.1 3.9 2.62 2.01 1.62 1.16 0.94 TL Tm = 3.5 15.8 10.4 7.8 5.1 3.9 2.59 1.98 1.61 1.15 0.93 TL Tm = 4.0 15.7 10.5 7.8 5.1 3.9 2.60 2.00 1.63 1.13 0.91 TL Tm = 4.5 15.6 10.4 7.8 5.1 3.8 2.54 1.98 1.60 1.13 0.92 TL Tm = 5.0 15.7 10.3 7.8 5.1 3.8 2.56 1.98 1.61 1.15 0.93 TL Tm = 5.5 15.5 10.3 7.8 5.1 3.9 2.55 1.98 1.62 1.17 0.92 TL Tm = 6.9 15.5 10.3 7.6 5.1 3.9 2.52 1.98 1.60 1.17 0.92 TL Tm = 6.5 15.5 10.3 7.8 5.2 4.0 2.60 2.00 1.62 1.18 0.93 TL Tm = 7.0 15.6 10.3 7.8 5.2 3.9 2.56 2.00 1.62 1.18 0.92 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Minimum IAE – Hill and Venable (1989). Model: Method 1 x1 K m τm Tm Ti Comment x 2Tm Load disturbance 1 model = . 1 + sTL x1 values (previous page) and x 2 values (this page) deduced from graphs; robust to ±30% process model parameter changes. 0.05 0.07 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0 TL Tm = 0.05 1.0 1.0 1.0 1.0 1.0 0.92 0.90 0.88 0.80 0.74 TL Tm = 0.1 1.0 1.0 1.0 1.0 1.0 0.92 0.90 0.88 0.80 0.74 TL Tm = 0.3 3.4 1.8 1.3 1.0 1.0 0.93 0.88 0.86 0.79 0.73 TL Tm = 0.5 4.4 2.8 2.0 1.7 1.2 1.00 0.93 0.87 0.78 0.72 TL Tm = 0.7 4.8 3.3 2.5 1.7 1.4 1.10 1.00 0.90 0.79 0.72 TL Tm = 1.0 5.4 3.8 3.0 2.2 1.7 1.32 1.11 0.99 0.82 0.74 TL Tm = 1.2 5.6 4.0 3.2 2.3 1.8 1.42 1.19 1.06 0.85 0.76 TL Tm = 1.5 6.0 4.2 3.4 2.5 2.0 1.54 1.32 1.13 0.91 0.80 TL Tm = 2.0 6.3 4.6 3.7 2.6 2.2 1.68 1.42 1.26 1.00 0.86 TL Tm = 2.5 6.7 4.7 3.8 2.7 2.3 1.76 1.53 1.32 1.06 0.90 TL Tm = 3.0 6.9 4.9 3.9 2.8 2.3 1.84 1.55 1.38 1.11 0.96 TL Tm = 3.5 7.1 5.1 4.0 3.0 2.4 1.91 1.62 1.43 1.16 0.99 TL Tm = 4.0 7.3 5.2 4.1 3.0 2.5 1.93 1.63 1.46 1.20 1.04 TL Tm = 4.5 7.4 5.3 4.2 3.1 2.5 2.01 1.68 1.50 1.21 1.05 TL Tm = 5.0 7.5 5.4 4.3 3.1 2.5 2.02 1.70 1.51 1.23 1.06 TL Tm = 5.5 7.6 5.4 4.2 3.2 2.6 2.04 1.72 1.52 1.23 1.09 TL Tm = 6.0 7.7 5.4 4.4 3.2 2.6 2.08 1.73 1.56 1.25 1.10 TL Tm = 6.5 7.9 5.5 4.3 3.2 2.6 2.05 1.74 1.56 1.26 1.10 TL Tm = 7.0 7.9 5.5 4.5 3.2 2.7 2.10 1.75 1.57 1.27 1.12 37 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 38 Rule Kc Ti Comment Minimum IAE – Marlin (1995), pp. 301–307. Model: Method 1; K c , Ti deduced from graphs; robust to ±25% process model parameter changes. 1.4 K m 0.24τ m τ m Tm = 0.11 1.8 K m 0.52τ m τ m Tm = 0.25 Minimum IAE – Huang et al. (1996). Model: Method 1 Minimum IAE – Shinskey (1996), p. 38. Model: Method 1 Minimum IAE – Edgar et al. (1997), pp. 8-14, 8-15. Model: Method 1 Minimum IAE – Edgar et al. (1997), p. 8-15. 10 Kc = 1.4 K m 0.75τ m τ m Tm = 0.43 1.0 K m 0.68τ m τ m Tm = 0.67 0. 8 K m 0.71τ m τ m Tm = 1.0 0.55 K m 0.60τ m τ m Tm = 1.50 0.45 K m 0.54τ m τ m Tm = 2.33 0.35 K m 0.49τ m τ m Tm = 4.0 0. 1 ≤ τm ≤1 Tm Kc Ti 0.95Tm K m τ m 3.4 τ m τ m Tm = 0.1 0.95Tm K m τ m 2.9 τ m τ m Tm = 0.2 0. 4 K m 0.5τ m Time delay dominant. 0.94Tm K m τ m 4τ m Time constant dominant. 0.67Tm K m τ m 3.5τ m Model: Method 2 10 −0.9077 −0.063 τ τ 1 τ 6.4884 + 4.6198 m + 0.8196 m − 5.2132 m Km Tm Tm Tm + τ 1 − 7.2712 m Km Tm 0.5961 τm − 0.7241e Tm , 2 3 4 τm τm τm τm Ti = Tm 0.0064 + 3.9574 − 10.7619 + 9.4348 − 6.4789 Tm Tm Tm Tm 5 6 τm τm . − 2.2236 + Tm 7.5146 Tm Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum IAE – Arrieta Orozco (2003). Model: Method 10 Minimum IAE – Shinskey (2003). Minimum IAE – Harriott (1988). Model: Method 1; coefficients of K c , Ti deduced from graphs. Kc 11 Kc Minimum IAE – Shinskey (1994), p. 167. Minimum IAE – Shinskey (1996), p. 121. Minimum ISE – Hazebroek and Van der Waerden (1950). 11 Kc = Ti 0.1 ≤ τm Tm ≤ 1.2 0.74Tm K m τm 4.06τm x1K u x 2Tu Model: Method 2 x1 x2 τm Tm x1 x2 τm Tm 0.55 0.52 0.50 0.91 0.81 0.64 0.1 0.2 0.5 0.43 0.42 0.45 0.33 1.0 2.0 0.5K u x 2Tu τm Tm x2 τm Tm x2 τm Tm 0.80 0.1 0.77 0.2 0.58K u 0.64 0.53 0.5 1.0 0.43 2.0 0.81Tu τ m Tm = 0.2 0.54 K u 0.66Tu τ m Tm = 0.5 0.48K u 0.47Tu τ m Tm = 1 0.46K u 0.37Tu τ m Tm = 2 Ku 3.05 − 0.35(Tu τm ) 12 Ti Model: Method 1 0.55K u 0.78Tu Model: Method 1; τm Tm = 0.2 . 13 1.43Tm Model: Method 2; τ m Tm < 0.2 . Kc 0.4749 1.1251 τ T 1 . , Ti = Tm − 0.2551 + 1.8205 m 0.4485 + 0.6494 m T Km τm m 2 T T Ti = Tu 0.87 − 0.855 u + 0.172 u . τm τm τ Tm 13 0.74 + 0.3 m . Kc = K m τ m Tm 12 Comment Also given by Arrieta Orozco and Alfaro Ruiz (2003). x2 Minimum IAE – Shinskey (1988), p. 148. Model: Method 1 Ti 39 b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 40 Rule Hazebroek and Van der Waerden (1950) – τm continued. Tm Minimum ISE – Haalman (1965). Model: Method 1 Minimum ISE – Murrill (1967), pp. 358–363. Model: Method 5 Minimum ISE – Frank and Lenz (1969). Model: Method 1 Ti x1Tm K m τm x 2τm x1 x2 0.80 7.14 0.83 5.00 0.89 3.23 0.67Tm Kmτm 1.305 Tm K m τ m 0.959 T 1 x 1 + x 2 m K m τm Coefficient values: x1 τm x2 Tm 0.7 1.0 1.5 0.96 1.07 1.26 τm Tm 2.44 1.85 1.41 x1 x2 2.0 1.46 1.18 3.0 1.89 0.95 5.0 2.75 0.81 M max = 1.9; A m = Tm 2.36; φ m = 50 0 . Tm τ m 0.492 Tm 0.739 0. 1 ≤ τm ≤ 1.0 Tm T τm x 1 + x 2 m x 2 τm Representative coefficient values (deduced from graph): τ m Tm x1 x2 τ m Tm x1 x2 0.2 0.4 0.6 0.8 Minimum ISE – Yu (1988). Model: Method 1 Comment Kc 0.2 0.3 0.5 14 FA 0.53 0.58 0.61 0.62 14 0.82 0.88 0.97 1.05 Kc 1.0 1.2 1.4 0.954 τm Tm 1.12 1.17 1.20 τm ≤ 0.35 Tm Ti K e −sτ L 0.921 TL Load disturbance model = L . Kc = 1 + sTL K m Tm T T Ti = m L 0.430 Tm 0.63 0.63 0.62 0.181− 0.205 − 0.49 τm Tm τ τm T m m T m 0.639 , −1.214 , τ TL ≤ 2.310 m + 0.077 . Tm Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Minimum ISE – Yu (1988) – continued. Minimum ISE – Zhuang (1992), Zhuang and Atherton (1993). Kc Ti 16 Kc Ti 17 Kc τm Tm > 0.35 Ti 1.279 Tm K m τ m 0.945 Tm τ m 0.535 Tm 0.586 1.346 Tm K m τ m 0.675 Tm τ m 0.552 Tm 0.438 1.157 TL Kc = K m Tm 0.32 0.35 0.37 0.38 τ − 0.045 + 0.344 m Tm τ m T m − 0.065 + 0.234 0.77 0.80 0.86 0.93 τm Tm τ m T m 1.0 1.2 1.4 1.1 ≤ τm ≤ 2. 0 Tm 0.04 + 0.067 τm Tm τ T m m 0.38 0.39 0.39 1.00 1.05 1.10 −1.014 , − 2.532 τm Tm − 0.292 τm T m 0.899 , 2.310 τm T + 0.077 ≤ L ≤ 1 . Tm Tm −1.047 , T T Ti = m L 0.347 Tm 1.289 TL Kc = K m Tm τm ≤ 1. 0 Tm T τm x 1 + x 2 m x 2 τm T T Ti = m L 0.359 Tm 17 0. 1 ≤ Representative coefficient values (deduced from graph): τ m Tm x1 x2 τ m Tm x1 x2 0.2 0.4 0.6 0.8 1.07 TL Kc = K m Tm τm ≤ 0.35 Tm 15 T 1 x 1 + x 2 m K m τm Minimum ITAE – Frank and Lenz (1969). Model: Method 1 16 Comment Model: Method 1 or Method 5 or Method 6 or Method 15. 0.977 0.680 τ Tm τ m 0.859 Tm 0.1 ≤ m ≤ 1.0 T m K m τm 0.674 Tm Minimum ITAE – Murrill (1967), pp. 358–363. Model: Method 5 15 Ti 41 − 0.889 T T , Ti = m L 0.596 Tm −1.112 0.372 τm Tm τm − 0.094 Tm τ − 0.44 m T m τm T m 0.898 , 1< 0.46 . TL ≤ 3. Tm b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 42 Rule Kc 0.598 TL Kc = K m Tm 0.272 − 0.254 τm Tm τ m T m x 2 (τm + Tm ) τm Tm x1 x2 τm Tm 0.67 1.00 1.50 0.94 0.66 0.54 Ti 0.76 0.71 0.65 2.33 0.47 0.59 4.00 0.42 0.52 9.00 0.375 0.47 Model: Method 1; τm ≤ 0.35 . Tm Ti Ti , m T m 0.787 TL Kc = K m Tm 0.084 + 0.154 τm Tm τ m T m 0.304 τm − 0.112 Tm τ m T m 0.878 TL Kc = K m Tm 0.172 − 0.057 τm Tm τ T m m , TL τ ≤ 2.385 m + 0.112 . Tm Tm , τ − 5.809 m + 0.241 Tm τ m T m 0.901 , 2.385 τm T + 0.112 ≤ L ≤ 1 . Tm Tm −1.042 , T T Ti = m L 0.431 Tm 21 0.196 −1.055 T T Ti = m L 0.425 Tm 20 x2 −1.341 τ − 0.011−1.945 m Tm τ 0.735 TL Kc = K m Tm x1 τm Tm > 0.35 Ti T T Ti = m L 0.805 Tm 19 Comment Ti Minimum ITAE with x1 K m OS ≤ 8% – Fertik x1 x2 τm (1975). T m Model: Method 3; 5.0 0.53 x 1 , x 2 deduced from 0.11 0.25 2.6 0.74 graphs. 0.43 1.45 0.80 18 Minimum ITAE – Kc Yu (1988). 19 Kc Load disturbance 20 K L e −sτ L Kc . model = 1 + sTL 21 Kc 18 FA − 0.909 T T , Ti = m L 0.794 Tm τ − 0.148 m − 0.365 Tm τ 0.228 m T m τm Tm − 0.257 τm T m 0.901 , 1< 0.489 . TL ≤ 3. Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum ITAE – Arrieta Orozco (2003). Model: Method 10. Minimum ITAE – Arrieta Orozco (2006), pp. 90–92. Minimum ITSE – Frank and Lenz (1969). Model: Method 1 Kc 22 23 Kc = 0.1 ≤ Ti τm ≤ 1.2 Tm Also given by Arrieta Orozco and Alfaro Ruiz (2003). 2 ≤ Am ≤ 4 ; Model: Method 10 Kc Ti T 1 x 1 + x 2 m Km τm T τm x 1 + x 2 m x2 τm 23 Representative coefficient values (deduced from graph): τ m Tm x1 x2 τ m Tm x1 x2 0.2 0.4 0.6 0.8 22 Comment Ti Kc 43 0.46 0.50 0.53 0.55 T 1 0.2607 + 0.6470 m Km τm 0.84 0.89 0.97 1.04 1.0 1.2 1.4 0.55 0.54 0.54 1.09 1.15 1.20 0.1789 τ . , Ti = Tm − 1.5926 + 2.9191 m T m 1.1055 −1.5345 + 0.2865 A m − 0.0389 A m 2 τm 1 2 , Kc = 1.2585 − 0.6179A m + 0.0764A m + x1 Km Tm 2 x1 = 0.8653 − 0.1783A m + 0.0127 A m , 0.1 ≤ τm ≤ 2.0 ; Tm 7.7448 − 4.4925 A m + 0.6454 A m 2 τ 2 , Ti = Tm 24.5244 − 17.4081A m + 2.7267 A m + x 2 m Tm 2 x 2 = −21.8710 + 16.5760A m − 2.6258A m , 0.1 ≤ τm < 1.0 ; Tm 1.3703 − 0.2639 A m − 0.0110 A m 2 τm 2 , Ti = Tm 1.1097 − 0.2739A m + 0.0272A m + x 2 Tm 2 x 2 = 1.1821 − 0.3504A m + 0.0417 A m , 1.0 ≤ τm ≤ 2.0 . Tm b720_Chapter-03 Rule Minimum ISTSE – Zhuang (1992), Zhuang and Atherton (1993). Minimum ISTSE – Zhuang (1992), Zhuang and Atherton (1993). Minimum ISTES – Zhuang (1992), Zhuang and Atherton (1993). Nearly minimum IAE, ISE, ITAE – Hwang (1995). Model: Method 44 Nearly minimum IAE and ITAE – Hwang and Fang (1995). Model: Method 30 25 26 27 FA Handbook of PI and PID Controller Tuning Rules 44 24 17-Mar-2009 Kc Comment Ti 1.015 Tm K m τ m 0.957 Tm τ m 0.667 Tm 0.552 1.065 Tm K m τ m 0.673 Tm τ m 0.687 Tm 0.427 0. 1 ≤ τm ≤ 1. 0 Tm 1.1 ≤ τm ≤ 2. 0 Tm Model: Method 1 or Method 5 or Method 6 or Method 15. 24 Model: Method 1 Ti K c 25 Model: Method 37 Ti Kc 1.021 Tm K m τ m 0.953 Tm τ m 0.629 Tm 0.546 1.076 Tm K m τ m 0.648 Tm τ m 0.650 Tm 0.442 0. 1 ≤ τm ≤ 1. 0 Tm 1.1 ≤ τm ≤ 2. 0 Tm Model: Method 1 or Method 5 or Method 6 or Method 15. Decay ratio = 0.15; 26 K µ K ω τ (1 − µ1 )K u c 2 u u 0.1 ≤ m ≤ 2.0 . Tm 27 Ti Kc Decay ratio = 0.12; τ 0.1 ≤ m ≤ 2.0 . Tm 1.892K m K u + 0.244 0.706K m K u − 0.227 τm K u , Ti = K c = 0.7229K K + 1.2736 Tu , 0.1 ≤ T ≤ 2.0 . + 3 . 249 K K 2 . 097 m m u m u ∧ ∧ τm 4.126K m K u − 2.610 ∧ 5.352K m K u − 2.926 ∧ K , = Kc = T u i T u , 0.1 ≤ T ≤ 2.0 . ∧ ∧ 5.848K K u − 1.06 5.539K K u + 5.036 m m m µ1 = ( ) ( ) 1.14 1 − 0.482ωu τm + 0.068ω2u τ2m 0.0694 − 1 + 2.1ωu τm − 0.367ω2u τ2m , µ2 = . K u K m (1 + K u K m ) K u K m (1 + K u K m ) 2 τ τ (K c / K u ωu ) K c = 0.515 − 0.0521 m + 0.0254 m 2 K u , Ti = 2 T Tm m 0.0877 + 0.0918 τm − 0.0141 τm 2 Tm Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Comment Minimum error: step load change – Gerry (2003). 0.3 K m 0.42τ m Model: Method 1; τm Tm ≥ 5 . x1 K m Minimum IAE or ISE – ECOSSE Team τ m Tm x1 (1996a). 0.11 1.1 Model: Method 1; 0.25 1.8 coefficients of K c , Ti 0.43 1.1 deduced from graphs. 0.67 1.0 1.0 0.8 1.50 0.59 2.33 0.42 4.0 0.32 28 Minimum IE – Devanathan (1991). Model: Method 1 x 2 (τ m + Tm ) Coefficient values: x2 τ m Tm 0.23 0.23 0.72 0.72 0.70 0.67 0.60 0.53 0.11 0.25 0.43 0.67 1.0 x1 x2 5.8 3.1 2.1 1.7 0.91 0.5 0.6 0.7 0.8 0.9 x 2 Tu Kc x 2 coefficient values (deduced from graphs): τ m Tm 0.2 0.4 0.6 x2 τ m Tm x2 τ m Tm x2 0.35 0.8 0.23 1.4 0.29 1.0 0.21 1.6 0.26 1.2 0.19 1.8 Minimum performance index: servo tuning x1 K m x 2 (τ m + Tm ) 0.18 0.18 0.17 Minimum IAE – Gallier and Otto (1968). Model: Method 58 Representative coefficient values (deduced from graphs): τ m Tm x1 x2 τ m Tm x1 x2 Minimum IAE – Rovira et al. (1969). Model: Method 5 0.758 Tm K m τ m 0.053 0.11 0.25 12 6.4 2.9 0.861 0.16 ξ cos tan −1 1.57 D R − D R 28 Kc = . −1 6.28Tm K m cos tan Tu 0.95 0.91 0.84 0.67 1.50 4.0 Tm τ 1.020 − 0.323 m Tm 1.15 0.65 0.45 0.75 0.66 0.55 0.1 ≤ τm ≤ 1.0 Tm 45 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 46 Rule Kc Comment Ti Minimum IAE – x1 K m x 2Tm Murata and Sagara x1 x2 τm Tm x1 x2 τm Tm (1977). 2.7 0.9 0.2 0.9 1.3 0.8 Model: Method 2; 1.5 1.1 0.4 0.8 1.4 1.0 x1 , x 2 deduced from 1.1 1.2 0.6 graphs. Minimum IAE – Bain 1 τm Tm > 0.2 and Martin (1983). 0.3 + 0.3 Tm + 0.4 τ m T Km τm m Model: Method 1 Minimum IAE – 1.4 K m 0.72τ m τm Tm = 0.11 Marlin (1995), K c , Ti deduced from graphs; robust to ±25% process model pp. 301–307. parameter changes. Model: Method 1 Minimum IAE – τ 0.1 ≤ m ≤ 1 29 Huang et al. (1996). T Kc i Tm Model: Method 1 Minimum IAE – Model: Method 1; 0.6Tm Smith and Corripio T 0 .1 ≤ τm Tm ≤ 1.5 . m K mτm (1997), p. 345. Minimum IAE – Minimum IAE = 0.59Tm Huang and Jeng T 2.1038τ m ; m Kmτm (2002). τ m Tm < 0.2 . Model: Method 1 29 Kc = τ 1 τ − 13.0454 − 9.0916 m + 0.3053 m Km Tm Tm −1.0169 τ + 1.1075 m Tm + 3.5959 τ 1 − 2.2927 m Km Tm 3.6843 τm + 4.8259e Tm , 2 3 4 τ τ τ τ Ti = Tm 0.9771 − 0.2492 m + 3.4651 m − 7.4538 m + 8.2567 m Tm Tm Tm Tm 5 6 τm τm . + 1.1496 + Tm − 4.7536 Tm Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum IAE – Arrieta Orozco (2003). Model: Method 10 Minimum IAE – Tavakoli and Fleming (2003). Model: Method 1 Minimum IAE – Tavakoli et al. (2006). Minimum IAE – Harriott (1988). Ti coefficients deduced from graph. Kc 30 Comment Ti 0.1 ≤ Ti Kc τm ≤ 1.2 Tm Also given by Arrieta Orozco and Alfaro Ruiz (2003). 31 0.1 ≤ Ti Kc τm ≤ 10 Tm Minimum A m = 6dB ; minimum φ m = 60 0 . Kc Ti Model: Method 42; A m ≥ 3 ; φ m ≥ 60 0 . 0.5K u x 2Tu Model: Method 1 32 x2 τm Tm x2 τm Tm x2 τm Tm 1.61 0.94 0.2 0.5 0.61 1.0 0.48 2.0 1.2099 0.8714 Tm τm 1 . Kc = 0.2438 + 0.5305 , Ti = Tm 0.9377 + 0.4337 Km τm Tm 1 T τ 31 m Kc = + 0.3047 , Ti = Tm 0.4262 m + 0.9581 . 0.4849 Km τm Tm 30 32 Kc = 1 Tm + 0.071 , Ti = min[Tm + 0.143τ m ,9τ m ] . 0.5 K m τm 47 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 48 Rule Small IAE – Hwang (1995). Model: Method 44 Minimum ISE – Zhuang (1992), Zhuang and Atherton (1993). Model: Method 1 or Method 5 or Method 6 or Method 15. Minimum ISE – Khan and Lehman (1996). Model: Method 1 Minimum ITAE – Rovira et al. (1969). Model: Method 5 33 17-Mar-2009 µ1 = ( 33 Kc Ti Comment (1 − µ1 )K u K c µ 2 K u ωu Decay ratio = 0.1; τ 0.1 ≤ m ≤ 2.0 . Tm 0.980 Tm K m τ m 0.892 1.072 Tm K m τ m 0.560 Tm τ 0.690 − 0.155 m Tm Tm τ 0.648 − 0.114 m Tm 0.01 ≤ τm Tm ≤ 0.2 35 Kc Ti 0.2 ≤ τm Tm ≤ 20 0.586 Tm K m τ m Tm 0.916 τ 1.030 − 0.165 m Tm 0.1 ≤ τm ≤ 1.0 Tm ) ) 1.11 1 − 0.467ωu τm + 0.0657ω2u τ2m , K u K m (1 + K u K m ) ( τm ≤ 2.0 Tm Ti µ2 = µ1 = 1.1 ≤ Kc 1.07 1 − 0.466ωu τm + 0.0667ω2u τ2m , K u K m (1 + K u K m ) ( τm ≤ 1.0 Tm 34 µ2 = µ1 = 0.1 ≤ ) ( ) 0.0328 − 1 + 2.21ωu τm − 0.338ω2u τ2m , r = 0.5; K u K m (1 + K u K m ) ( ) 0.0477 − 1 + 2.07ωu τm − 0.333ω2u τ2m , r = 0.75; K u K m (1 + K u K m ) 1.14 1 − 0.466ωu τm + 0.0647ω2u τ2m , K u K m (1 + K u K m ) ( ) 0.0609 − 1 + 1.97ωu τm − 0.323ω2u τ2m , r = 1.0; K u K m (1 + K u K m ) r = parameter related to the position of the dominant real pole. µ2 = 34 35 0.7388 0.3185 Tm 0.7388Tm + 0.3185τ m K c = + K , Ti = τ m − 0.0003082T + 0.5291τ . T τ m m m m m 0.808 0.511 0.808Tm + 0.511τm − 0.255 Tm τm 0.255 Tm . Kc = + − , Ti = τm τm Tm 0.095Tm + 0.846τm − 0.381 Tm τm Tm τ m K m b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti 49 Comment Minimum ITAE with x1 K m x 2 (τm + Tm ) OS ≤ 8% – Fertik x1 x2 x1 x1 x2 τm τm τm x2 (1975). T T T m m m Model: Method 3; x 1 , x 2 deduced from 0.11 3.05 0.95 0.67 0.71 0.76 2.33 0.42 0.59 0.25 1.70 0.88 1.00 0.55 0.71 4.00 0.39 0.52 graphs. 0.43 1.05 0.82 1.50 0.47 0.65 9.00 0.365 0.47 Minimum ITAE – τ 0.1 ≤ m ≤ 1.2 36 Arrieta Orozco Ti Kc Tm (2003). Also given by Arrieta Orozco and Alfaro Ruiz (2003). Model: Method 10 Minimum ITAE – 2 ≤ Am ≤ 4 ; 37 Arrieta Orozco Ti Kc Model: Method 10 (2006), pp. 90–92. τ Minimum ITAE – − 0.9707 m Tm 38 Model: Method 1 Barberà (2006). 0 . 9894 e Kc 36 37 T 1 Kc = 0.1440 + 0.5131 m Km τm 1.5426 . , Ti = Tm 0.9953 + 0.2073 τ m T m 1.0382 −1.0871+ 0.0048 A m + 0.0047 A m 2 τm 1 2 , Kc = 0.9639 − 0.4436A m + 0.0496A m + x1 Km Tm 2 x1 = 1.3296 − 0.4355A m + 0.0509A m , 0.1 ≤ τm Tm ≤ 2.0 ; 6.1845 − 3.4585 A m + 0.4764 A m 2 τm 2 , Ti = Tm 2.7836 − 1.2511A m + 0.1692A m + x 2 Tm 2 x 2 = 0.8298 − 0.1193A m + 0.0083A m , 0.1 ≤ τm Tm < 1.0 ; 1.6705 − 0.2509 A m − 0.0447 A m 2 τm 2 , Ti = Tm 2.0174 − 0.5089A m + 0.0311A m + x 2 Tm 2 x 2 = 1.0065 − 0.4874A m + 0.0889A m , 1.0 ≤ τm Tm ≤ 2.0 . 38 Kc = 1 . 2 τm τ + 2.0214 m + 0.00911 K m 1.5853 Tm Tm b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 50 Rule Kc Ti Comment Smith (2002). 0.586Tm K m τm Tm Model: Method 1 Approximately minimum ITAE. Minimum ISTSE – Model: Method 1; 1 Tm + 0.2 Fukura and Tamura 0.56 Tm + 0.36τm 0.05 ≤ τm Tm ≤ 1 . Km τm (1983). 0.921 Minimum ISTSE – τ Tm 0.712 Tm 0.1 ≤ m ≤ 1.0 Zhuang (1992), T 0 . 968 − 0 . 247 τ T m m m K m τm Zhuang and Atherton (1993). 0.559 τ Tm 0.786 Tm 1.1 ≤ m ≤ 2.0 Model: Method 1 or T 0 . 883 − 0 . 158 τ T m m m K m τm Method 5 or Method 6 or Method 15. 39 Minimum ISTSE – Model: Method 1; Ti Kc Zhuang (1992). 0.1 ≤ τm Tm ≤ 2.0 . Minimum ISTSE – Zhuang (1992), Zhuang and Atherton (1993). Minimum ISTSE – Majhi (2005). Model: Method 52 0.361K u 41 Ti Model: Method 1 Ti Model: Method 37 40 Kc 0.712 Tm K m τ m 0.921 0.712Tm 0.694 − 0.182 τm Tm 0.1 ≤ τm ≤ 1.0 Tm 0.780 Tm K m τ m 0.543 0.780Tm 0.683 − 0.120 τm Tm 1.1 ≤ τm ≤ 2.0 Tm 0.673 Tm K m τ m 0.331 0.673Tm 0.511 − 0.066 τm Tm 2.1 ≤ τm ≤ 3.5 Tm 4.264 − 0.148K m K u K u , Ti = 0.083(1.935K m K u + 1)Tu . K c = 12.119 − 0.432K m K u τ 40 Ti = 0.083(1.935K m K u + 1)Tu , 0.1 ≤ m ≤ 2.0 . Tm 39 41 ∧ ∧ τm ∧ 1.506K m K u − 0.177 ∧ Kc = K u , Ti = 0.055 3.616K m K u + 1 T u , 0.1 ≤ T ≤ 2.0 . ∧ 3.341K K u + 0.606 m m b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum ISTES – Zhuang (1992), Zhuang and Atherton (1993). Model: Method 1 or Method 5 or Method 6 or Method 15. Kc 0.569 Tm K m τ m 0.951 0.628 Tm K m τ m 0.583 Nearly minimum IAE and ITAE – Hwang and Fang (1995). Tm τ 1.023 − 0.179 m Tm Tm τ 1.007 − 0.167 m Tm 0.1 ≤ τm ≤ 1.0 Tm 1.1 ≤ τm ≤ 2.0 Tm Model: Method 30; 0.1 ≤ τ m Tm ≤ 2.0 ; decay ratio = 0.03. Minimum performance index: other tuning Model: Method 30; 43 Ti 0.1 ≤ τ m Tm ≤ 2.0 ; Kc decay ratio = 0.05. Kc Ti x1 K m Tm 42 Simultaneous servo/regulator tuning – Hwang and Fang (1995). 44 Minimum performance index – Wilton (1999). Model: Method 1; coefficients of K c estimated from graphs. Comment Ti Coefficient values: b τ m Tm τ m Tm x1 x1 b 0.1 0.1 0.1 0.1 750.0 17.487 4.8990 2.1213 0.0001 0.04 0.2 1 0.2 0.2 0.2 0.2 0.2 3.3675 2.2946 1.7146 1.1313 0.8764 0.008 0.04 0.2 1 5 0.4 0.4 0.4 0.4 0.4 1.6837 1.5297 1.2247 0.9192 0.7668 0.008 0.04 0.2 1 5 0.5 0.5 0.5 0.5 150.01 7.1386 2.6944 1.1314 0.0001 0.04 0.2 1 2 τ τ (K c / K u ωu ) K c = 0.438 − 0.110 m + 0.0376 m 2 K u , Ti = . 2 T Tm m 0.0388 + 0.108 τm − 0.0154 τm 2 Tm Tm 2 τ τ (K c / K u ωu ) 43 K c = 0.46 − 0.0835 m + 0.0305 m 2 K u , Ti = . 2 T T m m 0.0644 + 0.0759 τm − 0.0111 τm 2 Tm Tm 42 44 Performance index = ∫ [e (t) + bK (y(t) − y ) ]dt . ∞ 0 2 2 m ∞ 2 51 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 52 Rule Wilton (1999) – continued. Minimum ITAE – ABB (2001). Model: Method 7 Ream (1954). Model: Method 1; decay ratio ≈ 33% . Bryant et al. (1973). Model: Method 1 Chiu et al. (1973). Model: Method 1 Mollenkamp et al. (1973). Model: Method 1 Kc Ti x1 K m Tm Comment τ m Tm x1 b τ m Tm x1 b 0.6 0.6 0.6 0.6 0.6 1.1225 1.1218 0.9798 0.7778 0.6572 0.008 0.04 0.2 1 5 0.8 0.8 0.8 0.8 0.8 0.8980 0.9178 0.7348 0.7071 0.6025 0.008 0.04 0.2 1 5 1.0 1.0 72.004 1.6657 0.0001 0.2 1.0 1.0 3.6713 0.04 0.8485 1 τm < 0.5 Tm 0.8591 Tm K m τ m 0.977 0.68 τ 1.4837Tm m Tm Direct synthesis: time domain criteria 1.56 K m 0.368(τm + Tm ) τm Tm = 1.12 0.85 K m 0.454(τm + Tm ) τm Tm = 2.86 0.37Tm K m τm Tm 0.40Tm K m τm Tm undefined x 2K mτm Critically damped dominant pole. Damping ratio (dominant pole) = 0.6. G c = 1 (Tis ) τ m Tm Coefficient values (deduced from graph): x2 τ m Tm x2 0.33 0.40 0.50 0.67 0.33 0.40 0.50 0.67 12.5 11.1 11.1 9.1 7.1 6.3 5.6 4.3 1.0 2.0 10.0 6.7 4.2 3.0 Critically damped dominant pole. 1.0 2.0 10.0 3.2 2.1 1.6 Damping ratio (dominant pole) = 0.6. λ variable; suggested values: 0.2, 0.4, 0.6, 1.0. λ = 1.0 , OS < 10%, Model: Method 11 (Pomerleau and Poulin, 2004). Appropriate for Tm T ‘small τ m ’. m K (T + τ ) λTm K m (1 + λτ m ) m CL m Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Comment 0.44Tm Kmτm Tm 0.04 ≤ τ m Tm ≤ 1.4 ; deduced from graph. 0.43Tm K mτm Tm 1% overshoot – servo; τm Tm > 0.2 . 1 Km Tm 53 undefined 1 (x 2 τm ) G c = 1 Tis Keviczky and Csáki x 2 coefficient values (estimated from graph): (1973); 0% 5% 10% 15% 20% OS values in first τ m Tm OS row; τ m Tm values 0.1 30 22 16 12 10 0.2 18 11 9 7.5 6 in first column; 0.5 8 6 5 4 3.5 Km = 1. 1.0 5 4 3.5 3 2.5 Model: Method 1 2.0 4 3 2.5 2.2 2 5.0 3 2.4 2 1.8 1.6 10.0 2.5 2.1 1.8 1.7 1.5 5% overshoot – servo 0.52Tm K m τm Tm 0.04 ≤ τm Tm ≤ 1.4 – Smith et al. (1975). Model: Method 1. Also given by Bain and Martin (1983). 1% overshoot – servo – Smith et al. (1975). Model: Method 1 Bain and Martin (1983). Model: Method 1 Kc Ti ‘Very conservative tuning for small delay’. Model: Method 1 0.5Tm K m τm Tm OS = 10% 0.5Tm K m τm Tm Model: Method 1 Kuwata (1987). Suyama (1992). Model: Method 1 5% overshoot – servo – Smith and Corripio (1997), p. 346. Vítečková (1999), Vítečková et al. (2000a) (Model: Method 9 – Vítečková et al., 2000b). 45 Kc = 45 φ m = 60 0 ; 0.25 ≤ τm Tm ≤ 1 (Voda and Landau, 1995). x1Tm K m τm x1 OS (%) 0.368 0.514 0.581 0.641 0 5 10 15 Tm Coefficient values: OS (%) x1 0.696 0.748 0.801 0.853 20 25 30 35 x1 OS (%) 0.906 0.957 1.008 40 45 50 0.4(Tm + τm ) 0.5 − , Ti = 1.25x1 - 0.25(Tm + τm ) , with K m (Tm + τm − x1 ) K m x1 = (Tm + τm )2 − 0.267τm 2 (3Tm + τm ) . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 54 Rule Okada et al. (2006). Model: Method 1 x1 Comment Kc Ti x1Tm K m τm Tm ξ Coefficient values: ξ x1 ξ x1 x1 ξ x1 ξ 0.368 1.0 0.454 0.8 0.578 0.6 0.765 0.4 1.06 0.2 0.407 0.9 0.510 0.7 0.661 0.5 0.896 0.3 1.29 0.1 Mann et al. (2001). 0.51Tm K m τm Tm 0 < τm Tm < 2 Model: Method 37 or 46 Ti Kc Method 41. Closed loop response overshoot (step input) < 5%. 47 undefined G c = 1 (Tis ) Ti Vítečková and 48 Ti Kc Víteček (2002). Model: Method 1 For 0.05 < τ T < 0.8 , T = T ; overshoot is under 2% m m i m (Vítečková and Víteček, 2003). Pinnella et al. (1986). Model: Method 24 46 undefined 49 Gc = Ki s Ki Critically damped servo response. 1 T Tm Tm 0.56 + 1.02 m − − x1 + x 2 , 1.0404 τm τm τm Km 0.56 + 1.02(Tm τm ) − (Tm τm )(1.0404(Tm τm ) − x1 ) + x 2 τ Tm ; Ti = m − 1 0.04 − 1.02(Tm τm ) + (Tm τm )(1.0404(Tm τm ) − x1 ) + x 2 Tm x 1 = 0.4, x 2 = 3.68, 1 < τ m Tm < 2 ; x 1 = 0.8, x 2 = 3.28, 2 < τ m Tm < 4 ; Kc = x 1 = 1.2, x 2 = 2.88, 4 < τ m Tm . 47 48 Ti = − Km 2 2 1+ 0.25 τ m − 0.5 τ m −1 1 T 0.25 1 0.5 Tm Tm Tm e m + − − + − + 0 . 25 0 . 5 2 2 τ τ τ T τ T m m m m m m Kc = − [ ] 1 τm Tm x12 + (2Tm + τm )x1 + 1 e τ m x1 , Km x1 = − 49 Ki = . 1 4K m τ m 2 2− τm τm + Tm Tm 2 2 0.5 − + τ m Tm 2 τm 2 + 0.25 Tm 2 τ T x 2 + (2Tm + τm )x1 + 1 , Ti = − m m2 1 . x1 (τm Tm x1 + Tm + τm ) τm τm 2 2 0.5 2 − Tm + Tm τ τ Tm 2 − m + m + 1 e . Tm Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Comment Schneider (1988), Bekker et al. (1991), Khan and Lehman (1996). Model: Method 1 Bekker et al. (1991). Model: Method 1; ξ = 1 ; Tv = valve opening time. 0.368Tm K m τm Tm ξ =1 0.403Tm K m τm Tm ξ = 0.6 1.571Tm K m τm Tm ξ = 0.0 0.368Tm K m τm Tm Regulator – Gorecki et al. (1989). Model: Method 1 Hang et al. (1991). Model: Method 1 50 Kc = τm > 0.5Tm or T v < 20Tm . τm < 0.5Tm or 50 Kc 51 Kc Tv > 20Tm . Ti Pole is real and has maximum attainable multiplicity. Ti 0.16 ≤ τm Tm < 0.96 Ti undefined 53 55 52 Kc Servo response: 10% overshoot, 3% undershoot. 0.311Tm . τm T Tv − 0.0760 ln v − 0.2685 + 0.1026 0.1092 ln 0.0121 ln T T T τm m m m Tv K m τm Tm τ m 2 2 2 + τ m − 2 − τ m 2T τm 2 Tm 2 Tm 51 − 1 e m Kc = 2 + , 2Tm K m τm 2 52 G c = 1 (T i s ) , Ti = τ 1 + m 2Tm . Ti = τm 2 3 2 2 τm τ m τm τ m τm + + − 2 + 2+ 3 + 2T 2Tm Tm 2Tm 2Tm m 0.5K m τm . 2 1 + (τ 2T )2 − 1 e m m 1+ τ m −1− τ m 2T 2 Tm m τm 12 + 2 11(τm Tm ) + 13 11 + 13 ( ) τ − 37 T 4 4 5 T m m K , T = 0.2 53 m + 1Tu . Kc = u i 15 τm 6 11(τm Tm ) + 13 − 4 37 + 14 15 Tm 37(τm Tm ) − 4 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 56 Rule Kc McAnany (1993). Model: Method 29 Sklaroff (1992), Åström and Hägglund (1995). 54 0.5Tm K m τm 55 Kc 2 56 x 2 Tm τ m 2K m Kamimura et al. (1994). Model: Method 1; servo tuning. Zhang (1994), Smith (2002). Model: Method 1 Kc = Kc Ti Comment Ti TCL = 1.67Tm Model: Method 32 or 3 Method 33. T m K m (1 + 3τ m Tm ) pp. 250–251, 253. Honeywell UDC 6000 controller. 0.5556Tm K m τm 5τ m τm Tm < 0.33 Fruehauf et al. (1993). Model: Method 2 Nomura et al. (1993). Model: Method 1 Gonzalez (1994) – pp. 114–115. Model: Method 1 54 17-Mar-2009 Tm τm Tm ≥ 0.33 Ti 0.01 ≤ τm Tm ≤ 10 0.5τm δ0 = 0.21, 0.5, 0.7 (pp. 142–143). 57 Kc Ti 10% overshoot. 58 Kc Ti 0% overshoot. 59 Kc Ti ‘Quick’ response: ‘negligible’ overshoot. Tm K m τm Tm ‘Good’ servo response; τ m is ‘small’. (1.44Tm + 0.72Tm τm − 0.43τm − 2.14) , T = K m (5.56 + 2τm + τm 2 ) . ( K m 0.72τm + 0.36τm 2 + 2 ) i 4Tm + 1.28τm − 2.4 0.5x1 − 0.1(Tm + τm ) 55 Kc = , Ti = 1.25x1 − 0.25(Tm + τm ) , with K m (Tm + τm − x1 ) 2 2 x1 = 0.2τm + 0.4τmTm + Tm . 56 x2 = − 2x1 + τm 4 x12 τm 2 + 2 1 , x1 = , Tm τm 1 + (2π loge [b a ])2 b a = desired closed loop response decay ratio. 57 Kc = 58 Kc = 59 Kc = 0.5(Tm + τm ) 0.5τm (2Tm + τm ) + 0.10 , Ti = Tm + τm − . K m [0.5τm (2Tm + τm ) + 0.10] Tm + τm 0.375(Tm + τm ) 0.5τm (2Tm + τm ) + 0.0781 , Ti = Tm + τm − . K m [0.5τm (2Tm + τm ) + 0.0781] Tm + τm 0.425(Tm + τm ) 0.5τm (2Tm + τm ) + 0.083 , Ti = Tm + τm − . K m [0.5τm (2Tm + τm ) + 0.083] Tm + τm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Ti Comment Kc Ti ξ = 0.9; 0.2 ≤ τ m Tm ≤ 1 . 0.5 K m 0.5(Tm + τ m ) ‘Normal tuning’. 1 Km 0.7(Tm + τ m ) ‘Rapid tuning’. Kc Davydov et al. (1995). Model: Method 6 Kuhn (1995). Model: Method 17 Khan and Lehman (1996). Model: Method 1 Abbas (1997). Model: Method 1 60 61 Kc Ti 0.01 ≤ τm Tm ≤ 0.2 62 Kc Ti 0.2 ≤ τm Tm ≤ 20 63 Kc Tm + 0.5τ m 0 ≤ V ≤ 0.2 ; 0.1 ≤ τm Tm ≤ 5.0 . τm Tm Model: Method 1; ξ = 0.707 64. Valentine and Chidambaram (1997a). Bi et al. (1999). τm 1 0.43 − 0.97 Km Tm 0.5064Tm K m τm Tm Model: Method 16 Bi et al. (2000). 0.5064Tm K m τm 1.9747 K m Tm Model: Method 16 Ettaleb and Roche (2000). 1 Km Tm + τ m Model: Method 34; TCL = Tm + τ m . Ti x1 > 0 Prokop et al. (2000). Model: Method 1 60 Kc = 61 Kc = 65 Kc 0.413 + 0.8 τ 1 , Ti = 0.153 m + 0.362 Tm . T τm m K m 1.905 + 0.826 Tm τ Tm 0.3852 0.723 + − 0.404 m2 , K m τm Tm Tm Ti = τm − 0.404τm 2 + 0.723Tm τm − 0.3852Tm 2 − 0.525τm 2 + 0.4104Tm τm − 0.00024Tm 2 62 Kc = 0.404Tm + 0.256τm − 0.1275 Tm τm Tm 0.404 0.256 0.1275 + − , T = τm . i K m τm Tm τ T 0.0808Tm + 0.719τm − 0.324 Tm τm m m 63 Kc = 0.148 + 0.186(Tm τm ) . K m (0.497 − 0.464V 0.590 ) 64 ±1% TS = 2.5Tm , 0.1 ≤ τ m Tm ≤ 0.4 ; ±1% TS = 5Tm , 0.5 ≤ τ m Tm ≤ 0.9 . 65 1.045 Kc = 57 ( 2 ) Tm τm Tm x1 + 2Tm x1 − 1 , Ti = K m (τm + Tm )(1 + x1τm ) K m 2 τ m 2 K m 2 x1 + + Tm x1 − 1 Tm Tm x1 (Tm + τ m ) . . b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 58 Rule Kc Chen and Seborg (2002). Chidambaram (2002), p. 155. Skogestad (2003). Model: Method 56 Comment Kc Ti Model: Method 1 67 Kc 0.413Tm + 0.8τm Model: Method 1 0.5Tm K m τm min (Tm ,8τ m ) Ti = 8τm (Faanes and Skogestad, 2004). 0 ≤ τm Tm ≤ ∞ Kc Ti Tm (x1K m τm ) Tm 68 69 Gorez (2003). Model: Method 1 Ti 66 Klán (2000). Model: Method 18 66 17-Mar-2009 70 (1 − x 2 )τm + Tm Tm x1K m τm τm ≤ τm ≤1 τ m + Tm 2 Kc = T τ + 2Tm TCL − TCL 1 Tm τm + 2Tm TCL − TCL 2 , , Ti = m m 2 Tm + τm Km (TCL + τm ) Tise − τ m s y 0 < TCL < Tm + Tm 2 + Tm τ m , . = 2 d desired K c (TCLs + 1) 67 2 τm 1 τm . Kc = 3.113 − 5.73 + 3.24 Km Tm Tm 68 Kc = 2 1 2τm (Tm + τm ) , T = τm + Tm 1 + 1 + 2 τm − τ . 1− i τ +T m K m 1 + 1 + 2τ 2 (τ + T )2 2 m m m m m 69 x1 = 2 2 1 1.5M s − 2 τm 5τ m 3− + 2 M s (M s − 1) 0.32M s (M s − 1) τm + Tm ( ) T M τ + m m s [ ] 2.5τ m 0.5 1 − − M s (M s − 1) ( ) T M τ + m m s ( 2 M s 2 −1 ) ; 0≤ 2 2 τm τm − 0.4 M s − 0.4 M s 0.1M s τ + Tm τ m + Tm . 70 x1 = 7.5 − m , x 2 = 1 − 0.5 Ms −1 1 − 0.4 M s 1 − 0.4 M s τm ≤ τm . τ m + Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Comment Klán and Gorez (2005a). Klán and Gorez (2005b). 71 Kc Ti Model: Method 1 72 Kc Ti Model: Method 1; 0 ≤ τm Tm ≤ ∞ . Skoczowski (2004). 73 Kc Tm Model: Method 1 74 Ti Model: Method 1; 0.1 ≤ τm Tm ≤ 1 . Sree et al. (2004), Chidambaram et al. (2005). Foley et al. (2005). 0.9179 Tm K m τm Haeri (2005). Model: Method 1 McMillan (2005), pp. 60–61. Model: Method 13 71 Kc = 0.14Tm e K m τm 0.8915 75 Kc Ti Model: Method 1 76 Kc 0.96Tm + 0.46τ m 0 ≤ τm Tm ≤ 4 0.25 K m Tm + 0.25τ m τm Tm > 1 2 τm τm 1.96 +1.65 τ m + Tm τ m + Tm , Ti = 5.38τ m e 2 τm τm − 3.94 +1.80 τ m + Tm τ m + Tm 59 . 2 0.5 τm (τm + Tm )2 + Tm 2 . T = Kc = 1 + 1 − , i K m τm + Tm 2(τm + Tm ) 2T ln (1 OS) 73 K c ≤ m 1 + 2x12 − 2 x1 1 + x12 , with x1 ≈ . 2 2 τm π + [ln (1 OS)] 72 74 2 τm τ τm , 0.1 ≤ m ≤ 0.4 ; − + Ti = Tm 10.59 2 . 3588 0 . 8985 T T T m m m 4 3 2 τ τ τ τ Ti = Tm 0.7719 m − 3.6608 m + 6.5791 m − 5.1652 m + 2.8059 , Tm Tm Tm Tm 0.4 ≤ (1 + 2T ω ) 1 + T ω , T = 1 + 2T ω )] . ω [T + τ (1 + T ω λK ω [T + τ (1 + T ω )] 2 75 Kc = m Kc = 2 m 2 90 0 m m m 2 90 0 2 m 76 2 90 0 2 i 90 0 1 6.282 . 0.407 + 1.195 K m 1 + 14.956(τm Tm ) 2 m 90 0 m m 2 90 0 2 m 2 90 0 τm ≤ 1.5 . Tm b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 60 Rule Trybus (2005). Model: Method 1 x1 Kc Ti Comment 0.6Tm K m τm 0.8τm + 0.5Tm x1Tm K m τm Tm Servo response: small overshoot, short settling time. 77 τm Tm ≤ 2 OS x1 OS x1 OS OS x1 x1 OS x1 OS 0.34 0% 0.54 5% 0.64 10% 0.74 15% 0.82 20% 0.90 25% x1 K m 0.5τm τm Tm ≥ 2 x1 Xu et al. (2005). OS x1 OS x1 OS OS x1 x1 OS x1 OS 0.17 0% 0.27 5% 0.32 10% 0.37 15% 0.41 20% 0.45 25% Model: Method 1 (0.3K m τm ) Tm 3.33τm 0.4 K m 79 Ti Kukal (2006). Model: Method 1 undefined 80 Ti 40% overshoot; unit step servo response. 0% overshoot; unit step servo response. 0.01 ≤ τm Tm ≤ ∞ Ti 0.01 ≤ τm Tm ≤ ∞ Mesa et al. (2006). Model: Method 1 Tm + 0.5(1 − x1 )τm K m τm (x1 + 1) Tm + 0.5(1 − x1 )τm Sample x1 values: 0.3, 0.5, 0.9. 78 Cuesta et al. (2006). Model: Method 1 81 Ti Kc Kc 77 The author quotes the following source for this rule: Findeisen, W. (Ed.), Control Engineering Handbook (2nd Edition). WNT, Warsaw, 1974 (in Polish). 1 100τm + 6.25Tm T (100τm + 6.25Tm )(1000τm + 463Tm ) 78 Kc = , Ti = m . (100τm + 46.25Tm )(5.2τm + 684Tm ) K m 100τm + 46.25Tm 79 Ti = 0.4Tm (1000τm + 463Tm ) . (5.2τm + 684Tm ) 80 Gc = 2 T T 1 , Ti = K m 2 m + 4 m + 1 e − x1 , x1 = τ Tis τm m 81 Kc = 2 T 1 Tm + 1 − 2 m e x 1 , x1 = 8 τ K m τm m 4(Tm τm )2 + 1 − 1 2(Tm τm ) 8(Tm τm )2 + 1 − 1 2(Tm τm ) 3 −1 . −2; 2 1.5 T T T T 24 m + 4 m + 8 m + 1 + 8 m + 1 τ τm τm τm Ti = τm m 3 T 8 m + 1 τ m . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Skogestad (2006a). Skogestad (2006b). Model: Method 1 Visioli (2006), p. 157. Vítečková (2006). Hougen (1979), p. 335. Model: Method 1; coefficients of K c estimated from graphs. Latzel (1988). Gp = Gm ; Model: Method 1 Latzel (1988). Gp = K pe − sτ p (1 + sTp1 ) 2 ; Model: Method 2 Latzel (1988). Gp = K pe − sτ p (1 + sTp1 )3 ; Model: Method 2 82 Kc ≥ ∆d 0 ∆y max 82 Kc 1 61 Comment Ti min[Tm ,4(TCL + τm )] Model: Method 1 min[Tm , 4Tm K m ] Slow control; ‘acceptable’ disturbance rejection performance. Model: Method 1 0.3Tm K m τm min[Tm ,8τm ] ≥ 0.368Tm K m τm Model: Method 9 Tm Direct synthesis: frequency domain criteria Maximise crossover x1 K m Tm frequency. τ m Tm x1 τ m Tm x1 τ m Tm x1 0.1 7.0 0.2 3.5 0.3 2.2 0.4 1.8 0.785Tm K m τm 0.5 0.6 0.7 1.4 1.1 1.0 0.8 0.9 1.0 0.8 0.75 0.7 Tm φm = 450 0.524Tm K m τm Tm φm = 600 0.45Tm K m (τm + 0.03Tm ) 0.57Tm φm = 450 0.30Tm K m (τm + 0.03Tm ) 0.57Tm φm = 600 0.42Tm K m (τm + 0.10Tm ) 0.53Tm φm = 450 0.28Tm K m (τm + 0.10Tm ) 0.53Tm φm = 600 ; also given by Klein et al. (1992). , τm ≤ TCL ≤ Tm ∆y max − τm ; ∆d 0 = maximum magnitude of a K m ∆d 0 sinusoidal disturbance over all frequencies, ∆y max = maximum controlled variable change, corresponding to the sinusoidal disturbance. b720_Chapter-03 Rule Kc Regulator – Gorecki et al. (1989). Model: Method 1 Regulator – Gorecki et al. (1989). Model: Method 1 Somani et al. (1992). Model: Method 1; suggested A m : 1.75 ≤ A m ≤ 3.0 . Kc = Comment Ti Kc Ti undefined 2K m (τm + Tm ) 83 x 2τm x 1 K m A m 84 τ m Tm x1 0.0375 0.1912 0.3693 0.5764 0.8183 1.105 1.449 36.599 7.754 4.361 3.055 2.369 1.950 1.672 Coefficient values: x2 τ m Tm 3.57 3.33 3.13 2.94 2.78 2.63 2.50 x1 1.869 2.392 3.067 3.968 5.263 7.246 10.870 1 1 + 3(Tm τm ) + 6(Tm τm )2 + 6(Tm τm )3 , Km 4 1 + 3(Tm τm ) + 3(Tm τm )2 [ ] τm 0.9806 < 1 , Kc ≈ Tm K mAm 1 + 3(Tm τm ) + 6(Tm τm )2 + 6(Tm τm )3 [ 3 1 + 2(Tm τm ) + 2(Tm τ m )2 4 ( 1.886 + (4τ T ) − 1.373) 2 m Kc = ] . + 1 or m 2 Kc ≈ 85 x2 1.476 2.38 1.333 2.27 1.226 2.17 1.146 2.08 1.086 2.00 1.042 1.92 1.011 1.85 0.1 ≤ τm Tm ≤ 1 Ti Kc Ti = τm For Low frequency part of magnitude Bode diagram is flat. G c = 1 (Tis ) Low frequency part of the magnitude Bode diagram is flat. 85 84 FA Handbook of PI and PID Controller Tuning Rules 62 83 17-Mar-2009 2 T T τ 0.9806 0.9806 1 + 1.886 m ; for m > 1 , K c ≈ 1 + 8.669 m . Tm KmAm KmAm τm τm 5Tm 1 Tm . 0.80 + 1.33 , Ti = K mAm τm 1.04[0.80 + 1.33(Tm τm )]2 − 1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 86 Lee et al. (1992). Model: Method 1 x1 = 0.5 + 0.207 Hang et al. (1993a), (1993b), pp. 61–65. O’Dwyer (2001a). Model: Method 1; τm > 0.3 Tm (Yang and Clarke, 1996). O’Dwyer (2001a). Model: Method 1 2.42 86 Kc = Kc Ti Comment Ti M s = 1.25 τm , τm Tm < 2.42 ; x1 = 1 , τm Tm ≥ 2.42 . Tm ωp Tm 87 Model: Method 37 or Method 41. Ti KmAm x 1Tm K mτm A m = π 2x 1 ; Tm φ m = π 2 − x 1 .88 Representative results φm x1 x1 Am Am φm 1.048 1.5 30 0 0.524 3 60 0 0.7854 2 450 0.393 4 x1 τm Tm Tm K u Tu 2 2 Tu + 4 π Tm 2 63 67.50 A m = π 2x 1 ; φm = π 2 − x 1 . τ τm τ T τ − 4 + 1.21 m + 4.4x1 m m + 2 − 2.2 x1 m τm Tm Tm Tm Tm , τm τm + 1.21 K m 4 + 4.4 x1 Tm Tm τ τm τ T τ − 4 + 1.21 m + 4.4 x1 m m + 2 − 2.2 x1 m τm Tm Tm Tm Tm Ti = . τm Tm τm + 2 − 2.2 x1 0.96 + 2.42 τm Tm Tm τ 1 87 Ti = ; m ≤ 0.3 (Yang and Clarke, 1996). 2 Tm 4ωp τ m 1 2ωp − + π Tm 2.42 88 Given A m , ISE is minimised when φ m = 68.8884 − 34.3534 A m + 9.1606 τm Tm for servo tuning (Ho et al., 1999); 2 ≤ A m ≤ 5 ; 0.1 ≤ τ m Tm ≤ 1 ; Given A m , ISE is minimised when φ m = 45.9848A m 0.2677 (τ m Tm )0.2755 for regulator tuning (Ho et al., 1999); 2 ≤ A m ≤ 5 ; 0.1 ≤ τ m Tm ≤ 1 . b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 64 Rule Kc x1 Am 0.50 3.14 0.61 2.58 0.67 2.34 0.70 2.24 Chen et al. (1999a). r1ωr Tm K m Symmetrical optimum principle – Voda and Landau (1995). Model: Method 1 90 φm Ms 61.4 0 1.00 0.72 2.18 48.7 0 1.30 55.0 0 1.10 0.76 2.07 46.50 1.40 51.6 0 1.20 0.80 1.96 0 1.50 50.0 0 1.26 44.1 Model: Method 1 3.5 G p ( jω1350 ) 4.6 ω1350 τm ≤ 0.1 Tm 1 4 2.828 G p ( jω1350 ) ω1350 89 0.1 < τm ≤ 0.15 Tm Kc Ti 0.15 < τm Tm ≤ 1 0.1751 1 ω1350 τm > 2 ; A m > 3. Tm G p ( jω1350 ) 1 ωr − 1.273τm ωr + (1 Tm ) Kc = φm Ti 90 Ti = Tm Coefficient values: Ms x1 Am 1 Friman and Waller (1997). Model: Method 1 Comment Ti x1Tm K m τm Chen et al. (1999b). Model: Method 1 89 FA 2 1 4.6 G p ( jω1350 ) − 0.6K m , ωr = , Ti = π 2 r1 A m − 1 . 4 τ m (r1 A m )2 − 1 1.15 G p ( jω1350 ) + 0.75K m [ ω135 0 2.3 G p ( jω1350 ) − 0.3K m ]. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Schaedel (1997), Henry and Schaedel (2005) – optimal servo response; T1 , T2 given in footnote 91. Ti 0.5Ti K m (T1 − Ti ) T1 − 2T2 0.375Ti K m (T1 − Ti ) T1 − 91 2 T1 = Tm 1 + 3.45 T1 = τ am 2 T2 T1 Ti 2 2 4T2 T 1 − 2 22 T1 T1 2 3.86T2 2 1.46T2 2 Ti 1 1 − 0 . 69 − 1 T1 T12 Km T 2 93 Kc 94 Kc Tm + τ m , T2 2 = 1.25Tm τ am + 1 + 3.45 2 + τ m , T2 = Tm τ am + a Tm − 0.7 τ m 0.7Tm 2 τ am 93 94 Minimum ITAE. Minimum φ m = 50 0 . Tm τ m + 0.5τ m 2 , 0 < τma ≤ 0.104 ; Tm Tm + 0.5τ m 2 , a Tm − 0.7 τ m 2 92 ‘Normal’ design – Butterworth filter. 2 3.11T2 2 1.43T2 2 ‘Sharp’ design – 1 Ti 1 − Tschebyscheff filter 0.7 − 1 2 T1 T1 K m T2 (0.1 dB). Tm 0.7Tm 92 2 1 Ti 0 . 5 − 1 Km T 2 Leva (2001). Model: Method 1 91 ‘Normal’ design – Butterworth filter. Model: Method 25 ‘Sharp’ design – Tschebyscheff filter (0.5dB). Model: Method 25 Minimum ITAE (Henry and Schaedel, 2005). Model: Method 26 2 2 0.375Ti K m (T1 − Ti ) Henry and Schaedel (2005) – optimal regulator response. Model: Method 26; T1 , T2 given in footnote 91; τam Tm < 0.2 . Comment Kc T Ti = −0.64T1 + 1.64T1 1 − 1.2 2 . T1 0.6981Tm 20Tm 20 . K c = min , , ωc = τ + 5Tm K τ K ( τ + 5 T ) m m m m m m 0.6981Tm 50Tm 20 . K c = min , , ωc = τ + 5Tm m K m τm K m (τm + 5Tm ) τ am > 0.104 . Tm 65 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 66 Rule Kc Potočnik et al. (2001). Model: Method 48 Cluett and Wang (1997), Wang and Cluett (2000), p. 177. Model: Method 1; τ 0.01 < m ≤ 10 . Tm Ti Comment 95 Kc Ti τm Tm ≤ 0.1 96 Kc Ti 0.1 < τm Tm < 0.15 97 Kc Ti 0.1 < τm Tm < 1 98 Kc Ti 99 Kc Ti 100 Kc Ti 7.50 ≤ A m ≤ 8.28 , 78.50 ≤ φm ≤ 79.70 . 4.35 ≤ A m ≤ 5.02 , 70.7 0 ≤ φm ≤ 74.00 . TCL = 1.33τ m , 3.29 ≤ A m ≤ 3.89 , 64.40 ≤ φm ≤ 70.60 . 101 95 Kc = 2.74 ≤ A m ≤ 3.30 , Ti Kc 57.80 ≤ φm ≤ 68.50 . 0.5 , G p ( jω1350 ) G p jω135 0 1.14 + 13.24 Km ( ) ( 96 Kc = ( K m − 2 G p jω1350 [ ) ( ) 2 G p jω1350 G p jω1350 + 17.40 Ti = 1 + 2.48 . 2 ω135 0 Km Km 4 ) 2K m K m − 2 G p ( jω135 0 ) ( ] , Ti = [K + 2 G (jω ) ][2 G (jω ) − K ] . ω G ( jω ) [ G ( jω ) − K ] m p 135 0 ) p p 135 0 135 0 p m 135 0 m 135 0 ( ) G p jω135 0 G jω 0 2K m , Ti = 1 0.75 + 1.15 p 135 . 0.75 + 1.15 Km Km ω135 0 ω135 0 0.099508τm + 0.99956Tm 0.019952τm + 0.20042Tm 98 τm , TCL = 4τ m . Kc = , Ti = K m τm 0.99747 τm − 8.7425.10−5 Tm 97 Kc = 99 Kc = 0.16440τm + 0.99558Tm 0.055548τm + 0.33639Tm τm , TCL = 2τ m . , Ti = K m τm 0.98607 τm − 1.5032.10− 4 Tm 100 Kc = 0.20926τm + 0.98518Tm 0.092654τm + 0.43620Tm τm . , Ti = K m τm 0.96515τm + 4.2550.10− 3 Tm 101 Kc = 0.24145τm + 0.96751Tm 0.12786τm + 0.51235Tm τm , TCL = τ m . , Ti = K m τm 0.93566τm + 2.2988.10− 2 Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Comment 2.39 ≤ A m ≤ 2.93 , Cluett and Wang (1997), Wang and Cluett (2000) – continued. 102 Kc Ti 103 Kc Ti 38.10 ≤ φm ≤ 66.30 . Modulus optimum principle – Cox et al. (1997). Model: Method 20 Smith (1998). Model: Method 1 104 Kc Ti τm Tm ≤ 1 0.5Tm K m τm Tm τm Tm > 1 0.35 Km 0.42τ m 6 dB gain margin – dominant delay process. Tm 1.3 ≤ M s ≤ 2 Wang et al. (2000a). Model: Method 37 or 38. 105 Kc 67 49.60 ≤ φm ≤ 67.10 . 2.11 ≤ A m ≤ 2.68 , 102 Kc = 0.26502τ m + 0.94291Tm 0.16051τm + 0.57109Tm τ m , TCL = 0.8τ m . , Ti = K m τm 0.89868τ m + 6.9355.10 − 2 Tm 103 Kc = 0.19067 τm + 0.61593Tm 0.28242τm + 0.91231Tm , Ti = τm , TCL = 0.67τ m . K m τm 0.85491τm + 0.15937Tm 104 Kc = 0.5 Tm 3 + Tm 2 τ m + 0.5Tm τ m 2 + 0.167 τm 3 , 2 2 3 K m Tm τm + Tm τm + 0.667 τm T 3 + Tm 2 τm + 0.5Tm τm 2 + 0.167 τm 3 . Ti = m 2 2 Tm + Tm τm + 0.5τm 105 1.508 Tm K c = 1.451 − . Ms K m τm b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 68 Rule Kc Wang and Shao (2000a). Model: Method 1 Chen and Yang (2000). 106 0.7Tm K m τm Comment Ti 1.5 ≤ x1 ≤ 2.5 Model: Method 8 Tm M s = 1.26 ; A m = 2.24 ; φ m = 50 0 . 0.25Tm K m τm 0.8Tm 0.2 ≤ τm Tm < 1 0.1Tm 0.15 + K m τm K m 0.3τ m + 0.5Tm τm >1 Tm 0.15 K m 0.3τ m τm Tm → ∞ Hägglund and Åström (2002). Model: Method 1 Leva et al. (2003). Model: Method 1 Kc = Kc Ti A m > x1 ; φm > cos −1 (1 x1 ) (Wang and Shao, 1999b). Hägglund and Åström (2002). Model: Method 5; M s = 1.4 . 106 17-Mar-2009 107 Kc Ti 108 Kc Tm M s = 1.4 ; τm ≥ 0.5 . Tm 1 1 , f 2 (ω90 0 ) − λf1 (ω90 0 ) ω90 0 f1 ( ω900 ) = (1 + ω Km 2 90 0 Tm ) 2 1.5 [(T + {1 + ω m − f 2 ( ω900 ) = − 1 1 + ω900 2 [(T + {1 + ω T m 2 2 900 Tm 2 }τ m ) sin( − ω900 τ m − tan −1 ω900 Tm ) (1 + ω 2 900 Km 2 Tm 2 900 ) 1.5 [ω T cos(−ω τ − tan ω T )], 2 900 m 900 2 −1 m Tm }τ m ) cot( − ω900 τ m − tan −1 ω900 Tm ) m [ 2 2 ] 2 2 2 2 m ω900 Tm 2 1 + ω900 2 Tm 2 ; 2 ω90 0 Tm + (1 + ω90 0 Tm )τm cos θ + (1 + 2ω90 0 Tm ) sin θ 3 900 ] − Ti = ] 2 − ω90 0 Tm cos θ + ω90 0 [Tm + (1 + ω90 0 Tm )τm ] sin θ , θ = − ω900 τ m − tan −1 ω900 Tm . 107 108 Kc = 6.8τ m Tm T 1 0.14 + 0.28 m , Ti = 0.33τ m + . Km τm 10τ m + Tm T (0.5π − φm ) 5Tm K c = minimum m , ; minimum φ m , maximum ωc . K m τm K m TCL b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Leva et al. (2003) – continued. Mataušek and Kvaščev (2003). Model: Method 1 Kristiansson and Lennartson (2002b). Model: Method 1 110 Comment Kc Tm minimum φ m = 50 0 . x 1Tm K mτm Tm x 1 = 0.5π − φ m .110 109 Kristiansson (2003). Model: Method 12 Kristiansson and Lennartson (2006). 109 Ti Kc 111 Kc 0.57Tm + 0.29τm 0.2 ≤ τm Tm ≤ 5 112 Kc Ti τm Tm < 1 113 Kc 0.60Tm + 0.28τm τm Tm ≥ 1 114 Kc 1.25 Tm 1.6 ≤ M s ≤ 1.9 115 Kc 1.4 τ m + 0.8Tm τm Tm < 0.2 69 Model: Method 12; 1.65 ≤ M max ≤ 1.80 . 0.698Tm τm + 5Tm 5x1Tm K c = minimum , , 4 ≤ x1 ≤ 10 . , TCL = x1 K m τm K m (τm + 5Tm ) ‘Acceptable’ values of 0 x1 : 0.5 ≤ x 1 ≤ 3 . ‘Recommended’ values of x1 : 0 0.32 ≤ x 1 ≤ 0.54 ( 59 ≤ φ m ≤ 72 ). Choose x 1 = 0.32 when large variations in process parameters are expected; choose x 1 = 0.54 to give 6% ≤ OS < 13% and 1.4 < M s ≤ 2 . 111 112 113 Kc = 0.57(0.57 + 0.29[τm Tm ]) , 1.65 ≤ M max ≤ 1.75 . K m ([τm Tm ] − 0.055) 2 τm 0.57 τm , Kc = 0.37 + 0.94 − 0.42 K m ([τm Tm ] − 0.055) Tm Tm Kc = 0.57(0.60 + 0.28[τm Tm ]) . K m ([τm Tm ] − 0.055) ω150 0 0.075 . 0.2 + K m K150 0 K m + 0.05 0.57(0.8Tm + 1.4τm ) 115 Kc = . K m Tm ([τm Tm ] − 0.055) 114 Kc = [ ] 2 τ τ Ti = Tm 0.37 + 0.94 m − 0.42 m . Tm Tm b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 70 Rule Kc Kristiansson and Lennartson (2006) – continued. Bai and Zhang (2007). Model: Method 59 Leva et al. (1994). Ti Comment 116 Kc τ m + Tm 0.2 ≤ τm Tm ≤ 1 117 Kc 0.6τ m + 1.2Tm τ m Tm ≥ 1 0.526Tm K m τm Tm φm = 59.80 . Tan et al. (1996). Hägglund and Åström (2004). Model: Method 1 Kc Ti Model: Method 54 119 Kc Ti Model: Method 36 120 Kc Ti 1.920 < φ < 2.356 ; M s = 1.2 . 121 Kc Ti φ = 2.09 radians (good choice). 0.57(Tm + τm ) . K m Tm ([τm Tm ] − 0.055) Kc = 117 Kc = 0.57(1.2Tm + 0.6τ m ) . K m Tm ([τ m Tm ] − 0.055) 118 Kc = ωcn Ti Km 1 + ωcn 2Tm 2 2 1 + ωcn Ti , Ti = 2 [ ] tan φm − 1.571 + τm ωcn + tan −1 (ωcn Tm ) ωcn with ωcn = Kc = A m = 2.98 , 118 116 119 FA ( ) x1Ti ωφ 1 + x1Tmωφ 2 ( A m 1 + x1Ti ωφ ) 2 , Ti = 2.82 − φm − tan −1[(Tm τm )(1.571 − φm )] . τm 1 , ωφ < ω u . x1ωφ tan − tan x1Tm ωφ − x1τm ωφ − φ [ ] −1 x1 = 0.8, τm Tm < 0.5 ; x1 = 0.5, τm Tm > 0.5 . 120 121 Kc = Kc = 6.283(1.72φ − 2.55) 2.01 − 0.80φ , Ti = . 2 G p jωφ G p jωφ 1 + (15.45 − 6.30φ) G p jωφ 1 + (3.44φ − 5.20 ) ωφ Km Km ( ) 0.33 ( ) G jω 1 + 2.26 p φ G p jωφ Km ( ) ( ) ( ) , Ti = 6.61 ( ) 2 G jω 1 + 2.01 p φ ωφ Km . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Hägglund and Åström (2004) – continued. Hägglund and Åström (2004). Xu et al. (2004). Model: Method 37 Ti Comment 122 Kc Ti 2.094 < φ < 2.443 ; M s = 1.4 . 123 Kc Ti φ = 2.269 radians (reasonable choice). 124 Kc Ti 2.356 < φ < 2.705 ; M s = 2.0 . 125 Kc Ti 126 Kc Ti Tm x1K m τm Tm φ = 2.53 radians (good choice). Model: Method 5; ‘AMIGO’ tuning rule. 1.5 ≤ x1 ≤ 2.5 x1 = 2 implies A m > 2, φm > 600 (Xu and Shao, 2004b). Huang et al. (2005a). Model: Method 51 0.50Tm + 0.22 τ m Kmτm 0.9Tm + 0.4τ m Huang and Jeng (2005). 0.495Tm + 0.22τm K m τm 0.9Tm + 0.4 τ m 122 123 Kc = Kc = Model: Method 1 6.283(1.72φ − 3.05) 2.50 − 0.92φ , Ti = . 2 G p jωφ G p jωφ 1 + (10.75 − 4.01φ) G p jωφ 1 + (3.44φ − 6.10 ) ωφ Km Km ( ) 0.41 ( ) ( ) , Ti = 5.36 . 2 G p jωφ 1 + 1.71 ωφ Km 6.283(0.86φ − 1.50) 3.43 − 1.15φ 124 , Ti = . Kc = 2 G p jωφ G p jωφ 1 + (11.30 − 4.01φ) G p jωφ 1 + (1.72φ − 2.90) ωφ Km Km 4 . 25 0 . 52 125 , Ti = . Kc = 2 G p jωφ G p jωφ 1 + 1.15 G p jωφ 1 + 1.45 ωφ Km Km ( ) G jω 1 + 1.65 p φ G p jωφ Km ( ) ( ) ( ) 126 Kc = ( ) ( ) ( ) ( ) ( ) 0.15 τm Tm Tm 6.7 τm Tm 2 + 0.35 − , Ti = 0.35τm + 2 . 2 K m (τm + Tm ) K m τm Tm + 2Tm τm + 10τm 2 71 b720_Chapter-03 Rule Kc Åström and Hägglund (2006), pp. 228–229. Åström and Hägglund (2006), pp. 258–260. Model: Method 5; detuned ‘AMIGO’ tuning rule. Clarke (2006). Model: Method 27 K c (1) = Ti Comment (1) Model: Method 5; ‘AMIGO’ tuning rule; M s = M t = 1.4 . (1) Ti ( 2) Ti ( 2) τm Tm > 0.11 ( 3) Ti ( 3) ( 4) Ti ( 4) τm < 0.11 Tm 127 Kc 128 Kc 129 Kc 130 Kc Tm τm 0.5 T 1+ m Km τm τ m Tm Tm 13τ m Tm 2 0.15 (1) + 0.35 − , T = 0 . 35 τ + . i m K m (τ m + Tm )2 K m τ m Tm 2 + 12Tm τ m + 7 τ m 2 K c (1) 128 129 FA Handbook of PI and PID Controller Tuning Rules 72 127 17-Mar-2009 Kc ( 2) = x1K c K c ( 3) ≥ (1) , x1 = 0.5,0.1 or 0.01; Ti ( 2) = Ti (1) K c ( 2) Ti (1) K m + 1 − M s −1 K c (1) K c ( 2) K m + 1 − M s −1 2K c (1) (τm + Tm ) 1 + −1 , 2 (1) (1) −1 M s M s M s + M s − 1 Ti 1 − M s + K c K m ( ) Ti (3) = Ti (1) 130 K c ( 4) < . − K c (3) K c (1) K m + 1 − M s 1 Kc (1) K c K m + 1 − M s −1 ( 3) . 2K c (1) (τm + Tm ) 1 + −1, 2 (1) 1 (1) Ms M s M s + M s − 1 Ti 1 − M s − + K c K m ( ) Ti ( 4) = 2K c (1) K m τm + Tm . 2 2 M s M s + M s + 1 K c ( 4) K m + 1 − M s −1 ( ) b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule 131 Ti Comment Kc Ti 0.2 ≤ τm Tm ≤ 10 ; Kc Ti Model: Method 1 x1Tm K m τm Ti Model: Method 40 Kc King (2006). Model: Method 1 131 Lavanya et al. (2006b). Padhy and Majhi (2006b). 132 Kc = 133 73 1.2 ≤ M s ≤ 2.0 . 2 3 4 τ τm τ τ 1 + x 2 log10 m + x 3 log10 m + x 4 log10 m + x 5 , x1 log10 T T T K m m Tm m m 2 2 x1 = −0.2236M s + 1.247 M s − 0.9838 , x 2 = 0.738M s − 3.8471M s + 3.0252 , 2 2 x 3 = −0.945M s + 4.8577 M s − 3.8272 , x 4 = 0.605M s − 3.2148M s + 2.5509 , 2 x 5 = −0.2998M s + 1.6155M s − 1.276 ; 2 τ 3 τ τ Ti = Tm x 6 m + x 7 m + x 8 m + x 9 , Tm Tm Tm x 6 = −0.001003M s − 0.001185 , x 7 = 0.0155M s + 0.0264 , 2 x 8 = −0.0772M s + 0.197 M s − 0.0677 , x 9 = 0.1257 M s + 0.8159 . 132 Kc = 33.3 Km Kc = Tm K m τm ∧ 2 ∧ 2 1 + ωu Tm 2 1 + ωu TCL 2 , τm > 1 ; ∧ 2 ∧ 2 1 + ωu Tm 2 1 + ωu TCL 2 , τm < 1 ; Ti = ∧ tan − τm ωu + π − φm , φm measured in radians. ω 1 ∧ u Tm x1 K m τm 2φm + π(A m − 1) 133 . x1 = ; Ti = A m x1 A m x1 1 2 Am2 − 1 + 1 − τm π Tm ( ) b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 74 Rule Kc Ti Comment Robust Tm λK m Tm 2Tm + τ m 2λ K m Tm + 0.5τ m Tm K m (τ m + λ) Tm Tm + 0.5τ m K m (λ + τ m ) Tm + 0.5τ m λ ≥ 1.7 τ m , λ > 0.1Tm . Rivera et al. (1986). λ ≥ 1 . 7 τ , λ > 0 . 2 T (Luyben, 2001); λ = 2τ (minimum ISE – Model: Method 1 m m m de Oliveira et al., 1995); λ = max[2.3τm ,0.15Tm ] (Visioli, 2005). Chien (1988). Model: Method 1 Brambilla et al. (1990). Model: Method 1 134 λ ≥ 1.7 τ m , λ > 0.1Tm . 134 No model uncertainty: λ = τ m , 0.1 ≤ τ m Tm ≤ 1 ; λ = [1 − 0.5 log10 (τ m Tm )]τ m , 1 < τ m Tm ≤ 10 . λ = Tm (Chien, 1988); (‘aggressive’ tuning) (Åström and Hägglund, 2006). λ = 3Tm (‘robust’ tuning) (Åström and Hägglund, 2006). λ = 2τ m (Bialkowski, 1996); (‘aggressive, less robust’ tuning) (Gerry, 1999). λ > Tm + τ m (Thomasson, 1997). 0.45τm ≤ λ ≤ 0.8τm (Huang et al., 1998). λ = 2(Tm + τ m ) (‘more robust’ tuning) (Gerry, 1999). λ = a. max(Tm , τ m ) : a > 3 (‘slow’ design), 2 ≤ a ≤ 3 (‘normal’ design), 1 ≤ a < 2 (‘fast’ design), a < 1 (‘faster’ design) (Andersson (2000), p. 11). Model: Method 10. 0.1Tm ≤ λ ≤ 0.5Tm , τm Tm ≤ 0.25 ; λ = 1.5(τm + Tm ),0.25 < τm Tm ≤ 0.75 ; λ = 3(τm + Tm ), τm Tm > 0.75 (Leva, 2001). Tm ≤ λ ≤ τm or τm ≤ λ ≤ Tm (Smith, 2002). λ = 1.7Tm (Faanes and Skogestad, 2004). λ = max[0.25τ m ,0.2T m ] (Leva, 2007). λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning), λ > [Tm ,8τm ] (‘moderate’ tuning), λ > [10Tm ,80τm ] (‘conservative’ tuning) (Arbogast et al., 2006). b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Pulkkinen et al. (1993). 0.7Tm K m τm λ (τ m + Tm ) x1 K m x 2 Tm Comment Ti Model: Method 1 Ogawa (1995). τm Model: Method 1; x1 x2 coefficients of K c , Ti Tm 0.9 1.3 deduced from graphs. 0.5 1.0 0.6 1.6 0.5 0.7 1.3 1.0 0.47 1.7 0.5 0.47 1.3 1.0 0.36 1.8 0.5 0.4 1.3 1.0 0.33 1.8 Thomasson (1997). τm Model: Method 1 2K (τ + λ) τm Tm x1 x2 2.0 10.0 2.0 10.0 2.0 10.0 2.0 10.0 0.45 0.4 0.4 0.35 0.32 0.3 0.3 0.29 2.0 7.0 2.2 7.5 2.4 8.5 2.4 9.0 Chen et al. (1997). Model: Method 31 minimum (Tm ,6τ m ) m 0.5τ m m 0.5Tm K m max( τ m ,0.5) λ = TCL . τ m > 0.5 3 τ m < 0.5 Desired closed loop 2 τm 2(λ + τ m ) K m (λ + τ m ) 20% uncertainty in process parameters. 33% uncertainty in process parameters. 50% uncertainty in process parameters. 60% uncertainty in process parameters. τ m >> Tm ; Lee et al. (1998). Model: Method 1 Tm + Isaksson and Graebe (1999). Model: Method 1 Chun et al. (1999). Model: Method 43 Tm + 0.25τ m K mλ Tm + 0.25τ m Tm > τ m ; λ not specified. Tm (τ m + 2λ ) − λ2 K m (τ m + λ ) Tm (τ m + 2λ ) − λ2 τ m + Tm Representative λ = 0.4Tm . Tm K m (λ + τm ) Tm x1 K m x 2 (τm + Tm ) 2 Zhong and Li (2002). Model: Method 1 Wang (2003), p. 14. Model: Method 1. Regulator; x1 , x 2 deduced from graphs; robust to ±25% uncertainty in process model parameters. Tm + τm2 2(λ + τ m ) response = e − τ ms . (λs + 1) ∆τm ≤ λ ≤ 1.4∆τm ; x1 x2 τ x1 x2 τ 5.1 4.5 3.6 2.9 2.3 1.8 1.4 0.23 0.34 0.44 0.50 0.57 0.60 0.63 0.05 0.11 0.18 0.25 0.33 0.43 0.54 1.1 0.9 0.8 0.8 0.7 0.6 0.5 0.65 0.64 0.63 0.62 0.61 0.58 0.56 0.67 0.82 1.0 1.22 1.5 1.9 2.3 ∆τ m = time delay uncertainty. τ = τm Tm τ x1 x2 0.4 0.4 0.4 0.4 0.3 0.50 0.50 0.46 0.44 0.42 3.0 4.0 5.7 9.0 19.0 75 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 76 Rule Kc Wang (2003), p. 13. Model: Method 1. Servo; ±10% overshoot, step input. x1 , x 2 deduced from graphs; robust to ±25% uncertainty in process model parameters x1 K m Wang (2003), pp. 50–54. Model: Method 1. Servo; ±10% overshoot, step input. Wang (2003), pp. 50–54. Model: Method 1. Regulator; x1 deduced from graph; robust to ±25% uncertainty in process model parameters. Åström and Hägglund (2006), p. 187. Comment Ti x 2 (τm + Tm ) τm Tm x1 x2 τm Tm x1 x2 τm Tm 0.79 0.05 0.82 0.11 0.81 0.18 0.70 0.25 0.75 0.33 0.72 0.43 0.73 0.54 Tm K m (τ m + λ) 0.81 0.68 0.65 0.59 0.50 0.43 0.38 0.67 0.63 0.63 0.60 0.55 0.51 0.47 Tm 0.67 0.82 1.0 1.22 1.5 1.9 2.3 0.34 0.31 0.29 0.29 0.27 0.43 0.43 0.39 0.39 0.38 3.0 4.0 5.7 9.0 19.0 x1 x2 2.94 2.73 2.20 1.65 1.29 1.04 0.85 λ = 0.0132(τm + Tm ) , τm Tm ≤ 0.05 ; λ = 0.286τm + 0.0034(τm + Tm ) , 0.05 ≤ τm Tm ≤ 19.0 ; λ = 0.27(τm + Tm ) , τm Tm ≥ 19.0 ; robust to ±25% uncertainty in process model parameters. Tm λ = x1 (τm + Tm ) Tm K m (τ m + λ) x1 τm Tm x1 0.05 0.08 0.11 0.11 0.18 0.12 0.25 0.13 0.33 0.15 Tm K m (τ m + λ) 0.43 0.54 0.67 0.82 1.00 0.16 0.18 0.19 0.21 0.22 x1 0.01 0.03 0.04 0.06 0.07 τm Tm τm Tm x1 τm Tm 1.22 0.25 4.0 1.5 0.26 5.7 1.9 0.27 9.0 2.3 0.27 19.0 3.0 Model: Method 1; delay dominated max [Tm ,3τm ] processes. λ = Tm (aggressive tuning); λ = 3Tm (robust tuning). Ultimate cycle McMillan (1984). Model: Method 2 or Method 55. 135 Kc Ti Tuning rules developed from K u , Tu . 0.65 T 1.881 Tm 1 135 m = τ + T 1 . 67 1 Kc = , i m 0.65 . K m τm T Tm + τm m 1 + Tm + τm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 77 Comment Ti x 1K u x 2 Tu Kuwata (1987). x1 x2 x1 x2 x1 x2 τm τm τm Model: Method 1; T T Tu x 1 , x 2 deduced from u u 0.29 0.84 0.28 0.24 0.24 0.35 0.225 0.17 0.43 graphs. 0.27 0.46 0.30 0.23 0.20 0.38 0.225 0.155 0.45 0.25 0.31 0.33 0.225 0.18 0.40 0.225 0.115 0.48 0.654 Boe and Chang τ (1988). Quarter decay ratio. 1.935Tm m 0.54 K u Tm Model: Method 1 τ 0.876 Claimed equivalent to Hill and Adams 1.36 − 0.245 m 0.961 Tm Tm Ziegler–Nichols (1988). 0 . 83 τ e m K m τm ultimate cycle Model: Method 1 method. Geng and Geary 0.71Tm τm < 0.167 (1993). 3.33τ m K mτm Tm Model: Method 1 Luyben (1993). 0.225K u 0.415Tu τm Tm = 0.05 Model: Method 1 ^ ^ Perić et al. (1997). Model: Method 49 Matsuba et al. (1998). Model: Method 1 Huang et al. (2005a). Model: Method 51 K u tan 2 φ m 0.1592 Tu tan φ m 1 + tan 2 φ m 0.99 Tm K m τ m 136 0.9 Kc τ 2.63τ m m Tm Ti 0.097 0.1 ≤ τm ≤ 1.0 Tm 2 ^ 0.9 K u − 1 0.55 136 + Kc = 0 . 4 , 2 K m ^ −1 π − tan K u − 1 2 2 ^ ^ −1 Ti = 0.159 Tu 0.9 K u − 1 + 0.4π − tan K u − 1 . ^ b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 78 1 + Td s 3.3.2 Ideal PID controller G c (s) = K c 1 + Ti s E(s) R(s) + U(s) 1 K c 1 + + Td s Ti s − Y(s) K m e − sτ 1 + sTm m Table 10: PID controller tuning rules – FOLPD model G m (s) = Rule Callender et al. (1935/6). Model: Method 1 Ziegler and Nichols (1942). Model: Method 2 Chien et al. (1952) – regulator. Model: Method 2 Chien et al. (1952) – servo. Model: Method 2 Cohen and Coon (1953). Model: Method 2 Kc 1 1.066 K mτm Ti K m e − sτ m 1 + sTm Comment Td Process reaction 0.353τ m or 1.418τ m 0.47τ m τm = 0.3 Tm x1Tm K m τm 2τ m 0.5τ m 0.95Tm K mτm 2.38τ m 0.42τ m 1.2 ≤ x1 ≤ 2 ; quarter decay ratio. 0% overshoot; 0.1 < τm Tm < 1 . 1.2Tm K mτm 2τ m 0.42τ m 20% overshoot; 0.1 < τm Tm < 1 . 0.6Tm K mτm Tm 0.5τ m 0% overshoot; 0.1 < τm Tm < 1 . 0.95Tm K mτm 1.36Tm 0.47τ m 20% overshoot; 0.1 < τm Tm < 1 . Ti 0.37 τm 1 + 0.19(τm Tm ) 2 Kc 1 Decay ratio = 0.043; period of decaying oscillation = 6.28τ m . 2 Kc = 2.5(τm Tm ) + 0.46(τm Tm ) 2 1 T . 1.35 m + 0.25 , Ti = Tm Km τm 1 + 0.61(τm Tm ) Quarter decay ratio; 0.1 < τm Tm ≤ 1 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Comment Td Ti 0.950 0.738 Three constraints K c K m Td Tm τ m 1.370 Tm = 0.5 ; 3 method – Murrill T Tm d 1.351 Tm (1967), p. 356. K m τ m 0.1 < τm Tm < 1 . Model: Method 5 Quarter decay ratio; minimum integral error (servo mode). Åström and Model: 3 Km T90% 0.5τ m Hägglund (1988), Method 28 pp. 120–126. T90% = 90% closed loop step response time (Leeds and Northrup Electromax V controller). Parr (1989), Model: Method 2 1.25Tm K m τm 2.5τ m 0.4 τ m p. 194. Borresen and Model: Method 2 Tm K m τm 3τ m 0.5τ m Grindal (1990). 4 Sain and Özgen Model: Ti Td Kc (1992). Method 29 Connell (1996), Approximate 1.6Tm p. 214. quarter decay 1.6667 τ m 0.4 τ m K mτm Model: Method 2 ratio. x1 K m x 2τm x 3τ m Hay (1998), pp. Coefficient values – servo tuning: 197,198. τ T x x2 x3 τm Tm x1 x2 x3 m m 1 Model: 0.125 6.5 9.0 0.47 0.125 5.0 9.8 0.34 Method 1; 5.0 6.5 0.45 0.167 3.7 7.0 0.32 coefficients of 0.167 0.25 3.5 4.2 0.40 0.25 2.5 4.8 0.30 K c , Ti , Td 0.5 1.8 2.2 0.35 0.5 1.1 2.5 0.27 deduced from Coefficient values – regulator tuning: graphs. τm Tm x1 x2 x3 τm Tm x1 x2 x3 0.125 0.167 0.25 0.5 3 4 τ Td = 0.365Tm m Tm Kc = 9.8 7.2 4.5 2.2 2.0 1.8 1.6 1.4 0.52 0.50 0.45 0.35 0.1 0.125 0.167 0.25 0.5 10.0 8.0 6.0 4.0 2.0 2.2 2.2 2 1.8 1.4 0.35 0.35 0.34 0.32 0.28 0.950 . 0.8647Tm + 0.226τm 0.0565Tm 1 T 0.6939 m + 0.1814 , Ti = , Td = . T T Km τm m + 0.8647 0.8647 m + 0.226 τm τm 79 b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 80 Rule Hay (1998), p. 200. Model: Method 2; coefficients of K c , Ti , Td deduced from graphs. Comment Kc Ti Td x1 K m x 2 τm x 3τm x1 x2 x3 4.7 2.4 2.8 1.5 1.8 0.5 1.2 0.8 1.0 0.7 1.2Tm K m τm 0.87 1.87 0.95 1.55 1.00 1.33 1.00 1.15 0.95 1.00 1.32 1.28 1.07 1.0 0.90 0.78 0.78 0.64 0.72 0.54 τm Tm 0.3 0.4 0.5 0.6 0.7 Moros (1999). Model: Method 1 1.2Tm K m τm 2τ m 0.42τ m 2τ m 0.44τ m Lipták (2001). 1.6τ m 0.6τ m 2.4τm 0.38τm Chidambaram (2002), p. 154. Alfaro Ruiz (2005a). ControlSoft Inc. (2005). PMA (2006). Model: Method 2 5 FA Kc = 0.85Tm K m τm 5 Kc Regulator. Servo. Regulator. Servo. Regulator. Servo. Regulator. Servo. Regulator. Servo. Attributed to Oppelt. Attributed to Rosenberg. Model: Method 2 Model: Method 1. ‘Improved’ Ziegler–Nichols method. x1Tm K m τm 2.5τm 0.4τm 2.0 ≤ x1 ≤ 3.33 ; Model: Method 2 6 Ti Td 1.0 ≤ x1 ≤ 1.67 ; Kc Model: Method 1 7 Model: 2 Km τ m + Tm Td Method 11 2τm 0.59Tm 2τm K m τm Alternative. τm 1 Tm + 0.45 . 1.8 K m τm τ τ Tm 0.40 + 0.153 m Tm 2.50 + 0.957 m Tm Tm x T 6 , Td = . K c = 1 1.56 m + 0.68 , Ti = ( ) 1 + 0 . 787 τ T Km τm ( 1 0 . 787 T + τ m m m m) 7 τ T τ T Td = max m , m (slow loop); Td = min m , m (fast loop). 3 6 3 6 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Td Comment Minimum performance index: regulator tuning 0.921 0.749 Minimum IAE – Tm τ m 1.435 Tm 8 Model: Method 5 Murrill (1967), Td K m τm 0.878 Tm pp. 358–363. Minimum IAE – 12.52 K m 0.20Tm 0.04Tm τm Tm = 0.10 Lopez (1968), 2.68 K m 0.69Tm 0.22Tm τm Tm = 0.50 p. 50. 1.47 K m 1.04Tm 0.42Tm τm Tm = 1.00 Minimum IAE – Marlin (1995), pp. 301–307. Model: Method 8 Minimum IAE – Peng and Wu (2000). Model: Method 21 1.3 K m 0.7 τ m 0.01τ m τ m Tm = 0.43 1.0 K m 0.68τ m 0.05τ m τ m Tm = 0.67 0.85 K m 0.65τ m 0.08τ m τ m Tm = 1.0 0.65 K m 0.65τ m 0.10τ m τ m Tm = 1.5 0.45 K m 0.55τ m 0.14τ m τ m Tm = 2.33 0.4 K m 0.50τ m 0.21τ m τ m Tm = 4.0 Coefficients of K c , Ti , Td estimated from graphs; robust to ±25% process model parameter changes. 6.558K m 0.925Tm 0.061Tm τ m Tm = 0.1 2.311K m 0.677Tm 0.189Tm τ m Tm = 0.2 1.479 K m 0.633Tm 0.233Tm τ m Tm = 0.3 1.117 K m 0.647Tm 0.250Tm τ m Tm = 0.4 0.947 K m 0.705Tm 0.268Tm τ m Tm = 0.5 0.825K m 0.752Tm 0.289Tm τ m Tm = 0.6 0.731K m 0.792Tm 0.312Tm τ m Tm = 0.7 0.695K m 0.871Tm 0.316Tm τ m Tm = 0.8 0.630K m 0.895Tm 0.326Tm τ m Tm = 0.9 0.651K m 1.030Tm 0.306Tm τ m Tm = 1.0 1.137 8 τ Td = 0.482Tm m Tm , 0.1 ≤ τm ≤1. Tm 81 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 82 Rule 9 Minimum IAE – Kosinsani (1985), pp. 46–48. Model: Method 1 Kc Ti Td Comment Kc Ti Td Load disturbance model = 1 . 1 + sTL 0.05 ≤ τm T ≤ 1.0 ; 0.05 ≤ L ≤ 7.0 . Tm Tm Values of x1 to x 15 for varying τm Tm provided. The controller tuning formula is not recommended when τm Tm = 0.4, TL Tm > 6.5 ; τm Tm = 0.5, TL Tm > 5.5 ; τm Tm = 0.75, TL Tm > 5.0 ; τm Tm = 1.0, TL Tm > 4.5 . 0.75 1.0 x1 27.567 18.817 13.984 9.3318 7.0210 4.7110 3.6156 2.9398 2.0164 1.5805 x2 3.2174 1.2714 7.6166 4.9233 3.9864 2.9383 2.2643 1.8975 2.0290 0.6154 x3 -2.3336 -1.8823 6.8896 4.5267 3.2537 2.0947 1.5546 1.2108 0.7548 0.5345 x4 -31.416 -31.416 -1.9E-6 -3.4E-7 -1.0E-5 -5.6E-8 -9.0E-9 -9.5E-9 0.4277 0.3603 x5 -7.2662 -6.2990 -9.9212 -0.0978 1.9228 1.6537 2.3143 1.5861 1.5845 -0.6779 x6 7.8891 3.7533 1.1676 0.9631 1.1989 0.5403 2.5070 2.6381 0.6400 1.1821 x7 8.1491 8.0893 8.5751 6.3996 4.9687 4.0498 7.8855 0.7385 2.5128 2.6522 x8 -0.4706 -0.5880 -0.6459 -0.6104 -0.4955 -0.5196 -0.0233 -0.2809 -0.1774 -0.0445 x9 14.020 11.353 5.2382 1.7098 0.6395 0.1405 1.7941 3.5610 0.0825 x 10 -3.6987 -3.4514 -1.7079 -1.3466 -1.3080 -0.2366 -0.5171 -0.5509 -0.0100 -0.4917 τm Tm 9 0.05 0.075 0.1 0.15 0.2 0.3 0.4 0.5 0.6528 x 11 4.6405 3.5079 3.1794 3.5195 7.5219 1.8090 0.5537 0.1952 1.3773 x 12 -0.8986 -0.4818 -0.2947 -0.4508 -2.7213 -4.4931 -1.8157 -2.5837 -4.7205 -2.3611 x 13 0.0270 -1491.0 -4829.2 -1224.6 -13635 -31129 -31539 -40901 -52408 x 14 -0.0090 1491.1 4829.3 1224.7 52408 59325 x 15 -4.1759 6.2E-7 3.8E-7 3.2E-6 5.0E-7 3.9E-7 5.6E-7 5.7E-7 6.8E-7 7.5E-7 13635 31130 31539 40901 0.5675 -59324 T T 1 x1 − e − x 2 TL Tm x 3 cosx 4 L + x 5 sin x 4 L , Km Tm Tm Tm , Td = Tm x13 + x14e x15 TL Tm . Ti = TL x 8 Tl Tm x 10 TL Tm x6 + x7 1 − e sin x11 + x 9e + x12 Tm Kc = ( ) ( ) b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Comment Td Ti Load disturbance Minimum IAE – x1 K m x 2Tm x 3Tm 1 model = . Hill and Venable 1 + sTL (1989). Model: Method 1 x1 values (this page), x 2 values (p. 84) and x 3 values (p. 85) deduced from graphs: robust to ±30% process model parameter changes. 0.05 0.07 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0 τm Tm TL Tm = 0.05 19.0 11.8 8.6 TL Tm = 0.1 21.2 13.6 9.8 TL Tm = 0.3 22.0 14.6 10.9 TL Tm = 0.5 22.2 14.7 11.0 TL Tm = 0.7 22.1 14.7 TL Tm = 1.0 22.0 14.8 TL Tm = 1.2 22.2 TL Tm = 1.5 5.5 4.2 2.88 2.20 1.80 1.32 1.08 6.1 4.5 3.00 2.26 1.83 1.33 1.09 7.1 5.3 3.44 2.58 2.06 1.40 1.12 7.2 5.5 3.66 2.76 2.22 1.51 1.19 11.0 7.3 5.6 3.76 2.85 2.33 1.59 1.24 11.1 7.4 5.7 3.81 2.94 2.39 1.66 1.32 14.9 11.2 7.4 5.7 3.84 2.95 2.41 1.70 1.35 22.4 15.0 11.3 7.5 5.8 3.88 2.98 2.43 1.72 1.37 TL Tm = 2.0 22.5 15.0 11.2 7.5 5.8 3.91 3.00 2.45 1.74 1.40 TL Tm = 2.5 22.6 15.2 11.2 7.6 5.9 3.93 3.01 2.46 1.75 1.40 TL Tm = 3.0 22.8 15.0 11.2 7.7 5.9 3.95 3.02 2.48 1.76 1.40 TL Tm = 3.5 22.7 15.1 11.0 7.5 5.9 3.94 3.02 2.48 1.75 1.40 TL Tm = 4.0 22.7 15.0 10.9 7.6 5.9 3.94 3.02 2.48 1.76 1.40 TL Tm = 4.5 22.7 15.1 11.2 7.6 5.9 3.95 3.03 2.48 1.77 1.41 TL Tm = 5.0 22.6 15.1 11.2 7.5 5.9 3.94 3.03 2.50 1.78 1.42 TL Tm = 5.5 22.7 15.1 11.3 7.6 5.9 3.94 3.03 2.51 1.79 1.42 TL Tm = 6.0 22.6 15.0 11.5 7.6 5.9 3.94 3.03 2.51 1.80 1.44 TL Tm = 6.5 22.6 15.0 11.3 7.5 5.8 3.94 3.04 2.52 1.80 1.45 TL Tm = 7.0 22.7 14.9 11.5 7.6 5.9 3.95 3.05 2.54 1.81 1.46 83 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 84 Rule Kc Comment Td Ti Load disturbance Minimum IAE – 1 x1 K m x 2Tm x 3Tm model = . Hill and Venable 1 + sTL (1989). Model: Method 1 x1 values (p. 83), x 2 values (this page) and x 3 values (p. 85) deduced from graphs: robust to ±30% process model parameter changes. 0.05 0.07 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0 τm Tm 1.0 1.0 1.0 1.0 1.0 0.92 0.90 0.90 0.82 0.78 TL Tm = 0.1 1.0 1.0 1.0 TL Tm = 0.3 10.5 6.6 4.6 1.0 1.0 0.92 0.90 0.90 0.82 0.78 2.3 1.1 1.00 0.96 0.92 0.82 0.79 TL Tm = 0.5 10.7 7.0 5.1 3.2 2.3 1.48 1.18 1.04 0.86 0.80 TL Tm = 0.7 10.4 TL Tm = 1.0 9.7 7.0 5.2 3.4 2.7 1.80 1.40 1.20 0.95 0.82 6.7 5.1 3.5 2.8 2.00 1.62 1.40 1.06 0.92 TL Tm = 1.2 10.3 7.0 5.4 3.8 2.9 2.14 1.75 1.48 1.13 0.96 TL Tm = 1.5 11.0 7.6 5.8 4.0 3.1 2.32 1.86 1.59 1.20 1.00 TL Tm = 2.0 12.3 8.4 6.5 4.6 3.6 2.60 2.10 1.79 1.30 1.06 TL Tm = 2.5 13.1 9.0 7.0 4.9 3.8 2.80 2.26 1.90 1.40 1.15 TL Tm = 3.0 13.7 9.6 7.3 5.2 4.1 3.00 2.39 2.01 1.50 1.22 TL Tm = 3.5 14.1 9.8 7.8 5.4 4.2 3.12 2.50 2.13 1.57 1.26 TL Tm = 4.0 14.6 10.2 8.0 5.6 4.4 3.21 2.59 2.20 1.61 1.30 TL Tm = 4.5 15.0 10.5 8.0 5.7 4.5 3.32 2.66 2.23 1.63 1.33 TL Tm = 5.0 15.4 10.8 8.2 5.9 4.7 3.40 2.72 2.30 1.65 1.35 TL Tm = 5.5 15.6 10.9 8.3 6.0 4.8 3.45 2.76 2.33 1.67 1.37 TL Tm = 6.0 15.9 11.1 8.3 6.1 4.9 3.50 2.79 2.36 1.70 1.39 TL Tm = 6.5 16.1 11.2 8.6 6.2 5.0 3.54 2.80 2.38 1.70 1.40 TL Tm = 7.0 16.4 11.5 8.5 6.2 5.0 3.57 2.83 2.40 1.72 1.40 TL Tm = 0.05 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Comment Td Load disturbance Minimum IAE – 1 x1 K m x 2Tm x 3Tm model = . Hill and Venable 1 + sTL (1989). Model: Method 1 x1 values (p. 83), x 2 values (p. 84) and x 3 values (this page) deduced from graphs: robust to ±30% process model parameter changes. 0.05 0.07 0.1 0.15 0.2 0.3 0.4 0.5 0.75 1.0 τm Tm TL Tm = 0.05 0.022 0.034 0.045 0.067 0.089 0.13 0.17 0.21 0.30 0.39 TL Tm = 0.1 0.020 0.030 0.042 0.064 0.088 0.13 0.17 0.21 0.30 0.40 TL Tm = 0.3 0.027 0.038 0.050 0.068 0.081 0.13 0.17 0.21 0.31 0.41 TL Tm = 0.5 0.027 0.039 0.051 0.074 0.094 0.13 0.17 0.22 0.32 0.41 TL Tm = 0.7 0.026 0.038 0.050 0.075 0.097 0.14 0.18 0.22 0.32 0.41 TL Tm = 1.0 0.024 0.037 0.050 0.073 0.097 0.14 0.19 0.23 0.33 0.42 TL Tm = 1.2 0.026 0.038 0.051 0.076 0.100 0.14 0.19 0.23 0.34 0.43 TL Tm = 1.5 0.027 0.040 0.052 0.079 0.104 0.15 0.20 0.24 0.34 0.44 TL Tm = 2.0 0.028 0.041 0.057 0.083 0.110 0.16 0.21 0.26 0.37 0.46 TL Tm = 2.5 0.028 0.042 0.058 0.085 0.113 0.17 0.22 0.27 0.38 0.48 TL Tm = 3.0 0.029 0.045 0.059 0.088 0.118 0.17 0.23 0.28 0.40 0.50 TL Tm = 3.5 0.029 0.045 0.062 0.090 0.120 0.18 0.23 0.28 0.41 0.52 TL Tm = 4.0 0.029 0.046 0.064 0.090 0.122 0.18 0.24 0.29 0.42 0.52 TL Tm = 4.5 0.030 0.045 0.062 0.092 0.123 0.18 0.28 0.30 0.42 0.52 TL Tm = 5.0 0.030 0.044 0.062 0.095 0.125 0.18 0.24 0.30 0.42 0.52 TL Tm = 5.5 0.030 0.047 0.062 0.094 0.126 0.19 0.24 0.30 0.42 0.52 TL Tm = 6.0 0.031 0.048 0.061 0.094 0.128 0.19 0.25 0.30 0.42 0.52 TL Tm = 6.5 0.031 0.048 0.063 0.095 0.133 0.19 0.25 0.30 0.42 0.52 TL Tm = 7.0 0.031 0.049 0.062 0.095 0.130 0.19 0.25 0.30 0.42 0.52 85 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 86 Rule Kc Minimum IAE – Arrieta Orozco (2006). 10 Kc Ti Td Ti Td Comment 0.1 ≤ τm ≤ 2.0 Tm pp. 23–24. Model: Method 10 Minimum IAE – τ 0.1 ≤ m ≤ 1.0 Pessen (1994). 0.7 K u 0.4Tu 0.149Tu Tm Model: Method 1 0.749 0.921 Modified Tm τ m 3 Tm 11 minimum IAE – Model: Td 0.878 Tm Method 14 Cheng and Hung K m τ m (1985). 1.006 0.1 ≤ τ 0.945 0.771 Minimum ISE – m Tm ≤ 1 ; τ Tm τ m 1.495 Tm 0.56Tm m Murrill (1967), Model: Method 5 K m τm 1.101 Tm Tm pp. 358–363. Minimum ISE – 13.41 K m 0.14Tm 0.05Tm τm Tm = 0.10 Lopez (1968), 2.84 K m 0.55Tm 0.28Tm τm Tm = 0.50 p. 50. 1.52 K m 0.86Tm 0.56Tm τm Tm = 1.00 0.970 0.753 τ 1.473 Tm Tm τ m 0.1 ≤ m ≤ 1.0 12 Minimum ISE – T T d m 1115 . Zhuang (1992), K m τ m Tm Zhuang and 0.735 0.641 τ Tm τ m Atherton (1993). 1.524 Tm 1.1 ≤ m ≤ 2.0 13 T T d m K m τm 1.130 Tm Model: Method 1 or Method 5 or Method 6 or Method 15. 10 T 1 Kc = 0.2068 + 1.1597 m Km τm 1.0158 0.5022 , Ti = Tm − 0.2228 + 1.3009 τm , T m τ Td = 0.3953Tm m Tm 1.137 11 12 13 τ Td = 0.482Tm m Tm τ Td = 0.550Tm m Tm 0.948 τ Td = 0.552Tm m Tm 0.851 . . . 0.8469 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Minimum ISE – 14 Ho et al. (1998). Ti Kc Model: Method 1 0.947 0.738 Minimum ITAE Tm τ m 1.357 Tm – Murrill (1967), K m τm 0.842 Tm pp. 358–363. Minimum ITAE 11.80 K m 0.22Tm – Lopez (1968), 2.59 K m 0.75Tm p. 50. 1.39 K m 1.15Tm Minimum ITAE – Arrieta Orozco (2006), pp. 90–92. 14 16 0.116 τm 1.0722 φm − Kc = 0.8432 K m Am Tm −0.908 1.2497Tm φm A m 0.2099 Td = 300 ≤ φm ≤ 600 . Td 0.1 ≤ τm Tm ≤ 1 ; Model: Method 5 0.03Tm τm Tm = 0.10 0.19Tm τm Tm = 0.50 0.37Tm τm Tm = 1.00 Td 2 ≤ Am ≤ 4 ; Model: Method 10 15 1.0082 , Ti = 2 ≤ Am ≤ 5 , Td Ti Kc Comment Td Ti τm T m 0.3678 0.4763Tm φm − Given A m , ISE is minimised when φ m = 46.5489A m 87 Am 0.2035 0.0961 , 0.328 1.0317 τm T m , 0. 1 ≤ τm ≤ 1.0 ; Tm (τ m Tm )0.3693 ; 2 ≤ A m ≤ 5 ; 0.1 ≤ τ m Tm ≤ 1.0 (Ho et al., 1999). 15 16 τ Td = 0.381Tm m Tm 0.995 . Tuning rule continued onto p. 88. −0.8757 − 0.0851A m + 0.0120 A m 2 τm 1 2 , Kc = 0.6791 − 0.2272A m + 0.0253A m + x1 Km Tm 2 x1 = 1.9742 − 0.7126A m + 0.0813A m , 0.1 ≤ τm Tm ≤ 2.0 ; −5.1852 + 4.4910 A m − 0.7891A m 2 τm 2 , Ti = Tm 7.1461 − 4.8748A m + 0.6843A m + x 2 Tm 2 x 2 = −5.6764 + 4.7562A m − 0.6850A m , 0.1 ≤ τm Tm < 1.0 ; 0.6969 + 0.0948 A m − 0.0231A m 2 τm 2 , Ti = Tm 0.2931 + 0.2270A m − 0.0427 A m + x 2 Tm 2 x 2 = 0.9924 − 0.2202A m + 0.0217 A m , 1.0 ≤ τm Tm ≤ 2.0 . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 88 Rule Kc Minimum ISTSE – Zhuang (1992), Zhuang and Atherton (1993). Ti Td 17 Kc Ti Td 18 Kc Ti Td Comment Model: Method 1 or Method 5 or Method 6 or Method 15. Model: Method 1 Ti 0.144Tu K 19 20 c 0.960 Model: Method 37 τ 0.1 ≤ m ≤ 1.0 Tm ∧ Ti Kc 0.15 T u 0.746 Tm τ m 1.531 Tm 23 Minimum ISTES Td K τ 0 .971 Tm m m – Zhuang (1992), 0.705 0.597 Zhuang and τ Tm τ m 1.1 ≤ m ≤ 2.0 Atherton (1993). 1.592 Tm 24 T T d m K m τm 0.957 Tm Model: Method 1 or Method 5 or Method 6 or Method 15. ( 1.8232 − 0.5092 A m − 0.0721A m 2 2 τm ( ) ) Td = Tm 0.4718 − 0.1251A m + 0.0202A m Tm Kc = 1.468 Tm K m τm 0.970 , Ti = 0.730 Tm τm 0.942 Tm 0.725 τm < 1.0 ; Tm , 1.0 ≤ τm ≤ 2.0 . Tm , 0. 1 ≤ τm ≤ 1.0 . Tm 1.4576 − 0.4119 A m + 0.0813A m 2 τ Td = Tm 0.5312 − 0.1662A m + 0.0271A m 2 m Tm 17 , 0.1 ≤ τ , Td = 0.443Tm m Tm 0.939 0.847 0.598 τ τ , 1.1 ≤ m ≤ 2.0 , Td = 0.444Tm m Tm Tm τ 4.434K m K u − 0.966 1.751K m K u − 0.612 19 Kc = K u , Ti = Tu , 0.1 ≤ m ≤ 2.0 . Tm 5.12K m K u + 1.734 3.776K m K u + 1.388 Tm τm 0.957 Tm 18 Kc = 1.515 Tm K m τm 20 Kc = τ 6.068K m K u − 4.273 ∧ 1.1622K m K u − 0.748 ∧ K u , Ti = T u , 0. 1 ≤ m ≤ 2. 0 . ∧ ∧ Tm 5.758K m K u − 1.058 2.516K m K u − 0.505 , Ti = ∧ 23 24 τ Td = 0.413Tm m Tm ∧ 0.933 τ Td = 0.414Tm m Tm . 0.850 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Td Comment Minimum error – step load change – Gerry (1998). Nearly minimum IAE, ISE, ITAE – Hwang (1995). Model: Method 44; decay ratio = 0.15. 0.3 Km 0.5τ m Tm τm Tm > 5 ; Model: Method 1 Hwang and Fang (1995). 25 26 Kc Ti 26 Kc Ti 27 Kc Ti 28 Kc Ti 29 Kc 25 3 ≤ ε1 < 20 Nearly minimum IAE, ITAE. Model: Method 30; decay ratio = 0.12. Td [ ] 2 2 0.674 1 − 0.447ωH1τm + 0.0607ωH1 τm , K c = K H1 − K m (1 + K H1K m ) τ K c (1 + K H1K m ) Ti = , 0. 1 ≤ m ≤ 2. 0 . 2 2 Tm ωH1K m 0.0607 1 + 1.05ωH1τm − 0.233ωH1 τm K c = K H1 − [ ( 2 2 0.778 1 − 0.467ωH1τm + 0.0609ωH1 τm K m (1 + K H1K m ) [ [ ) ] , ( K c (1 + K H1K m ) ωH1K m 0.0309 1 + 2.84ωH1τm − 0.532ωH12 τm 2 ] 2 ω τ 1.31(0.519 ) H1 m 1 − 1.03 ε1 + 0.514 ε1 , K c = K H1 − K m (1 + K H1K m ) K c (1 + K H1K m ) ( ωH1K m 0.0603(1 + 0.929 ln[ωH1τm ]) 1 + 2.01 ε1 − 1.2 ε12 ] 2 2 1.14 1 − 0.482ωH1τm + 0.068ωH1 τm , K c = K H1 − K m (1 + K H1K m ) Ti = 29 2.4 ≤ ε1 < 3 ε1 ≥ 20 Ti Ti = 28 ε1 < 2.4 0.471K u K m ωu Ti = 27 89 ( K c (1 + K H1K m ) ωH1K m 0.0694 − 1 + 2.1ωH1τm − 0.367ωH12 τm 2 2 τ τ K c = 0.802 − 0.154 m + 0.0460 m K u , Tm Tm Kc Ti = , K u ωu 0.190 + 0.0532(τm Tm ) − 0.00509(τm Tm )2 [ ] 2 τ τ Ku τ Td = 0.421 + 0.00915 m − 0.00152 m , 0.1 ≤ m ≤ 2.0 . T Tm T ω K m m u c ). ). ). b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 90 Rule Kc O’Connor and Denn (1972). Model: Method 1; K m = Tm . Comment Td Ti ∞ ∫[ Minimum 0.5 x 2 y( t ) + (du ( t ) dt ) 2 2 ] dt . 0 30 x 2 τ m 2 Syrcos and Kookos (2005). Model: Method 1 Minimum IE – Devanathan (1991). Model: Method 1; coefficients of K c , Ti provided are deduced from graphs. 31 τm Tm < 0.2 x1τm 2 + 2 x1 ( 2 + x1 ) τ m x 2 τm + 2 x1 2τ m x 2 τm x 2 + 2 x1 Ti Td Kc 2 2 ≤ τm Tm ≤ 5 Minimum performance index with constraints. 32 0.32Tu 0.08Tu x 2 Tu x 3Tu Kc τ m Tm Coefficient values: x3 τ m Tm x2 x2 0.2 0.4 0.6 0.8 1.0 0.31 0.08 1.2 0.26 0.29 0.07 1.4 0.26 0.29 0.07 1.6 0.24 0.29 0.07 1.8 0.24 0.28 0.07 Minimum performance index: servo tuning x1 K m x 2 (Tm + τ m ) x 3 (Tm + τ m ) x3 0.07 0.07 0.06 0.06 Minimum IAE – Representative coefficient values (deduced from graphs): Gallier and Otto τ T x1 x2 x3 τ m Tm x1 x2 x3 (1968). m m Model: 0.053 12.3 0.95 0.0215 0.67 1.45 0.75 0.155 Method 1 0.11 6.8 0.93 0.041 1.50 0.85 0.63 0.21 0.25 3.4 0.88 0.083 4.0 0.57 0.53 0.19 0.869 Minimum IAE – T m 1.086 Tm 33 Model: Method 5 Rovira et al. τ Td K m τm 0.740 − 0.13 m (1969). Tm 30 x1 = x 2 τm 2 − 2(τm Tm ) + 2 (τm Tm )2 + 2τm 2 x 2 2 + (τm Tm ) ; recommended 0. 6 τm 2 ≤ x2 ≤ 1 τm 2 . 31 Kc = 2.2 Tm 1 Tm τm T T 0 . 777 0 . 45 , 0 . 31 0 . 6 , + = + T T 0 . 44 0 . 56 = − i d m m τ . Tm Km τm m 32 Kc = ξ cos tan −1 (1.57 D R − (0.16 D R )) . K m cos tan −1 (6.28Tm Tu ) 33 [ [ τ Td = 0.348Tm m Tm ] 0.914 , 0.1 < ] τm ≤1. Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum IAE – Marlin (1995), pp. 301–307. Model: Method 1 Minimum IAE – Sadeghi and Tych (2003). Minimum IAE – Arrieta Orozco (2006). Wang et al. (1995a). Kc Ti Td Comment 1.3 K m 0.93τ m 0.02τ m τ m Tm = 0.25 1.0 K m 0.80τ m 0.05τ m τ m Tm = 0.43 0.8 K m 0.71τ m 0.06τ m τ m Tm = 0.67 0.7 K m 0.73τ m 0.09τ m τ m Tm = 1.0 0.55 K m 0.64τ m 0.12τ m τ m Tm = 1.5 0.48 K m 0.56τ m 0.15τ m τ m Tm = 2.33 0.4 K m 0.52τ m 0.21τ m τ m Tm = 4.0 91 Coefficients of K c , Ti , Td estimated from graphs; robust to ±25% process model parameter changes. Model: Method 1 1.45933τ m 34 Ti τ τm Kc 0.1 ≤ m ≤ 1.0 . + 3.27873 T Tm m 35 Kc Ti Td pp. 23–24. Model: Method 10 36 Kc Tm + 0.5τ m 0.5Tm τ m Tm + 0.5τ m 0.05 < τm Tm < 6; Model: Method 1 34 Kc = 1 T τ 0.26266 + 0.82714 m , Ti = Tm 0.28743 m + 1.35955 . K m τm T m 35 Kc = 0.8456 0.9971 τ T 1 , , Ti = Tm 0.9781 + 0.3723 m 0.3295 + 0.7182 m T Km τm m τ Td = 0.3416Tm m Tm 0.9414 , 0.1 ≤ τm ≤ 2.0 . Tm x2 x1 + (Tm + 0.5τm ) τ m Tm 36 , x1 = 0.7645 , x 2 = 0.6032 , Minimum IAE; Kc = K m (Tm + τm ) x1 = 0.9155 , x 2 = 0.7524 , Minimum ISE. b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 92 Rule Kc Comment Td Ti 0.897 τ 1.048 Tm 0.1 ≤ m ≤ 1.0 37 Minimum ISE – T T d Tm i Zhuang (1992), K m τ m 0.567 Zhuang and τ 1.1 ≤ m ≤ 2.0 Atherton (1993). 1.154 Tm 38 T Ti d Tm K m τ m Model: Method 1 or Method 5 or Method 6 or Method 15. Minimum ISE – 2 ≤ Am ≤ 5 , 39 Ho et al. (1998). T T Kc i d 300 ≤ φm ≤ 600 . Model: Method 1 Minimum ISE – 40 Model: Method 1 Xing et al. Ti Td Kc (2001). Minimum ISE – 1.93576τ m τ 0.1 ≤ m ≤ 1.0 ; 41 Sadeghi and Ti τm Kc Tm + 3.83528 Tych (2003). Tm Model: Method 1 37 τ Ti = , Td = 0.489Tm m τm Tm 1.195 − 0.368 Tm 0.888 τ , Td = 0.490Tm m τm Tm 1.047 − 0.220 Tm 0.708 Tm 38 Ti = 39 Kc = Tm 0.9471 . . 1.0264 τm , 0.0845 T Am m τ 0.0211Tm [1 + 0.3289A m + 6.4572φm + 25.1914(τm Tm )] Ti = , 0.1 ≤ m ≤ 1.0 ; Tm 1 + 0.0625A m − 0.8079φm + 0.347(τm Tm ) 0.0821 Tm 1.8578 φm 0.9087 K m Am τm , Td = 0.4899Tm φm 0.1457 Given A m , with 2 ≤ A m ≤ 5 and 0.1 ≤ τ m Tm ≤ 1.0 , ISE is minimised when φ m = 29.7985 + 40 62.1189 40.3182 τ m 76.2833τ m + − (Ho et al., 1999). Am Tm A m Tm 2 τm τm , Kc = , Ti = Tm 1.126 + 0.168 + 0.043 2 Tm Tm τm τm − 0.293 0.022 + 1.558 T Tm m Km [ ] Td = Tm 0.005 + 0.202(τm Tm ) − 0.016(τm Tm ) . 41 Kc = 1 T τ 0.40455 + 0.96441 m , Ti = Tm 0.43770 m + 1.39588 . K m τm T m 2 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 0.85 Minimum ITAE 0.965 Tm – Rovira et al. Km τm (1969). Minimum ITAE 43 – Wang et al. Kc (1995a). Minimum ITAE 44 – Sadeghi and Kc Tych (2003). 0.855 Modified 1.2 Tm minimum ITAE K m τm – Cheng and Hung (1985). 0.855 Modified 0.965 Tm minimum ITAE – Smith (2003). K m τ m Minimum ITAE 1.034Tm 0.8733 – Zhang et al. K m τm 0.8823 (2005). 42 Ti Td Comment 42 Ti Td Model: Method 5 Tm + 0.5τ m 0.5Tm τ m Tm + 0.5τ m Model: Method 1 Ti 1.55237 τ m τm + 4.00000 Tm 0.1 ≤ τm Tm ≤ 1; Model: Method 1 ξ = 0.707; Ti Td Model: Method 14 1.26Tm 0.308τm Model: Method 1 Td Model: Method 1 45 46 Ti τ Tm Ti = , Td = 0.308Tm m 0.796 − 0.1465(τm Tm ) Tm 0.929 , 0. 1 ≤ τm ≤1. Tm 0.5307 0.7303 + (Tm + 0.5τm ) τm Tm τ 43 , 0.05 < m < 6 . Kc = Tm K m (Tm + τm ) 1 T τ 0.20360 + 0.80451 m , Ti = Tm 0.29349 m + 1.29110 . Km τm Tm 44 Kc = 45 Ti = 46 Ti = 1.6013Tm τ Tm , Td = 0.308Tm m 0.796 − 0.147(τm Tm ) Tm 0.9011 τm 0.0884 0.929 . , Td = 0.3088Tm 0.1131τm 0.9047 , 0.02 ≤ τm Tm ≤ 4 . 93 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 94 Rule Kc Minimum ITAE – Arrieta Orozco (2006), pp. 90–92. Minimum ITAE – Barberà (2006). 47 17-Mar-2009 Ti Td Comment Td 2 ≤ Am ≤ 4 ; Model: Method 10 Td Model: Method 1 47 Kc Ti 48 Kc −1.1734 1.3167e τm Tm −1.3612 + 0.2192 A m − 0.0342 A m 2 τm 1 2 , Kc = 1.5985 − 0.7918A m + 0.1088A m + x1 Km Tm 2 x1 = 0.7655 + 0.0142A m − 0.0253A m , 0.1 ≤ τm Tm ≤ 2.0 ; −4.9021+ 4.2515 A m − 0.7519 A m 2 τm 2 , Ti = Tm − 0.8444 + 1.4211A m − 0.2677 A m + x 2 Tm 2 x 2 = 2.3292 − 1.4155A m + 0.2428A m , 0.1 ≤ τm Tm < 1.0 ; 4.9240 −1.8437 A m + 0.1863A m 2 τm 2 , Ti = Tm 1.3028 + 0.1528A m − 0.0837 A m + x 2 Tm 2 x 2 = 0.2535 − 0.1997 A m + 0.0678A m , 1.0 ≤ τm Tm ≤ 2.0 . ( 1.0743 − 0.2489 A m + 0.0595 A m 2 2 τm ( 2 τm ) Td = Tm 1.1385 − 0.5597 A m + 0.0839A m Tm ) , 0.1 ≤ −0.7838 +1.1971A m − 0.1797 A m 2 Td = Tm 1.1532 − 0.5945A m + 0.0929A m Tm 1 48 Kc = , K m 2.2132(τm Tm )2 + 1.0802(τm Tm ) + 0.001045 [ , 1.0 ≤ τm < 1.0 ; Tm τm ≤ 2.0 . Tm ] τ τ τ Td = 2.2132 m + 1.0802 m + 0.001045 0.2371 − 0.2088 m . T T T m m m 2 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Minimum ISTSE – Zhuang (1992), Zhuang and Atherton (1993). Ti Td 49 Kc Ti Td 50 Kc Ti Td 95 Comment 0. 1 ≤ τm ≤ 1.0 Tm 1.1 ≤ τm ≤ 2. 0 Tm Model: Method 1 or Method 5 or Method 6 or Method 15. 51 Model: Method 1 0.509 K u 0.125Tu Ti ∧ 0.604 K u Minimum ISTSE – Xing et al. (2001). 53 Kc Ti 0.130 T u Model: Method 37 Ti Td Model: Method 1 52 ∧ 0.904 τ 0.968 Tm 0.1 ≤ m ≤ 1.0 54 Minimum ISTES T Ti d Tm – Zhuang (1992), K m τ m 0.583 Zhuang and τ 1.1 ≤ m ≤ 2.0 Atherton (1993). 1.061 Tm 55 Td T T i m Km τm Model: Method 1 or Method 5 or Method 6 or Method 15. Kc = 1.042 Tm K m τm 0.897 49 Kc = 1.142 Tm K m τm 0.579 50 51 Ti = 0.051(3.302K m K u + 1)Tu , 1 ≤ K m K u ≤ 16 , 0.1 ≤ τm Tm ≤ 2.0 . 52 53 τ Tm , Td = 0.385Tm m 0.987 − 0.238(τm Tm ) Tm 0.906 , Ti = τ Tm , Td = 0.384Tm m 0.919 − 0.172(τm Tm ) Tm 0.839 , Ti = . . ∧ ∧ Ti = 0.04 4.972K m K u + 1 T u , 0.1 ≤ τm Tm ≤ 2.0 . 2 τm τm , , Ti = Tm 1.154 + 0.193 Kc = + 0.042 2 Tm Tm τm τm − 0.272 0.019 + 1.439 T Tm m Km [ ] Td = Tm 0.003 + 0.236(τm Tm ) − 0.022(τm Tm ) . 54 Ti = τ Tm , Td = 0.316Tm m 0.977 − 0.253(τm Tm ) Tm 55 Ti = τ Tm , Td = 0.315Tm m 0.892 − 0.165(τm Tm ) Tm 0.892 . 0.832 . 2 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 96 Rule Kc Minimum ISTES – Xing et al. (2001). Xing et al. (2001). Model: Method 1 Kc = Ti Td Comment Model: Method 1 56 Kc Ti Td 57 Kc Ti Td Kc Ti 59 Kc Ti 60 Kc Ti Minimise ∞ 6 2 ∫ t e (t)dt . 0 Nearly minimum IAE, ISE, ITAE – Hwang (1995). Model: Method 44 56 17-Mar-2009 58 Km 0.028 + 1.254(τm Tm ) − 0.239(τm Tm )2 ε1 < 2.4 0.471K u K m ωu 2.4 ≤ ε1 < 3 3 ≤ ε1 < 20 , [ ] T = T [0.002 + 0.274(τ T ) − 0.032(τ T ) ]. Ti = Tm 1.191 + 0.243(τm Tm ) + 0.042(τm Tm ) , 2 2 d 57 58 Kc = m m m m m 2 1.329 + 0.413 τm + 0.027 τm , T T , = i m 2 T T τ τ m m 0.030 + 1.036 m − 0.210 m Tm Tm K c = K H1 − Km [ 2 2 0.822 1 − 0.549ωH1τm + 0.112ωH1 τm K m (1 + K H1K m ) ] ωH1K m 0.0142 1 + 6.96ωH1τm − 1.77ωH12 τm 2 ). [ K c (1 + K H1K m ) ( ] 2 2 0.786 1 − 0.441ωH1τm + 0.0569ωH1 τm K c = K H1 − , K m (1 + K H1K m ) Ti = 60 2 ] , Ti = 59 [ Td = Tm 0.014 + 0.246(τm Tm ) − 0.031(τm Tm ) . [ ] 1.28(0.542 )ωH1τ m 1 − 0.986 ε1 + 0.558 ε12 , K c = K H1 − K m (1 + K H1K m ) Ti = ( K c (1 + K H1K m ) ωH1K m 0.0172 1 + 4.62ωH1τm − 0.823ωH12 τm 2 K c (1 + K H1K m ) ( ωH1K m 0.0476(1 + 0.996 ln[ωH1τm ]) 1 + 2.13 ε1 − 1.13 ε12 ). ). b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 61 Hwang (1995) – continued. Kc Ti Td Comment Ti 0.471K u K m ωu ε1 ≥ 20 97 Decay ratio = 0.1; 0.1 ≤ τm Tm ≤ 2.0 . Nearly minimum IAE, ITAE – Hwang and Fang (1995). 62 Ti Kc Model: Method 30 Td 0.1 ≤ τm Tm ≤ 2.0 ; decay ratio = 0.03. 63 Minimum performance index: other tuning Ti Td Kc Hwang and Fang Nearly minimum IAE, ITAE – simultaneous servo/regulator tuning. (1995). Decay ratio = 0.05. Model: Method 30 Servo or regulator tuning Model: Tm – minimum IAE 0.36 + 0.76 τ Method 46; 0.47Tm τ m 0.47τ m + Tm m – Huang and τm Tm < 0.33 0.47τ m + Tm K m Jeng (2003). 61 [ ] 2 2 1.14 1 − 0.466ωH1τm + 0.0647ωH1 τm K c = K H1 − , K m (1 + K H1K m ) Ti = 62 [ ( ] K c = 0.537 − 0.0165(τm Tm ) + 0.00173(τm Tm ) K u , Ti = 2 [ Kc K c (1 + K H1K m ) ωH1K m 0.0609 − 1 + 1.97ωH1τm − 0.323ωH12 τm 2 K u ωu 0.0503 + 0.163(τm Tm ) − 0.0389(τm Tm )2 ], 2 τ Ku τ Td = 0.350 − 0.0344 m + 0.00644 m . Tm Tm ωu K c 63 [ ] K c = 0.713 − 0.176(τm Tm ) + 0.0513(τm Tm ) K u , Ti = 2 [ Kc K u ωu 0.149 + 0.0556(τm Tm ) − 0.00566(τm Tm )2 ], 2 τm K u τm τm Td = 0.371 − 0.0274 + 0.00557 ω K , 0.1 ≤ T ≤ 2.0 . Tm T m m u c ). b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 98 Rule Huang and Jeng (2003) – continued. Kc 64 Kc Ti Td Comment Ti Td τm Tm ≥ 0.33 x 1 τ m + x 2 Tm x 3 τ m + x 4 Tm x 6 Tm τ m τm Åström and τ m + α 5 Tm K mτm τ m + x 7 Tm Hägglund (2004). Model: x1 x2 x3 x4 x5 x6 x7 Method 1 0.057 0.139 0.400 0.923 0.012 1.59 4.59 Coefficient values given. M max = 1.1 0.103 0.261 0.389 0.930 0.040 1.62 4.44 M max = 1.2 0.139 0.367 0.376 0.900 0.074 1.66 4.39 M max = 1.3 0.168 0.460 0.363 0.871 0.111 1.70 4.37 M max = 1.4 0.191 0.543 0.352 0.844 0.146 1.74 4.35 M max = 1.5 0.211 0.616 0.342 0.820 0.179 1.78 4.34 M max = 1.6 0.227 0.681 0.334 0.799 0.209 1.81 4.33 M max = 1.7 0.241 0.740 0.326 0.781 0.238 1.84 4.32 M max = 1.8 0.254 0.793 0.320 0.764 0.264 1.87 4.31 M max = 1.9 0.264 0.841 0.314 0.751 0.288 1.89 4.30 M max = 2.0 Direct synthesis: time domain criteria Tm τ m T 1 0.5 + m Van der Grinten Step disturbance. Tm + 0.5τ m τ m + 2Tm Km τm (1963). 65 Stochastic Model: Method 1 Ti Td Kc disturbance. 64 Kc = 0.8194Tm + 0.2773τm K m τm 0.9738Tm 0.0262 , Ti = 1.0297Tm + 0.3484τm , 0.4575Tm + 0.0302τm τm . Td = 1.0297Tm + 0.3484τm 65 Kc = e − ωd τ m T 1 − 0.5e − ωd τ m + m e − ωd τ m , τm Km Ti = ( ) 1 − 0.5e− ωd τ m Tm τm T 1 − 0.5e− ωd τ m + m e− ωd τ m , Td = − ωd τ m τm 1 − 0.5e − ωd τ m + Tm τm e − ωd τ m e ( ) . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Td Comment 66 Regulator – Ti Td τm Tm < 2 Kc Gorecki et al. (1989). Pole is real and has maximum attainable multiplicity. Model: Method 1 Saito et al. 0.315Tm τ m τm Tm < 1 (1990). 0.215τ m + Tm 0.315τ m + Tm 0.315τ m + Tm Model: 1.37 K m τ m 67 1 ≤ τm Tm < 10 Td Method 22 68 Suyama (1992). OS=10% Tm + 0.309 τ m Td Kc Model: Method 1 Fruehauf et al. 0.56Tm K m τm ≤ 0.5τ m 5τ m τm Tm < 0.33 (1993). ≤ 0.5τ m 0.5Tm K m τm Tm τm Tm ≥ 0.33 Model: Method 2 69 Nomura et al. ‘Kitamori’. Tm + 0.315τm Td Kc (1993). Model: 70 ‘Butterworth’. Tm + 0.3109τm Td Kc Method 1; τ 0.01 ≤ m ≤ 10 . 71 Minimum ITAE. Tm Tm + 0.3610τm Td Kc 66 τm 2 τ τ x1 − 3− 2 Tm 2Tm τ 6 + m x1 − 9 − m − m e , K m τm 2Tm 2Tm 2Tm Kc = Ti = τm [6 + (τm 2Tm )]x1 − 9 − (τm 2Tm ) − (τm 2Tm )2 [21 + 3(τm Tm ) + (τm 2Tm )2 ]x1 − 36 − 4.5(τm Tm ) − 1.5(τm Tm )2 − (τm 2Tm )3 , Td = 0.5τm 2 τ , x 1 = 3 + m . 2 [6 + (τm 2Tm )]x1 − 9 − 0.5(τm Tm ) − (τm 2Tm ) 2Tm x1 − 1 2 67 Td = 0.315τm Tm + 0.003τm . 0.315τm + Tm 68 Kc = 2.236Tm τm 1 Tm . + 0.2236 , Td = 0.7236 7.236Tm + 2.236τm Km τm 69 Kc = 0.0027 τm + 0.315Tm 0.73 T 0.315 + m , Td = τm . τm 0.315τm + Tm K m 70 Kc = 0.0082τm + 0.3109Tm 0.73 T 0.3109 + m , Td = τm . τm 0.3109τm + Tm Km 71 Kc = 0.0690τm + 0.3610Tm 0.73 T 0.3610 + m , Td = τm . τm 0.3610τm + Tm K m 99 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 100 Rule Kc Ti Td Comment 72 Kc Tm + 0.4860τm Td ‘Binomial’. 73 Kc Juang and Wang (1995). 74 Kc Ti Td Abbas (1997). 75 Kc Tm + 0.5τ m Tm τ m 2Tm + τ m Model: Method 1 Tm + τm K mTm τm 4Tm τ m Tm + τ m Tm τ m Tm + τ m Model: Method 1 Nomura et al. (1993) – continued. Gonzalez (1994), p. 119. Model: Method 1 Camacho et al. (1997). 72 Kc = 73 Kc = Kc = Tm Ti x 2 ( 2 Kc = m 1 4 x1 = 0.2 ; x 3 = 0.21 (p. 121). Model: Method 1 TCL = x1Tm . − x 3Tm + x 32Tm 2 + Tm (τm − Ti ) − x1Ti 2 2 K m 1 + x 2 x1Ti 2 ) , x2 = Tm (τm − Ti ) − x1Ti 2 1 1 + (2π log e [b a ]) 2 , x1 + (τm Tm ) + 0.5(τm Tm )2 K m (x1 + (τm Tm )) 2 b = desired closed loop response decay ratio. a x1 + (τm Tm ) + 0.5(τm Tm ) , x1 + (τm Tm ) 2 , Ti = Tm Td = 75 4x1 ] τ [1 − 1 − 4x ] 0.1329τm + 0.4860Tm 0.73 T 0.4860 + m , Td = τm . τm 0.4860τm + Tm Km with x 3 = 74 [ τ m 1 − 1 − 4 x1 0.5Tm τm 2 (x1 + (τm Tm ) − 0.5x1 (τm Tm )) (x1 + (τm Tm ))(x1 + (τm Tm ) + 0.5(τm Tm )2 ) 1.002 τ 0.177 + 0.348(Tm τm ) , 0 ≤ V ≤ 0. 2 , 0. 1 ≤ m ≤ 5. 0 . 0.713 Tm K m (0.531 − 0.359V ) , 0 < x1 < 1 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Cluett and Wang (1997), Wang and Cluett (2000). Model: Method 1; τ 0.01 ≤ m ≤ 10 . Tm 76 Kc = Kc = Kc = Kc Ti Td TCL = 4τ m 77 Kc Ti Td TCL = 2τm 78 Kc Ti Td TCL = 1.33τ m 79 Kc Ti Td TCL = τ m 80 Kc Ti Td TCL = 0.8τ m 81 Kc Ti Td TCL = 0.67τ m −0.0069905τm + 0.029480Tm τm , 7.97 ≤ A m ≤ 8.56 , 79.67 0 ≤ φm ≤ 79.700 . 0.029773τm + 0.29907Tm 0.16440τm + 0.99558Tm 0.055548τm + 0.33639Tm τm , , Ti = K m τm 0.98607 τm − 1.5032.10− 4 Tm Kc = Kc = Kc = −0.030407τ m + 0.27480Tm τm , 3.30 ≤ A m ≤ 3.56 , 65.860 ≤ φm ≤ 68.500 . 0.25285τm + 1.0132Tm 0.26502τm + 0.94291Tm 0.16051τm + 0.57109Tm τm , , Ti = K m τm 0.89868τm + 6.9355.10− 2 Tm Td = 81 −0.024442τm + 0.17669Tm τm , 3.81 ≤ A m ≤ 4.16 , 69.840 ≤ φm ≤ 70.700 . 0.17150τm + 0.80740Tm 0.24145τm + 0.96751Tm 0.12786τm + 0.51235Tm τm , , Ti = K m τm 0.93566τm + 2.2988.10− 2 Tm Td = 80 −0.016651τm + 0.093641Tm τm , 4.84 ≤ A m ≤ 5.31 , 73.90 ≤ φm ≤ 74.140 . 0.093905τm + 0.56867Tm 0.20926τm + 0.98518Tm 0.092654τm + 0.43620Tm τm , , Ti = K m τm 0.96515τm + 4.2550.10− 3 Tm Td = 79 Comment 76 Td = 78 Td 0.099508τm + 0.99956Tm 0.019952τm + 0.20042Tm τm , , Ti = K m τm 0.99747 τm − 8.7425.10−5 Tm Td = 77 Ti 101 −0.035204τm + 0.38823Tm τm , 3.00 ≤ A m ≤ 3.19 , 60.600 ≤ φm ≤ 67.210 . 0.33303τm + 1.1849Tm 0.19067 τm + 0.61593Tm 0.28242τm + 0.91231Tm τm , , Ti = K m τm 0.85491τm + 0.15937Tm Td = −0.039589τm + 0.51941Tm τm , 2.80 ≤ A m ≤ 2.95 , 52.350 ≤ φm ≤ 66.450 . 0.40950τm + 1.3228Tm b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 102 Rule Kc Valentine and Chidambaram (1997a). Morilla et al. (2000). Model: Method 33 Mann et al. (2001). Model: Method 37 or Method 44. 82 17-Mar-2009 Ti Td Comment Ti Td ξ = 0.707 ; Model: Method 1 82 Kc 83 τm x1τm x 1 = 0. 1 ; Kc 1 + 1 − 4 x1 1 + 1 − 4 x1 0.2 ≤ x 3 ≤ 0.5 . 84 Kc Ti Td OS < 5%; 0 < τ m Tm < 1 . 85 Kc Ti Td OS < 5%; 1 ≤ τ m Tm < 2 . Kc = 2 τ τ 1 τ 0.746 − 1.242 ln m , Ti = Tm 0.3106 + 1.372 m − 0.545 m , K m Tm Tm Tm Td = Tm . 16.015Tm − 11.702τm 2 ±1% TS = 2.5Tm , 0.1 ≤ τm Tm ≤ 0.5 ; ±1% TS = 3.0Tm , τm Tm = 0.6 ; ±1% TS = 3.75Tm , τm Tm = 0.7; ±1% TS = 4.3Tm , τm Tm = 0.8; ±1% TS = 5.0Tm , τm Tm = 0.9 . 83 Kc = − x 3Tm + x 32Tm 2 + Tm (τm − Ti ) − x1Ti 2 Ti x 2 , x2 = K m [2 x 3 + x 2 (τm − Ti )] Tm (τm − Ti ) − x1Ti 2 with x 3 = 84 1 1 + (2π log e [b a ]) 2 , b = desired closed loop response decay ratio. a [0.770 + 0.245(τ T ) ]T , T = 1.262 + 0.147 τ 0.854 Kc = m m m m i K m τm 0.854 Tm , T m Td = 85 0.770 + 0.245(τm Tm )0.854 τm . [0.603 + 0.275(T τ ) ]T , T = 1.1618 + 0.165 T T , 2.4 2 .4 Kc = 0.262 + 0.147(τm Tm )0.854 m K m τm m m i m τ m Td = m 0.1618 + 0.165(Tm τm )2.4 0.603 + 0.275(Tm τm )2.4 τm . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Chen and Seborg 86 K c (2002). τ Chidambaram −1.763 m Tm (2002), p. 155. 4.105 e Km Vítečková and Víteček (2002). Model: Method 1 86 88 Ti Td Comment Ti Td Model: Method 1 Ti Td Model: Method 1 Ti Td 87 Kc For 0.05 < τm Tm < 1.6 , Ti = 1.2Tm ; overshoot is under 2% (Vítečková and Víteček, 2003). ( ) 1 2Tm τm + 0.5τm 2 (3TCL + 0.5τm ) − 2TCL 3 − 3TCL 2 τm , Km 2TCL3 + 3TCL 2 τm + 0.5τm 2 (3TCL + 0.5τm ) Kc = (2T τ + 0.5τ )(3T + 0.5τ ) − 2T 2 Ti = m m m CL 3 CL 2 − 3TCL τm , 3TCL τm Tm + 0.5Tm τm (3TCL + 0.5τm ) − 2(Tm + τm )TCL 2 (2T τ + 0.5τ )(3T + 0.5τ ) − 2T 2 m m 88 m τm (2Tm + τm ) 2 Td = 87 103 m CL m CL 3 2 − 3TCL τm 3 T s(1 + 0.5τms )e y = i ; K c (TCLs + 1)3 d desired − sτ m . 2 τ Tm τ Ti = Tm 0.311 + 1.372 m − 0.545 m , Td = . Tm 16.016 − 11.702(τm Tm ) Tm Kc = [ ( ] ) eτ m x1 3 0.5 2 3 2 2 τm Tm x1 + 3τm Tm + τm x1 + τm x1 − 1 , x 1 = − − + τ m Tm Km ( ) τ 2T x 3 + 3τ T + τm 2 x12 + τm x1 − 1 Ti = −2 m m 13 2 m m , 2 x1 τm Tm x1 + 2τm Tm + τm ( ) ( 3 τm2 + ) ) 0.25 Tm 2 ; τ 2T x 2 + 4τm Tm + τm 2 x1 + 2τm + 2Tm Td = −0.5 m 2 m 1 3 . 2 2 τm Tm x1 + 3τm Tm + τm x1 + τm x1 − 1 ( b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 104 Rule Kc Gorez (2003). (1 − x1 )τm + Tm Model: Method 1 x 2 K m τm Sree et al. (2004). Model: Method 1 Kukal (2006). 89 17-Mar-2009 For 0 ≤ Ti Comment Td (1 − x1 )τm + Tm (1 − x1 )τmTm (1 − x1 )τm + Tm 90 Kc Tm + 0.5τ m Td 91 Kc Ti Td 0.1 ≤ τm Tm ≤ 1 92 Kc Ti Td Model: Method 1 τm τ T + Tm 2 + 0.5τm 2 , ≤ τ m , x1 = m m τ m + Tm (τm + Tm )2 x2 = For τ m ≤ 1 (1.3M s 2 − 1) 1.56τm 2 (0.5τm + 3Tm ) + ; 2M s (M s − 1) 2M s (M s − 1) (τm + Tm )3 τm 0.65M s τ T + Tm 2 + 0.5τm 2 , x2 = ≤ 1 , x1 = m m . τ m + Tm Ms − 1 (τm + Tm )2 90 Kc = 0.5τm (Tm + 0.1667 τm ) 1 Tm + 0.5 , Td = . Tm + 0.5τm K m τm 91 Kc = 1.377 Tm K m τm 92 Kc = 2 12(Tm τm )2 + 1 − 1 x1 T 1 T 6 12 m + 1 − 18 m − 1 + e , Km 2Tm τm τm τm x1 = 89 0.8422 τ , Ti = 1.085Tm m Tm 12(Tm τm )2 + 1 − 1 2(Tm τm ) − 3 ; Ti = τm 4 0.4777 τ , Td = Tm 0.3899 m + 0.0195 . Tm x 2 + x 3 12(Tm τm )2 + 1 3 27(2(Tm τm ) + 1)4 , 2 T T T T x 2 = 864 m + 432 m + 252 m + 72 m + 5 , τm τm τm τm 3 2 T T T x 3 = 216 m + 60 m + 18 m + 5 ; τm τ τ m m Td = τm 3 2 2 T 2 T T T T 72 m + 8 m + 10 m + 1 + 12 m + 1 12 m + 1 τm τm τm τm τm 2 T 3 Tm Tm m + + + 4 108 36 20 5 τ τm τm m . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Ti Td Comment 1.282Tm 0.321Tm τm > 0.368 Tm Kc Vítečková (2006). Model: Method 9 93 Gorecki et al. (1989). 94 Kc Kc Direct synthesis: frequency domain criteria Model: Method 1 Ti Td Regulator – low frequency part of magnitude Bode diagram is flat. 95 x1Td Td Kc Li et al. (1994). Model: Method 48 ∧ 4hx 2 πA p 93 Kc ≥ 1.282Tm . K m (2.718τm − Tm ) 94 Kc = 1 , 2K m (τm Ti )(1 + (Tm τm )) − 2K m Ti = τm Td = τm 95 Kc = ∧ 0.318 T u x 2 given in footnote 95. 0.080 T u 7 + 42(Tm τm ) + 135(Tm τm )2 + 240(Tm τm )3 + 180(Tm τm )4 , 1 + 7(Tm τ m ) + 27(Tm τm )2 + 60(Tm τm )3 + 60(Tm τm )4 . [ 15[2(Tm τm ) + 1]1 + 3(Tm τm ) + 6(Tm τm )2 ] 7 + 42(Tm τm ) + 135(Tm τm )2 + 240(Tm τm )3 + 180(Tm τm )4 4h πµ x 2 cos φm 2 Ap − µ µ 2 4 + − 0.5υ + υ2 + 2 x 1 υ + υ + (4 x1 ) , µ = hysteresis value, 2 tan φm − µ A p 2 − µ 2 4 2 . Td = 0.080 T u υ + υ + , υ = 2 x1 2 1 + µ Ap − µ The values of x 1 and x 2 are given below. ∧ φm x1 150 2.8 200 3.2 250 3.5 300 3.8 350 4.0 400 4.2 450 4.6 500 4.9 550 5.2 600 5.4 650 5.5 x2 0.5 0.6 0.6 0.6 0.7 0.7 0.8 0.8 0.8 0.9 0.9 105 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 106 Rule Kc 96 Lee et al. (1992). Model: Method 1 Kc Td Comment Ti Td M s = 1.25 1.1 97 , τm τ < 2.52 ; x1 = 1 , m ≥ 2.52 . Tm Tm x 2τm Kc x 3τ m τ m Tm x2 Coefficient values: x3 τ m Tm x2 x3 0.0375 0.1912 0.3693 0.5764 0.8183 1.105 1.449 3.57 3.33 3.13 2.94 2.78 2.63 2.50 0.71 0.67 0.63 0.59 0.56 0.53 0.50 2.38 2.27 2.17 2.08 2.00 1.92 1.85 0.48 0.45 0.43 0.42 0.40 0.38 0.37 1.869 2.392 3.067 3.968 5.263 7.246 10.870 [3.08(T τ ) − 4 + 9.92x (T τ ) + 0.77(4 + 2(τ T ))(T τ ) ] , K [1.54(τ T ) + 4 + 4.96 x (τ T ) ] 0.1 Kc = Ti τ x1 = 0.55 + 0.163 m Tm Somani et al. (1992). Model: Method 1; suggested A m : 1.75 ≤ A m ≤ 3.0 . 96 17-Mar-2009 m 0.55 m 1 m m 0.1 m m 0.9 m m 0.45 m 1 m m m m 0 .1 0.55 τ 3.08(Tm τ m ) − 4 + 9.92 x1 (Tm τm ) Ti = Tm 0.5 m + , 1 .1 1.54(4 + 2(τm Tm ))(Tm τm ) Tm Tm Td = . 2Tm 1.54(Tm τm )1.1 (4 + 2(τm Tm )) + 0.1 0.55 τm 3.08(Tm τm ) − 4 + 9.92 x1 (Tm τm ) 97 For τm 0.7809 < 1 , Kc ≈ Tm K mAm 4 ( 0.803 + 4(τ T ) + 0.896) 2 m 2 Kc ≈ + 1 or m 2 T T τ 0.7809 0.7809 1 + 0.803 m ; K c ≈ 1 + 0.455 m , m > 1 . T τ τ KmAm K A m m m m m b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Ti Td Comment Minimum ISTSE 98 mK cos(φ ) u m – Zhuang and Atherton (1993). 99 Model: Method 1 mK u cos(φ m ) Ti Ti x1 A m = 2, φ m = 60 0 . Chidambaram (1995a). Ti Td Model: Method 1 98 Kc 100 Kc ( ) tan (φm ) + 4 + tan 2 (φm ) x1 m = 0.614(1 − 0.233e −0.347 K m K u ) , φ m = 33.8 0 1 − 0.97e −0.45K m K u , 0.1 ≤ τm Tm ≤ 2.0 , x1 = 0.413(3.302K m K u + 1) , Ti = x1 99 100 2ωu , Model: Method 1. ∧ ∧ ∧ m = 0.613(1 − 0.262e −0.44 K m K u ) , φ m = 33.2 0 1 − 1.38e −0.68 K m K u , x1 = 1.687 K m K u , 0.1 ≤ τm Tm ≤ 2.0 , Model: Method 37. Kc ≈ Ti ≈ Td ≈ 1 K mAm 1.17 ( 2 or 0.85 1 + 4.45 Tm ; + 0 . 85 2 τ KmAm m 4.45 + 4(τm Tm ) − 2.11 5Tm 5Tm or ; 2 2 Tm 3.42 − 0.15 4.45 + 0.85 − 1 2 τm 4.45 + 4(τm Tm ) − 2.11 ( 3.42 ) ) 0.8Tm 1.17 2 ( 3.42 −1 + 0 . 85 2 4.45 + 4(τm Tm ) − 2.11 ) or 0.8Tm 2 T 4.45 m − 0.15 τm . 107 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 108 Rule Chidambaram (1995a) – continued. Kc Ti Td Comment x1 K m A m x2τm x 3τ m Alternative. τm Tm x1 x2 0.062 29.770 0.083 22.100 0.150 12.470 0.196 9.670 0.290 6.686 0.339 5.784 1.20Tm K m τm 2.34 0.37 2.33 0.37 2.29 0.37 2.27 0.36 2.23 0.36 2.21 0.35 2.4 τ m Chidambaram (1995a). 1.03Tm K m τm Model: Method 1 0.90Tm K m τm 101 Åström and Hägglund (1995), pp. 208–210. τ 0.14 ≤ m ≤ 5.5. Tm 102 104 Coefficient values: x3 τm Tm Kc Kc 0.389 0.543 1.03 2.92 3.69 x1 x2 x3 5.104 3.781 2.262 1.209 1.105 2.19 2.14 2.00 1.72 1.67 0.35 0.34 0.32 0.28 0.27 0.38τ m A m = 1.5 2.4 τ m 0.38τ m A m = 1.75 2.4 τ m 0.38τ m A m = 2.0 Ti Td M s = 1.4 103 Ti Td Ti Td 105 Td Ti M s = 2.0 Model: Method 5 or Method 17. Ku cos φ m Am Tan et al. (1996). Model: Method 36 Kφ 107 Am 101 x1Td 106 Td Ti x1 chosen arbitrarily. Td Arbitrary A m , φ m at ωφ . 1 Tuning rules converted from formulae developed for G c (s) = K c [1 − α ] + + Tds . T s i 2 2 2 1.52e −8.22 τ +10.1τ Tm , Ti = 2.08τme −2.32 τ +1.4 τ , Td = 2.23τm e −0.55τ − 6.9 τ . K m τm 102 Kc = 103 Ti = 0.184Tm e 2.98τ + 0.7 τ , Td = 0.193Tm e 4.82 τ − 7.6 τ . 104 Kc = 105 Ti = 0.06Tme 4.45τ −1.549 τ , Td = 0.345Tm e −2.75τ −1.151τ . 106 Td = Tu 2 2 2 107 2 2 1.85e −8.95 τ + 9.851τ Tm , Ti = 0.704τm e −0.85τ − 0.879 τ , Td = 3.91τme −2.55τ − 0.491τ . K m τm 2 Ti = tan φm + (4 x1 ) + tan 2 φm 4π ( r2 K φ ωu 2 − ωφ 2 ωu ωφ 2 2 2 2 ) K u − r2 K φ 2 , Td = ; recommended A m = 2 , φ m = 450 . ωu K u 2 − r2 2 K φ 2 ( r2 K φ ωu 2 − ωφ 2 Ku ) , r = 0.1 + 0.9 K . 2 φ b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Friman and Waller (1997). Model: Method 1; Am > 2 Kc Ti Td 0.25K u 1.1879Tu 0.2970Tu 2.6064 ω1500 0.6516 ω1500 0.4830 ( G p jω1500 ) 109 Comment τ m Tm < 0.25 , φ m > 60 0 . τm ≤ 2.0, Tm 0.25 ≤ φ m > 450 . Branica et al. (2002). Model: Method 1; M s = 1.8 108 Kc Ti Td 109 Kc Ti Td Kc Ti Td x 2 − x1 x 2K m (x 2 − x1 )τm (1 − x1 ) T (x 2 − x1 ) m 110 111 Gorez (2003). Model: Method 1 Åström and Hägglund (2006), pp. 261–262. 108 109 110 111 112 Kc = 113 Kc (1) Ti (1) T 1 T 0.021 + 0.788 m , Ti = 1.534τ m m τ τm K m m 0.326 T 1 T 0.292 + 0.671 m , Ti = 1.239τm m Kc = τ τm K m m 0.554 T 1 T 0.338 + 0.623 m , Ti = 1.228τm m τ τm Km m 0.480 Kc = (1) τm ≤ 0.5 Tm 0.5 < τm ≤ 1.1 Tm 1.1 < τm ≤ 2.0 Tm 112 Model: Method 5; M s = M t = 1. 4 . T , Td = 0.233τ m m τm T , Td = 0.230τm m τm T , Td = 0.231τm m τm 0.055 . 0.069 . 0.120 . 1 Tuning rules converted from formulae developed for G c (s) = K c [1 − α ] + + Tds . T s i For 0 ≤ τm τ T + Tm 2 + 0.5τm 2 ≤ τ m , x1 = m m τm + Tm (τm + Tm )2 x2 = For τ m ≤ 113 Td 0.1 ≤ K c (1) = 1 (1.3M s 2 − 1) 1.56τm 2 (0.5τm + 3Tm ) + ; 2M s (M s − 1) 2M s (M s − 1) (τm + Tm )3 τm 0.65M s τ T + Tm 2 + 0.5τm 2 . ≤ 1 , x1 = m m , x2 = 2 τ m + Tm Ms − 1 (τm + Tm ) 0.4τm + 0.8Tm 0.5Tm τm 1 Tm (1) (1) τm , Td = . 0.2 + 0.45 , Ti = τm 0.3τm + Tm Km τm + 0.1Tm b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 110 Rule Kc Åström and Hägglund (2006), pp. 261–262. 114 Kc ( 2) Ti Td Comment ( 2) Td ( 2) τm Tm > 0.11 Ti Model: Method 5; detuned ‘AMIGO’ tuning rule. (5) (5) τm Tm < 0.11 115 T T K ( 5) i c 114 Kc ( 2) = Kc (3) ( 2) K c Td + ( 2) (1) K c Td (1) (K (1) − Kc c ( 4) d ), K ( 3) c = x1K c ( 4) , x1 = 0.5,0.1 or 0.01; K c ( 4) = τmTm Tm 0.15 + 0.35 − ; K m (τm + Tm )2 K m τm Kc 115 FA Ti ( 2) Ti ( 4) Kc (5) = Kc ( 2) ( 2) (1) ( 4) Kc − Kc + ( 3) (1) (1) (1) ( 4) Ti K c Td Ti Ti Kc ( 3) ( 2) K c Td = 0.35τm + = Kc 13τm Tm Tm 2 + 12Tm τm + 7 τm (5) ( 6) K c Td + 2 (5) K c (1)Td (1) (K (1) c − Kc ; Td 2 (7) , Ti ( 3) ( 2) = ), K K c(7) ≥ = Ti (6) K m + 1 − Ms ( 4) K c Ti Kc ( 4) K c K m + 1 − Ms (1) c ( 4) ( 3) 0.1K c Td (7) ( 3) (1) or Td K c ( 2) = x1K c ( 4) ( 2) Ti (5) Td ( 5) = ( 6) (5) K c Td (1) = 0.1K c Td Kc (5) (1) (5) (5) or Td (5) = , (1) = 0.01K c Td (1) . K c ( 2) 2K c ( 4 ) (τm + Tm ) 1 + −1 ; 2 ( 4) ( 4) −1 M s M s M s + M s − 1 Ti 1 − M s + K c K m (1) (7) Kc − Kc + (6) (1) (1) (1) ( 4) Ti K c Td Ti Ti Kc −1 , x1 = 0.5,0.1 or 0.01; ( K c ( 4) Kc −1 , Ti ( 6) = Ti 0.01K c (1) Td (1) K c ( 5) . ( 4) K c ( 6) Ti ( 4 ) ) K m + 1 − M s −1 K c ( 4 ) K c ( 6 ) K m + 1 − M s −1 ; b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Åström and Hägglund (2006), pp. 261–262 – continued. 116 Kc ( 8) = Kc K c (10) < Ti (8 ) = (9) + ( 8) 116 Kc K c Td (8 ) (8 ) (1) (1) K c Td Ti Td Comment (8 ) Td (8 ) τm Tm < 0.11 (10 ) , x1 = 0.5,0.1 or 0.01; Ti (K (1) c − Kc (10 ) ), K (9 ) c = x1K c 2K c ( 4 ) (τm + Tm ) 1 + −1 ; 2 ( 4) ( 4) −1 M s M s M s + M s − 1 Ti 1 − M s + K c K m ( ) K c (8 ) , (9) ( 8) (8 ) (1) (10 ) Kc K c Td K c K c + − Ti (9) K c (1) Td (1) Ti (1) Ti ( 4) Ti (9) = 2K c (9) K m Td (8 ) (1) = 111 0.1K c Td Kc ( 8) (1) or Td ( 8) (1) = 0.01K c Td Kc (8 ) (1) . τm + Tm ; 2 2 M s M s + M s + 1 K c (9) K m + 1 − M s −1 ( ) b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 112 Rule Kc King (2006). Model: Method 1 117 Kc Ti Td Ti Td Comment 0.2 ≤ τm ≤ 10 ; Tm 1.4 ≤ M s ≤ 1.9 . Robust Rivera et al. Tm + 0.5τ m (1986). K m (λ + 0.5τ m ) Model: Method 1 Brambilla et al. 118 Tm + 0.5τ m (1990). K m (λ + τ m ) Model: Method 1 117 Tm + 0.5τ m λ > 0.1Tm , Tm τ m 2Tm + τ m λ ≥ 0.8τ m . 0.5τ m ≤ λ ≤ 3τ m (Zou et al., 1997). Tm + 0.5τ m Tm τ m 2Tm + τ m 0.1 ≤ τm ≤ 10 Tm No model uncertainty: λ ≈ 0.35τ m . 4 3 2 τm τm τm 1 + x 2 log10 Kc = x1 log10 T + x 3 log10 T K m Tm m m + τm 1 + x 5 , x 4 log10 K m T m x1 = −0.724M s 2 + 2.767 M s − 1.987 , x 2 = 0.329M s 2 − 2.513M s + 1.707 , x 3 = −0.509M s 2 + 3.697 M s − 2.9131 , x 4 = 1.166M s 2 − 5.389M s + 4.264 , x 5 = −0.753M s 2 + 3.216M s − 2.407 ; τ 2 τ Ti = Tm x 6 m + x 7 m + x 8 , x 6 = −0.0486M s 2 + 0.1603M s − 0.1327 , Tm Tm x 7 = 0.2061M s 2 − 0.7414M s + 0.9609 , x 8 = −0.1759M s 2 + 0.5994M s + 0.4937 ; 2 τ 3 τm τm m + x12 , + + Td = Tm x 9 x x 10 11 T T Tm m m x 9 = −0.0165M s 2 + 0.0518M s − 0.0387 , x10 = 0.1733M s 2 − 0.5468M s + 0.3938 , x11 = −0.2998M s 2 + 0.7705M s − 0.1951 , x 12 = −0.0326M s 2 + 0.1475M s − 0.1426 . 118 0.1Tm ≤ λ ≤ 0.5Tm , τm Tm ≤ 0.25 ; λ = 1.5(τm + Tm ),0.25 < τm Tm ≤ 0.75 ; λ = 3(τm + Tm ), τm Tm > 0.75 (Leva, 2001). b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Lee et al. (1998). Model: Method 1 Kc Ti Td Ti 119 Td K m (λ + τ m ) Comment TCL = λ ; recommended TCL = 0.5τm (Lee et al., 2006). Kc Ti Td Tm K m (λ + 0.5τ m ) Tm 0.5τ m 120 Gerry (1999). Model: Method 11 Ti 113 λ = 2τ m (aggressive, less robust tuning); λ = 2(Tm + τ m ) (more robust tuning). Gong (2000). Model: Method 1 121 Kc Wang (2003), x1 K m pp. 16–17. τm Tm x1 Model: 0.05 13.2 Method 1. 0.11 5.3 Regulator; 0.18 4.3 x1 , x 2 , x 3 0.25 3.3 deduced from 2.6 graphs; robust to 0.33 0.43 2.0 ±25% 0.54 1.6 uncertainty in 0.67 1.4 process model 0.82 1.3 parameters. 1.0 1.1 119 Ti = Tm + 120 Kc = x 2 (τm + Tm ) 0.3866τm τ 1 + 0.3866 m Tm x 3 (τm + Tm ) x2 x3 τm Tm x1 x2 x3 0.18 0.33 0.43 0.49 0.51 0.56 0.59 0.61 0.61 0.61 0 0 0 0 0.044 0.074 0.091 0.106 0.120 0.124 1.2 1.5 1.9 2.3 3.0 4.0 5.7 9.0 19.0 0.9 0.8 0.7 0.6 0.5 0.5 0.4 0.4 0.4 0.60 0.59 0.57 0.56 0.51 0.50 0.47 0.46 0.46 0.136 0.139 0.144 0.146 0.147 0.147 0.148 0.149 0.149 τm 2 τm 2 τ 1 − m . , Td = 2(λ + τm ) 2(λ + τm ) 3Ti Ti T 2 + x1τm − 0.5τm 2 , Ti = Tm + x1 − CL , K m (2TCL + τm − x1 ) 2TCL + τm − x1 Tm x1 − Td = Tm + 0.3866τm 0.167 τm 3 − 0.5x1τm 2 T 2 + x1τm − 0.5τm 2 2TCL + τm − x1 − CL , Ti 2TCL + τm − x1 2 τm T − x1 = Tm − Tm 1 − CL e Tm . Tm 121 Kc = τ 0.7556Tm 1 + 0.3866 m . K m τm Tm b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 114 Rule Kc Ti x 2 (τm + Tm ) Wang (2003), x1 K m pp. 15–16. τm Tm x1 Model: 0.05 2.89 Method 1. 0.11 2.61 Servo; ±10% 2.28 overshoot, step 0.18 0.25 1.96 input; x1 , x 2 , x 3 0.33 1.58 deduced from 1.40 graphs; robust to 0.43 0.54 1.19 ±25% 0.67 1.04 uncertainty in 0.82 0.90 process model 1.0 0.85 parameters. Shamsuzzoha et 122 al. (2006c). Kc Model: Method 1 122 FA Kc = x 3 (τm + Tm ) x2 x3 τm Tm x1 x2 x3 0.81 0.86 0.86 0.87 0.85 0.84 0.78 0.77 0.73 0.74 0 0 0 0.034 0.043 0.066 0.075 0.094 0.100 0.120 1.2 1.5 1.9 2.3 3.0 4.0 5.7 9.0 19.0 0.72 0.68 0.56 0.50 0.43 0.38 0.34 0.32 0.29 0.68 0.65 0.60 0.55 0.52 0.48 0.44 0.44 0.40 0.130 0.142 0.149 0.140 0.137 0.136 0.122 0.116 0.096 Ti Td Ti λ2 + x1τm − 0.5τm 2 , Ti = Tm + x1 − , K m (2λξ + τm − x1 ) 2λξ + τm − x1 Tm x1 − Td = Comment Td 0.167 τm 3 − 0.5x1τm 2 λ2 + x1τm − 0.5τm 2 2λξ + τm − x1 , Ti 2λξ + τm − x1 ( ) λ2 − 2λξTm + Tm 2 e − τ m Tm x1 = Tm 1 − . 2 Tm λ , ξ not defined. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Comment Td λ = x1Tm 123 Ti Td Kc Shamsuzzoha Coefficients of x1 (deduced from graph): and Lee (2007a). Model: Method 1 M s = 1.6 M s = 1.7 M s = 1.8 M s = 1.9 M s = 2.0 123 Kc = 0.11 0.10 0.08 0.07 0.06 0.20 0.17 0.15 0.14 0.12 τm = 0.1Tm 0.35 0.31 0.27 0.25 0.22 τm = 0.2Tm 0.46 0.41 0.37 0.34 0.30 τm = 0.3Tm 0.55 0.49 0.44 0.40 0.36 τm = 0.4Tm 0.62 0.55 0.50 0.46 0.42 τm = 0.5Tm 0.68 0.60 0.55 0.51 0.47 τm = 0.6Tm 0.72 0.65 0.59 0.55 0.50 τm = 0.7Tm 0.77 0.69 0.63 0.59 0.53 τm = 0.8Tm 0.81 0.72 0.66 0.62 0.56 τm = 0.9Tm 0.84 0.76 0.69 0.65 0.59 τm = 1.0Tm 0.91 0.82 0.75 0.70 0.64 τm = 1.25Tm τm = 0.05Tm 0.97 0.87 0.80 0.75 0.70 τm = 1.5Tm undefined 0.91 0.84 0.78 0.73 τm = 1.75Tm undefined 0.96 0.87 0.81 0.76 τm = 2.0Tm Ti 3λ2 + 2x 2 τm − 0.5τm 2 − x 2 2 , Ti = Tm + 2 x 2 − , K m (3λ + τm − 2x 2 ) 3λ + τm − 2 x 2 2Tm x 2 + x 2 2 − Td = x 2 = Tm 1 − λ3 + 0.167 τm3 − x 2 τm 2 + x 2 2 τm 3λ2 + 2x 2 τm − 0.5τm 2 − x 2 2 3λ + τm − 2 x 2 , Ti 3λ + τm − 2 x 2 3 λ − τ m Tm 1 − e T . m 115 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 116 Rule Kc Ti Td Comment Tuning rules developed from K u , Tu . Ultimate cycle McMillan (1984). Model: Method 2 or Method 55. Kuwata (1987). x 1 , x 2 deduced from graphs. Kc Ti 0.25Ti x 1K u x 2 Tu x 3 Tu 124 x1 x2 x3 τm Tu x1 x2 x3 τm Tu x1 x2 x3 τm Tu 0.44 0.80 0.08 0.28 0.37 0.26 0.06 0.39 0.28 0.18 0.03 0.50 0.42 0.43 0.08 0.32 0.33 0.22 0.05 0.43 0.40 0.32 0.07 0.35 0.30 0.18 0.04 0.47 Geng and Geary τm 0.94Tm < 0.167 (1993). 2τ m 0.5τ m Tm Kmτm Model: Method 1 Wojsznis and 1.3 ≤ x1 ≤ 1.6 ; Tm τ m Tm + 0.5τm Blevins (1995). Tm + 0.5τ m 2Tm + τ m Model: x1K m τm Method 45 Perić et al. ^ Recommended (1997). x1Td Td 125 K u cos φ m Model: x1 = 4 . Method 49 Matsuba et al. τ 0.1 ≤ m ≤ 1.0 126 (1998). Ti Td Kc Tm Model: Method 1 127 Wojsznis et al. Model: Method 1 x1Ti K u cos φ m Ti (1999). A m 0.65 T 1.415 Tm 1 124 m = τ + T 1 Kc = , i m 0.65 . K m τm T Tm + τm m 1 + Tm + τm 125 Td = tan φm + 126 1.31 Tm 0.9 ^ T 4 + tan 2 φm u . 4π x1 2.19 Tm ≤ Kc ≤ K m τm 0.9 T , Ti = 1.59τm m τm T 127 Ti = tan φ m + 4 x1 + tan 2 φ m u . 4πx1 K m τm 0.097 T , Td = 0.40τm m τm 0.097 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Td Comment Ti 0.15Ti Default values. 0.38K u 1.2Tu 0.18Tu 0.27 K u 0.87Tu 0.13Tu Kc Wojsznis et al. (1999) – continued. Ti 0.5K u cos φ m 128 φ m = 450 ; τ m Tm < 0.2 . Am = 3 , φ m = 330 ; τ m Tm ≈ 0.2 5 . Wojsznis et al. (1999). Model: Method 45 Gu et al. (2003). 128 129 130 129 Ti Kc 0.125Ti 0.3 ≤ x1 ≤ 0.4 ; 0.25 ≤ x 2 ≤ 0.4 ; 0.2 ≤ 130 Ti Kc τm ≤ 0.7 . Tm Model: Method 1 Td Ti = 0.53Tu tan φm + 0.6 + tan 2 φm . 0.6 0.6 . K c = K u x 2 + 0.251.0 − x1 − , T T x = + i u 1 T T − u − 7.0 − u − 7.0 τ τ 1+ e m 1+ e m Representative tuning rule: 0.6K u ≤ K c ≤ K u , Ti = 0.5Tu , Td = 0.125Tu , with K u ≈ Tu ≈ 2π τm 3 + 5τm 2Tm + 12τm Tm 2 + 12Tm3 [ ( K m τm τm 2 + 5τm Tm + 8Tm 2 ( ) , ) τm K m K u τm τm 2 + 6τm Tm + 12Tm 2 + τm 3 + 6τm 2Tm + 18τm Tm 2 + 24Tm 3 ( 12(K m K u + 1) τm 2 + 4τm Tm + 6Tm 2 ) ]. 117 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 118 3.3.3 Ideal controller in series with a first order lag 1 1 G c (s) = K c 1 + + Td s Tf s + 1 Ti s R(s) E(s) − + 1 K c 1 + + Td s Ti s 1 1 + Tf s U(s) Y(s) K m e − sτ 1 + sTm m Table 11: PID controller tuning rules – FOLPD model G m (s) = Rule Kc Lim et al. (1985). Schaedel (1997). Model: Method 25 1 2 Kc Kc Ti K m e − sτ m 1 + sTm Comment Td Direct synthesis: time domain criteria 0 Model: Method 1 Tm Direct synthesis: frequency domain criteria 2 2 Tf = Td N ; T1 − T2 T2 2 T33 − 2 5 ≤ N ≤ 20 . T1 − Td T1 T2 2 1 Kc = TCL 2 Tm . , Tf = 2ξTCL 2 + τ m K m (2ξTCL 2 + τm ) 2 Kc = 0.375Ti Tm , T1 = + τm , K m (T1 + (Td N ) − Ti ) 1 + 3.45 τam Tm ( 2 T2 = 1.25Tm τ am + T1 = 0.7Tm 2 Tm − 0.7 τ am T3 3 = ( τam Tm 2 τ + 0 . 5 τ , T = 0 , 0 < ≤ 0.104 ; 3 m m Tm 1 + 3.45 τam Tm ( 0.7Tm 2 + τ m , T2 = Tm τ am + ) 2 Tm − 0.7 τ am + 0.5τ m 2 , ) + T τ τ + 0.35T τ 0.952Tm 2 τ am − 0.1 Tm − 0.7 τ am ) a m m m 2 2 m a Tm − 0.7 τ m m + 0.167τ m 3 , τ am > 0.104 . Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kristiansson and Lennartson (2002b). Kc = Ti Td Comment Kc Ti Td Model: Method 1 3 0.2 ≤ τm Tm ≤ 5 , 1.65 ≤ M max ≤ 1.85 . Kristiansson and Lennartson (2006). 3 Kc 4 Kc Ti Td Model: Method 1 5 Kc Ti Td Model: Method 12 0.05 ≤ τm Tm ≤ 2 , 1.65 ≤ M max ≤ 1.85 . [ ] 1.8 0.13 + 0.5(τm Tm ) − 0.04(τm Tm )2 , τ 0.12 K m m − 0.05 0.7 + (τm Tm ) + 0.05 Tm 2 τ τ 0.12 Ti = 2Tm 0.13 + 0.5 m − 0.04 m 0.7 + (τm Tm ) + 0.05 Tm Tm 2 2 2 τm τm τm τm 0.5Tm 0.13 + 0.5 − 0.04 0.9Tm 0.13 + 0.5 − 0.04 Tm Tm Tm Tm ,T = . Td = f τm 0.12 Tm ,25 − 0.05 min 5 + 2 0.7 + (τm Tm ) + 0.05 τm Tm 0 . 54 T ( ) 0 . 12 + τ 0 . 12 4 m m 0.7 + , Ti = 0.6(τm + Tm )0.7 + , Kc = τm τm τ + 0.05 + 0 . 05 K m τm m − 0.05 T T m m T m Td = 5 Kc = 0.15(τm + Tm ) 0.12 0.7 + ( ) τ T + 0 . 05 m m [ , Tf = 0.081(τm + Tm )2 τ , 0.7 ≤ m ≤ 4 . τm Tm T − 0.05 min 5 + 2 m ,25 T τ m m 2[1.1(Tm τm ) − 0.1] 0.06 + 0.68(τm Tm ) − 0.12(τm Tm )2 [ K m 0.68 + 0.84(τm Tm ) − 0.25(τm Tm ) 2 [ ] [0.68 + 0.84(τ T ) − 0.25(τ T ) ] , 2Tm 0.06 + 0.68(τm Tm ) − 0.12(τm Tm ) ] ], 2 Ti = 2 m m m m 2 2 τ τ τ τ Td = 0.5Tm 0.06 + 0.68 m − 0.12 m 0.68 + 0.84 m − 0.25 m , Tm Tm Tm Tm [ ] T 0.06 + 0.68(τm Tm ) − 0.12(τm Tm ) [1.1(Tm τm ) − 0.1] Tf = m . min{5 + 2(Tm τm ),25} 2 2 119 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 120 Rule Kristiansson and Lennartson (2006). Kc Ti Td Comment Kc Ti Td Model: Method 12 6 0.7 < τm Tm < 2 ; 1.65 ≤ M max ≤ 1.8 . Robust Tm + 0.5τ m K m (λ + τ m ) Horn et al. (1996). Model: Method 1 Tm + 0.5τ m Tm τ m 2Tm + τ m Tf = λτ m , 2(λ + τ m ) λ ∈ [τ m , Tm ] ( ) λ = max 0.25τ m ,0.2Tm (Luyben, 2001); λ > 0.25τ m (Bequette, 2003); λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning), λ > [Tm ,8τm ] (‘moderate’ tuning), λ > [10Tm ,80τm ] (‘conservative’ tuning) (Cooper, 2006b). 7 H ∞ optimal – Tm + 0.5τ m τ m Tm Kc τ m + 2Tm Tan et al. (1998b). λ = 2 : ‘fast’ response; λ = 1 : ‘robust’ tuning; λ = 1.5 : recommended. Model: Method 1 Rivera and Jun Tm τmλ Tf = ; (2000). 0 Tm K m (2 τ m + λ ) 2τ m + λ Model: Method 1 λ not specified. T 0.6667(Tm + τm )1.1 m − 0.1 0.6667(Tm + τm ) τm 6 Kc = , Ti = , 2 2 τ τ τ τ 0.68 + 0.84 m − 0.25 m K m τm 0.68 + 0.84 m − 0.25 m T Tm Tm Tm m (Tm + τ m )2 1.1 Tm − 0.1 Tf = 7 Kc = τm T 9Tm 5 + 2 m τm 2 , T = 0.1667(τ + T )0.68 + 0.84 τm − 0.25 τm . d m m T Tm m τm 0.265λ + 0.307 Tm τ + 0.5 , Tf = 5.314λ + 0.951 . Km m b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Td Comment Kc Ti Td Model: Method 1 Tf = Td . Tm + 0.5τ m K m (λ + τ m ) Tm + 0.5τ m Tm 2Tm + τ m Lee and Edgar (2002). Huang et al. (2002). Model: Method 1 8 Tf = λτ m 2(λ + τ m ) λ = 0.65τ m : minimum ISE; λ = 0.5τ m : overshoot to a step input ≤ 5% , A m = 3 . In general, A m = 2(λ τ m + 1) . Zhang et al. (2002b). 9 Vilanova (2006). Model: Method 1 10 Tang et al. (2007). Model: Method 1 121 Kc Tm + 0.5τ m Tm τ m 2Tm + τ m Kc Tm + 1.75τm 1.72τm Tm Tm + 1.75τm τm + 2Tm K m (4λ + τm ) 0.5τm + Tm Tm τm τm + 2Tm Model: Method 1 φm ≥ 450 Tf = 2λ2 4λ + τm Labelled the H ∞ Smith predictor; 0.3 ≤ λ τm ≤ 0.8 . τm + 2Tm 2K m (λ + τm ) 0.5τm + Tm Tm τm τm + 2Tm Tf = λτ m 2(λ + τm ) Labelled the H 2 Smith predictor or the Dahlin controller; 0.3 ≤ λ τm ≤ 0.8 . 2Tm + τm 3K m τm 8 Kc = 2 Kc = 10 Kc = Recommended λ = 0.5 ; Tf = 0.167 τm . 2 τm 2 1 3Tm − τm Tm + τm , + τm + K m (λ + τm ) 2(λ + τm ) 6(λ + τm ) 4(λ + τm )2 Ti = Tm + 9 Tm τm τm + 2Tm Tm + 0.5τm 2 2 τm 3Tm − τm τm 3Tm − τm τm + τm + , Td = τm + . 2 2(λ + τm ) 6(λ + τm ) 4(λ + τm ) 6(λ + τm ) 4(λ + τm )2 0.1Tm τ m 1 Tm + 0.5τm λ2 , Tf = . or Tf = 2λ + 0.5τ m τ m + 2Tm K m 2λ + 0.5τm 0.377(Tm + 1.75τm ) , Tf = 1.72τm . K m τm b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 122 Td s 1 + 3.3.4 Controller with filtered derivative G c (s) = K c 1 + Ti s 1 + s Td N R(s) E(s) − + T s 1 d + K c 1 + Td Ti s 1+ s N U(s) m Table 12: PID controller tuning rules – FOLPD model G m (s) = Rule Kc Ti Y(s) K m e − sτ 1 + sTm K m e − sτ m 1 + sTm Comment Td Minimum performance index: regulator tuning Ti Td 1 K Minimum IAE – c Alfaro Ruiz Model: Method 10; N = 10. (2005a). Minimum IAE – Ti Td 2 Kc Arrieta Orozco Model: Method 10; N = 10. Also given by Arrieta Orozco and Alfaro (2003). Ruiz (2003). 1 Kc = T 1 0.2068 + 1.1597 m Km τm 1.0158 0.5022 , Ti = Tm − 0.2228 + 1.3009 τm , T m τ Td = 0.3953Tm m Tm 2 Kc = 0.8469 , 0.05 ≤ τm ≤ 2.0 . Tm 0.9946 0.4796 T 1 , Ti = Tm − 0.2512 + 1.3581 τm , 0.1050 + 1.2432 m T Km τm m τ Td = Tm − 0.0003 + 0.3838 m Tm 0.9479 , 0.1 ≤ τm ≤ 1.2 . Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Td Comment Ti Td 3 τ Kc 0.1 ≤ m ≤ 1.2 Minimum ITAE Tm – Arrieta Orozco Model: Method 10 ; N = 10. Also given by Arrieta Orozco and Alfaro (2003). Ruiz (2003). Minimum performance index: servo tuning Tavakoli and Minimum IAE. Ti 4 Kc Tavakoli (2003). 0.0111Tm Minimum ISE. Ti 5 Model: Kc Method 1; τ 0.1 ≤ m ≤ 2 ; T m 1 0 . 8 τ m 0.3 + Tm τ 0.06 Minimum ITAE. m τm τm K m τm N = 10 + 0.1 + 0.04 Tm Tm 3 T 1 Kc = 0.1230 + 1.1891 m Km τm 1.0191 0.4440 , Ti = Tm − 0.3173 + 1.4489 τm , T m 0.9286 τ (Arrieta Orozco, 2003); Td = Tm 0.0053 + 0.3695 m Tm τ Td = Tm − 0.0053 + 0.3695 m Tm 4 0.9286 (Arrieta Orozco and Alfaro Ruiz, 2003). τ 0.3 m + 1.2 1 1 0.06 Tm 2 Kc = , Ti = τm . τm τm K m τm + 0.2 + 0 . 08 + 0 . 04 Tm Tm Tm τ 0.3 m + 0.75 1 1 T 5 . m , Ti = 2.4τm Kc = τm K m τm + 0.05 + 0 . 4 T T m m 123 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 124 Rule Kc Ti Comment Td Minimum IAE – Ti Td 6 Kc Alfaro Ruiz Model: Method 10; N = 10. (2005a). Minimum IAE – Ti Td 7 Kc Arrieta Orozco Model: Method 10; N = 10. Also given by Arrieta Orozco and Alfaro (2003). Ruiz (2003). Minimum ITAE T Td 8 i Kc – Arrieta Orozco Model: Method 10; N = 10. Also given by Arrieta Orozco and Alfaro (2003). Ruiz (2003). 6 Kc = T 1 0.3295 + 0.7182 m Km τm 0.9971 0.8456 , Ti = Tm 0.9781 + 0.3723 τm , T m τ Td = 0.3416Tm m Tm 7 Kc = 0.9414 Kc = τm ≤ 2.0 . Tm 0.9597 1.0092 T 1 , Ti = Tm 1.0068 + 0.3658 τm , 0.2268 + 0.8051 m T τm Km m τ Td = Tm − 0.0146 + 0.3500 m Tm 8 , 0.05 ≤ T 1 0.1749 + 0.8355 m Km τm 0.8100 , 0.1 ≤ τm ≤ 1.2 . Tm 0.9462 0.6884 , Ti = Tm 0.9581 + 0.3987 τm , T m 0.7417 τm , 0.1 ≤ τm ≤ 1.2 . Td = Tm − 0.0169 + 0.3126 Tm T m b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc K c = x1 Comment 9 12 13 1.473 Tm K m τm T τ Ti = x1 m m 1.115 Tm Td Minimum performance index: servo/regulator tuning Td τ Kc 0.1 ≤ m ≤ 1.0 Ti T 10 11 m Kc Td Minimum ISE – Arrieta and Vilanova (2007). Model: Method 10; N = 10. 9 Ti 125 Kc Kc 0.970 0.753 τ Td = 0.550x1Tm m Tm Td Ti 14 τm ≤ 2.0 Tm 0.897 + (1 − x1 ) 1.048 Tm K m τ m + (1 − x1 ) Tm ; 1.195 − 0.368(τm Tm ) 0.948 Td 1.1 ≤ ; τ + 0.489(1 − x1 )Tm m Tm 0.888 ; x1 = 0.5778 − 0.5753(τm Tm ) + 0.5528(τm Tm ) . 2 10 11 12 Kc = 1.473x1 + 1.048(1 − x1 ) Tm τ Km m K c = x1 1.524 Tm K m τm Tm τm 1.130 Tm 0.735 0.641 τ Td = 0.552x1Tm m Tm 14 . τ Td = [0.550x1 + 0.489(1 − x1 )]Tm m Tm Ti = x1 13 0.970 x 1 + 0.897 (1− x 1 ) Kc = 0.948 x 1 + 0.888(1− x 1 ) . + (1 − x1 ) 1.154 Tm K m τ m + (1 − x1 ) Tm 0.851 0.567 1.047 − 0.220 ; τ + 0.490(1 − x1 )Tm m Tm 1.524 x1 + 1.154((1 − x1 ) Tm τ Km m ; τm Tm 0.708 T ; x1 = 0.2382 + 0.2313 m τm 0.735 x 1 + 0.567 (1− x 1 ) τ Td = [0.552 x1 + 0.490(1 − x1 )]Tm m Tm . 0.851x 1 + 0.708(1− x 1 ) . 7.1208 . b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 126 Rule Kc Minimum ISTSE – Arrieta and Vilanova (2007). Model: Method 10; N = 10. 15 17-Mar-2009 K c = x1 1.468 Tm K m τm T τ Ti = x1 m m 0.942 Tm 15 Kc 16 Kc 18 Kc 19 Kc 0.970 0.725 τ Td = 0.443x1Tm m Tm Ti Td Ti Td 17 Ti τm ≤ 1.0 Tm 1.1 ≤ τm ≤ 2.0 Tm Td 0.897 + (1 − x1 ) 1.042 Tm K m τ m + (1 − x1 ) Tm ; 0.987 − 0.238(τm Tm ) 0.939 0.1 ≤ Td Td 20 Comment ; τ + 0.385(1 − x1 )Tm m Tm 0.906 ; x1 = 0.5778 − 0.5753(τm Tm ) + 0.5528(τm Tm ) . 2 16 17 18 Kc = 1.468x1 + 1.042(1 − x1 ) Tm τ Km m K c = x1 1.515 Tm K m τm Tm τm 0.957 Tm 0.730 0.598 τ Td = 0.444x1Tm m Tm 20 . τ Td = [0.443x1 + 0.385(1 − x1 )]Tm m Tm Ti = x1 19 0.970 x 1 + 0.897 (1− x 1 ) Kc = + (1 − x1 ) + (1 − x1 ) 0.847 0.939 x 1 + 0.906 (1− x 1 ) . 1.142 Tm K m τ m 0.579 Tm 0.919 − 0.172 ; τm Tm τ + 0.384(1 − x1 )Tm m Tm 1.515x1 + 1.142(1 − x1 ) Tm τ Km m ; 0.839 T ; x1 = 0.2382 + 0.2313 m τm 0.730 x 1 + 0.579 (1− x 1 ) τ Td = [0.440x1 + 0.384(1 − x1 )]Tm m Tm . 0.847 x 1 + 0.839 (1− x 1 ) . 7.1208 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Minimum ISTES – Arrieta and Vilanova (2007). Model: Method 10; N = 10. 21 K c = x1 1.531 Tm K m τm T τ Ti = x1 m m 0.971 Tm 21 Kc 22 Kc 24 Kc 25 Kc 0.960 0.746 τ Td = 0.413x1Tm m Tm Ti Td Ti Td 23 Ti 26 0.1 ≤ τm ≤ 1.0 Tm 1.1 ≤ τm ≤ 2.0 Tm Td 0.904 + (1 − x1 ) 0.968 Tm K m τ m + (1 − x1 ) Tm ; 0.977 − 0.253(τm Tm ) 0.933 Comment Td Td 127 ; τ + 0.316(1 − x1 )Tm m Tm 0.892 ; x1 = 0.5778 − 0.5753(τm Tm ) + 0.5528(τm Tm ) . 2 22 23 24 Kc = 1.531x1 + 0.968(1 − x1 ) Tm τ Km m K c = x1 1.592 Tm K m τm Tm τm 0.957 Tm 0.705 0.597 τ Td = 0.414x1Tm m Tm 26 . τ Td = [0.413x1 + 0.316(1 − x1 )]Tm m Tm Ti = x1 25 0.960 x 1 + 0.904 (1− x 1 ) Kc = 0.933 x 1 + 0.892 (1− x 1 ) . 0.583 + (1 − x1 ) 1.061 Tm K m τ m + (1 − x1 ) T , x1 = 0.2382 + 0.2313 m τ τm 0.892 − 0.165 m Tm 0.850 Tm τ + 0.315(1 − x1 )Tm m Tm 1.592 x1 + 1.061(1 − x1 ) Tm τ Km m ; 0.832 . 0.705 x 1 + 0.583(1− x 1 ) τ Td = [0.414 x1 + 0.315(1 − x1 )]Tm m Tm . 0.850 x 1 + 0.832 (1− x 1 ) . 7.1208 ; b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 128 Rule Kc Davydov et al. (1995). Model: Method 6 Kuhn (1995). Model: Method 17; N = 5. Schaedel (1997). Ti 27 Kc 28 Kc Comment Td Direct synthesis: time domain criteria Ti 0.25Ti ξ =0.9; 0 . 2 ≤ τm Tm ≤ 1. T 0.4T i i 1 Km 0.66(τ m + Tm ) 0.167(τ m + Tm ) ‘Normal tuning’. 2 Km 0.8(τ m + Tm ) 0.194(τ m + Tm ) ‘Rapid tuning’. T2 2 T33 − T1 T2 2 Model: Method 25 29 2 2 T1 − T2 Kc T1 − Td 27 Kc = 1 τ , Ti = 0.186 m + 0.532 Tm , N = K m . T K m (1.552(τm Tm ) + 0.078) m 28 Kc = 1 τ , Ti = 0.382 m + 0.338 Tm , N = K m . T K m (1.209(τm Tm ) + 0.103) m 29 Kc = Tm 0.375Ti , T1 = + τm , K m (T1 + (Td N ) − Ti ) 1 + 3.45 τam Tm ( 2 T2 = 1.25Tm τ am + T1 = 0.7Tm 2 Tm − 0.7 τ am T3 3 = N= ( Tm τa τm + 0.5τm 2 , T3 = 0 , 0 < m ≤ 0.104 ; a Tm 1 + 3.45 τm Tm ( 0.7Tm 2 + τ m , T2 = Tm τ am + ) 2 Tm − 0.7 τ am a m m m Td a + τm − Ti − + 2 (a + τm − Ti )2 + 0.375 TiTd 4 + 0.5τ m 2 , ) + T τ τ + 0.35T τ 0.952Tm 2 τ am − 0.1 Tm − 0.7 τ am ) 2 2 m + 0.167τ m 3 , Tm − 0.7τ am m τ am > 0.104 . Tm , 0.05 ≤ a ≤ 0.2 , K v = prescribed Kv overshoot in the manipulated variable for a step response in the command variable. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Goodwin et al. (2001). Latzel (1988). Model: Method 1 Latzel (1988). Model: Method 2 30 Kc = Ti = Ti Td p. 426; Model: Method 1 Direct synthesis: frequency domain criteria 0.84Tm 0.067Tm φm = 450 Td Kc 0.84Tm 0.067Tm φm = 600 33 Kc 0.76Tm 0.13Tm φm = 450 34 Kc 0.76Tm 0.13Tm φm = 600 Kc 31 Kc 32 (4ξTCL 2 + τm )(τm + 2Tm ) − 4TCL 2 2 ( K m 16ξ2TCL 2 2 + 8ξTCL 2 τm + τm 2 ), (4ξTCL 2 + τm )(τm + 2Tm ) − 4TCL 2 2 ( ( 2 16ξ 2TCL 2 2 + 8ξTCL 2 τm + τm 2 2 Td = 16ξ 2TCL 2 + 8ξTCL 2 τm + τm 2 ) [(4ξT ) (4ξT CL 2 + τ m ), )2 τ m Tm − 2TCL2 2 (4ξTCL2 + τ m )(τ m + 2Tm ) + 8TCL2 4 ] , (4ξTCL2 + τ m )3 [(4ξTCL2 + τ m )(τ m + 2Tm ) − 4TCL2 2 ] CL 2 + τ m N= Comment Ti 30 129 4ξTCL 2 + τm 2TCL 2 2 Td with G CL (s) = 1 TCL 2 2 s 2 + 2ξTCL 2 s + 1 31 Kc = 0.66Tm , N = 8.38; G p = G m K m (τm − 0.02Tm ) 32 Kc = 0.44Tm , N = 8.38; G p = G m K m (τm − 0.02Tm ) 33 Kc = K pe p 0.60Tm , N = 6.19; G p = . K m (τm − 0.07Tm ) (1 + sTp1 ) 2 34 Kc = K pe p 0.40Tm , N = 6.19; G p = . K m (τm − 0.07Tm ) (1 + sTp1 ) 2 − sτ − sτ . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 130 Rule Latzel (1988). Model: Method 2 Kc Ti Td Comment 35 Kc 0.74Tm 0.14Tm φm = 450 36 Kc 0.74Tm 0.14Tm φm = 600 Tm τ m 2Tm + τ m N = 10. Robust Chien (1988). Model: Method 1 Tm + 0.5τ m K m (λ + 0.5τ m ) Tm + 0.5τ m λ ∈ [τm , Tm ] , Chien and Fruehauf (1990); λ > [0.8τm ,0.2Tm ] , Yu (2006), p. 19. Morari and Zafiriou (1989). Model: Method 1 Gong et al. (1996). Maffezzoni and Rocco (1997). 37 Kc Tm + 0.5τ m 38 Kc Tm + 0.3866τ m 39 Kc Ti λ > 0.25τ m , Tm τ m 2Tm + τ m λ > 0.2Tm . 3 ≤ N ≤ 10 ; 0.3866Tm τ m Tm + 0.3866τ m Model: Method 1 Model: Method 14 Td − sτ 35 Kc = K pe p 0.58Tm , N = 5.83; G p = K m (τm − 0.08Tm ) (1 + sTp1 )3 36 Kc = K pe p 0.39Tm , N = 5.83; G p = K m (τm − 0.08Tm ) (1 + sTp1 )3 37 Kc = 2T (λ + τ m ) 1 Tm + 0.5τm , N= m . λ (2Tm + τ m ) K m λ + τm 38 Kc = 39 Kc = − sτ (0.1388 + 0.1247 N )Tm + 0.0482 Nτ m τ . Tm + 0.3866τm , λ= m 0.3866( N − 1)Tm + 0.1495Nτ m K m (λ + 1.0009τm ) 2Tm (τm + x1 ) + τm 2 2K m (τm + x1 )2 τm τm 2 [2Tm (τm + x1 ) − τm x1 ] , Td = , 2(τm + x1 ) 2(τm + x1 ) 2Tm (τm + x1 ) + τm 2 2 , Ti = Tm + 2Tm (τ m + x 1 ) − τ m x 1 [ ] τ N= . x 1 = 0.25τ m , uncertainty on τ m ; x1 > 0.1Tm , m < 0.4 , 2 Tm x 1 2Tm (τ m + x 1 ) + τ m [ ] uncertainty on the minimum phase dynamics within the control band. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 40 Gong et al. (1998). Model: Method 1; λ = x1τm ; x1 values deduced from graph. Kc = 41 Comment Ti Td N = 3, 5 or 10. x1 τm Tm x1 τm Tm x1 τm Tm 1 2 3 4 1 2 3 4 1 2 3 4 0.46 0.44 0.43 0.41 0.41 0.40 0.39 0.38 0.37 0.36 0.36 0.35 5 6 7 8 5 6 7 8 5 6 7 8 0.41 0.40 9 10 N = 3. 0.38 0.37 9 10 N = 5. 0.35 0.35 9 10 N = 10. Kc Ti Td 42 Kc Ti Td 43 Kc Ti Td 41 Robust to 20% change in plant parameters. Robust to 30% change in plant parameters. Robust to 40% change in plant parameters. (Tm + 0.3866τm )(λ + τm ) − (0.3866λ − 0.1247τm )τm , K m (λ + τm )2 Ti = Tm + 0.3866τm − Td = Td 0.63 0.57 0.51 0.48 0.52 0.48 0.45 0.43 0.42 0.40 0.38 0.38 Kasahara et al. (1999). Model: Method 1; N = 10; 0 < τm Tm < 1 . 40 Kc Ti 131 0.3866λ − 0.1247 τm τm , λ + τm 0.3866Tm τ m 0.3866λ − 0.1247 τ m − τm . Tm + 0.3866τ m λ + τm K c = 0.6K u [0.718 − 0.057(τm Tm )] , Ti = 0.5Tu [0.296 + 0.838(τm Tm )] , Td = 0.125Tu [0.635 − 1.091(τm Tm )] . 42 K c = 0.6K u [0.690 − 0.076(τm Tm )] , Ti = 0.5Tu [0.274 + 0.788(τm Tm )] , Td = 0.125Tu [0.526 − 0.057(τm Tm )] . 43 K c = 0.6K u [0.331 + 0.149(τm Tm )] , Ti = 0.5Tu [0.041 + 0.857(τm Tm )] , Td = 0.125Tu [0.495 − 0.064(τm Tm )] . b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 132 Rule Kc Kasahara et al. (1999) – continued. Leva and Colombo (2000). 44 Kc 45 Kc Leva (2001). Model: Method 1 46 Kc Leva et al. (2003), (2006). 47 Kc 48 Kc Ti Td Ti Td Comment Robust to 50% change in plant parameters. λ not specified; Model: Method 1 2 Zhang et al. (2002b). Model: Method 1 44 17-Mar-2009 Tm + τm 2(λ + τ m ) Td Tm + τm2 2(λ + τ m ) Td Tm + τm2 2(λ + τ m ) Td Model: Method 1 τm Tm Td − 2Ti N λ > 0.0735τm for stability (Ou et al., 2005c). Ti K c = 0.6K u [0.195 + 0.266(τm Tm )] , Ti = 0.5Tu [− 0.013 + 0.731(τm Tm )] , Td = 0.125Tu [0.489 − 0.065(τm Tm )] . 45 46 Kc = 2Tm (λ + τ m ) Tm (λ + τm ) 1 2(λ + τm )Tm + τm ,N= . , Td = 2 Km 2( λ + τ m ) 2 2(λ + τm )Tm + τm λ 2(λ + τ m )Tm + τ m 2 Kc = 2Tm (λ + τ m ) 1 2(λ + τm )Tm + τm , N= −1, Km 2( λ + τ m ) 2 λ 2(λ + τ m )Tm + τ m 2 2 [ ] 2 2 Td = τm 2 2 [ 2λTm + (2Tm − λ )τm [ 2(λ + τm ) 2Tm (λ + τm ) + τm 2 [ ] ] ] . 0.1T ≤ λ ≤ 0.5T , τ T ≤ 0.25 ; m m m m λ = 1.5(τ m + Tm ) , 0.25 < τ m Tm ≤ 0.75 ; λ = 3(τ m + Tm ) , τm Tm > 0.75 . 47 Kc = 2 τm Tm (λ + τm ) 1 2(λ + τm )Tm + τm , Td = , Km 2( λ + τ m ) 2 2(λ + τm )Tm + τm 2 2Tm (λ + τ m ) [ ] 2 N= [ λ 2(λ + τ m )Tm + τ m 2 ] . 0.1T ≤ λ ≤ 0.5T , τ T ≤ 0.33 ; m m m m λ = 1.5(τ m + Tm ) , 0.33 < τm Tm ≤ 3 ; λ = 3(τ m + Tm ) , τm Tm > 3 . 48 Kc = λ2 0.5τm + Tm − (Td N ) T T or N = 10. , Ti = 0.5τm + Tm − d , d = K m (2λ + 0.5τm ) N N 2λ + 0.5τm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Vilanova and Balaguer (2006). Ti K m (x1 + x 2 ) 50 Tang et al. (2007). Model: Method 1; Td N = Tf . 49 Kc Ti 49 Ti Ti x2 Td Comment x1 + x 3 N x1 + x 2 Model: Method 1 133 Td Labelled the H ∞ Smith predictor. 51 Kc Ti Td Labelled the H 2 Smith predictor or the Dahlin controller. 2(3Tm + τm ) 9K m τm Tm + 0.333τm (6Tm − τm )τm 6(3Tm + τm ) Recommended λ = 0.5 ; Tf = 0.167 τm . x + x3 T x x + x2 , N= m 1 1 Ti = Tm + τm + x 3 − x1 − x 2 1 −1 . Ti τm x1 + x 3 x1 + x 2 For minimum ISE, servo response: x 3 = τm ; for minimum IE, regulator response: x 2 = τm ; x1 = real constant. (τm + 2Tm )(4λ + τm ) − 4λ2 , T = 0.5τ + T − 2λ2 , i m m 4λ + τ m K m (4λ + τm )2 2 Tm τm (4λ + τm ) 2λ 2λ2 λ Td = − T = , ; 0.3 ≤ ≤ 0.8 . f 2 4λ + τm τm (τm + 2Tm )(4λ + τm ) − 4λ 4λ + τm (τ + 2Tm )(λ + τm ) − λτm , T = 0.5τ + T − λτm , 51 Kc = m i m m 2(λ + τm ) 2K m (λ + τm )2 λτ m Tm τ m (λ + τ m ) λτ m λ ,T = − Td = ≤ 0.8 . ; 0.3 ≤ (τ m + 2Tm )(λ + τ m ) − λτ m 2(λ + τm ) f 2(λ + τm ) τm 50 Kc = b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 134 1 1 + Td s 3.3.5 Classical controller G c (s) = K c 1 + Ti s 1 + Td s N E(s) R(s) − + U(s) 1 1 + Td s K c 1 + Td Ti s s 1 + N Y(s) K m e − sτ 1 + sTm m Table 13: PID controller tuning rules – FOLPD model G m (s) = Rule Kc Kraus (1986). Model: Method 23 0.833Tm K m τm Witt and Waggoner (1990). Comment Td Process reaction 1.5τ m 0.25τm N=∞ Foxboro EXACT controller pre-tuning. Also given by Hang et al. (1993b), p. 76. x1Tm K m τm τm τm 10 ≤ N ≤ 20 ; 0.6 ≤ x1 ≤ 1.0 . Model: Method 2. ‘Equivalent to’ Ziegler and Nichols (1942). 1 Ti Td 10 ≤ N ≤ 20 Kc Witt and Waggoner (1990). 1 Ti K m e − sτ m 1 + sTm Model: Method 2. ‘Preferred’ tuning. Tuning ‘equivalent to’ Cohen and Coon (1953). 1.350 Kc = τ Tm T τ + 0.25 + m 0.7425 + 0.0150 m + 0.0625 m τm τm Tm Tm 2K m Ti = 2 , Tm τ τ T T 1.350 m + 0.25 − m 0.7425 + 0.0150 m + 0.0625 m τm τm Tm Tm 2 , Tm Td = T T 1.350 m + 0.25 + m τm τm τ τ 0.7425 + 0.0150 m + 0.0625 m Tm Tm 2 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 2 Witt and Kc Waggoner (1990) – continued. St. Clair (1997), Tm K m τm p. 21. Model: Method 0.5Tm K m τm 2; N ≥ 1 . Harrold (1999). 1 Km Model: 0.5 K m Method 11; 0.25 K m N not specified. Tan et al. 0.6Tm (1999a), p. 25. K mτm Shinskey (2000). Model: Method 2 Ti Td Ti Td 135 Comment ‘Alternate’ tuning. 10 ≤ N ≤ 20 . 5τ m 0.5τ m 5τ m 0.5τ m Tm ≤ 0.25Tm ‘Aggressive’ tuning. ‘Conservative’ tuning. τ m Tm ≤ 0.25 τ m Tm ≈ 0.5 τ m Tm ≥ 1 0.889Tm K mτm τm τm N =∞; Model: Method 2 1.75τ m 0.70τ m τm Tm = 0.167 ; N not specified. x1Tm K m τm x 2τm x 3τ m N=∞ O’Dwyer Representative results: equivalent to Chien et al. (1952) – regulator. (2001b). x2 x3 Model: Method 2 x 1 0.7236 1.8353 0.5447 0% overshoot; 0.1 < τ m Tm < 1 . 0.84 1.350 2 Kc = 1.4 0.6 20% overshoot; 0.1 < τ m Tm < 1 . τ Tm T τ + 0.25 − m 0.7425 + 0.0150 m + 0.0625 m τm τm Tm Tm 2K m Ti = 2 , Tm τ τ T T 1.350 m + 0.25 + m 0.7425 + 0.0150 m + 0.0625 m τm τm Tm Tm Td = 2 , Tm τ τ T T 1.350 m + 0.25 − m 0.7425 + 0.0150 m + 0.0625 m τm τm Tm Tm 2 . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 136 Rule O’Dwyer (2001b). Model: Method 2; N =∞. Faanes and Skogestad (2004). Minimum ITAE with OS ≤ 8% – Fertik (1975). Model: Method 3; N = 11. Kc Ti Td Comment 3 Kc Ti Td 0% overshoot; τ m Tm < 0.5 . 4 Kc Ti Td 20% overshoot; τ m Tm < 0.7234 Representative results: equivalent to Chien et al. (1952) – servo. Model: 0.47Tm Method 1; τ τ m m K m τm N = 10. Minimum performance index: regulator tuning 8.1 K m 0.29(τm + Tm ) 0.13(τm + Tm ) τm Tm = 0.11 3.8 K m 1.85 K m 0.93 K m 0.56 K m 0.46(τm + Tm ) 0.23(τm + Tm ) 0.46(τm + Tm ) 0.40(τm + Tm ) 0.50(τm + Tm ) 0.40(τm + Tm ) 0.33(τm + Tm ) 0.44(τm + Tm ) τm Tm = 0.25 τm Tm = 0.43 τm Tm = 0.67 τm Tm = 1.00 Coefficients of K c , Ti , Td deduced from graphs and converted to this equivalent controller architecture. 0.982 Minimum IAE – N = 8; 0.817 Tm 5 Huang and Chao Model: Method 1 Td Ti K m τm (1982). Minimum IAE – Model: Method 6 Kaya and Scheib 5; N = 10; Ti Td Kc (1988). 0 < τm Tm ≤ 1 . T 2τ 0.3Tm 2τ T 2τ 1 − 1 − m Ti = m 1 + 1 − m Td = m 1 − 1 − m K m τm Tm 2 Tm 2 Tm , , τ τ 0.475Tm 4 Kc = 1 + 1 − 1.3824 m Ti = 0.68Tm 1 + 1 − 1.3824 m K m τm Tm Tm , , 3 Kc = τ Td = 0.68Tm 1 − 1 − 1.3824 m T m 5 6 τ Ti = 0.903Tm m Tm Kc = 0.780 0.98089 Tm K m τm τ , Td = 0.602Tm m Tm 0.76167 0.954 . 1.05211 , Ti = Tm τm 0.91032 Tm τ , Td = 0.59974Tm m Tm 0.89819 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum IAE – Witt and Waggoner (1990). Kc Ti Td 7 Kc Ti Td 8 Kc Ti Td Comment 0.1 < τm Tm ≤ 0.258 ; 10 ≤ N ≤ 20 ; Model: Method 2. 0.95Tm K m τ m 1.43τ m 0.52τ m τ m Tm = 0.2 1.17 τ m 0.48τ m τ m Tm = 0.5 1.03τ m 0.40τ m τ m Tm = 1 0.77τ m 0.35τ m τ m Tm = 2 Minimum IAE – Shinskey (1996), 0.96Tm K m τ m p. 119. 1.43τ m 0.52τ m Model: Method 1; τ m Tm = 0.2 . Minimum IAE – Edgar et al. 0.94Tm K m τ m (1997), pp. 8-14, 8-15. 1.5τ m 0.55τ m Minimum IAE – Shinskey (1988), 0.95Tm K m τ m p. 143. 1.14Tm K m τ m Model: Method 1 1.39Tm K m τ m 7 ‘Preferred’ tuning. K c = 0.718 Tm K m τm 0.921 1 + 1 − 1.693 τm T m Model: Method 1; Time constant dominant; N = 10. −1.886 , 1.137 Ti = 8 137 ‘Alternate’ tuning. K c = 0.718 Tm K m τm τ 0.964Tm m Tm 1.886 τ 1 − 1 − 1.693 m Tm 0.921 1 − 1 − 1.693 τm T m 1.137 , Td = τ 0.964Tm m Tm 1.886 τ 1 − 1 + 1.693 m Tm 1.886 . τ 1 + 1 + 1.693 m Tm −1.886 , 1.137 Ti = τ 0.964Tm m Tm 1.137 , Td = τ 0.964Tm m Tm 1.886 τ 1 + 1 − 1.693 m Tm . b720_Chapter-03 Rule Kc Minimum IAE – Edgar et al. 0.88Tm K m τ m (1997), p. 8–15. Minimum IAE – Smith and Tm Corripio (1997), K mτm p. 345. Minimum IAE – 0.89Tm K m τm Shinskey (2003). 0.56K u Minimum IAE – 0.49 K u Shinskey (1988), 0.51K u p. 148. Model: Method 1 0.46K u Minimum IAE – K u τm Shinskey (1994), 3τ − 0.32T m u p. 167. Minimum ISE – 1.101 T 0.881 m Huang and Chao K m τ m (1982). Minimum ISE – Kaya and Scheib (1988). 11 Kc Minimum ITAE 0.745 T 1.036 m – Huang and K m τm Chao (1982). 10 11 12 FA Handbook of PI and PID Controller Tuning Rules 138 9 17-Mar-2009 Comment Ti Td 1.8τ m 0.70τ m Tm 0.5τ m 1.75τm 0.70τ m Model: Method 2 0.39Tu 0.14Tu τ m Tm = 0.2 0.34Tu 0.14Tu τ m Tm = 0.5 0.33Tu 0.13Tu τ m Tm = 1 0.28Tu 0.13Tu τ m Tm = 2 9 Ti 0.14Tu 10 Ti Td Ti 12 Td Td Ti Model: Method 2; N = 10. Model: Method 1; τ 0.1 ≤ m ≤ 1.5 . Tm Model: Method 1; N not specified. Model: Method 1; N = 8. Model: Method 5; 0 < τm Tm ≤ 1 ; N = 10. Model: Method 1; N = 8. T Ti = Tu 0.15 u − 0.05 . τ m τ Ti = 1.134Tm m Tm Kc = 0.883 1.11907 Tm K m τm τ Ti = 0.771Tm m Tm τ , Td = 0.563Tm m Tm 0.89711 0.595 , Ti = 0.881 . Tm τm 0.7987 Tm 0.9548 1.006 τ , Td = 0.597Tm m Tm . τ , Td = 0.54766Tm m Tm 0.87798 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Minimum ITAE – Kaya and Scheib (1988). Minimum ITAE – Witt and Waggoner (1990). 13 Kc Ti Td Comment Ti Td 0 < τm Tm ≤ 1 N = 10; Model: Method 5. 14 Kc Ti Td Preferred tuning. 15 Kc Ti Td Alternate tuning. Model: Method 2; 10 ≤ N ≤ 20 . 0.738 0.947 Minimum ITAE Tm τ m 1.357 Tm 16 – Ou and Chen Td 0.842 Tm K m τm (1995). 0.907 Minimum ITSE – 17 Huang and Chao 0.994 Tm Td Ti Km τm (1982). 1.06401 13 Kc = 0.77902 Tm K m τm 14 Kc = 0.679 Tm K m τm , Ti = Tm τm 1.14311 Tm 0.947 1 + 1 − 1.283 τm T m Ti = 15 0.679 Tm Kc = K m τm 0.947 1 − 1 − 1.283 τm T m Ti = 16 17 139 τ Td = 0.381Tm m Tm τ Ti = 1.032Tm m Tm 0.995 0.869 , N=± 0.70949 Model: Method 57 Model: Method 1; N = 8. 1.03826 τ , Td = 0.57137Tm m Tm . −1.733 , 0.1 < τ m < 0.379 ; Tm τ 0.762Tm m Tm 0.995 1.733 τ 1 − 1 − 1.283 m Tm , Td = τ 0.762Tm m Tm 0.995 1.733 . τ 1 + 1 + 1.283 m Tm −1.733 , τ 0.762Tm m Tm 0.995 1.733 τ 1 − 1 + 1.283 m Tm , Td = 0.75K m K cTd , N>0. Ti + K m K c (Td + Ti − τm − Tm ) τ , Td = 0.570Tm m Tm 0.884 . τ 0.762Tm m Tm 0.995 1.733 τ 1 + 1 − 1.283 m Tm . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 140 Rule Kc Minimum ITAE with OS ≤ 8% – Fertik (1975). Model: Method 3; N = 11. 5.0 K m Ti Td Comment Minimum performance index: servo tuning 0.63(τm + Tm ) 0.13(τm + Tm ) τm Tm = 0.11 2.4 K m 1.2 K m 0.69 K m 0.51 K m 0.57(τm + Tm ) 0.23(τm + Tm ) 0.51(τm + Tm ) 0.33(τm + Tm ) 0.46(τm + Tm ) 0.40(τm + Tm ) 0.40(τm + Tm ) 0.44(τm + Tm ) τm Tm = 0.25 τm Tm = 0.43 τm Tm = 0.67 τm Tm = 1.00 Coefficients of K c , Ti , Td deduced from graphs and converted to this equivalent controller architecture. 1.00 Minimum IAE – Model: 0.673 Tm 18 Huang and Chao Method 1; T Ti d K m τm (1982). N = 8. 1.04432 Minimum IAE – 0.65 Tm Kaya and Scheib K m τm (1988). Minimum IAE – Smith and Corripio (1997), p. 345. 0.83Tm K mτm Ti Td Tm 0.5τ m 19 1.032 18 Ti = 19 Ti = τ Tm , Td = 0.502Tm m τ Tm 0.148 m + 0.979 T m . 1.08433 τ , Td = 0.50814Tm m τm Tm 0.9895 + 0.09539 Tm Tm . Model: Method 5; 0 < τm Tm ≤ 1 ; N = 10. Model: Method 1; τ 0.1 ≤ m ≤ 1.5 ; Tm N not specified. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Minimum IAE – Witt and Waggoner (1990). Kc = 1.086 Tm K m τm Td Comment 20 Kc Ti Td Preferred tuning. 21 Kc Ti Td Alternate tuning. 0.1 ≤ τm Tm ≤ 1 ; 10 ≤ N ≤ 20 ; Model: Method 2. Minimum ISE – 0.787 T 0.964 m Huang and Chao K m τ m (1982). 20 Ti 22 0.869 1 + 1 − 1.392 τm T m Ti = Ti 0.914 Kc = 1.086 Tm K m τm 0.869 τm 0.74 − 0.13 , Tm τ 0.696Tm m Tm τ 1 − 1 − 1.392 m Tm 1 − 1 − 1.392 τm T m Ti = 0.914 0.869 1.028 T Ti = 1.678Tm m τm τ 0.696Tm m Tm τ 1 + 1 − 1.392 m Tm 0.869 0.914 . τm 0.74 − 0.13 Tm τm 0.74 − 0.13 , Tm τ 0.696Tm m Tm τ 1 + 1 − 1.392 m Tm τ , Td = 0.594Tm m Tm 0.914 τm 0.74 − 0.13 T m 0.869 0.971 . 0.914 , τm 0.74 − 0.13 T m Td = 22 Model: Method 1; N = 8. Td Td = 21 141 τ 0.696Tm m Tm τ 1 − 1 − 1.392 m Tm 0.869 0.914 τm 0.74 − 0.13 Tm . b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 142 Rule Kc Minimum ISE – 23 Kaya and Scheib Kc (1988). Model: Method 5 0.995 Minimum ITAE 0.684 Tm – Huang and K m τm Chao (1982). Minimum ITAE 25 – Kaya and Kc Scheib (1988). Model: Method 5 0.85 Minimum ITAE 0.965 Tm – Ou and Chen K m τm (1995). 1.03092 23 17-Mar-2009 Kc = 0.71959 Tm K m τm , Ti = Ti Td Ti Td Ti Td Ti Td 24 26 Td Ti Comment τm ≤ 1; Tm N = 10. 0< Model: Method 1; N = 8. τ 0 < m ≤ 1; Tm N = 10. Model: Method 57 τ , Td = 0.54568Tm m τm Tm 1.12666 − 0.18145 Tm Tm 0.86411 . 1.049 24 τ Tm Ti = , Td = 0.491Tm m τm Tm + 0.986 0.114 Tm 25 Kc = 26 Ti = 1.12762 Tm K m τm 0.80368 , Ti = 1.0081 τ , Td = 0.42844Tm m τm Tm 0.99783 + 0.02860 Tm Tm τ , Td = 0.308Tm m τ Tm 0.796 − 0.1465 m Tm Tm . . 0.929 , N=± 0.75K m K cTd , N>0. Ti + K m K c (Td + Ti − τm − Tm ) b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum ITAE – Witt and Waggoner (1990). Kc 0.965 Tm Kc = K m τm Td Comment 27 Kc Ti Td Preferred tuning. 28 Kc Ti Td Alternate tuning. 0.1 ≤ τm Tm ≤ 1 ; 10 ≤ N ≤ 20 ; Model: Method 2. 1.00 Minimum ITSE – 0.718 Tm Huang and Chao K m τm (1982). 27 Ti 29 0.85 1 + 1 − 1.232 τm T m Ti = Ti 0.929 0.965 Tm Kc = K m τm 0.85 τm 0.796 − 0.1465 , Tm τ 0.616Tm m Tm τ 1 − 1 − 1.232 m Tm 1 − 1 − 1.232 τm T m Ti = 0.929 0.85 , τ 0.616Tm m Tm τ 1 + 1 − 1.232 m Tm 0.85 0.929 . τm 0.796 − 0.1465 T m τm 0.796 − 0.1465 , Tm τ 0.616Tm m Tm τ 1 + 1 − 1.232 m Tm 0.85 τ Tm , Td = 0.596Tm m τ Tm 0.063 m + 1.061 T m 0.929 , τm 0.796 − 0.1465 T m 1.028 Ti = 0.929 τm 0.796 − 0.1465 T m Td = 29 Model: Method 1; N = 8. Td Td = 28 143 . τ 0.616Tm m Tm τ 1 − 1 − 1.232 m Tm 0.85 0.929 τm 0.796 − 0.1465 T m . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 144 Rule Kc x1Tm K m τm Tsang et al. (1993). Model: Method 19 x1 ξ 1.6818 1.3829 1.1610 0.0 0.1 0.2 x1 ξ Comment Td Ti Direct synthesis: time domain criteria Tm 0.25τ m Coefficient values: N = 2.5. x1 ξ x1 ξ 0.9916 0.3 0.6693 0.6 0.8594 0.4 0.6000 0.7 0.7542 0.5 0.5429 0.8 Coefficient values: N = ∞ . x1 ξ x1 ξ x1 ξ 0.4957 0.4569 0.9 1.0 x1 1.8194 0.0 1.0894 0.3 0.7482 0.6 0.5709 1.5039 0.1 0.9492 0.4 0.6756 0.7 0.5413 1.2690 0.2 0.8378 0.5 0.6170 0.8 Tsang and Rad 0.809Tm K m τm Tm 0.5τ m (1995). Overshoot = 16%; N = 5; Model: Method 15. ξ 0.9 1.0 Smith and 0.5τ m 0.5Tm K m τm Tm Corripio (1997), Servo – 5% overshoot; N not specified; Model: Method 1. p. 346. 2 Schaedel (1997). T T1 2 − 2T2 2 T1 − 2 30 Model: N not defined. Kc T1 Method 25 Wang et al. Tm 1 (2002). Labelled ‘IMC’. 0.5τ m τ τ Model: K m 1 + m K m 1 + m 2Tm 2Tm Method 1; N = 10. 30 Kc = T1 = T2 0.375 T1 − 2 T1 T 2 − 2T 2 2 T 2 2 2 Km 1 + 2 − T1 − 2T2 N T1 Tm 1 + 3.45 T1 = 0.7Tm τ am Tm 2 Tm − 0.7 τ am + τ m , T2 2 = 1.25Tm τ am + , Tm 1 + 3.45 2 + τ m , T2 = Tm τ am + 0.7Tm 2 Tm − 0.7 τ am τ am Tm τ m + 0.5τ m 2 , 0 < + 0.5τ m 2 , τ am Tm τ am > 0.104 . Tm ≤ 0.104 ; b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Wang et al. (2002) – continued. Kc 0.65 Tm K m τ m 145 Ti Td Comment 31 Ti Td Labelled ‘IAE setpoint’. 1.04432 32 Kc Ti Td 33 Kc Ti Td 34 Kc Ti Td 35 Kc Ti Td 36 Kc Ti Td Labelled ‘IAE load’. Labelled ‘ISE setpoint’. Labelled ‘ISE load’. Labelled ‘ITAE setpoint’. Labelled ‘ITAE load’. Åström and 1 Tm Hägglund (2006), K 0.5τ + λ Model: Method 1 Tm 0.5τm m m pp. 188–189. λ = Tm (aggressive tuning), λ = 3Tm (robust tuning). N = ∞ . 31 32 33 34 35 36 1.08433 τ , Td = 0.50814 m Ti = τm Tm 0.9895 + 0.09539 Tm Tm 0.98089 Tm K m τ m 0.76167 0.71959 Tm Kc = K m τ m 1.03092 1.11907 Tm K m τ m 0.89711 1.12762 Tm Kc = K m τ m 0.80368 Kc = Kc = Kc = 0.77902 Tm K m τ m 1.06401 . 1.05211 τ , Ti = 1.09851Tm m Tm τ , Td = 0.59974Tm m Tm 0.89819 . τ , Ti = , Td = 0.54568 m τm Tm 1.12666 − 0.18145 Tm 0.86411 0.95480 0.87798 Tm τ , Ti = 1.2520Tm m Tm τ , Td = 0.54766Tm m Tm . . 1.0081 τ , Ti = , Td = 0.42844 m τm Tm 0.9978 + 0.02860 Tm Tm τ , Ti = 0.87481Tm m Tm 0.70949 . 1.03826 τ , Td = 0.57137Tm m Tm . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 146 Rule Hougen (1979) – p. 335. O’Dwyer (2001a). Model: Method 1 Huang et al. (2005a). Model: Method 51 Kc 37 Td Ti Comment Direct synthesis: frequency domain criteria Tm 0.45τm Kc Model: Method 1; maximise crossover frequency; 10 ≤ N ≤ 30 . x1Tm K m τm Tm N Tm A m = π 2 x 1 N ; φ m = 0.5π − x 1 N ; representative result below. 0.079Tm K mτm 0.1Tm Tm 0.65Tm K m τm Tm 0.4 τ m A m = 2.0; φ m = 450 . N = 20, A m = 2.7 , φ m = 650 . The above is a representative tuning rule (also given by Huang et al. (2003), N not defined, Model: Method 37). Other rules may be deduced from a graph provided by the authors for other A m , φ m values. Huang and Jeng (2005). 0.65Tm K m τm Harris and Tyreus (1987). Model: Method 1 Tm K m (τ m + λ ) τm N = 40; Model: Method 1 0.5τ m N = (τ m + λ ) λ Tm Robust Chien (1988). Model: Method 1 Tm λ ≥ maximum of [0.45τ m ,0.1Tm ] Tm K m (λ + 0.5τ m ) Tm 0.5τ m 0.5τ m K m (λ + 0.5τ m ) 0.5τ m Tm Tm 0.55τm 0.5τ m 1.1Tm Tm Chien (1988). K (λ + 0.5τ ) m m Model: Method 1 0.5τ m K m (λ + 0.5τ m ) λ ∈ [τ m , Tm ] (Chien and Fruehauf, 1990); N = 10. λ ∈ [τ m , Tm ] (Chien and Fruehauf, 1990); N = 11. Tuning rules converted to this equivalent controller architecture. 37 K c = 9.0 K m , τm Tm = 0.1 ; K c = 4.2 K m , τm Tm = 0.2 ; K c = 2.8 K m , τm Tm = 0.3 ; K c = 2.1 K m , τm Tm = 0.4 ; K c = 1.7 K m , τm Tm = 0.5 ; K c = 1.45 K m , τm Tm = 0.6 ; K c = 1.2 K m , τm Tm = 0.7 ; K c = 1.1 K m , τm Tm = 0.8 ; K c = 0.95 K m , τm Tm = 0.9 ; K c = 0.85 K m , τm Tm = 1.0 ; values deduced from graph. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Zhang et al. 38 (1996b). Kc Model: 39 Kc Method 47 40 Zhang et al. Kc (2002b). Shi and Lee 41 (2002). Kc Model: Method 1 Lee and Shi 42 (2002). Kc Model: Method 1 Faanes and 0.77Tm K m τm Skogestad 0.5Tm K m τm (2004). N = 10. Tm Tang et al. K m (2λ + 0.5τm ) (2007). τm Model: Method 1 K m (4λ + τm ) Comment Ti Td 0.5τ m Tm Tm 0.5τ m ≤ 1. 2 τ m . Tm 0.5τ m Model: Method 1 Tm ωg τ m 1 tan ωg 2 Suggested ωg = 0.6 τm . Tm 43 Ti 0.5τ m Tm 0.5τ m 8τm 0.5τ m Tm 0.5τm 0.5τm Tm 0.2τm ≤ λ Suggested ωg = 0.6 τm . Model: Method 1 Tf = 2λ2 , 4λ + τm 0.3 ≤ λ ≤ 0.8 . τm Tf = λτ m , 2(λ + τm ) 0.3 ≤ λ ≤ 0.8 . τm Labelled the H ∞ Smith predictor. Tm K m (λ + τm ) Tm 0.5τm τm 2K m (λ + τm ) 0.5τm Tm Labelled the H 2 Smith predictor or the Dahlin controller. T (2λ + 0.5τ m ) 0.5τm , N= m . K m (0.5τm + 2λ ) λ2 38 Kc = 39 Kc = 0.5τ m (2λ + 0.5τ m ) Tm , N= . K m (0.5τm + 2λ ) λ2 40 Kc = Tm λ2 . , N = 10 or N = 0.5τ m (2λ + 0.5τ m ) K m (2λ + 0.5τm ) 41 Kc = ωg τ m 2ω bw ωbw Tm . tan , N = 1+ ωg K m 1 + 2ωbw ωg tan 0.5ωg τm 2 42 Kc = ωg τ m 2ω bw ωbw Tm . tan , N = 1+ ωg K m 1 + 2ωbw ωg tan 0.5ωg τm 2 43 Ti = [ ( [ ( ) ( ) ( T 1 Tm . , 1 < x1 < 1 + m + x1 τm ωbw τm )] )] 147 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 148 Rule Kc Ti Td Comment Tang et al. (2007) – continued. 0.667Tm K m τm Tm 0.5τm 0.333 K m 0.5τm Tm Recommended λ = 0.5 ; Tf = 0.167 τm . Pessen (1994). Model: Method 1 0.35K u Ultimate cycle 0.25Tu 0.25Tu 44 Huang et al. (2005a). Model: Method 51 Kc Ti N =∞; 0.1 ≤ τm Tm ≤ 1 . Td N = 20, A m = 2.7 , φ m = 650 . The above is a representative tuning rule; other rules may be deduced from a graph provided by the authors for other A m , φ m values. 2 ^ K u − 1 2 ^ 0.65 ^ 44 , Ti = 0.159 Tu K u − 1 , Kc = 2 Km ^ −1 π − tan K u −1 ^ Td = 0.064 Tu π − tan −1 2 ^ K u − 1 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 149 3.3.6 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N R(s) E(s) + − 2 T s 1 d b f 0 + b f 1s + b f 2 s K c 1 + + 2 Ti s T 1 + a f 1s + a f 2 s 1+ d s N U(s) K m e − sτ 1 + sTm m Table 14: PID controller tuning rules – FOLPD model G m (s) = Rule Tsang et al. (1993). Model: Method 19 Kristiansson (2003). Model: Method 12 1 Kc Ti Y(s) K m e − sτ m 1 + sTm Comment Td Direct synthesis: time domain criteria 0 x1Tm K m τm Tm 2 2 b f 0 = 1 , b f 1 = 0.5τm , b f 2 = 0.0833τm , a f 1 = 0.2 τ m , a f 2 = 0.01τ m . x1 ξ x1 ξ 1.851 1.552 1.329 0.0 1.160 0.3 0.841 0.6 0.622 0.1 1.028 0.4 0.768 0.7 0.553 0.2 0.925 0.5 0.695 0.8 Direct synthesis: frequency domain criteria 0.9 1.0 1 Kc ξ x1 Ti Equations continued into the footnote on the next page; 2 T τ τ 21.1 m − 0.1 0.06 + 0.68 m − 0.12 m Tm τm Tm Kc = , 2 τ τ K m 0.68 + 0.84 m − 0.25 m Tm Tm ξ x1 Td 0.05 ≤ τm ≤2; Tm 1.6 ≤ M s ≤ 1.9 . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 150 Rule Kristiansson (2003) – continued. Shamsuzzoha and Lee (2007b). Kc Ti Td 2 Kc Ti Td 3 Kc 0.5τm 0.167 τm Comment Model: Method 1 2 2 τ τ τ τ 2Tm 0.06 + 0.68 m − 0.12 m 0.5Tm 0.06 + 0.68 m − 0.12 m Tm Tm Tm Tm Ti = , Td = , 2 2 τm τm τ τ m m 0.68 + 0.84 0.68 + 0.84 − 0.25 − 0.25 Tm Tm Tm Tm N = ∞ , b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , a f 1 = 0.8Tf , a f 2 = Tf 2 with Tf = 2 Kc = Td = [ Kc = 2 2 0.6667(Tm + τm )[1.1(Tm τm ) − 0.1] 0.6667(Tm + τm ) , Ti = , 2 2 τm τm τ τ m m 0.68 + 0.84 K m τm 0.68 + 0.84 − 0.25 − 0.25 Tm Tm Tm Tm 0.1667(Tm + τm ) , N = ∞ , b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , a f 1 = 0.8Tf , 2 τm τ m 0.68 + 0.84 − 0.25 Tm Tm a f 2 = Tf 2 with Tf = 3 ] Tm 0.06 + 0.68(τm Tm ) − 0.12(τm Tm ) [1.1(Tm τm ) − 0.1] . min{5 + 2(Tm τm ),25} (Tm + τm )2 [1.1(Tm τm ) − 0.1] ; 0.7 < τ m < 2 ; 1.6 ≤ M ≤ 1.9 . s Tm 9Tm (5 + 2(Tm τm )) 2 τm λ − τ m Tm e , bf 2 = 0 , , N = ∞ , b f 0 = 1 , b f 1 = Tm 1 − 1 − 2K m (2λ − b f 1 + τm ) Tm af1 = 0.5b f 1τm + λτ m + λ2 − Tm , a f 2 = 0 ; λ = x1Tm , λ values deduced from a graph 2λ − b f 1 + τm for M s = 1.4, 1.5, 1.6, 1.8 and 1.9; λ values defined for 0.01 ≤ τm Tm ≤ 5 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Td Comment Model: Method 1; λ > τ m , λ < Tm . 151 Robust Horn et al. (1996). Lee and Shi (2002). Model: Method 1 Shamsuzzoha et al. (2004). Model: Method 1 Shamsuzzoha and Lee (2006a). Model: Method 1 4 Kc Tm + 0.5τ m τ m Tm τ m + 2Tm 5 Kc Tm x1 0 6 Kc Tm + 0.5τm Tm τm 2Tm + τm 0.5τm K m (λ + τm ) 0.5τm 0.167 τm af1 = 1 < x1 < 2Tm τm λ not explicitly specified. 2 τmλ τm λ , af 2 = , b f 0 = 1 , b f 1 = Tm , 2(τm + λ ) 12(τm + λ ) b f 2 = 0 , N = ∞ . λ = x1τm ; x1 = 1.360 , M s = 1.4 ; x1 = 0.677 , M s = 1.6 ; x1 = 0.372 , M s = 1.8 ; x1 = 0.337 , M s = 2.0 . Yu (2006), p. 18. Model: Method 1 4 Kc = 0.6K u Ultimate cycle 0 0.5Tu 0.2 < τm Tm < 2 b f 0 = 0 , b f 1 = 0.125Tu , b f 2 = 0 , a f 1 = 0.0125Tu , a f 2 = 0 . 2Tm + τm λ2 τm + 2Tm τm (τm − λ ) 2λ (2Tm − λ ) , bf 0 = 1 , bf 1 = + , (τ m + 2λ ) 2(2λ + τm − b f 1 )K m Tm (τm + 2λ ) bf 2 = 0 , a f 1 = 2λτ m + 2λ2 + b f 1 τ m λ2 τ m , af 2 = , N =∞. 2(2λ + τ m − b f 1 ) 2(2λ + τ m − b f 1 ) ωbw Tm ; N = ∞ ; b f 0 = 1 , b f 1 = Tm + 0.5τm ; 2ωbw ωg τm K m (ωbw x1τm + 1)1 + tan ωg 2 τ + 2ωbw Tm τm + 2Tm Tm τm 0. 6 ; af 2 = ; ωg = b f 2 = 0.5τm Tm , a f 1 = m . 2(ωbw x1τm + 1) 2(ωbw x1τm + 1) τm 5 Kc = 6 Kc = 2 2Tm + τm 2T (2ξλ + τm ) + τm − 2λ2 , bf 0 = 1 ; bf 1 = m , bf 2 = 0 , 2K m (2ξλ + τm − bf 1 ) 2(τm + Tm ) af1 = 2λξτ m + 2λ2 + bf 1τm λ2 τm , af 2 = , N =∞. 2(2ξλ + τm − b f 1 ) 2(2ξλ + τm − bf 1 ) b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 152 3.3.7 Two degree of freedom controller 1 T s T s β 1 d d E (s) − K c α + R (s) + U (s) = K c 1 + Td Td Ti s 1+ s 1+ s N N β T s d α + K Td c 1+ s N E(s) + R(s) − T s 1 d + K c 1 + Td Ti s + s 1 N − U(s) Y(s) K m e − sτ 1 + sTm m + Table 15: PID controller tuning rules – FOLPD model G m (s) = K m e − sτ m 1 + sTm Rule Kc Hiroi and Terauchi (1986) – regulator. τ 0.1 < m < 1 . Tm 0.95Tm K mτm 2.38τ m 0.42τ m 0% overshoot. 1.2Tm K mτm 2τ m 0.42τ m 20% overshoot. VanDoren (1998). 1.5Tm K m τm Ti Td Comment Process reaction ABB (2001). OMEGA Books (2005). α = 0.6 , β = 1 ; Model: Method 2. 2.5τ m 0.4 τ m Model: Method 2 α = 0 , β =1, N = ∞ . 1.25Tm K m τm 2τ m 0.5τ m Model: Method 2 α = 0 , β =1, N = ∞ . Tm K m τm 2.5τ m 0.4 τ m Model: Method 2 α = 0 , β =1, N = ∞ . Minimum performance index: regulator tuning Minimum IAE – τ m Tm = 0.1 ; Shinskey (1996), 1.32Tm K m τ m 1.80τ m 0.44τ m Model: Method 1 p. 117. α = 0 , β =1, N = ∞ . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Ti Td Comment 1.77 τ m 0.41τ m τ m Tm = 0.2 1.43τ m 0.41τ m τ m Tm = 0.5 1.17 τ m 0.37τ m τ m Tm = 1 0.92τ m 0.32τ m τ m Tm = 2 1.8τ m 0.45τ m Model: Method 1 2.1τ m 0.63τ m Model: Method 2 Kc Minimum IAE – 1.32Tm K m τ m Shinskey (1988), p. 143. Model: 1.35Tm K m τ m Method 1; 1.49Tm K m τ m α = 0 , β =1, 1.82Tm K m τ m N =∞. Minimum IAE – 1.30Tm K m τ m Edgar et al. 1.33Tm K m τ m (1997), pp. 8–14, 8–15. α = 0 , β =1, N = ∞ . 2 ∞ du Kotaki et al. Minimise (ISTSE+w.ISTC), ISTC = dt , w is chosen so that 0 dt (2005a). Model: Method 1 ISTSE = 0.6 . p = % allowed perturbation on individual model w.ISTC parameters; α = 1 , β = 1 , N = ∞ . ∫ x1 K m x1 Kotaki et al. (2005a), (2005b). Model: Method 1; coefficients of K c , Ti , Td and p obtained from plots. 5.9988 3.0722 1.2389 0.6885 0.4305 x1 K m x 2Tm x 3Tm τm Tm p 0.3338 0.0018 0.1 0.5594 0.0013 0.2 0.7901 0.00072 0.4 0.9000 0.00007 0.6 0.8715 0.00026 0.8 x 2Tm x 3Tm 29 30 44 61 89 x2 x3 x1 x2 x3 p 6.0 5.4 4.2 3.3 2.3 1.7 1.1 0.6 4.1 3.1 2.3 1.7 1.4 1.2 1.0 0.8 0.5 0.35 0.35 0.4 0.4 0.5 0.5 0.55 0.6 0.55 0.55 0.6 0.6 0.6 0.65 0.65 0.65 0.65 0 0 0 0 0 0 0 0 0.01 0 0 0 0 0 0 0 0 29 33 42 50 63 72 83 100 20 30 42 53 61 67 73 81 100 τm Tm 0.1 0.2 153 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 154 Rule Kotaki et al. (2005a), (2005b) – continued. Kc Ti Td x1 K m x 2Tm x 3Tm x1 2.3 1.7 1.4 0.9 0.8 0.7 0.6 0.4 0.4 2.0 1.7 1.4 1.0 0.8 0.7 0.5 0.4 1.7 1.1 0.8 0.7 0.6 0.4 0.4 1.5 1.0 0.8 0.6 0.6 0.5 0.4 0.4 x2 x3 0.85 0.02 0.9 0 0.8 0 0.75 0 0.8 0 0.75 0 0.75 0 0.75 0 Kotaki et al. (2005a). 0.6 0 Kotaki et al. (2005b). 1.0 0.14 1.0 0 1.0 0 1.0 0 0.9 0 0.9 0 0.85 0 0.7 0 1.1 0.24 1.2 0 1.1 0 1.05 0 1.05 0 0.90 0 0.85 0 1.25 0.32 1.35 0 1.25 0 1.2 0 1.1 0 1.05 0 0.95 0 0.95 0 Comment p 20 30 41 55 61 74 82 100 τm Tm 0.4 100 20 30 40 50 61 74 82 100 21 30 43 52 66 83 100 22 30 42 55 60 74 83 100 0.6 0.8 1.0 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kotaki et al. (2005b). Model: Method 1; α = 1, β = 1 , N =∞. Kc 2.0 7.5 30 180 40 200 800 1900 800 2000 8000 14000 2000 8000 20000 38000 4000 20000 80000 20000 38000 80000 95000 Kc 1.2787 Kc = 0.5249 Tm 1 + K m τm ∞ ∫ (du dt ) dt . p = % allowed 2 0 perturbation on individual model parameters. p w p τm Tm w 1 Comment Td Minimise (ISTSE+w.ISTC), ISTC = Madhuranthakam et al. (2008). Model: Method 1 1 Ti 33 42 50 63 30 42 53 61 31 40 55 61 30 41 50 61 30 43 52 35 44 55 60 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.8 0.8 0.8 1.0 1.0 1.0 1.0 Ti 155 τm Tm 750 3800 42000 72 83 100 0.1 0.1 0.1 8000 20000 60000 73 81 100 0.2 0.2 0.2 40000 56000 160000 74 82 100 0.4 0.4 0.4 80000 105000 200000 73 81 100 0.6 0.6 0.6 95000 165000 250000 190000 400000 450000 66 83 100 74 99 100 0.8 0.8 0.8 1.0 1.0 1.0 Td 0.1 ≤ τm ≤2 Tm τm , , Ti = τm 2.1356 − 1.9167 τm + Tm 2 τm τm Td = Tm 1.1321 0 . 1788 + τ + T , N = 10, α = 0 , β = 1 . τ m + Tm m m b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 156 Rule Minimum ISE – Argelaguet et al. (1997), (2000). Kc Ti Comment Td Minimum performance index: servo tuning 2Tm + τ m Tm τ m Model: Method 1 Tm + 0.5τ m 2K m τ m 2Tm + τ m α = 1, β = 1 , N = ∞ . Madhuranthakam et al. (2008). Model: Method 1 2 Ti Kc x1 K m 1.2299 Kc = 0.4967 Tm 1 + τm K m τm ≤2 Tm Minimum performance index: other tuning x 2Tm x 3Tm N=∞ Araki (1985) – τm servo/regulator α x1 x 2 x3 β T m optimisation. Model: Method 1 12.5 0.22 0.04 0.68 0.75 0.1 6.1 0.41 0.08 0.63 0.70 0.2 4.1 0.57 0.11 0.62 0.70 0.3 3.1 0.71 0.15 0.59 0.69 0.4 2.5 0.83 0.18 0.58 0.69 0.5 x1 K m x 2Tm Taguchi et al. τm (1987) – α x1 x 2 x3 β Tm servo/regulator optimisation. 6.30 0.40 0.08 0.60 0.63 0.2 Model: Method 1 3.20 0.69 0.16 0.56 0.60 0.4 2.18 0.92 0.23 0.51 0.57 0.6 1.67 1.10 0.30 0.47 0.54 0.8 1.38 1.25 0.36 0.43 0.52 1.0 1.18 1.38 0.42 0.39 0.51 1.2 1.05 1.50 0.48 0.36 0.50 1.4 Minimum IAE – 0.77 K u 0.48Tu Shinskey (1988), 0.70K u 0.42Tu p. 148. Model: Method 1; 0.66K u 0.38Tu α = 0 , β =1, 0.60K u 0.34Tu N =∞. 2 0.1 ≤ Td x1 x2 x3 α β τm Tm 2.11 0.94 0.21 0.56 0.69 0.6 1.82 1.05 0.24 0.54 0.69 0.7 1.61 1.13 0.28 0.51 0.68 0.8 1.44 1.22 0.31 0.48 0.69 0.9 1.33 1.26 0.34 0.49 0.66 1.0 x 3Tm N=∞ x1 x2 x3 α β τm Tm 0.95 1.61 0.54 0.34 0.50 1.6 0.87 1.72 0.59 0.31 0.51 1.8 0.81 1.82 0.64 0.30 0.52 2.0 0.71 2.07 0.75 0.26 0.57 2.5 0.64 2.32 0.85 0.24 0.63 3.0 0.56 2.82 1.00 0.20 0.73 4.0 0.51 3.33 1.09 0.15 0.78 5.0 0.11Tu τ m Tm = 0.2 0.12Tu τ m Tm = 0.5 0.12Tu τ m Tm = 1 0.12Tu τ m Tm = 2 1.317 T , Ti = 0.6739τm 1 + m τm , 2 τm τm + Td = Tm 1.138 0 . 1992 τ + T , N = 10, α = 0 , β = 1 . τm + Tm m m b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum IAE – Shinskey (1994), p. 167. Taguchi and Araki (2000). Model: Method 1 3 Kc Ti Kc 0.125Tu τm 2 157 Td Comment 0.12Tu Model: Method 1 α = 0 , β =1, N = ∞ . 4 Kc Ti 0 5 Kc Ti Td τ m Tm ≤ 1.0 Servo/regulator optimisation; overshoot (servo step) ≤ 20% . 6 Model: Method 7 Ti Td K ABB (2001). τm Tm ≥ 0.5 . c α = 0 , β =1, N = ∞ . x1 K m x 2Tm Taguchi and α x1 x 2 x3 β Araki (2002) – servo/regulator 4.38 1.13 0.091 0.03 0.38 optimisation. 6.36 0.44 0.078 0.62 0.71 Model: Method 1 6.32 0.40 0.081 0.61 0.64 N=∞ x 3Tm x4 x1 x2 x3 α β x4 0.1 6.68 0.31 0.089 0.65 0.55 1.5 0.5 7.18 0.23 0.097 0.68 0.38 2.0 1.0 Load disturbance model = K m e −sτ m (1 + sx 4Tm ) ; τm Tm = 0.2 . 3 Kc = T T Ku Ku , u < 2.7 ; K c = , u ≥ 2.7 . 3.73 − 0.69(Tu τm ) τ m 2.62 − 0.35(Tu τm ) τ m 4 Kc = 1 0.7382 , 0.1098 + K m (τm Tm ) − 0.002434 ( ) Ti = Tm 0.06216 + 3.171(τm Tm ) − 3.058(τm Tm ) + 1.205(τm Tm ) , 2 α = 0.6830 − 0.4242(τm Tm ) + 0.06568(τm Tm ) . 3 2 5 Kc = 1 1.224 , 0.1415 + K m (τm Tm ) − 0.001582 2 3 τ τ τ Ti = Tm 0.01353 + 2.200 m − 1.452 m + 0.4824 m , Tm Tm Tm 2 τ τ Td = Tm 0.0002783 + 0.4119 m − 0.04943 m , N fixed (but not specified); Tm Tm 2 α = 0.6656 − 0.2786 6 Kc = 1.3570 Tm K m τm 2 τ τ τm τ + 0.03966 m , β = 0.6816 − 0.2054 m + 0.03936 m . Tm Tm Tm Tm 0.947 τ , Ti = 1.1760Tm m Tm 0.738 τ , Td = 0.3810Tm m Tm 0.995 . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 158 Rule Tavakoli et al. (2005). Model: Method 1 Minimum ITAE – Setiawan et al. (2000). Model: Method 1 7 Kc Ti Td Comment Kc Ti 0 M max = 2 Servo/regulator optimisation – minimum IAE; 0.1 ≤ τm Tm ≤ 2 . x1Tm K m τm τm Tm x1 x2 0.5 0.75 1.0 1.5 2.0 0.554 0.570 0.594 0.645 0.720 1.805 1.373 1.117 0.822 0.679 Shen (2002). Model: Method 5; N = ∞ ,β = 0. 8 Kc Kuwata (1987). 9 Kc 0 x 2τm Coefficient values: x1 x2 τm Tm τm Tm x1 x2 3 4 5 6 7 8 9 10 1.740 1.930 2.120 2.410 0.413 0.398 0.385 0.389 0.900 1.090 1.290 1.520 ∞ Minimise ∫ α =1 0.547 0.480 0.445 0.428 ∞ r ( t ) − y( t ) dt + 0 ∫ d( t) − y(t) dt ; M ≤ 2 . s 0 Ti Td 0 < τm Tm < ∞ Direct synthesis: time domain criteria Model: Method 1 0 Ti 7 Kc = 0.450(τ m Tm ) + 1.083 1 Tm τ , α = 0.417 − 0.225 m . 0.273 + 0.571 , Ti = Km τm (τ m Tm ) + 0.176 Tm 8 Kc = 2 τm τm Tm , + 11.15 exp 2.94 − 11.63 τ +T K m τm m m τm + Tm 2 τm τm , + 0.86 Ti = τm exp 1.88 − 3.63 τ +T m m τm + Tm 2 τm τm , Td = τm exp − 0.25 − 0.06 − 1.99 τm + Tm τm + Tm 2 τm τm . α = 1 − exp − 0.22 − 0.90 + 1.45 τ m + Tm τ m + Tm 9 Kc = 0.833τm (2Tm + τm ) 1.667 τm (2Tm + τm ) 0.8(Tm + τm ) 1 − , Ti = 1 − K m (Tm + τm − x1 ) K m Tm + τm (Tm + τm )2 with x1 = (Tm + τm )2 − 0.267τm 2 (3Tm + τm ) , α = 1 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Kuwata (1987), Suyama (1993). 10 Shigemasa et al. (1987), Suyama (1993). α = 1, β = 1 , N =∞. 11 Kc Ti Td Ti Td 159 Comment Model: Method 1 th Kc Closed loop transfer function is 4 order. Model: Method 1 Ti Td Closed loop transfer function is 4th order Butterworth. 12 Ti Td Kc Toshiba AdTune th Closed loop transfer function is 4 order ITAE. TOSDIC 211D8 13 Ti Td product. Kc Closed loop transfer function is 4th order Bessel. 14 Ti Td Kc Closed loop transfer function is 4th order Binomial. 10 Kc = 3 2 2 3 1 0.2τm + 1.2τm Tm + 5.4τm Tm + 9.6Tm , α = 1, β = 1 , N = ∞ ; 2 K m τm (τm + 3Tm ) Ti = 1.667 τm 3 τm + 3Tm 2 (τ m + 3Tm ) − 1.389τm , τm + 2Tm (τm + 2Tm )4 Td = 11 Kc = 1.3319(τm + 2Tm )3 − 2 K m τm (τm + 3Tm ) Kc = 2.6242(τm + 2Tm )3 − 2 K m τm (τm + 3Tm ) Kc = 0.9450(τm + 2Tm )3 − 2 K m τm (τm + 3Tm ) Kc = 0.5622(τm + 2Tm )3 − 2 K m τm (τm + 3Tm ) 1.5095(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm ) 1.3319(τm + 2Tm )3 − τm (τm + 3Tm )2 2.3130(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm ) . 2.6242(τm + 2Tm )3 − τm (τm + 3Tm )2 . 3 1 τ + 3Tm 2 (τ m + 3Tm ) , Ti = 3.3333τm m − 3.527 τm , Km τm + 2Tm (τm + 2Tm )4 Td = τm (τm + 3Tm ) 14 . 3 1 τ + 3Tm 2 (τ m + 3Tm ) , Ti = 1.8898τm m − 0.720τm , Km τm + 2Tm (τm + 2Tm )4 Td = τm (τm + 3Tm ) 13 1.2(τm + 2Tm )3 − τm (τm + 3Tm )2 3 1 τ + 3Tm 2 (τ m + 3Tm ) , Ti = 2.2532τm m − 1.692τm , Km τm + 2Tm (τm + 2Tm )4 Td = τm (τm + 3Tm ) 12 Tm 2 τm (τm + 3Tm ) 1.3444(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm ) 1.9450(τm + 2Tm )3 − τm (τm + 3Tm )2 . 3 1 τ + 3Tm 2 (τ m + 3Tm ) , Ti = 5.3337 τm m − 9.488τm , Km τm + 2Tm (τm + 2Tm )4 Td = τm (τm + 3Tm ) 1.1244(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm ) 0.5622(τm + 2Tm )3 − τm (τm + 3Tm )2 . b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 160 Rule Kc Kasahara et al. (1997). Model: Method 1 Chien et al. (1999). Kc = Ti Ti Kc Ti 17 Kc Ti Td 18 Kc Ti 0 16 Comment Td Kc 15 Nomura et al. (1993), Kuwata (1987). 15 17-Mar-2009 Model: Method 1; τ 0.01 ≤ m ≤ 10 . Tm 0 Closed loop transfer function is 4th order. Model: Method 1 T + τ m − x1 0.8(Tm + τm )(Tm + 0.5τm ) 1 , − , Ti = 2.5(1.25x1 − 0.25) m K m τm (Tm + 0.333τm ) Km Tm + τ m with x1 = 0.2τm 2 + 0.4τmTm + Tm 2 ; α = 1 . 16 Kc = 17 Kc = (1 + (τm Tm ))2 − 1.667 τm (1 + 0.5(τm Tm )) , 1.667 K m τm (1 + 0.5(τm Tm )) 2 3.333τm (1 + 0.5(τm Tm )) 10(1 + 0.5(τm Tm )) , α = 1. Ti = 1 − 2 1 + (τm Tm ) (3Tm + τm )(1 + (τm Tm )) 3 2 2 3 1 0.2τm + 1.2τm Tm + 5.4τm Tm + 9.6Tm , 2 K m τm (τm + 3Tm ) Ti = 1.667 τm 3 τm + 3Tm 2 (τ m + 3Tm ) − 1.389τm , τm + 2Tm (τm + 2Tm )4 Tm 2 τm (τm + 3Tm ) 2 τ m + 3Tm 2 1 , N =∞, Td = , β = 1 − 0.261τm 3 2 τ + 2 T T 1.2(τm + 2Tm ) − τm (τm + 3Tm ) m i Td m −1 τ + 3Tm τ + 3Tm τ 2 (τ + 3Tm )3 1.667 τ m m − 1.389 m m α = 1 − 0.723τ m m . (τ m + 2Tm )4 τ m + 2Tm τ m + 2Tm 18 Kc = − TCL 2 2 + 1.414TCL 2Tm + τm Tm ( K m TCL 2 2 + 1.414TCL 2 τm + τm 2 , Ti = 2 ) − TCL 2 + 1.414TCL 2Tm + Tm τm ; Tm + τm Underdamped system response: ξ = 0.707 ; τ m > 0.2Tm ; α = 1 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Chien et al. (1999). Chidambaram (2000a). Kasahara et al. (2001). Model: Method 1 19 Kc Ti Td Comment Ti Td Model: Method 1 161 α = 1, β = 1 , N = ∞ . Tm Ti 20 TCL 2 + Ti τm Ti 0 Model: Method 1; choose TCL , ξ . 21 Kc Ti Td OS = 0% 22 Kc Ti Td OS = 10% α = 1, β = 1 , N = ∞ . 19 Underdamped system response: ξ = 0.707 . τ m > 0.2Tm . Kc = 1.414TCL 2Tm + τmTm + 0.25τm 2 − TCL 2 2 ( K m TCL 2 2 + 0.707TCL 2 τm + 0.25τm 2 2 Ti = ) , 2 1.414TCL 2Tm + τmTm + 0.25τm − TCL 2 , Tm + 0.5τm 0.707TmTCL 2 τm + 0.25Tm τm 2 − 0.5τmTCL 2 2 Td = Tm τm + 0.25τm 2 + 1.414TCL 2Tm − TCL 2 2 . 2 Tm τm + 2TCL ξ − TCL T − ξTCL K K [T + τm ] − Ti , α= i or α = c m i . Tm + τm Ti 2K c K m Ti 20 Ti = 21 Kc = 0.281(τm + 2Tm )3 K m τm (τm + 3Tm ) 2 − 3 1 τ + 3Tm 2 (τ m + 3Tm ) , Ti = 5.33τm m − 18.98τm , Km τm + 2Tm (τm + 2Tm )4 Td = τm (τm + 3Tm ) 22 Kc = 0.562(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm ) 0.281(τm + 2Tm )3 − τm (τm + 3Tm )2 . 1.066Tm − 0.467 τm 4.695τm (1.066Tm − 0.467 τm ) , Ti = , K m τm τm + 2Tm 0.331Tm − 0.334τm . Td = τm 1.066Tm − 0.467 τm b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 162 Rule Ogawa and Katayama (2001). Srinivas and Chidambaram (2001). Srinivas and Chidambaram (2001). Chidambaram (2002), pp. 157–158. 23 17-Mar-2009 Kc 23 Kc Ti Td Comment Ti Td N = 10 Minimum ISE: servo. α = 1 , β = 1 ; Model: Method 1. 24 0.9Tm K mτm 25 3.33τ m 0 Ti Td 2τ m 0.5τ m Kc 26 1.2Tm K m τm Desired closed loop response = e −sτCL . Kc = Model: Method 1 Model: Method 1; N=∞. Model: Method 2; β =1, N = ∞ . 1 (τCL Tm ) − 2(TCL Tm ) + 4 , K m (τCL Tm ) + 2(TCL Tm ) (1 + sTCL ) ((τ T ) + 2(TCL Tm ))((τCL Tm ) − 2(TCL Tm ) + 4 ) , Ti = Tm CL m 2(τCL Tm ) + 4 (τCL Tm )((τCL Tm ) + 4(TCL Tm ) − 2(τCL Tm )2 ) ; 0.01 ≤ TCL ≤ 1.00 ; Td = Tm ((τCL Tm ) + 2(TCL Tm ))((τCL Tm ) − 2(TCL Tm ) + 4 ) Tm 2 2 τ τ TCL τ = −0.1902 CL + 0.6974 CL + 0.007393 , 0.05 ≤ CL ≤ 1.00 . Tm T T Tm m m 24 Sample K c , Ti taken by the authors correspond to the process reaction curve tuning rule of Ziegler and Nichols (1942); α = 25 this tuning rule. Sample K c , Ti , Td taken by the authors correspond to the process reaction tuning rule of Ziegler and Nichols (1942); α = β= 26 τm 1 τ − = 0.30 − 1.11 m for Ti K c K m Tm τm 1 τ − = 0.5 − 0.83 m , Ti K c K m Tm 2 1 Tm τ − m = −0.16 for this tuning rule. τm − Td K c K m 2Ti α = 0.6762 − 0.4332(τm Tm ) , τm Tm ≤ 0.8 ; or α = 0.207 + 3.149(τm Tm ) − 8.405(τm Tm )2 + 8.431(τm Tm )3 , τm Tm < 0.5 ; α = 0.9416 − 0.44(τm Tm ) , 0.5 ≤ τm Tm ≤ 0.9 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 27 Kc Haeri (2002). Ozawa et al. (2003). 27 Kc = Td Comment Ti Td Model: Method 1 α = 0 , β =1, N = ∞ . Kc Ti Td x1 K m x 2 Tm x 3Tm 28 Kim et al. (2004). Model: Method 1 Ti 163 τ m Tm Model: Method 1; 0% ≤ OS ≤ 10%. Coefficient values: x2 x3 x1 α β 0.1 12.5 0.22 0.04 0.68 0.75 0.2 6.1 0.41 0.08 0.63 0.70 0.3 4.1 0.57 0.11 0.62 0.70 0.4 3.1 0.71 0.15 0.59 0.69 0.5 2.5 0.83 0.18 0.58 0.69 0.6 2.11 0.94 0.21 0.56 0.69 0.7 1.82 1.05 0.24 0.54 0.69 0.8 1.61 1.13 0.28 0.51 0.68 0.9 1.44 1.22 0.31 0.48 0.69 1.0 1.33 1.26 0.34 0.49 0.66 OS < 20% (servo step); settling time ≤ that of corresponding rule of Chien et al. (1952); a cost function is minimised. 1 6.84 + 0.64 , 0.7 3.5 K m 1 + 2.33(τm Tm ) + 7.82(τm Tm ) [ ] T = τ [2.15 − 0.76(τ T ) + 0.33(τ T ) ] , 0.29 < τ T < 4 ; Ti = τm 0.95 + 2.58(τm Tm ) + 3.57(τm Tm ) , 0 < τm Tm ≤ 0.29 , 2 2 i m m m m m m m 2.8 τm τm τm τ , 0 < m ≤ 1.79 ; + 3.94 − 4.65 Td = τm 0.29 T +τ T +τ T T + τ m m m m m m m 2 [ Td = τm 0.87(τm Tm ) − 0.49(τm Tm ) + 0.09(τm Tm ) 2 1.6193(τm + 2Tm ) 2.8 ] , 1.79 < τ T < 4 . m m 3 1 τm + 3Tm 2 (τ m + 3Tm ) 28 Kc = − , T = 2 . 7051 τ − 1 . 6706 τ , i m m τm + 2Tm (τm + 2Tm )4 K m τm (τm + 3Tm )2 K m 3 Td = τm (τm + 3Tm ) 1.7793(τm + 2Tm )2 − (τm + Tm )(τm + 3Tm ) 1.6193(τm + 2Tm )3 − τm (τm + 3Tm )2 , α = 1, β = 1 , N = ∞ . b720_Chapter-03 Rule Kotaki et al. (2005a). Model: Method 1; α = 1, β = 1 , N =∞. Kc = Kc Ti x1 K m x 2Tm Td x 3Tm Closed loop transfer function is 4 order; p = % allowed perturbation on individual model parameters. p x1 x2 x3 τm Tm 6.8016 3.2091 1.4202 0.8298 0.5383 0.3661 29 Kc 30 Kc 31 0.3482 0.0243 0.1 32 0.6001 0.0453 0.2 36 0.9000 0.0754 0.4 46 1.0194 0.0850 0.6 59 1.0293 0.0656 0.8 77 0.9685 0.0042 1.0 100 Direct synthesis: frequency domain criteria 0 Ti M s = 1.25 Ti 0 0.7Tm 0 Kc 0.4Tm K mτm 0.1 ≤ τm ≤2 Tm α = 0.5; Model: Method 5 or 17. ( K m 4 + 4 x1x 2 (τm Tm ) + x 2 2 (τm Tm )2 ) , 2 x 2 2 (τm Tm ) − 4 + x 2 (x 2 (τm Tm ) + 4x1 )((τm Tm ) + 2 − 2 x1x 2 (τm Tm )) 2 x 2 2 (τm Tm ) − x 2 4 (τm Tm )2 + 2 x 2 2 ((τm Tm ) + 2 − 2 x1x 2 (τm Tm )) x1 = 0.5 + 0.25 x2 2 , 0.486 + 0.675(Tm τm ) τm τ ; < 1.91 ; x1 = 1 , m ≥ 1.91 ; x 2 = 1 + 0.0972(τm Tm ) Tm Tm , α = 0.488 − 0.985(τm Tm ) , τm Tm ≤ 0.495 ; α = 0 , τm Tm > 0.495 . 2 2 2 0.29e −2.7 τ + 3.7 τ Tm , Ti = 8.9τm e−6.6 τ + 3.0 τ or 0.79Tm e −1.4 τ+ 2.4 τ , K m τm 2 α = 1 − 0.81e 0.73τ+1.9 τ , M s = 1.4. 2 31 Model: Method 5 or 17; τ 0.14 ≤ m ≤ 5.5. Tm 2 x 2 2 (τm Tm ) − 4 + x 2 (x 2 (τm Tm ) + 4 x1 )((τm Tm ) + 2 − 2x1x 2 (τm Tm )) Ti = Tm Kc = Comment th Lee et al. (1992). Model: Method 1 Dominant pole design – Åström and Hägglund (1995), pp. 204– 208. Modified Ziegler–Nichols – Åström and Hägglund (1995), p. 208. 30 FA Handbook of PI and PID Controller Tuning Rules 164 29 17-Mar-2009 2 0.78e −4.1τ + 5.7 τ Tm , α = 1 − 0.44e 0.78 τ−0.45τ , M s = 2.0. Kc = K m τm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Huang et al. (2000). Model: Method 1 Kc λ (Tm + 0.4τm ) K m τm α= Ti Td Comment Tm + 0.4 τ m 0.4Tm τ m Tm + 0.4τ m For λ = 0.80 , A m = 2.13 . Tm − 1 , β = 0 , N = min[20,20Td ] . Tm + ατ m Hägglund and Åström (2002). Model: Method 5; M s = 1.4; 0.35Tm 0.6 − K mτm Km 7τ m 0.35Tm 0.6 − K mτm Km 0.8Tm 0 < α < 0.5 . 0.25Tm K mτm 0.8Tm Hägglund and Åström (2002). 32 Leva and Colombo (2004). 33 τm < 0.11 Tm 0 0 Ti Kc 0.11 < τm < 0.17 Tm 0.17 < τm < 0.2 Tm τm Tm < 0.5 M s = 1.4; 0 < α < 0.5 , Model: Method 1. Robust Ti Kc Model: Method 50 α = 0 , β =1, N =∞. Td Tm + 0.5τm Tm + 0.5τm Tm τm Cooper (2006a). K (λ + 0.5τ ) 2Tm + τm m m Model: Method 1 λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning); λ > [Tm ,8τm ] (‘moderate’ tuning); λ > [10Tm ,80τm ] (‘conservative’ tuning). 32 Kc = 6.8τm Tm 1 T 0.14 + 0.28 m , Ti = 0.33τm + . τm 10τm + Tm K m 33 Kc = 2 2 1 τm τm , , Ti = Tm + Tm + K m (τm + λ ) 2(τm + λ ) 2(τm + λ ) Td = τm Tm (τm + λ ) 2Tm (τm + λ ) + τm − 2 2Tm (τ m + λ )2 λτ m ,N= −1 . 2(τm + λ ) λ 2Tm (τ m + λ ) + τ m 2 [ ] 3∆τ m , ∆τ m = maximum variation in the time delay, π with only variation in the time delay considered (Leva and Colombo, 2004); λ α not specified; β = 1 . λ ≥ ^ chosen so that nominal cut-off frequency = ωu (Leva, 2005). 165 b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 166 Rule Kc Ti Comment Td 34 Ti Td Kc Shamsuzzoha and Lee (2007a). M s = 1.6 M s = 1.7 M s = 1.8 M s = 1.9 M s = 2.0 Model: 0.11 0.10 0.08 0.07 0.06 Method 1; 0.20 0.17 0.15 0.14 0.12 N =∞, 0.35 0.31 0.27 0.25 0.22 α = 0.6 , β = 0 ; 0.41 0.37 0.34 0.30 coefficients of x1 0.46 (deduced from graph) indicated in columns. Kc = λ = x1Tm τm = 0.05Tm τm = 0.1Tm τm = 0.2Tm τm = 0.3Tm 0.55 0.49 0.44 0.40 0.36 0.62 0.55 0.50 0.46 0.42 τm = 0.5Tm 0.68 0.60 0.55 0.51 0.47 τm = 0.6Tm 0.72 0.65 0.59 0.55 0.50 τm = 0.7Tm 0.77 0.69 0.63 0.59 0.53 τm = 0.8Tm 0.81 0.72 0.66 0.62 0.56 τm = 0.9Tm 0.84 0.76 0.69 0.65 0.59 τm = 1.0Tm 0.91 0.82 0.75 0.70 0.64 τm = 1.25Tm 0.97 0.87 0.80 0.75 0.70 τm = 1.5Tm τm = 0.4Tm - 0.91 0.84 0.78 0.73 τm = 1.75Tm - 0.96 0.87 0.81 0.76 τm = 2.0Tm Ultimate cycle 0 x 2 Tu x 1K u Kuwata (1987); x 1 , x 2 deduced from graphs; α = 1. 34 FA x1 x2 τm Tu x1 x2 τm Tu x1 x2 τm Tu 0.37 0.32 0.29 0.26 0.03 0.07 0.08 0.09 0.25 0.28 0.30 0.33 0.24 0.22 0.21 0.20 0.10 0.11 0.12 0.13 0.35 0.38 0.40 0.43 0.20 0.20 0.14 0.14 0.45 0.48 2 2 3λ2 + 2x1τm − 0.5τm − x1 Ti , , Ti = Tm + 2 x1 − K m (3λ + τm − 2x1 ) 3λ + τm − 2x1 2 2Tm x1 + x1 − Td = x1 = Tm 1 − λ3 + 0.167 τm 3 − x1τm 2 + x12 τm 2 2 3λ2 + 2x1τm − 0.5τm − x1 3λ + τm − 2 x1 , Ti 3λ + τm − 2x1 (1 − (λ Tm ))3 e− τ T . m m b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Td 167 Comment Kuwata (1987); x 1K u x 2 Tu x 3 Tu x 1 , x 2 deduced x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm from graphs; Tu Tu Tu α = 1, β = 1 , 0.69 0.63 0.08 0.25 0.38 0.28 0.07 0.36 0.20 0.14 0.01 0.48 N =∞. 0.57 0.48 0.08 0.28 0.31 0.20 0.05 0.40 0.20 0.14 0 0.50 0.47 0.37 0.08 0.32 0.26 0.18 0.04 0.42 35 Hang and 0.5Tu 0.125Tu 0. 6 K u Åström (1988a). 36 Ti 0.125Tu 0. 6 K u Model: Method 2; τm Tm > 1.0 ; β = 1 , N = 10 . 0. 6 K u 0.335Tu 0.125Tu α = 0.2 . Hang et al. (1991) – servo. 37 0. 6 K u 0.5Tu 0.125Tu 38 0. 6 K u 0.222 K ' Tu 0.125Tu Hang and Cao (1996). Model: Method 37 0. 6 K u 39 Ti Td Model: Method 2; β = 1 , N = 10 . 0.1 ≤ τm < 0.5 ; Tm N = 10; β = 1 . τ τ 1.66τm τ m , < 0.3 ; α = 1 − 2OS − m , 0.3 ≤ m < 0.6 ; Tm Tm Tm Tm OS = % overshoot. τ τ τ τ 36 Ti = 0.51.5 − 0.83 m Tu . α = m − 0.6 , 0.6 ≤ m < 0.8 ; α = 0.2 , 0.8 ≤ m < 1.0 . Tm Tm Tm Tm 35 α = 1 − 2(OS − 0.1) − 37 0.16 ≤ τm 15 − K ' < 0.57 ; α = 1 − , 10% overshoot; Tm 15 + K ' α = 1− 38 39 0.57 ≤ 11[τ m Tm ] + 13 36 . , 20% overshoot with K ' = 2 ' 27 + 5K 37[τ m Tm ] − 4 τm 8 4 < 0.96 ; α = 1 − K ' + 1 , 20% overshoot, 10% undershoot. 17 9 Tm τ τ T Ti = 0.53 − 0.22 m Tu , Td = 0.53 − 0.22 m u . α equals 1.0, 0.2 and Tm 4 Tm 0.40(τ m Tm ) − 0.05(τ m Tm ) + 0.58 in different circumstances. 2 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 168 3.3.8 Two degree of freedom controller 2 T s 1 d 1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) , U(s) = K c 1 + + 0 Ti s T 1 + a f 1s 1+ d s N 1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4 F(s) = 1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4 R(s) F(s) + − T s 1 d 1 + b f 1s K c 1 + + Ti s Td 1 + a f 1 s 1+ s N + U(s) − K m e − sτ 1 + sTm m K0 Table 16: PID controller tuning rules – FOLPD model G m (s) = Rule Kc Ti Td Y(s) K m e − sτ m 1 + sTm Comment Direct synthesis: time domain criteria 0.375(τ m + 2Tm ) N=∞ Tm τ m Normey-Rico et Tm + 0.5τ m K mτm 2Tm + τ m al. (2000). a = 0 . 13 τ , b = 0 ; a = 0. 5τ m , b f 2 = 0. 2 τ m ; f1 m f1 f2 Model: Method 1 a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . Robust Lee and Edgar (2002). Model: Method 1 1 Kc 0 Ti a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u . 1 2 Kc = 1 + 0.25K u K mTi T − 0.25K u K m τm τm . , Ti = m + K m (λ + τm ) 1 + 0.25K u K m 2(λ + τm ) b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Lee and Edgar (2002). Model: Method 1 2 Kc Ti Td Comment Kc Ti Td N=∞ 169 a f 1 = Td , b f 1 = 0 ; a f 2 = 0 , b f 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u . Zhang et al. (2002c). 0.1τm ≤ λ1 ≤ τm , 0.1τm ≤ λ 2 ≤ τm . Tm K m (λ 2 + τm ) 0 Tm Model: Method 1 a f 1 = 0.5τm λ 2 (λ 2 + τm ) , b f 1 = 0.5τm ; a f 2 = λ1 , b f 2 = λ 2 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . 0.5τm 0.167 τm 0 ≤ x1 ≤ 1 Kc Shamsuzzoha and Lee (2007c). a f 2 = b f 1 , b f 2 = x1bf 1 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , Model: Method 1 b f 5 = 0 , K 0 = 0 . λ is given as a multiple of Tm (from a graph, for 3 0.01 ≤ τm Tm ≤ 5 ), for M s = 1.4,1.5,1.6,1.8 and 1.9. Zhang et al. (2003). Model: Method 1 4 Ultimate cycle 0.5Tu 0.125Tu 0.6K u N=∞ a f 1 = 0 , b f 1 = 0 ; a f 2 = Ti , b f 2 = x 2Ti ; a f 3 = Ti Td , b f 3 = x1Ti Td , a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . 2 Kc = 2 1 + 0.25K u K m Tm − 0.25K u K m τm τm + + Td , K m (λ + τm ) 1 + 0.25K u K m 2(λ + τm ) Ti = Tm − 0.25K u K m τm τm + + Td , 1 + 0.25K u K m 2(λ + τm ) 2 2 0.25K u K m τm τm (T − 0.25K u K m τm )τm 2 Td = τm − + + m 2 2(1 + 0.25K u K m )(λ + τm ) 2(1 + 0.25K u K m ) 6(λ + τm ) 4(λ + τm ) 3 Kc = τm λ − Tm τm e , b f 1 = Tm 1 − 1 − , Tm 2K m (2λ − b f 1 + τm ) af1 = 4 0.5b f 1τm + λτ m + λ2 − Tm , N = ∞ . 2λ − b f 1 + τ m T + K m K c [Ti − τm ] 2Tm Ti + 2K m K cTi [Td − τm ] + K m τm K c . , x2 = i 2K m K cTi Td K m K c Ti 2 x1 = 0.5 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 170 3.3.9 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) Y(s) Model − F2 (s) Table 17: PID controller tuning rules – FOLPD model G m (s) = K m e − sτ m 1 + sTm Rule Kc Hiroi and Terauchi (1986) – regulator. Model: Method 2; τ 0.1 < m < 1 . Tm 0.95Tm K m τm Process reaction 2.38τm 0.42τm 0% overshoot 1.2Tm K m τm 2τm 0.42τm 20% overshoot Ti Td Comment α = 0.6 ; β = 1 or β = −1.48 ; χ = 0 ; δ = 0.15 ; N not specified; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 ; af 4 = 0 , K0 = 0 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kaya and Scheib (1988). Model: Method 5; τ 0 < m ≤ 1. Tm Kc Comment Td Ti Minimum performance index: regulator tuning Minimum IAE. x1K c Kc 0 2 Minimum ISE. x1K c Kc 1 3 Minimum ITAE. x1K c Kc α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x 2 ; b f 5 = 0.1x 2 . x 1 Tm K m τ m Kaya and Scheib (1988). Model: Method 5 x2 Tm τ m x 3 Tm x4 0< 0 τm ≤1 Tm α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = x 5Tm (τm Tm )x 6 ; a f 3 = 0.1bf 3 ; a f 4 = 0 ; K 0 = 0. Coefficient values: x1 x2 x3 x4 x5 x6 Minimum IAE Minimum ISE Minimum ITAE 0.91 1.1147 0.7058 0.7938 0.8992 0.8872 0.8826 1 Kc = 1.31509 Tm K m τm Kc = 1.3466 Tm K m τm 0.9308 2 Kc = 1.3176 Tm K m τm 0.7937 3 1.01495 0.9324 1.03326 1.3756 , x1 = Tm τm 1.2587 Tm 1.25738 , x1 = Tm τm 1.6585 Tm , x1 = Tm τm 1.12499 Tm 1.00403 0.8753 0.99138 0.5414 0.56508 0.60006 τ , x 2 = 0.5655Tm m Tm 0.4576 τ , x 2 = 0.79715Tm m Tm 1.42603 0.7848 0.91107 0.971 . 0.41941 τ , x 2 = 0.79715Tm m Tm . 0.41941 . 171 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 172 Rule Kc Kaya and Scheib (1988). Model: Method 5; τ 0 < m ≤1 Tm 4 Kc 5 Kc 6 Kc Ti Comment Td Minimum performance index: servo tuning Minimum IAE. x1K c 0 Minimum ISE. x1K c Minimum ITAE. x1K c α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x 2 ; b f 5 = 0.1x 2 . x 1 Tm K m τ m Kaya and Scheib (1988). Model: Method 5 Tm x2 τ x3 − x4 m Tm 0< 0 τm ≤1 Tm α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = x 5Tm (τm Tm )x 6 ; a f 3 = 0.1bf 3 ; a f 4 = 0 ; K 0 = 0. Coefficient values: x1 x2 x3 x4 x5 x6 Minimum IAE Minimum ISE Minimum ITAE 0.81699 1.1427 0.8326 Minimum ISE – Zhuang (1992), Zhuang and Atherton (1993). 7 Kc 1.09112 0.99223 1.00268 Ti 8 Kc Ti 0.22387 0.35269 0.00854 0 0.44278 0.97186 0.35308 0.78088 0.44243 1.11499 0.1 ≤ τ m Tm ≤ 1 1.1 ≤ τm Tm ≤ 2 α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x1 ; b f 5 = 0.1x1 . Model: Method 1 or Method 5 or Method 6 or Method 15. 0.81314 4 Kc = 1.13031 Tm K m τm 5 Kc = 1.26239 Tm K m τm 6 1.004 0.9365 0.7607 , x1 = τ Tm , x 2 = 0.32175Tm m 5.7527 − 5.7241(τm Tm ) Tm , x1 = τ Tm , x 2 = 0.47617Tm m 6.0356 − 6.0191(τm Tm ) Tm 0.8388 0.98384 Tm Kc = K m τm 0.49851 , x1 = 0.17707 . 0.24572 . Tm , 2.71348 − 2.29778(τ m Tm ) x 2 = 0.21443Tm (τm Tm ) 0.16768 7 Kc = 1.260 Tm K m τm 8 Kc = 1.295 Tm K m τm 0.887 0.886 , Ti = τ Tm , x1 = 0.375Tm m 0.701 − 0.147(τm Tm ) Tm , Ti = τ Tm , x1 = 0.378Tm m 0.661 − 0.110(τm Tm ) Tm 0.619 . 0.756 . . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Kc Ti Kc Ti 9 Minimum ISTSE – Zhuang (1992), Zhuang and Atherton (1993). 10 Comment Td 0 173 0.1 ≤ τ m Tm ≤ 1 1.1 ≤ τm Tm ≤ 2 α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x1 ; b f 5 = 0.1x1 . Model: Method 1 or Method 5 or Method 6 or Method 15. 11 0.1 ≤ τ m Tm ≤ 1 Ti Kc 0 Minimum ISTES 12 Ti 1.1 ≤ τm Tm ≤ 2 Kc – Zhuang (1992), Zhuang and α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; Atherton (1993). b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x1 ; b f 5 = 0.1x1 . Model: Method 1 or Method 5 or Method 6 or Method 15. 13 Model: Method 1 Ti Kc 0 Minimum ISTSE ∧ ∧ 14 Model: Kc – Zhuang (1992), 0.271 K u T u K m Method 37 Zhuang and α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; Atherton (1993). b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 , K 0 = 1 ; b f 3 = 0 ; b f 4 = x1 ; b f 5 = 0.1x1 . 0.930 9 Kc = 1.053 Tm K m τm 10 Kc = 1.120 Tm K m τ m 11 Kc = 0.942 Tm K m τm 12 13 1.001 Tm Kc = K m τm Kc = τ Tm , x1 = 0.349Tm m 0.736 − 0.126(τm Tm ) Tm 0.907 τ Tm , x1 = 0.350Tm m 0.720 − 0.114(τm Tm ) Tm 0.811 Ti = , Ti = τ Tm , x1 = 0.308Tm m 0.770 − 0.130(τm Tm ) Tm , Ti = 0.625 0.933 0.624 τ Tm , Ti = , x1 = 0.308Tm m 0.754 − 0.116(τm Tm ) Tm . . 0.897 . 0.813 . 4.437 K m K u − 1.587 K u , Ti = 0.037(5.89K m K u + 1)Tu , x1 = 0.112Tu 8.024K m K u − 1.435 0.1 ≤ τm Tm ≤ 2.0 . ∧ 14 Kc = ∧ τ 2.354K m K u − 0.696 ∧ K u , x1 = 0.116 T u , 0.1 ≤ m ≤ 2.0 . ∧ Tm 3.363K m K u + 0.517 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 174 Rule Kc Ti Comment Td Minimum performance index: other tuning 15 Åström and Ti Td Kc Hägglund (2004). Model: Method 1 α = 0 , τm Tm ≤ 1 ; α = 1 , τm Tm > 1 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 ; K 0 = 0 . 16 Åström and Ti Td Kc Hägglund (2004). α = 0 , β = 1 ; χ = 0 ; δ = 0 ; 4 ≤ N ≤ 10 ; b f 1 = 0 , a f 1 = 0 , a f 2 = 0 ; Model: Method 56 ε = 0 ; φ = 0 ; ϕ = 0 ; b = 0 ; a = 2T N ; a = T 2 N 2 ; K = 0 . f2 15 Kc = f3 d f4 d 0 0.4τm + 0.8Tm 0.5τm Tm 1 Tm τm ; Td = . 0.2 + 0.45 , Ti = Km τm τm + 0.1Tm 0.3τm + Tm 2 a f 3 = 0.2τm , a f 4 = 0.01τm (Åström and Hägglund, 2004); a f 3 = 0.17 τm , a f 4 = 0 (Eriksson and Johansson, 2007); 2 a f 3 = 0.333δmax , a f 4 = 0.028δmax , τm < Tm , δmax = jitter margin (Eriksson and Johansson, 2007); a f 3 = 4.6δmax − 6τm (with a f 3 ≥ 1 ), a f 4 = (2.3δ max − 3τm ) , τm > Tm (Eriksson and Johansson, 2007). [ ] 1 T 0 . 5 T N + 16 m d Kc = 0.2 + 0.45 , Km τm + 0.5[Td N ] 0.4τm + 0.8Tm + 0.6[Td N ] Ti = (τm + 0.5[Td N ]) , τm + 0.1Tm + 0.55[Td N ] 2 Td = 0.5(τm + 0.5[Td N ])(Tm + 0.5[Td N ]) τ m >1. ; Tm 0.3τm + Tm + 0.65[Td N ] b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Majhi and Atherton (1999), Atherton and Boz (1998). Model: Method 1 Kc m m m α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; x1 (2Tm + τ m )3 x , b = 1 , b = x 5 , a = 0 . 1 − 1 − 3 f3 f5 f4 Tm K m 2Tm 2 τ m Coefficient values (deduced from graphs). Model: Method 1 x2 x3 x4 x1 x2 x3 x4 1.2 2.6 4.5 6.4 Minimum ISTE – 1.8 servo – Atherton 4.2 and Boz (1998). 6.9 10.0 Minimum ITSE – 1.8 servo – Majhi 4.2 and Atherton 6.9 (1999). 10.0 Minimum ISTSE 2.3 – servo – 5.2 Atherton and Boz 8.4 (1998). 12.4 Minimum ISTSE 2.3 – servo – Majhi 5.2 and Atherton 8.7 (1999). 12.4 17 Comment Td Direct synthesis: time domain criteria τm (2Tm + τm )3 x1 τmTm 17 ≤ 0.8 x2 0 2 T 2 T + τ m m m 2K τ T K0 = Minimum ISE – servo – Atherton and Boz (1998). Ti 0.58 0.91 0.89 0.98 0.17 0.22 0.25 0.26 0.17 0.22 0.25 0.26 0.08 0.11 0.14 0.13 0.08 0.11 0.12 0.13 -0.81 -0.55 -0.30 -0.22 -0.31 -0.16 -0.11 -0.08 -0.29 -0.16 -0.11 -0.08 -0.16 -0.10 -0.07 -0.04 -0.16 -0.10 -0.06 -0.04 2.36 2.49 2.18 2.23 1.08 0.96 0.95 0.92 1.05 0.96 0.95 0.92 0.71 0.65 0.68 0.63 0.69 0.65 0.62 0.64 8.5 10.8 13.3 16.0 13.0 16.2 20.3 24.0 13.0 16.8 20.3 1.02 1.03 1.02 1.00 0.28 0.31 0.29 0.30 0.28 0.27 0.29 -0.16 -0.14 -0.12 -0.09 -0.07 -0.06 -0.04 -0.04 -0.06 -0.05 -0.04 2.18 2.16 2.13 2.10 0.94 0.99 0.93 0.95 0.94 0.90 0.97 16.5 20.4 25.2 31.2 16.5 20.4 25.2 0.14 0.15 0.15 0.14 0.14 0.15 0.15 -0.04 -0.03 -0.02 -0.02 -0.03 -0.03 -0.02 0.64 0.65 0.64 0.61 0.64 0.67 0.65 Equations developed from information provided by Majhi and Atherton (1999). (2T + τ )2 m m x 4 − 2(Tm + τ m ) T Tm x5 = m . 3 2 (2Tm + τm ) x 3 −1− 2 2Tm τ m 175 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 176 Rule Kc 18 Chidambaram (2000a). Model: Method 1 Kc 19 1.2 τ m Comment Ti Td Ti Td 0.5τ m 0.15τ m K m Tm 2 K m Tm β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = α ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = K c ; bf 3 = 1 ; 0.8(Tm + 0.4 τ m ) Huang et al. Kmτm (2000). ( 0 . 6 T m + 0.4 τ m ) Model: Method 1 Kmτm bf 4 = 0 ; a f 5 = 1 . Regulator tuning. Tm + 0.4 τ m 0 Servo tuning. α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = af 3 = 18 Kc = 0.4Tm τm + af 3 , Tm + 0.4τm 0.4Tm τm ; af 4 = 0 ; K0 = 0 . (Tm + 0.4τm )min 20, 0.4Tm τm Tm + 0.4τm 2 τ 1 τ 4.114 − 6.575 m + 3.342 m , Km Tm Tm 2 τ K cTi τ Ti = Tm 0.311 + 1.372 m − 0.545 m , Td = ; [ Tm T 16 . 016 − 11.7(τm Tm )] m α = 0.207 + 3.149(τm Tm ) − 8.405(τm Tm ) + 8.431(τm Tm ) , τm Tm ≤ 0.5 , 2 3 α = 0.9416 − 0.44(τm Tm ) , 0.5 ≤ τm Tm ≤ 0.9 ; K 0 = K c . 19 α = 0.6762 − 0.4332 τ τm τ m 1. 2 τ m , . ≤ 0.8 ; b = 0.1493 , m = 0.9 ; K 0 = Tm Tm Tm K m Tm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 20 Lee et al. (1992). Model: Method 1 Ti Kc 177 Comment Td Direct synthesis: frequency domain criteria Ti Td M s = 1.25 β = 1 ; χ = 0 ; δ = 0 ; N not specified; b f 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0 . Åström and Hägglund (2006), pp. 252, 265. Model: Method 56 21 Kc Ti Td α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; 2 ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = x1 , a f 4 = 0.5x1 ; K 0 = 0 . Padhy and Majhi (2006a). Model: Method 39 22 Kc x1 1 + x1x 2 τm 0 α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 2Tm τm K m ; b f 3 = 1 ; b f 4 = 0.5τm ; a f 5 = 0 . 20 Kc = 2 x 2 2 (τm Tm ) − 4 + 8x1x 2 + 0.5(τm Tm )x 2 2 (4 + 2 τm Tm ) [ ] K m x 2 2 (τm Tm )2 + 4 + 4 x1x 2 (τm Tm ) , 2 τ 2 x 2 m − 4 + 8x1x 2 Tm τ Tm , Ti = Tm 0.5 m + , Td = T τ (4 + 2 τ m Tm )x 2 2 Tm m 4 + 2 m x 22 + 2 Tm τ m 2 x 2 2 [τ m Tm ] − 4 + 8x1x 2 x1 = 0.5 + 0.25 x2 2 , 0.416 + 0.956 Tm τm τm τ ; ≤ 2.51 ; x1 = 1 , m > 2.51 ; x 2 = 1 + 0.0498 τm Tm Tm Tm α = 1 − 0.517(τm Tm ) 0.279 , τm Tm ≤ 10.64 ; α = 0 , τm Tm > 10.64 . 0.4τm + 0.8Tm + 0.6 x1 1 Tm + 0.5x1 21 Kc = (τm + 0.5x1 ) , 0.2 + 0.45 , Ti = Km τm + 0.1Tm + 0.55x1 τm + 0.5x1 0.5(τm + 0.5x1 )(Tm + 0.5x1 ) T T Td = , d ≤ x1 ≤ d . 0.3(τm + 0.5x1 ) + (Tm + 0.5x1 ) 30 3 22 Kc = T − 0.7071 T τ [ T − 0.7071 T τ ] , x = and 2K τ (A − 1) 1.4142 T τ + 1 2φm + π(A m − 1) 2 m m m m m 1 m m m m m m 2φ + π(A m − 1) 2A m 2φm + π(A m − 1) x 2 = 0.7854A m m 1 − . 2 π 2 A m 2 − 1 2 A m − 1 ( ) ( ) b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 178 Rule Kc Ti Td Comment 0.1 ≤ τm ≤ 10 , 23 Eriksson and Ti Td Kc 0.1 ≤ Tm ≤ 10 . Johansson α = 1 , τ T ≤ 1 , α = 0 , τ T > 1 ; β = 1 ; χ = 0 ; δ =0 ; N =∞; (2007). m m m m Model: Method 2 bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0.1 ωg , a f 4 = 0.0025 ωg 2 , τm Tm ≤ 0.25 ; a f 3 = 0.2τm , 2 a f 4 = 0.01τm , τm Tm > 0.25 , ωg = gain crossover frequency; K0 = 0 . Robust Tm K m (0.5τ m + λ ) Harris and Tyreus (1987). Model: Method 1 0 Tm λ ≥ maximum [0.8τ m , Tm ] α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0.5τm ; a f 3 = x1b f 3 , x1 not specified; a f 4 = 0 ; K 0 = 0. 24 Ho et al. (2001). Model: Method 1 Kc 0 Tm 1.551Tm KmAmτm α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0.5τm ; a f 3 = x1b f 3 , x1 not specified; a f 4 = 0 ; K 0 = 0. 25 Lee and Edgar (2002). Model: Method 1 0 Ti Kc α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25Tu , b f 3 = 1 ; bf 4 = 0 ; a f 5 = 0 . 23 Kc = Ti = 0.01Tm (0.4Tm + 11) 1 0.4Tm − 0.04 , + 0.16 , Td = ( ( 0 . 4 Tm − 0.04 ) τm ) + 0.16 K m τm 100((0.4Tm − 0.04) τ m ) + 0.16 (0.35T + 4T + 50)τ − (0.11T − 1.5T + 1.5) 2 m 3 m m m 2 τm 2 . m Tm 24 Kc = , TCL = −0.5τ m + 0.25τ m 2 + ωg −2 , K m (0.5τm + TCL ) ωg = frequency at which the Nyquist curve has a magnitude of unity. 25 Kc = (1 + 0.25K u K m )Ti , T = Tm − 0.25K u K m τm + τm 2 . i K m (λ + τm ) 1 + 0.25K u K m 2(λ + τm ) b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 26 Lee and Edgar (2002). Model: Method 1 Kc Ti Td Ti Td Comment α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25Tu , bf 3 = 1 , bf 4 = 0 , a f 5 = 0 . 27 Visioli (2002). Model: Method 1 Tm + 0.5τ m Kc Tm τ m 2Tm + τ m α = 0 , β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0.05 ≤ τm ≤1 Tm Tm , 1 + K0 λτ m λτ m Tm ; ε = 0;φ = 0; ϕ = 0; af1 = + Tm , a f 2 = 2(λ + τm ) 2(λ + τm ) b f 2 = b f 1 , a f 3 = a f 1 , a f 4 = a f 2 ; for minimum IAE (regulator), K 0 = 0.2171(Tm τm ) 1.481 ; b f 3 = 1 ; b f 4 = 0 ; a f 5 = Tm . Tm Cooper (2006a), K (λ + 0.5τ ) Tm 0.5τm m m (2006b). α = −0.5τm Tm , χ = 0 , a f 1 = 0 (Cooper, 2006a); Model: Method 1 Tm λ (Cooper, 2006b). α = 0 , χ = −0.5τm Tm , a f 1 = 2(λ + τm ) Tm + 0.5τm K m (λ + 0.5τm ) Tm + 0.5τm α = 0 , χ = 0 , af1 = Tm τm 2Tm + τm Cooper (2006b) Tm λ (Cooper, 2006b). 2(λ + τm ) β = 1 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 2 = 0 ; a f 3 = 0 ; ε = α ; φ = χ ; ϕ = 0 ; bf 2 = 0 , a f 3 = a f 1 , a f 4 = 0 ; K 0 = 0 . λ > [0.1Tm ,0.8τm ] (‘aggressive’ tuning); λ > [Tm ,8τm ] (‘moderate’ tuning); λ > [10Tm ,80τm ] (‘conservative’ tuning). 26 2 K c = x1Ti , Ti = 1 + 0.25K u K m Tm − 0.25K u K m τm τm + , x1 = 1 + 0.25K u K m 2(λ + τm ) K m (λ + τm ) 3 4 2 0.25K u K m τm 2 τm τm τ (T − 0.25K u K m τm ) Td = x1 − + + m m . 2 2(1 + 0.25K K ) 6(λ + τ ) 4(λ + τ ) 2(1 + 0.25K u K m )(λ + τm ) u m m m 2Tm + τm 1 27 Kc = 2K m (λ + τm ) (1 + K 0 ) 179 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 180 3.4 FOLPD Model with a Zero G m (s ) = K (1 − sTm 4 )e −sτ m K m (1 + sTm 3 )e − sτm or G m (s) = m 1 + sTm1 1 + sTm1 1 3.4.1 Ideal PI controller G c (s) = K c 1 + Ti s E(s) R(s) − + 1 K c 1 + T is Y(s) U(s) Model Table 18: PI controller tuning rules – FOLPD model with a zero K (1 + sTm3 )e −sτ m K (1 − sTm 4 )e −sτ m G m (s ) = m or G m (s) = m 1 + sTm1 1 + sTm1 Rule Slätteke (2006). Model: Method 3 Kc Comment Ti Minimum performance index: regulator tuning Minimum IE, subject x1 (Tm1 + 0.333τm ) x T + x 4 τm x 2 τm 3 m1 to M s . K m Tm 3τm x 5Tm1 + x 6 τm x1 x2 x3 x4 x5 x6 Ms 0.09 0.16 0.23 0.28 4 5 1 1 6 23 100 88 1 4 17 15 1 9 11 12 21 105 94 85 1.1 1.2 1.3 1.4 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 1 3 3 Kc = Ti Model: Method 1 Tm1 K m (τm + Tm3 ) 2 Direct synthesis: frequency domain criteria Model: Method 1; Ti 4 Kc Tm1 > Tm 4 . Tm1 + 0.5τm (λ + τm )K m Marchetti and Scali (2000). Model: Method 1 2 x1Tm1 K mTm 4 4Tm1 Desbiens (2007), p. 90. Kc = Comment Direct synthesis: time domain criteria ‘Smaller’ τm Tm 4 . Kc Tm1 2 ‘Larger’ τm Tm 4 . Kc Chidambaram (2002), p. 157. Model: Method 1 Sree and Chidambaram (2003a). Slätteke (2006). Model: Method 3 1 Ti Tm1 + 0.5τm (λ + τm + 2Tm 4 )K m 1.571 ( ) . ( ) . K m τm 1 + 3.142 Tm 4 τm 2 2.358 K m τm 1 + 4.712 Tm 4 τm 2 Robust Tm1 + 0.5τm Negative zero. Tm1 + 0.5τm Positive zero. x1 = value of the ‘inverse jump’ of the closed loop system; x1 < K m Tm 4 Tm1 . Ti = (x1Tm1 Tm 4 )[Tm 4 (1 − x 2 ) − 0.5τm (1 + x 2 )] ; (x1Tm1 Tm 4 )(1 − x 2 ) − x 2 T 0.98x1Tm1 0.95x1Tm1 ≤ x2 ≤ , 0.1 ≤ m 4 ≤ 0.3 ; x1Tm1 + Tm 4 x1Tm1 + Tm 4 Tm1 0.3x1Tm1 T 0.1x1Tm1 ≤ x2 ≤ , m4 > 1 . x1Tm1 + Tm 4 Tm1 x1Tm1 + Tm 4 4 Kc = Ti ω Km 2 Tm1 ω2 + 1 , with ω given by solving (for (TiTm 4 ) ω4 + (Ti 2 + Tm 42 )ω2 + 1 φm = 58.13 ): 1.015 = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 4ω) − tan −1 (Tm1 ω) − τ m ω ; 2 Ti = Tm1 + 0.175τ m , τm Tm1 < 2 ; Ti = 0.65Tm1 + 0.35τ m , τ m T m1 > 2 . 181 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 182 3.4.2 Ideal controller in series with a first order lag 1 1 + Td s G c (s) = K c 1 + Ti s 1 + Tf s E(s) R(s) − + 1 1 K c 1 + + Td s Ti s 1 + Tf s U(s) Model Y(s) Table 19: PID controller tuning rules – FOLPD model with a zero K (1 + sTm 3 )e −sτm K (1 − sTm 4 )e −sτ m G m (s) = m or G m (s) = m 1 + sTm1 1 + sTm1 Rule Kc Ti Comment Td Direct synthesis: time domain criteria Sree and Chidambaram (2003a). Jyothi et al. (2001). Model: Method 1 1 Kc 2 Kc 3 Kc 0 Tm1 Model: Method 1 Direct synthesis: frequency domain criteria τm Tm3 < 1 0 Tm1 τm Tm3 > 1 Robust Chang et al. Tm1 (1997). K m (TCL + τ m ) Model: Method 2 1 Kc = 2 Kc = Tm1 T (λ − τ m ) . , Tf = m 4 K m (2Tm 4 + λ + τm ) 2Tm 4 + λ + τm 1.57Tm1 K m τm 3 Kc = Tm1 T 1 + 9.870 m3 τm 0.79Tm1 T K m τm 1 + 2.467 m3 τm 2 2 . . 0 Tf = Tm 3 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models K m e − sτ m 3.5 SOSPD model 2 Tm1 s 2 + 2ξ m Tm1s + 1 or 183 K m e −sτ m (1 + Tm1s )(1 + Tm 2 s ) 1 3.5.1 Ideal PI controller G c (s) = K c 1 + Ti s E(s) R(s) + − 1 K c 1 + Ti s U(s) Y(s) SOSPD model Table 20: PI controller tuning rules – SOSPD model K m e − sτ m K m e − sτ m or (1 + Tm1s)(1 + Tm 2 s) Tm1 2 s 2 + 2ξ m Tm1s + 1 Rule Kc Comment Ti Minimum performance index: regulator tuning Model: Method 1 x1 K m x 2Tm1 Minimum IAE – Lopez (1968), pp. 63, 69–74. Coefficients of K c and Ti deduced from graphs. These are representative results. 0.5 0.6 0.8 1.0 1.5 2.0 4.0 ξm x1 x2 τm Tm1 = 0.1 4.8 4.2 5.7 3.4 7.8 2.8 9.7 2.3 15 1.7 21 1.4 35 x1 x2 x1 x2 x1 x2 x1 x2 x1 x2 x1 x2 - τm Tm1 = 0.2 2.2 3.3 2.7 3.0 3.9 2.7 5.1 2.4 8.3 1.9 11.5 1.7 27 1.2 τm Tm1 = 0.5 0.76 2.1 1.0 2.3 1.6 2.4 2.1 2.4 3.7 2.4 5.4 2.4 12.5 2.4 τm Tm1 = 1.0 0.33 1.2 0.50 1.6 0.82 2.2 1.2 2.5 2.1 2.8 3.0 3.3 6.6 3.7 τm Tm1 = 2.0 0.23 1.2 0.34 1.6 0.52 2.2 0.70 2.8 1.15 3.8 1.65 4.4 3.4 5.9 τm Tm1 = 5.0 0.32 2.6 0.34 2.9 0.38 3.3 0.46 3.8 0.62 5.0 0.80 6.3 1.5 9.6 τm Tm1 = 10.0 0.34 5.0 0.35 5.3 0.38 5.6 0.39 5.9 0.46 7.1 0.52 8.3 0.90 - Minimum IAE – Murata and Sagara (1977). Model: Method 4; K c , Ti deduced from graphs. x1 K m Tm1 = Tm 2 x 2Tm1 x1 x2 τm Tm1 x1 x2 τm Tm1 2.6 1.3 1.0 2.5 2.5 2.6 0.2 0.4 0.6 0.8 0.7 2.7 2.8 0.8 1.0 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 184 Rule Minimum IAE – Harriott (1988). Model: Method 1; coefficients of Ti deduced from graphs. Minimum IAE – Harriott (1988). Model: Method 1; coefficients of Ti deduced from graphs. Minimum IAE – Harriott (1988). Model: Method 1 Minimum IAE – Shinskey (1994), p. 158. Minimum IAE – Shinskey (1996), p. 48. Model: Method 1; τm Tm1 = 0.2 . 1 17-Mar-2009 Kc Ti Comment 0.5K u 0.4Tu 0.1 ≤ τm Tm1 ≤ 2 . 0.5K u x 2Tu Alternative. Tm1 = Tm 2 ; x2 τm Tm1 x2 τm Tm1 1.10 0.85 1.45 1.10 1.75 1.25 0.1 0.2 0.1 0.2 0.1 0.2 0.65 0.52 0.75 0.55 0.90 0.5 1.0 0.5 1.0 0.5 0.5K u τm Tm1 x2 Tm 2 Tm1 = 0.2 Tm 2 Tm1 = 0.5 Tm 2 Tm1 = 1.0 x 2Tu τm Tm1 x2 1.35 0.1 0.65 0.5 Tm 2 Tm1 = 0.2 0.85 0.2 2.00 0.1 0.80 0.5 Tm 2 Tm1 = 0.5 1.10 0.2 0.60 1.0 2.25 0.1 1.00 0.5 Tm 2 Tm1 = 1.0 1.25 0.2 Disturbance input subsequent to the smaller time constant and delay terms. Tm1 = Tm 2 ; 0.45K u 1.1Tu τ T = 0.2 . m m1 Random load disturbance through a filter with time constant = 2Tm1 . Kc Ti Model: Method 1 0.77Tm1 K m τm 2.83(τ m + Tm 2 ) Tm 2 Tm1 = 0.1 1 0.70Tm1 K m τm 0.80Tm1 K m τm 0.80Tm1 K m τm 2.65(τ m + Tm 2 ) 2.29(τ m + Tm 2 ) 1.67(τ m + Tm 2 ) Tm 2 Tm1 = 0.2 Tm 2 Tm1 = 0.5 Tm 2 Tm1 = 1.0 3Tm1 − (τ m + Tm 2 ) Kc = , Ti = τ m 0.5 + 3.5 1 − e . Tm1 − τ + T K m (τm + Tm 2 ) 50 + 551 − e m m 2 100Tm1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum IAE – Huang et al. (1996). Model: Method 1 2 2 Kc Ti Kc Ti Comment 0 < Tm 2 Tm1 ≤ 1 ; 0.1 ≤ τm Tm1 ≤ 1 . Note: equations continued into the footnote on p. 186. τm τ T 1 T − 3.491 m 2 − 25.3143 m m2 2 Kc = 6.4884 + 4.6198 K m Tm1 Tm1 Tm1 + −0.9077 −0.063 0.5961 τm τm τ 1 0.8196 m 5 . 2132 7 . 2712 − − T T T Km m1 m1 m1 + 0.7204 1.0049 1.005 Tm 2 Tm 2 T 1 − 18.0448 m 2 13 . 9108 5 . 3263 + + T T T Km m1 m1 m1 + 0.8529 0.5613 T T τ Tm 2 τ m 1 0.4937 m 2 + 19.1783 m 2 m 12 . 2494 + Km Tm1 Tm1 Tm1 Tm1 τm + 0.557 1.1818 τ m Tm 2 τ T 1 8.4355 m m 2 17 . 6781 − Km Tm1 Tm1 Tm1 Tm1 + τ m Tm 2 Tm 2 τm 2 1 − 0.7241e Tm1 − 2.2525e Tm1 + 5.4959e Tm1 ; Km 2 τ T τ Ti = Tm1 0.0064 + 3.9574 m + 4.4087 m 2 − 6.4789 m Tm1 Tm1 Tm1 2 3 T τ τ T + Tm1 − 12.8702 m m2 2 − 1.5083 m 2 + 9.4348 m Tm1 Tm1 Tm1 2 2 3 T T τ τ T + Tm1 17.0736 m 2 m + 15.9816 m m 2 − 3.909 m 2 Tm1 Tm1 Tm1 Tm1 Tm1 4 3 2 2 τ T τ T τ + Tm1 − 10.7619 m − 10.864 m 2 m − 22.3194 m m 2 Tm1 Tm1 Tm1 Tm1 Tm1 3 4 5 T τ τ T + Tm1 − 6.6602 m m 2 + 6.8122 m 2 + 7.5146 m Tm1 Tm1 Tm1 Tm1 4 2 3 2 3 τ m Tm 2 Tm 2 τ m Tm 2 τ m + 11.1207 + 11.4666 + Tm1 2.8724 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 185 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 186 Rule Minimum IAE – Huang et al. (1996). Model: Method 1 3 Kc Ti Comment Kc Ti 0.05 ≤ τm Tm1 ≤ 1 . 0.4 ≤ ξ m ≤ 1 ; 4 5 6 τ T τ T + Tm1 − 1.2174 m m 2 − 4.3675 m 2 − 2.2236 m Tm1 Tm1 Tm1 Tm1 5 6 4 2 τ T T T τ + Tm1 − 0.112 m 2 m + 1.0308 m 2 − 1.9136 m m 2 Tm1 Tm1 Tm1 Tm1 Tm1 3 3 2 4 5 τ T τ T τ T + Tm1 − 3.4994 m m 2 − 1.5777 m m 2 + 1.1408 m m 2 . Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 3 Note: equations continued into the footnote on p. 187. τm τ 1 − 5.5175ξ m − 26.5149ξ m m Kc = − 10.4183 − 20.9497 Km Tm1 Tm1 1.4439 0.1456 0.3157 τ τm τm 1 42.7745 m + + 10 . 5069 15 . 4103 T T T Km m1 m1 m1 1 3.7057 4.5359 1.9593 + 34.3236ξ m − 17.8860ξ m − 54.0584ξ m Km + [ τ 1 22.4263ξ m m + T Km m1 ] −0.0541 τ + 2.7497ξ m m Tm1 4.7426 + T τ 1 1.8288 τ m − 17.1968 m ξ m 2.7227 + 1.0293ξ m m1 50.2197ξ m K m Tm1 τ m Tm1 + τ τm ξm m 1 − 16.7667e Tm1 + 14.5737e ξm − 7.3025e Tm1 , Km 2 τ τ τ Ti = Tm1 11.447 + 4.5128 m − 75.2486ξm − 110.807 m − 12.282ξ m m Tm1 Tm1 Tm1 3 2 τ τ + Tm1 345.3228ξ m 2 + 191.9539 m + 359.3345ξ m m Tm1 Tm1 4 τm τm 2 3 + Tm1 − 158.7611 ξ m − 770.2897ξ m − 153.633 Tm1 Tm1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Comment Minimum IAE – τm Tm1 = 0.2 , Shinskey (1996), 0.48K u 0.83Tu Tm 2 Tm1 = 0.2 . p. 121. Model: Method 1 Minimum ISE – x1 K m x 2τm McAvoy and Johnson Coefficient values: (1967). Model: Method 1; ξ m Tm1 x 1 x 2 ξ m Tm1 x 1 x 2 ξ m Tm1 x 1 x 2 τm τm τm coefficients of K c 1 0.5 0.8 1.82 4 0.5 4.3 3.45 7 0.5 7.8 3.85 and Ti deduced from 1 4.0 5.7 12.5 4 4.0 27.1 6.67 7 4.0 51.2 5.88 graphs. 1 10.0 13.6 25.0 3 2 τ τ τ + Tm1 − 412.5409ξ m m − 414.7786ξ m 2 m + 485.0976 m ξ m 3 Tm1 Tm1 Tm1 5 4 τ τ + Tm1 864.5195ξ m 4 + 55.4366 m + 222.2685ξ m m Tm1 Tm1 3 2 τ τ τ + Tm1 275.116ξ m 2 m + 205.2493 m ξ m 3 − 479.5627 m ξ m 4 Tm1 Tm1 Tm1 6 5 τ τ + Tm1 − 473.1346ξ m 5 − 6.547 m − 43.2822ξ m m + 99.8717ξ m 6 Tm1 Tm1 4 3 2 τ τ τ + Tm1 − 73.5666 m ξ m 2 − 56.4418 m ξ m 3 − 37.497 m ξ m 4 Tm1 Tm1 Tm1 τ + Tm1 160.7714 m ξ m 5 . T m1 187 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 188 Rule Kc Ti Comment Minimum ISE – Lopez (1968), pp. 63, 69–74. x1 K m x 2Tm1 Model: Method 1 Coefficients of K c and Ti deduced from graphs. These are representative results. 0.5 0.6 0.8 1.0 1.5 2.0 4.0 ξm x2 x1 x2 τm Tm1 = 0.1 6.5 4.8 7.7 4.2 10.5 3.4 13 2.9 20.5 2.1 28 1.6 x1 x2 x1 x2 x1 x2 x1 - - τm Tm1 = 0.2 3.1 4.2 3.9 3.8 5.4 3.3 7 x2 x1 x2 x1 2.9 11 2.4 15 2.0 33 1.4 τm Tm1 = 0.5 1.25 2.9 1.55 2.9 2.2 2.9 2.9 2.9 4.8 2.9 6.7 2.7 14.5 2.5 τm Tm1 = 1.0 0.63 2.2 0.83 2.5 1.2 2.9 1.65 3.1 2.7 3.7 3.8 3.8 τm Tm1 = 2.0 0.42 1.9 0.54 2.4 0.75 3.0 1.0 3.6 1.6 4.5 2.3 5.9 4.5 6.9 8 4.2 τm Tm1 = 5.0 0.45 3.5 0.48 3.8 0.56 4.4 0.65 5.3 0.85 6.7 1.05 7.7 2.0 11.6 τm Tm1 = 10.0 0.46 6.3 0.48 6.7 0.53 7.4 0.56 8.0 0.63 9.1 0.75 10.6 1.15 16.1 Minimum ITAE – Lopez (1968), pp. 63, 69–74. x1 K m Model: Method 1 x 2Tm1 Coefficients of K c and Ti deduced from graphs. These are representative results. 0.5 0.6 0.8 1.0 1.5 2.0 4.0 ξm x2 x1 x2 τm Tm1 = 0.1 2.9 3.0 3.8 2.7 5.5 2.3 7.4 2.0 11.5 1.5 16.5 1.2 x1 x2 x1 x2 x1 x2 x1 x2 x1 x2 x1 - 0.8 τm Tm1 = 0.2 1.35 2.4 1.9 2.4 2.8 2.2 3.8 2.0 6.5 1.7 9.5 1.5 23 1.1 τm Tm1 = 0.5 0.42 1.3 0.65 1.6 1.15 1.9 1.7 2.0 3.0 2.1 4.5 2.1 10.5 2.0 τm Tm1 = 1.0 0.21 0.8 0.34 1.2 0.64 1.8 0.93 2.1 1.7 2.7 2.5 2.9 5.6 3.4 τm Tm1 = 2.0 0.18 0.9 0.26 1.3 0.42 1.9 0.58 2.4 0.98 3.3 1.4 4.0 3.2 5.4 τm Tm1 = 5.0 0.25 2.3 0.29 2.5 0.34 2.9 0.38 3.4 0.52 4.1 0.66 5.6 1.4 8.7 τm Tm1 = 10.0 0.28 4.3 0.30 4.5 0.32 4.9 0.33 5.6 0.38 6.3 0.46 7.4 0.77 - Minimum ITAE – Lopez et al. (1969). Model: Method 8; coefficients of K c x1 K m ξm τm Tm1 x1 x 2Tm1 x2 and Ti deduced from 0.5 0.1 3.0 2.86 0.5 1.0 0.2 0.83 graphs. 0.5 10.0 0.3 4.0 Coefficient values: ξm τ m x1 x 2 Tm1 ξm τm Tm1 1 1 1 4 4 4 0.1 40.0 0.83 1.0 6.0 3.33 10.0 0.75 10.0 0.1 7.0 2.00 1.0 0.95 2.22 10.0 0.35 5.00 x1 x2 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum ITAE – Chao et al. (1989). Model: Method 1 Minimum ITAE – Hassan (1993). Model: Method 8 4 Kc = 4 Kc Ti Kc Ti Comment 0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 . 5 0.5 ≤ ξ m ≤ 2 , Ti Kc 0.09578 + 0.6206ξ m − 0.1069ξ m 2 τm 2ξ T Km m m1 0.1 ≤ τm Tm1 ≤ 4 . −1.108 + 0.2558ξ m − 0.0579 ξ m 2 , ( Ti = 2ξ m Tm1 1.31 − 0.0914ξ m + 0.07464ξ m 5 189 ) τm 2ξ T m m1 2 −0.6441+ 0.7626 ξ m − 0.1193ξ m 2 . Formulae correspond to graphs of Lopez et al. (1969). K c is obtained as follows: log[K m K c ] = −0.0099458 + 1.9743700ξm − 3.8984310 τm − 1.1225270ξ m 2 Tm1 2 τ τ 2 τ + 2.3493280 m + 1.9126490ξ m m + 0.0754568ξ m m T T T m1 m1 m1 2 2 τ 2 τ 3 − 0.5594610ξ m m − 0.7930846ξ m m + 0.2412770ξ m T T m1 m1 3 3 τ τ 2 τ − 0.3567954 m − 0.0024730ξ m m + 0.0478593ξ m m T T T m1 m1 m1 2 2 − 0.1354322ξ m 3 3 τm 3 τ 3 τ − 1.1756470ξ m m − 0.0096974ξ m m ; Tm1 T T m1 m1 Ti is obtained as follows: T τ log i = 0.9321658 − 0.7200651ξ m − 3.4227010 m + 0.0432084ξ m 2 T T m1 m1 2 τ τ 2 τ + 1.4965440 m + 5.3636520ξ m m − 0.0848431ξ m m T T T m1 m1 m1 2 2 τ 2 τ 3 − 1.6011510ξ m m − 2.1777800ξ m m + 0.0404586ξ m T T m1 m1 3 3 τ τ 2 τ − 0.1769621 m + 0.0912008ξ m m + 0.1501474ξ m m T T T m1 m1 m1 2 + 0.3033341ξ m 3 2 3 τm 3 τ 3 τ + 0.1753407ξ m m − 0.0622614ξ m m . Tm1 T T m1 m1 b720_Chapter-03 Rule Minimum ITSE – Chao et al. (1989). Model: Method 1 Nearly minimum IAE, ISE, ITAE – Hwang (1995). Kc = 6 Kc Ti Kc Ti Comment 0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 . 7 Decay ratio = 0.2; Model: Method 14 Ti Kc 0.3366 + 0.6355ξ m − 0.1066ξ m 2 τm 2ξ T Km m m1 −1.0167 + 0.1743ξ m − 0.03946 ξ m 2 , ( Ti = 2ξ m Tm1 1.9255 − 0.3607ξ m + 0.1299ξ m 7 FA Handbook of PI and PID Controller Tuning Rules 190 6 17-Mar-2009 0.2 ≤ τm Tm1 ≤ 2.0 , 0.6 ≤ ξ m ≤ 4.2 . [ ) τm 2ξ T m m1 2 −0.5848 + 0.7294 ξ m − 0.1136 ξ m 2 . ] 0.622 1 − 0.435ωH τ m + 0.052(ωH τm )2 K H , K c = 1 − K H K m (1 + K H K m ) Ti = [ ( 0.0697ωH K m 1 + 0.752ωH τm − 0.145ωH 2 τm 2 ] 0.724 1 − 0.469ωH τm + 0.0609(ωH τm )2 K H , K c = 1 − K H K m (1 + K H K m ) Ti = [ K c (1 + K H K m ) K c (1 + K H K m ) ( 0.0405ωH K m 1 + 1.93ωH τm − 0.363ωH 2 τm 2 ] ) , ε < 2.4 ; ) , 2.4 ≤ ε < 3 ; 1.26(0.506)ωH τ m 1 − 1.07 ε + 0.616 ε 2 K H , K c = 1 − K H K m (1 + K H K m ) K c (1 + K H K m ) Ti = , 3 ≤ ε < 20 ; 0.0661ωH K m (1 + 0.824 ln[ωH τm ]) 1 + 1.71 ε − 1.17 ε 2 [ ( ] 1.09 1 − 0.497ωH τm + 0.0724(ωH τm )2 K H , K c = 1 − K H K m (1 + K H K m ) Ti = ( ) K c (1 + K H K m ) 0.054ωH K m − 1 + 2.54ωH τm − 0.457ωH 2 τm 2 ) , ε > 20 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule 8 Kc Comment Ti Minimum performance index: servo tuning x1 K m x 2 (Tm1 + Tm 2 + τ m ) Tm1 = Tm 2 Minimum IAE – Gallier and Otto (1968). Model: Method 25. Representative coefficient values (deduced from graphs): τm 2Tm1 x1 x2 τm 2Tm1 x1 x2 Minimum IAE – Murata and Sagara (1977). Model: Method 4; K c , Ti deduced from graphs. Minimum IAE – Huang et al. (1996). Model: Method 1 x 2Tm1 0.053 3.6 0.11 2.3 0.25 1.35 x1 K m 1.35 1.17 0.93 0.67 1.50 4.0 0.76 0.72 0.53 0.60 0.43 0.51 Tm1 = Tm 2 x1 x2 τm Tm1 x1 x2 τm Tm1 1.4 1.0 0.8 2.3 2.3 2.4 0.2 0.4 0.6 0.6 0.5 2.4 2.5 0.8 1.0 8 Kc Ti 0 < Tm 2 Tm1 ≤ 1 ; 0.1 ≤ τm Tm1 ≤ 1 . Note: equations continued into the footnote on p. 192. 1 T τm τ T Kc = + 2.6647 m 2 + 9.162 m m2 2 − 13.0454 − 9.0916 K m Tm1 Tm1 Tm1 + −1.0169 3.5959 3.6843 τm τ τm 1 0.3053 m + − 1 . 1075 2 . 2927 T T T Km m1 m1 m1 + 0.8476 2.6083 2.9049 Tm 2 T Tm 2 1 − 31.0306 m 2 − + 13 . 0155 9 . 6899 T T T Km m1 m1 m1 + −0.2016 1.3293 T T τ Tm 2 τ m 1 − 0.6418 m 2 + 18.9643 m 2 m − 39 . 7340 τm Km Tm1 Tm1 Tm1 Tm1 + 0.801 3.956 τ T τ m Tm 2 1 28.155 m m 2 2 . 0067 − Km Tm1 Tm1 Tm1 Tm1 + τ m Tm 2 τm Tm 2 2 1 4.8259e Tm1 + 2.1137e Tm1 + 8.4511e Tm1 , Km 2 τ τ T T τ Ti = Tm1 0.9771 − 0.2492 m + 0.8753 m 2 + 3.4651 m − 3.8516 m m2 2 Tm1 Tm1 Tm1 Tm1 191 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 192 Rule Minimum IAE – Huang et al. (1996). Model: Method 1 9 Kc Ti Kc Ti Comment 0.4 ≤ ξ m ≤ 1 ; 0.05 ≤ τm Tm1 ≤ 1 . 2 3 2 T τ T τ + Tm1 7.5106 m 2 − 7.4538 m + 11.6768 m 2 m Tm1 Tm1 Tm1 Tm1 2 3 4 τ T τ T + Tm1 − 10.9909 m m 2 − 16.1461 m 2 + 8.2567 m Tm1 Tm1 Tm1 Tm1 3 2 2 3 τ T T τ τ T + Tm1 − 18.1011 m 2 m + 6.2208 m m 2 + 21.9893 m m 2 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 4 5 4 T τ T τ + Tm1 15.8538 m 2 − 4.7536 m + 14.5405 m 2 m Tm1 Tm1 Tm1 Tm1 2 3 2 3 τ T T τ + Tm1 − 2.2691 m 2 m − 8.387 m m 2 Tm1 Tm1 Tm1 Tm1 4 5 6 τ τ T T + Tm1 − 16.651 m m 2 − 7.1990 m 2 + 1.1496 m Tm1 Tm1 Tm1 Tm1 5 6 4 2 τ T T T τ + Tm1 − 4.728 m 2 m + 1.1395 m 2 + 0.6385 m m 2 Tm1 Tm1 Tm1 Tm1 Tm1 3 3 2 4 5 τ T τ T τ T + Tm1 1.0885 m m 2 + 3.1615 m m 2 + 4.5398 m m 2 . Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 9 Note: equations continued into the footnote on p. 193. Kc = + 3 τ τ τ 1 − 10.95 − 1.8845 m − 3.4123ξ m + 4.5954ξm m − 1.7002 m Km Tm1 Tm1 Tm1 2 τ τ 1 − 2.1324ξ m m − 14.4149ξ m 2 m − 0.7683ξ m 3 T T Km m1 m1 0.421 0.1984 1.8033 τm τ τm 1 7.5142 m 3 . 7291 5 . 3444 + + T T T Km m1 m1 m1 1 19.5419 1.0749 1.1006 + − 0.0819ξ m − 3.603ξ m + 7.1163ξ m Km + [ ] b720_Chapter-03 17-Mar-2009 Chapter 3: Controller Tuning Rules for Self-Regulating Process Models + −0.6753 −0.1642 τ τm 1 1.9948 τ m 3.206ξ m m 7 . 8480 11 . 3222 − ξ + ξ m m T T T Km m1 m1 m1 + τm ξm τm T 1 2.4239e Tm1 + 3.4137e ξm + 1.0251e Tm1 − 0.5593ξ m m1 , Km τm 2 τ τ Ti = Tm1 2.4866 − 23.3234 m + 5.3662ξm + 65.6053 m Tm1 Tm1 3 τ τ + Tm1 29.0062ξ m m − 24.1648ξ m 2 − 83.6796 m Tm1 Tm1 2 τm τ + 43.1477 m ξ m 2 + 51.9749ξ m 3 + Tm1 − 135.9699ξ m Tm1 Tm1 4 3 2 τm τm 2 τm + Tm1 86.0228 + 70.4553ξ m + 153.4877ξ m Tm1 Tm1 Tm1 5 τm τm 3 4 + Tm1 − 125.0112 ξ m − 68.5893ξ m − 62.7517 Tm1 Tm1 4 3 2 τm τm 2 τm ξm3 + Tm1 27.6178ξ m − 152.7422ξ m + 20.8705 Tm1 Tm1 Tm1 6 τm τm 4 5 + Tm1 54.0012 ξ m + 58.7376ξ m + 13.1193 Tm1 Tm1 5 4 τm τm 6 ξm 2 + Tm1 20.2645ξ m − 23.2064ξ m − 61.6742 Tm1 Tm1 3 2 τm τm τ 3 ξ m 4 + 20.4168 m ξ m 5 . + Tm1 136.2439 ξ m − 95.4092 Tm1 Tm1 Tm1 FA 193 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 194 Rule Kc Comment Ti x1 x 2τm Minimum ISE – Coefficient values: Keviczky and Csáki 0.1 0.2 0.5 1.0 ξ (1973). τ m Tm1 x1 x 2 x1 x 2 x1 x 2 x1 x 2 Model: Method 1; representative K c , Ti 0.1 0.26 1.5 0.55 2 2.2 3.5 6 6 0.2 0.21 1.3 0.4 1.5 1.5 2.8 3 4.5 coefficients estimated 0.5 0.15 1.2 0.25 1.3 0.8 2 2 3.5 from graph; K m = 1 . 1.0 0.12 1.1 0.2 1.2 0.65 1.8 1.2 3.3 2.0 0.1 1.0 0.16 1.2 0.4 1.9 0.9 3.5 5.0 0.1 1.2 0.16 1.4 0.4 3.0 0.65 5.0 10.0 0.15 1.6 0.2 2.0 0.45 5.5 0.6 7.5 Other coefficient values: τm τm x1 x2 ξ x1 x2 T T m1 m1 0.2 0.5 0.5 1.0 1.0 Minimum ITAE – Chao et al. (1989). Model: Method 1 Minimum ITSE – Chao et al. (1989). Model: Method 1 10 Kc = 30 13 30 7 14 10 15 14 30 14 30 5.0 5.0 10.0 5.0 10.0 Kc 2.0 2.0 5.0 5.0 10.0 10.0 15 30 17 32 19 34 x2 14 9 9 8 5 7 3 6.5 1.8 7 1.0 8 0.7 10 ξ 5.0 10.0 5.0 10.0 5.0 10.0 Ti 0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 . 11 Kc Ti 0.7 ≤ ξ m ≤ 3 ; 0.05 ≤ 0.5τm ξ m Tm1 ≤ 0.5 . 2 0.2763 + 0.3532ξ m − 0.06955ξ m τm 2ξ T Km m m1 ( −0.3772 − 0.2414 ξ m + 0.03342 ξ m 2 , Ti = 2ξ m Tm1 0.9953 + 0.07857ξ m + 0.005317ξ m 11 4 8 1.9 3.4 1.1 2.0 2.0 x1 2 0.3970 + 0.4376ξ m − 0.08168ξ m τm Kc = 2ξ T Km m m1 ( ) τm 2ξ T m m1 2 −0.3276 + 0.3226 ξ m − 0.05929 ξ m 2 . −0.5629 − 0.1187 ξ m + 0.01328ξ m 2 , ) τm Ti = 2ξ m Tm1 1.3414 − 0.1487ξ m + 0.07685ξ m 2 2ξ m Tm1 −0.4287 + 0.1981ξ m − 0.01978 ξ m 2 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc 12 Nearly minimum IAE, ISE, ITAE – Hwang (1995). Model: Method 14 12 195 Comment Ti Ti Kc 2 For ξ m ≤ 0.776 + 0.0568 τ τm + 0.18 m : Tm1 Tm1 Coefficient values: x3 x4 x5 x6 x7 -0.549 x9 0.122 x 10 0.0142 x 11 6.96 x 12 -1.77 x 13 0.786 x 14 -0.441 x 15 0.0569 x 16 0.0172 x 17 4.62 x 18 -0.823 x 19 1.28 x 20 0.542 x 21 -0.986 x 22 0.558 x 23 0.0476 x 24 0.996 x 25 2.13 x 26 -1.13 1.14 -0.466 0.0647 0.0609 1.97 -0.323 x1 x2 0.822 x8 Decay ratio = 0.1; 0.2 ≤ τm Tm1 ≤ 2.0 ; 0.6 ≤ ξ m ≤ 4.2 . [ ] x 1 + x 2ωH τm + x 3 (ωH τm )2 K H , K c = 1 − 1 K H K m (1 + K H K m ) Ti = [ ] K c (1 + K H K m ) ( x 4ωH K m 1 + x 5ωH τm + x 6ωH 2 τm 2 x 1 + x 8ωH τm + x 9 (ωH τm )2 K H , K c = 1 − 7 K H K m (1 + K H K m ) Ti = [ ] K c (1 + K H K m ) ( x10ωH K m 1 + x11ωH τm + x12ωH 2 τm 2 ) , ε < 2.4 ; ) , 2.4 ≤ ε < 3 ; x x ωH τ m 1 + x15 s + x16 ε 2 K H , K c = 1 − 13 14 K H K m (1 + K H K m ) Ti = [ K c (1 + K H K m ) , 3 ≤ ε < 20 ; x17 ωH K m (1 + x18 ln[ωH τm ]) 1 + x19 s + x 20 s 2 ( ] x 1 + x 22ωH τm + x 23 (ωH τm )2 K H , K c = 1 − 21 K H K m (1 + K H K m ) Ti = ( ) K c (1 + K H K m ) x 24ωH K m − 1 + x 25ωH τm + x 26ωH 2 τm 2 ) , ε > 20 . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 196 Rule Kc Comment Ti 2 Nearly minimum IAE, ISE, ITAE – Hwang (1995) – continued. For ξ m > 0.889 + 0.496 τ τm + 0.26 m : Tm1 Tm1 Coefficient values: x3 x4 x5 x6 x7 -0.541 x9 0.126 x 10 0.0078 x 11 8.38 x 12 -1.97 x 13 0.738 x 14 -0.415 x 15 0.0575 x 16 0.0124 x 17 4.05 x 18 -0.63 x 19 1.15 x 20 0.564 x 21 -0.959 x 22 0.773 x 23 0.0335 x 24 0.947 x 25 1.9 x 26 -1.07 1.07 -0.466 0.0667 0.0328 2.21 -0.338 x1 x2 0.794 x8 2 τ τ For ξ m > 0.776 + 0.0568 m + 0.18 m and Tm1 Tm1 2 τ τ ξ m ≤ 0.889 + 0.496 m + 0.26 m : Tm1 Tm1 Coefficient values: x3 x4 x5 x6 x7 -0.527 x9 0.11 x 10 0.009 x 11 9.7 x 12 -2.4 x 13 0.76 x 14 -0.426 x 15 0.0551 x 16 0.0153 x 17 4.37 x 18 -0.743 x 19 1.22 x 20 0.55 x 21 -0.978 x 22 0.659 x 23 0.0421 x 24 0.969 x 25 2.02 x 26 -1.11 1.11 -0.467 0.0657 0.0477 2.07 -0.333 x1 x2 0.789 x8 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Minimum ISTSE – Fukura and Tamura (1983). Model: Method 1 13 Comment Ti Tm 2 ≤ Tm1 ; Ti Kc 197 0.05 ≤ τm Tm1 ≤ 1 . Direct synthesis: time domain criteria 13.9 K m 0.140(τm + Tm1 ) τm Tm1 = 0.22 Ream (1954). Model: Method 1; Tm1 = Tm 2 . 0.268(τm + Tm1 ) 3.61 K m 0.368(τm + Tm1 ) 1.69 K m τm Tm1 = 2.11 0.462(τm + Tm1 ) 0.82 K m Bryant et al. (1973). Model: Method 1 τm Tm1 = 1.27 0.420(τm + Tm1 ) 1.14 K m Decay ratio ≈ 33% . τm Tm1 = 0.60 undefined 14 τm Tm1 = 3.92 G c = 1 (Tis ) ; x 2 K m Tm1 Tm1 > Tm 2 . 1 Tm1 1 1 Tm 2 13 + 0. 2 + Kc = 0.56 , 1.5 τm Km Tm1 Tm1 1 + 2 Tm 2 2 + 0.2 τm τm T 0.3 m1 τ m 3.6 Tm 2 1 1 Tm 2 1 + + Ti = [Tm1 + 0.36τm ] 1.5 τm T Tm1 Tm1 m 1 1 + 2 Tm 2 + 0.05 2 + 0.2 T m1 τm τm 14 0.25 . Representative x 2 values, summarised as follows, are deduced from graphs: τ m Tm1 x 2 value ξm 0.5 1.0 2.0 3.0 4.0 5.0 6.0 0.4 0.6 0.8 0.9 1.0 1.1 1.2 1.5 1.9 7.94 8.93 9.80 12.2 15.4 5.13 5.99 5.81 9.01 9.90 10.8 13.3 16.7 5.81 6.94 7.30 7.30 11.5 12.3 13.2 15.4 18.9 7.46 8.40 9.17 10.0 13.9 14.5 15.4 18.2 21.3 9.17 10.0 11.9 14.1 16.4 16.9 17.9 20.0 23.8 10.9 11.8 15.4 19.6 18.9 19.6 20.4 22.7 25.6 12.7 14.1 20.0 28.6 - b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 198 Rule Bryant et al. (1973) – continued. Kc Ti Comment x1 K m τ m Tm1 Tm1 > Tm 2 τ m Tm1 Coefficient values (deduced from graph): x1 τ m Tm1 x1 0.33 0.40 0.50 0.67 0.33 0.40 0.50 0.67 0.08 0.09 0.09 0.11 0.14 0.16 0.18 0.23 15 x 1 Kuwata (1987). Model: Method 1 15 FA 1.0 2.0 10.0 0.15 0.24 0.33 Critically damped dominant pole. 1.0 2.0 10.0 0.31 0.47 0.64 Damping ratio (dominant pole) = 0.6. Km x 2 K m Tm1 Tm1 > Tm 2 Tm1 = Tm 2 16 Kc Ti 17 Kc Ti Representative x 1 values, summarised as follows, are deduced from graphs: τ m Tm1 x 1 value ξm 1.0 2.0 3.0 4.0 5.0 0.4 0.006 0.076 0.103 0.115 0.6 0.039 0.093 0.114 0.122 0.8 0.055 0.110 0.127 0.114 0.093 0.9 0.139 0.147 0.122 0.093 0.070 Representative x 2 values, summarised as follows, are deduced from graphs: 6.0 0.122 0.120 0.075 0.052 τ m Tm1 x 2 value 5.0 6.0 0.4 5.81 7.46 9.17 10.9 0.6 5.13 6.94 8.40 10.0 11.8 0.8 5.99 7.30 9.17 11.9 15.4 0.9 5.81 7.30 10.0 14.1 19.6 0 . 4 2 T + τ 0 . 5 ( ) 16 m1 m Kc = − , Ti = 1.25x1 - 0.25(2Tm1 + τm ) , with K m (2Tm1 + τm − x1 ) K m 12.7 14.1 20.0 28.6 ξm 1.0 2.0 3.0 x1 = 17 Kc = 4.0 (2Tm1 + τm )2 − 0.267τm [6Tm12 + 6Tm1τm + τm 2 ] . 0.4(2ξ m Tm1 + τm ) 0. 5 − , Ti = 1.25x1 - 0.25(2ξm Tm1 + τm ) , with K m (2ξ m Tm1 + τm − x1 ) K m x1 = (2ξmTm1 + τm )2 − 0.267τm [6Tm12 + 6ξmTm1τm + τm 2 ] . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Rotach (1995). Model: Method 9 3.35 K m 3.32τm Pomerleau and Poulin (2004). Model: Method 6 Vítečková (2006). Model: Method 5 Damping factor for oscillations to a disturbance input = 0.75. Tm1 = Tm 2 ; Tm1 1.5Tm1 K m (Tm1 + τ m ) OS < 10%. 18 19 Kc ≥ 20 Five criteria are fulfilled. Ti Kc 1.571Tm1 . K m (2.718τm + 1.5Tm1 ) 2 x1 1 0.7Tm 2 (Tm1 + Tm 2 ) + Tm12 τ , Ti = (Tm1 + 0.7Tm 2 )1 + 0.1 m ; Kc = 1+ Km Tm1Tm 2 Tm1 + 0.7Tm 2 Tm1 x 1 values are obtained as follows: τ m Tm1 x 1 value 20 Tm1 = Tm 2 1.571Tm1 Kc Direct synthesis: frequency domain criteria Maximise crossover 19 frequency. Ti Kc Hougen (1979), pp. 345–346. Model: Method 1 Hougen (1988). Model: Method 1 18 Comment Tm 2 Tm1 0.1 0.2 0.3 0.4 0.5 0.1 0.3 0.5 1.0 0.31 0.6 0.65 0.7 0.19 0.42 0.5 0.6 0.15 0.35 0.45 0.5 τ m Tm1 0.11 0.28 0.46 0.45 0.09 0.23 0.32 0.38 Tm 2 Tm1 0.6 0.7 0.8 0.9 1.0 0.1 0.3 0.5 1.0 0.085 0.2 0.29 0.34 0.075 0.18 0.26 0.32 0.07 0.16 0.24 0.31 0.065 0.155 0.23 0.29 0.06 0.15 0.22 0.28 Equations for K c deduced from graph; T 0.73 log10 m 2 + 0.65 τm T 0.48 log10 m 2 + 0.05 τm 1 Kc = 10 Km 1 Kc = 10 Km Ti = Tm1 + 0.7Tm 2 + 0.12τm . Tm 2 Tm 2 0.61 log10 + 0.29 T T 1 τm , m 2 = 0.1 ; K c = 10 , m 2 = 0.2 ; Tm1 Tm1 Km 0.41 log10 − 0.06 T T 1 τm , m 2 = 0.5 ; K c = 10 , m2 = 1 . Tm1 Tm1 Km 199 b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 200 Rule Kc 1.75 ≤ A m ≤ 3.0 Comment Ti Kc Ti x1 K m A m x 2τm 21 Somani et al. (1992). Model: Method 1; suggested A m : τm Tm 2 τm Tm1 x1 Alternative. Coefficient values: τm τm x2 Tm 2 Tm1 0.1 0.147 11.707 12.5 0.1 0.440 10.091 8.33 0.1 0.954 10.315 6.25 0.1 1.953 11.070 5.00 0.1 4.587 12.207 4.17 0.1 13.89 13.337 3.70 0.125 0.121 11.318 12.5 0.125 0.404 8.613 8.33 0.125 0.897 8.510 6.25 0.125 1.842 8.997 5.00 0.125 4.219 9.840 4.17 0.125 11.628 10.707 3.70 0.25 0.064 14.494 11.11 21 FA 1.00 1.00 1.00 1.00 1.67 1.67 1.67 1.67 1.67 2.5 2.5 2.5 2.5 x1 0.082 12.389 0.437 3.462 1.016 2.370 2.075 2.087 0.169 6.871 0.581 2.773 1.242 1.930 2.445 1.624 4.651 1.957 0.156 7.607 0.338 4.015 0.559 2.797 1.182 1.848 x2 6.25 5.00 4.17 3.57 5.00 4.17 3.57 3.13 3.13 4.55 4.17 3.85 3.33 For 25Tm1Tm 2 Ti << 1 , 2 T T T 0.9806 Kc ≈ 1 + m1 2.944 − tan −1 m1 + m 2 τ m KmAm τm τ m 2 2 2 T T T 1 + m 2 2.944 − tan −1 m1 + m 2 or τ m τm τ m 2 T T T 0.9806 Kc ≈ 1 + m1 2.944 − m1 − m 2 τ KmAm τ τm m m 2 T 0.9806 For 25Tm1Tm 2 Ti >> 1 , K c ≈ 1 + m1 x 1 2 KmAm τm 2 2 2 T T T 1 + m 2 2.944 − m1 − m 2 ; τ τ τm m m 2 T 1 + m 2 x 1 2 , with τm 2 τ τ τ τ τ τ x 1 = 0.981 + 0.5 m + m + 0.945 + 0.25 m + m + 0.945 m + m T Tm 2 Tm1 Tm 2 Tm1 Tm 2 m1 2 τ τ τ τ τ τ + 0.5 m + m + 0.945 − 0.25 m + m + 0.945 m + m T Tm 2 Tm1 Tm 2 Tm1 Tm 2 m1 0.333 0.333 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Somani et al. (1992) – continued. Ti x1 K m A m x 2τm Coefficient values: τm τm x2 Tm 2 Tm1 τm Tm1 0.25 0.25 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5 0.256 6.504 0.667 5.136 1.406 4.961 3.021 5.174 6.452 5.502 0.062 14.918 0.376 4.353 0.909 3.274 1.880 3.025 3.401 3.038 Tan et al. (1996). Model: Method 2 x1 8.33 6.25 5.00 4.17 3.70 8.33 6.25 5.00 4.17 3.70 Ti Tm1 K m (Tm1 + τ m ) Tm1 −ωTm1 τm Schaedel (1997). Model: Method 1 22 Kc = Ti = ( ) x1Ti ωφ 1 + x1Tm1ωφ 2 ( A m 1 + x1Ti ωφ 5.0 5.0 5.0 5.0 5.0 10.0 10.0 10.0 10.0 10.0 Kc 22 Poulin et al. (1996). Model: Method 1 ) Comment Kc τm Tm 2 2 23 x1 x2 0.075 16.187 4.17 0.238 5.623 3.85 0.433 3.448 3.57 0.667 2.52 3.33 0.951 2.014 3.13 0.073 17.650 3.85 0.235 5.996 3.57 0.424 4.524 3.33 0.649 2.641 3.13 0.917 2.093 2.94 Tm1 = Tm 2 Tm1 > 5Tm 2 ; minimum φ m = 58 0 . Ti Minimum φ m = 250 24 Kc Ti ‘Normal’ design. 25 Kc Ti ‘Sharp’ design. , x1 = 0.8, 201 τm τ < 0.5 ; x1 = 0.5, m > 0.5 ; ωφ < ωu ; Tm1 Tm1 1 . x ω tan [− 2 tan x T ω − x τ ω − φ ] 1 φ −1 1 m φ 1 m φ m 4Tm1 (τm + Tm 2 ) 23 Ti = , 0.16Tm1 ≤ Tm 2 + τ m ≤ 0.3Tm1 ; ω = frequency 1.3Tm1 − 4(τm + Tm 2 ) where phase of G m ( jω)G c ( jω) is maximum; G m = − K m e − sτ m . (1 + Tm1s )(1 + Tm 2s ) (2ξmTm1 + τm )2 − 2(Tm12 + 2ξmTm1τm + 0.5τm 2 ) . 24 Kc = 0.5Ti , Ti = K m (2ξ m Tm1 + τm − Ti ) 25 Kc = 0.375 4ξm 2Tm12 + 2ξ mTm1τm + 0.5τm 2 − Tm12 , Km Tm12 + 2ξ m Tm1τm + 0.5τm 2 2 Ti = 2 2 2 4ξ m Tm1 + 2ξ m Tm1τm + 0.5τm − Tm1 . 2ξm Tm1 + τm b720_Chapter-03 Rule Leva et al. (2003). Model: Method 1 Comment Kc Ti Tm1 2 K m (Tm 2 + τ m ) 4(Tm 2 + τ m ) Hägglund and Åström (2004). Model: Method 1; Tm1 = Tm 2 . Kc = FA Handbook of PI and PID Controller Tuning Rules 202 26 17-Mar-2009 26 Kc Ti 27 Kc Ti 28 Kc Ti 29 Kc Ti 30 Kc Ti 2.01 − 0.80φ Tm1 >> Tm 2 + τ m ; symmetrical optimum principle. 1.920 < φ < 2.356 ; M s = 1.2 . φ = 2.09 radians (‘good’ choice). 2.094 < φ < 2.443 ; M s = 1.4 . φ = 2.269 radians (‘reasonable’ choice). 2.356 < φ < 2.705 ; M s = 2. 0 . 6.283(1.72φ − 2.55) , Ti = . 2 G p jωφ ωφ 1 + (3.44φ − 5.20 ) Km 0.33 6.61 27 Kc = , Ti = . 2 G p jωφ G p jωφ 1 + 2.26 G p jωφ ωφ 1 + 2.01 Km Km 2 . 50 0 . 92 − φ 6 .283(1.72φ − 3.05) 28 Kc = , Ti = . 2 G p jωφ G p jωφ 1 + (10.75 − 4.01φ) G p jωφ ωφ 1 + (3.44φ − 6.10 ) Km Km 0.41 5.36 29 Kc = , Ti = . 2 G p jωφ G p jωφ 1 + 1.65 G p jωφ ωφ 1 + 1.71 Km Km 3 . 43 1 . 15 − φ 6 .283(0.86φ − 1.50) 30 Kc = , Ti = . 2 G p jωφ G p jωφ 1 + (11.30 − 4.01φ) G p jωφ ωφ 1 + (1.72φ − 2.90) Km Km ( ) G jω 1 + (15.45 − 6.30φ) p φ G p jωφ Km ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Hägglund and Åström (2004) – continued. Desbiens (2007), p. 90. 31 Kc 32 Kc Ti Comment Ti φ = 2.53 radians (‘good’ choice). Tm1 > Tm 2 ; Model: Method 1 Ti Robust Tm1 + Tm 2 + 0.5τ m K m τ m (2λ + 1) Brambilla et al. (1990). Model: Method 1; λ values obtained from graph. 31 32 Kc = 0.1 0.2 0.5 1.0 2ξ mTm1 + 0.5τm K m (2λ + τm ) Kc = 0.52 ( ) G jω 1 + 1.15 p φ G p jωφ Km Ti Km ( ) 0.1 ≤ τm ≤ 10 Tm1 + Tm 2 Coefficient values: τm λ Tm1 + Tm 2 τm Tm1 + Tm 2 Marchetti and Scali (2000). Model: Method 1 Tm1 + Tm 2 + 0.5τ m , Ti = 3.0 1.8 1.0 0.6 2.0 5.0 10.0 λ 0.4 0.2 0.2 2ξ m Tm1 + 0.5τm 4.25 ( ) 2 G jω 1 + 1.45 p φ ωφ Km . (Tm1Tm 2 )2 ω6 + (Tm12 + Tm 22 )ω4 + ω2 , with ω given by solving (for Ti 2ω2 + 1 φm = 58.13 ): 1.015 = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 2ω) − tan −1 (Tm1 ω) − τ m ω . 2 T τ T τ Ti = Tm1 1 + 0.175 m + 0.3 m 2 + 0.2 m 2 , m < 2 ; Tm1 Tm1 Tm1 Tm1 2 Tm 2 τm Tm 2 τ m + 0.3 + >2. 0 . 2 Ti = Tm1 0.65 + 0.35 , Tm1 Tm1 Tm1 Tm1 203 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 204 Rule Kc Comment Ti Kristiansson (2003). Model: Method 1 33 Kc 34 Kc Panda et al. (2004). Model: Method 1 35 Kc Ti 37 Kc Ti x 1K u Ultimate cycle x 2 Tu Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 0.5 . Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 0.7 . Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 1.5 . 2ξ m Tm1 M max = 1.7 36 x1 x2 τm Tu x1 x2 τm Tu x1 x2 τm Tu 0.08 0.09 0.03 0.32 0.14 0.03 x 1K u 0.05 0.10 0.15 0.07 0.13 0.18 0.04 0.08 0.10 x 2 Tu 0.25 0.30 0.35 0.21 0.12 0.225 0.14 0.225 0.155 0.40 0.45 0.50 x1 x2 τm Tu x1 x2 τm Tu x1 x2 τm Tu 0.01 0.04 0.08 0.12 0.29 0.19 0.15 0.14 x 1K u 0.05 0.10 0.15 0.20 0.16 0.18 0.21 0.22 0.14 0.15 0.15 0.15 x 2 Tu 0.25 0.30 0.35 0.40 0.225 0.155 0.225 0.155 0.45 0.50 x1 x2 τm Tu x1 x2 τm Tu x1 x2 τm Tu 0.21 0.25 0.26 1.13 0.70 0.43 0.15 0.20 0.25 0.25 0.29 0.23 0.21 0.225 0.17 0.30 0.35 0.40 0.225 0.155 0.225 0.155 0.45 0.50 33 Kc = 2ξ m Tm1 1 T 1 x1 − x12 − 1 + , x1 = 1 + m 2 1 − 1 − . 2 K m τm τm M max M max 2 34 Kc = 2ξ m Tm1 T T T 2 1 + 0.1913 m 2 − 0.346 + 0.3826 m 2 + 0.0366 m 22 . K m τm τm τm τm 35 Kc = 4ξ T + ξ m τ m + 0.5Tm1 8ξm Tm1 + 2ξ m τm + Tm1 , Ti = m m1 . 2λK m 2ξ m 2 2 T λ ≥ maximum 0.828 + 0.812 m 2 + 0.172Tm1e − (6.9Tm 2 / Tm1 ) ,1.7 τm . T m1 Tm 2 0 . 558 T T m 2 m1 + 0.172Tm1e − (6.9Tm 2 / Tm1 ) + 0.828 + 0.812 0.5τ m Tm1 Tm1 + 1.208Tm 2 37 + , Kc = λK m λK m 36 Ti = 0.828 + 0.812 Tm 2 0.558Tm 2Tm1 + 0.172Tm1e − (6.9Tm 2 / Tm1 ) + + 0.5τ m . Tm1 Tm1 + 1.208Tm 2 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Ti Comment Kc 2Tm1 Tm1 = Tm 2 ; Model: Method 20 Kc Other tuning 1.95 tan (0.5φ) ωφ 0.413 G p ( jω2.27 ) 4.182 ω 2.27 Kc Thyagarajan and Yu (2003). 38 39 Hägglund and Åström (2004). Model: Method 1 3 T T 2 0.0200 m1 + 0.2714 m1 Tu Tu 38 . Kc = 2 Tm1 +1 K m 0.2947 Tu 2.31 39 Kc = . 2 1 + tan (0.5φ) G p ( jωφ ) [ ] Tm1 = Tm 2 ; Tm1 >> τ m . φ = 2.27 radians (‘reasonable’ choice). 205 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 206 1 + Td s 3.5.2 Ideal PID controller G c (s) = K c 1 + Ti s E(s) R(s) + − U(s) 1 K c 1 + + Td s Ti s Table 21: PID tuning rules – SOSPD model G m (s) = G m (s ) = Rule Auslander et al. (1975). 1 Kc 1 K m e − sτ m or (1 + sTm1 )(1 + sTm 2 ) K m e −sτ m Tm1 2 s 2 + 2ξ m Tm1s + 1 Ti Comment Td Process reaction x1 +2τm 0.25Ti Kc Y(s) SOSPD model Model: Method 3 Tm 2 Tm 1 1.2 Tm1Tm 2 Tm1 Tm 2 Tm1 −Tm 2 Tm 2 Tm1 −Tm 2 Kc = {τm + ln − T K m (Tm1 − Tm 2 ) Tm1 − Tm 2 Tm 2 Tm1 m1 Tm 2 Tm1 Tm 2 Tm1 −Tm 2 Tm 2 Tm1 −Tm 2 − (Tm1 − Tm 2 )2 Tm1 Tm1 2 1 − 1+ x1 = 2Tm1Tm 2 Tm1 − ln Tm1 − Tm 2 Tm 2 Tm1 Tm1 −Tm 2 Tm 2 Tm 2 Tm1 − Tm 2 Tm1 − Tm 2 Tm1 −Tm 2 Tm 2 Tm1 Tm1 − Tm 2 Tm1 Tm 2 Tm1 2 Tm 2 Tm1 −Tm 2 Tm 2 Tm1 −Tm 2 − T (Tm1 − Tm 2 ) Tm1 m1 1+ Tm1 Tm1 −Tm 2 Tm 2 Tm 2 Tm1 − Tm 2 Tm1 − Tm 2 Tm1 −Tm 2 Tm 2 Tm1 Tm1 − Tm 2 Tm1 . }−1 , b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Alfaro Ruiz (2005a). 2 ξm τm Tm1 = 0.1 Kc Ti Td Comment Kc Ti Td Model: Method 1 Tm1 = Tm 2 ; 1.0 ≤ x1 ≤ 1.67 . Minimum performance index: regulator tuning Representative 5.8 tuning values; 0.36Tu 0.21Tu Km Tm 2 = τ m 5 = 0.1Tm1 . 0.29Tu 0.29Tu Km Minimum IAE – Wills (1962b) – deduced from graph. Model: Method 1 Minimum IAE – Lopez (1968), pp. 79–90. 207 x1 K m x 2Tm1 Model: Method 1 x 3Tm1 Coefficients of K c , Ti and Td deduced from graphs. These are representative results. 0.5 0.6 0.8 x1 x2 x3 x1 x2 x3 x1 x2 x3 27 - 0.28 28 - 0.26 30 - 0.26 τm Tm1 = 0.2 9.5 0.61 0.48 10.5 0.59 0.45 11.5 0.57 0.41 τm Tm1 = 0.5 2.25 0.97 0.92 2.6 1.00 0.82 3.2 1.05 0.69 τm Tm1 = 1.0 0.78 1.11 1.40 1.0 1.25 1.20 1.4 1.47 0.94 τm Tm1 = 2.0 0.39 1.27 1.40 0.52 1.56 1.30 0.77 1.92 1.15 τm Tm1 = 5.0 0.42 2.7 1.40 0.45 2.9 1.50 0.53 3.0 1.70 τm Tm1 = 10.0 0.38 4.9 1.15 0.41 5.3 1.40 0.47 5.7 1.90 1.0 ξm 1.5 2.0 4.0 x1 x2 x3 x1 x2 x3 x1 x2 x3 x1 x2 x3 τm Tm1 = 0.1 31 - 0.24 35 - 0.22 40 - - 68 - - τm Tm1 = 0.2 13.0 0.59 0.43 16.0 0.56 0.30 19.5 0.56 0.25 47 0.53 τm Tm1 = 0.5 4.0 1.06 0.59 6.0 1.11 0.46 7.8 1.14 0.39 16.5 1.14 0.29 τm Tm1 = 1.0 1.85 1.56 0.82 3.0 1.75 0.68 4.3 1.85 0.60 9.0 1.92 0.52 τm Tm1 = 2.0 1.05 2.3 1.10 1.65 2.6 1.05 2.3 2.9 1.00 4.8 τm Tm1 = 5.0 0.62 3.6 1.90 0.85 4.3 2.10 1.1 4.8 2.15 2.15 6.5 τm Tm1 = 10.0 0.52 6.1 2.35 0.60 6.9 - 3.3 0.98 2.3 2.9 0.70 7.4 3.45 1.2 10.0 4.0 τ τ 16.57 + 20.83 m 2.65 + 3.33 m x 2 . 0 T T T 2 m1 m1 m1 K c = 1 0.70 + τ , T = τ . , Ti = m d τm τm m Km τm 1 11 . 36 1 11 . 36 + + Tm1 Tm1 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 208 Rule Kc Ti Td Comment Minimum IAE – Shinskey (1988), p. 151. Model: Method 2 0.62 K u 0.38Tu 0.15Tu Tm 2 = 0.25 Tm 2 + τ m 0.68K u 0.33Tu 0.19Tu Tm 2 = 0.5 Tm 2 + τ m 0.79 K u 0.26Tu 0.21Tu Tm 2 = 0.75 Tm 2 + τ m Ti Td Model: Method 1 Minimum IAE – Kang (1989). T 0.05 ≤ L ≤ 7.0. Tm1 Minimum ISE – Lopez (1968), pp. 79–90. ξm 3 Kc x1 , x 2 …. x14 , x15 values are specified on pp. 127–165 of Kang (1989), for τm Tm1 = 0.1, 0.2 ….. 0.9, 1.0 with Tm 2 Tm1 = 0.1 , 0.2…..0.9, 1.0. x1 K m Model: Method 1 x 3Tm1 Coefficients of K c , Ti and Td deduced from graphs. These are representative results. 0.5 0.6 0.8 x1 3 x 2Tm1 x2 x3 x1 x2 x3 x1 x2 x3 τm Tm1 = 0.1 29 0.21 0.34 30 0.21 0.33 31 0.21 0.31 τm Tm1 = 0.2 10.0 0.38 0.56 10.5 0.38 0.54 12.0 0.38 0.48 τm Tm1 = 0.5 2.4 0.67 1.10 2.8 0.69 1.10 3.2 0.74 0.83 τm Tm1 = 1.0 0.86 0.87 1.70 1.1 0.95 1.45 1.55 1.11 1.20 τm Tm1 = 2.0 0.45 1.1 2.00 0.57 1.3 1.80 0.85 1.6 1.60 τm Tm1 = 5.0 0.44 2.4 2.00 0.49 2.6 2.20 0.57 2.9 2.40 τm Tm1 = 10.0 0.45 5.3 1.50 0.48 5.6 1.85 0.53 5.9 2.50 Kc = Ti = T −x2 L 1 T T x1 − e Tm1 x 3 cos x 4 L − x 5 sin x 4 L , Km Tm1 Tm1 Tm1 T T x8 L x 10 m 2 T Tm1 + x 9e TL sin x11 L + x12 x 6 + x 7 1 − e T m1 1 Load disturbance model = . 1 + sTL , Td = T x 15 L 1 x13 + x14e Tm1 , Tm1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 209 Rule Kc Ti Td Comment Minimum ISE – Lopez (1968), pp. 79–90 – continued. ξm x1 K m x 2Tm1 x 3Tm1 Model: Method 1 Coefficients of K c , Ti and Td deduced from graphs. These are representative results. 1.0 x1 τm Tm1 = 0.1 1.5 x2 x3 x1 2.0 x2 x3 x1 x2 4.0 x3 x1 x2 x3 32 0.21 0.30 38 0.21 0.25 42 0.21 0.23 70 0.21 - τm Tm1 = 0.2 13 0.38 0.44 17 0.38 0.36 21 0.38 0.31 39 0.38 0.20 τm Tm1 = 0.5 4.2 0.77 0.74 6.2 0.80 0.57 8.5 0.82 0.50 17.5 0.83 0.37 τm Tm1 = 1.0 2.0 1.18 1.05 3.2 1.33 0.85 4.4 1.39 0.77 9.5 1.49 0.66 τm Tm1 = 2.0 1.1 2.6 1.2 τm Tm1 = 5.0 0.66 3.1 2.6 0.90 3.7 2.8 1.20 4.2 2.85 2.25 5.3 3.0 τm Tm1 = 10.0 0.58 5.9 3.0 0.67 6.3 4.0 0.80 6.7 - Minimum ITAE – Lopez (1968), pp. 79–90. x1 K m x 2Tm1 x 3Tm1 ξm 1.8 1.5 1.8 2.1 1.35 2.9 2.3 1.30 5.2 - 1.25 8.3 Model: Method 1 Coefficients of K c , Ti and Td deduced from graphs. These are representative results. 0.5 0.6 0.8 x1 x2 x3 x1 x2 x3 x1 x2 x3 τm Tm1 = 0.1 24 - 0.25 24 - 0.25 24 0.50 0.225 τm Tm1 = 0.2 8.8 0.71 0.44 9.2 0.74 0.41 10.0 0.74 0.35 τm Tm1 = 0.5 2.1 1.15 0.84 2.5 1.15 0.74 3.2 1.22 0.60 τm Tm1 = 1.0 0.72 1.22 1.20 0.93 1.39 1.00 1.35 1.61 0.82 τm Tm1 = 2.0 0.36 1.30 1.15 0.48 1.59 1.10 0.75 2.04 1.00 τm Tm1 = 5.0 0.40 2.7 1.20 0.44 2.9 1.30 0.52 3.2 1.45 τm Tm1 = 10.0 0.34 4.3 0.98 0.38 5.0 1.30 0.43 - 1.75 1.0 ξm x1 x2 1.5 x3 2.0 4.0 x1 x2 x3 x1 x2 x3 x1 x2 x3 τm Tm1 = 0.1 26 0.53 0.20 31 - 0.175 36 - 0.15 68 - 0.1 τm Tm1 = 0.2 11.3 0.77 0.32 14.5 0.71 0.25 18.2 0.68 0.205 36 0.57 0.14 τm Tm1 = 0.5 3.8 1.25 0.52 5.8 1.28 0.40 7.7 1.21 0.34 16.2 1.21 0.26 τm Tm1 = 1.0 1.8 1.75 0.70 2.9 1.92 0.58 4.0 2.00 0.54 8.7 2.13 0.46 τm Tm1 = 2.0 0.36 2.4 0.96 0.48 2.9 0.92 0.73 3.2 0.88 0.97 3.6 0.84 τm Tm1 = 5.0 0.40 3.6 1.6 0.44 4.5 1.7 0.53 5.3 1.85 0.58 6.9 2.0 τm Tm1 = 10.0 0.34 5.9 - - - 0.38 6.9 0.43 7.7 - 0.48 - b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 210 Rule Minimum ITAE – Lopez et al. (1969). Model: Method 23 Bohl and McAvoy (1976b). Minimum ITAE – Hassan (1993). 4 17-Mar-2009 Comment Kc Ti Td x1 K m x 2 Tm1 x 3 Tm1 Representative coefficient values (from graphs): x1 x2 x3 ξm τ m Tm1 25 0.7 0.35 25 1.8 9.0 0.5 1.3 5 0.5 1.7 2 0.25 1.2 1.0 0.2 0.7 0.45 0.5 0.5 0.5 1.0 1.0 4.0 0.1 1 10 0.1 1 1 4 Kc Ti Td Minimum ITAE; Model: Method 1 5 Kc Ti Td Model: Method 8 Tm 2 Tm1 = 0.12,0.3,0.5,0.7,0.9 ; τm Tm1 = 0.1,0.2,0.3,0.4,0.5 . 10 τ m T 10 τ m −1.2096 + 0.1760 ln 0.1044 + 0.1806 ln 10 m 2 − 0.2071 ln Tm1 10Tm 2 Tm1 Tm1 10.9507 10τm Kc = , K m Tm1 Tm1 10 τ m 10 Tm 2 10 τ m + 0.0081 ln 0.7750 − 0.1026 ln 0.1701+ 0.0092 ln T 10τm Tm1 10Tm 2 Tm1 m1 Ti = 0.2979Tm1 , Tm1 Tm1 10 τ m 10 Tm 2 10 τ m + 0.0128 ln 0.6025 − 0.0624 ln 0.4531− 0.0479 ln T 10τm Tm1 10Tm 2 Tm1 m1 Td = 0.1075Tm1 . Tm1 Tm1 5 Note: equations continued into the footnote on p. 211. 0.5 ≤ ξ m ≤ 2 ; 0.1 ≤ τm Tm1 ≤ 4 . K c is obtained as follows: log[K m K c ] = 1.9763370 − 0.6436825ξm − 5.1887660 τm + 0.4375558ξ m 2 Tm1 2 τ τ 2 τ + 2.9005550 m + 3.1468010ξ m m − 0.1697221ξ m m T T T m1 m1 m1 2 2 τ τ 3 − 0.8161808ξ m m − 1.2048220ξ m 2 m − 0.0810373ξ m T T m1 m1 3 3 τ τ 2 τ − 0.4444091 m + 0.0319431ξ m m + 0.1054399ξ m m T T T m1 m1 m1 2 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 2 + 0.1652807ξ m 3 3 τm 3 τ 3 τ + 0.1175991ξ m m − 0.0375245ξ m m ; Tm1 T T m1 m1 Ti is obtained as follows: T τ log i = −0.7865873 + 0.6796885ξm + 2.1891540 m − 0.3471095ξ m 2 T T m1 m1 2 τ τ 2 τ − 1.9003610 m − 0.7007801ξ m m + 0.3077857ξ m m T T T m1 m1 m1 2 2 τ τ 3 + 0.8566974ξ m m − 0.2535062ξ m 2 m + 0.0412943ξ m T T m1 m1 3 3 τ τ 2 τ + 0.3484161 m − 0.1626562ξ m m − 0.0661899ξ m m T T T m1 m1 m1 2 2 + 0.2247806ξ m 3 3 τm 3 τ 3 τ − 0.2470783ξ m m + 0.0493011ξ m m , Tm1 T T m1 m1 Td is obtained as follows: T τ log d = −0.6726798 − 0.2072477ξ m + 2.6826330 m + 0.0807474ξm 2 T T m1 m1 2 τ τ 2 τ − 1.7707830 m − 1.6685140ξ m m + 0.0845958ξ m m T T T m1 m1 m1 2 2 τ τ 3 + 0.7159307ξ m m + 0.5631447ξ m 2 m − 0.0225269ξ m T T m1 m1 3 3 τ τ 2 τ + 0.2821874 m − 0.0616288ξ m m − 0.0626626ξ m m T T T m1 m1 m1 2 − 0.0372784ξ m 3 2 3 τm 3 τ 3 τ − 0.0948097ξ m m + 0.0272541ξ m m . Tm1 T T m1 m1 211 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 212 Rule Minimum ITAE – Sung et al. (1996). 6 17-Mar-2009 Kc = 6 Kc Ti Td Comment Kc Ti Td Model: Method 11 2.001 0.766 τ T T 1 ξ m , m < 0.9 or + 2.189 m1 − 0.67 + 0.297 m1 Km Tm1 τm τm T τ 1 − 0.365 + 0.260 m − 1.4 + 2.189 m1 Km T τm m1 2 Kc = 0.766 τ ξ m , m ≥ 0. 9 ; Tm1 0.520 τ τ − 0.3 , m < 0.4 or Ti = Tm1 2.212 m Tm1 Tm1 2 τ τ Ti = Tm1{−0.975 + 0.910 m − 1.845 + x1} , m ≥ 0.4 with T T m1 m1 ξm − τ 2 0.15 + 0.33 m τ Tm1 x1 = 1 − e 5.25 − 0.88 m − 2.8 ; Tm1 Td = Tm1 ; ξm 1.171 0.530 − 1.121 T T 1 − e − 0.15 + 0.939(Tm1 τ m ) 1.45 + 0.969 m1 − 1.9 + 1.576 m1 τ τ m m 0.05 < τ τm ≤ 2 (Sung et al., 1996); 0 < m < 2 (Park et al., 1997). Tm1 Tm1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Nearly minimum IAE, ISE, ITAE – Hwang (1995). Model: Method 14 213 Comment Kc Ti Td 7 Kc Ti 8 Kc Ti 9 Kc Ti 3 ≤ ε 2 < 20 10 Kc Ti ε 2 ≥ 20 ε 2 < 2.4 2.4 ≤ ε 2 < 3 Td Decay ratio = 0.2; 0.2 ≤ τ m Tm1 ≤ 2.0 and 0.6 ≤ ξ m ≤ 4.2 . Minimum performance index: servo tuning Representative tuning values; 6.5 K m 1.45Tu 0.14Tu Tm 2 = τ m Minimum IAE – Wills (1962b) – deduced from graph. Model: Method 1 7 7 Km 0.95Tu [ ] ( ) 2 2 0.622 1 − 0.435ωH 2 τm + 0.052ωH 2 τm Kc = KH2 − , K m (1 + K H 2 K m ) K c (1 + K H 2 K m ) Ti = , ωH 2 K m 0.0697 1 + 0.752ωH 2 τm − 0.145ωH 2 2 τm 2 Td = 8 [ 1.45(1 + K u K m ) 1.16 1 − 1 − 0.612ω u τ m + 0.103ω u 2 τ m 2 . ε K m 2ωu 0 [ ( K c (1 + K H 2 K m ) ωH 2 K m 0.0405 1 + 1.93ωH 2 τm − 0.363ωH 2 2 τm 2 ] 1.26(0.506 )ωH 2 τ m 1 − 1.07 ε 2 + 0.616 ε 2 2 Kc = KH2 − , K m (1 + K H 2 K m ) [ ) ] Ti = 10 ( 2 2 0.724 1 − 0.469ωH 2 τm + 0.0609ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) Ti = 9 = 0.1Tm1 . 0.22Tu K c (1 + K H 2 K m ) ( ωH 2 K m 0.0661(1 + 0.824 ln[ωH 2 τm ]) 1 + 1.71 ε 2 − 1.17 ε 2 2 ] 2 2 1.09 1 − 0.497ωH 2 τm + 0.0724ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) Ti = ( K c (1 + K H 2 K m ) ωH 2 K m 0.054 − 1 + 2.54ωH 2 τ m − 0.457ωH 2 2 τ m 2 ). ). ). b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 214 Rule Minimum IAE – Gallier and Otto (1968). Model: Method 1 Minimum ITAE – Wills (1962a) – deduced from graph. Minimum ITAE – Sung et al. (1996). Kc x1 K m τm 2Tm1 Ti Td Comment 11 Td Tm1 = Tm 2 Ti Representative coefficient values – deduced from graphs: τm x1 x2 x3 x1 x2 x3 2Tm1 0.053 0.11 0.25 6.7 4.15 2.35 0.89 0.84 0.77 18 Km 12 Kc 0.24 0.24 0.24 0.67 1.50 4.0 1.05 0.70 0.52 ≈ Tu ≈ 0.2Tu Ti Td 11 Ti = x 2 (Tm1 + Tm 2 + τm ) , Td = x 3 (Tm1 + Tm 2 + τm ) . 12 Kc = 0.64 0.24 0.55 0.26 0.50 0.20 Tm1 = Tm 2 ; τ m = 0.1Tm1 ; Model: Method 1 Model: Method 11 −0.983 τ 1 ξ m , ξ m ≤ 0.9 or − 0.04 + 0.333 + 0.949 m Km Tm1 Kc = −0.832 τ τ 1 ξ m , ξ m > 0.9 . − 0.544 + 0.308 m + 1.408 m Km Tm1 Tm1 Ti = Tm1 [2.055 + 0.072(τm Tm1 )]ξ m , τ m Tm1 ≤ 1 or Ti = Tm1[1.768 + 0.329(τm Tm1 )]ξ m , τ m Tm1 > 1 . Td = Tm1 ; 1.090 (τ m Tm1 )1.060 ξ m − 0.870 1 − e 0.55 + 1.683 Tm1 τ m 0.05 < τ τm ≤ 2 (Sung et al., 1996); 0 < m < 2 (Park et al., 1997). Tm1 Tm1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Nearly minimum IAE, ISE, ITAE – Hwang (1995). Model: Method 14 13 Ti Kc Ti 15 Kc Ti 16 Kc Ti 17 Kc Ti 13 215 Comment Td 14 Decay ratio = 0.1; τ 0.2 ≤ m ≤ 2.0 ; Tm1 0.471K u K m ωu 0. 6 ≤ ξ m ≤ 4. 2 . [ ] 2 2 0.822 1 − 0.549ωH 2 τm + 0.112ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) Ti = K c (1 + K H 2 K m ) ( ωH 2 K m 0.0142 1 + 6.96ωH 2 τm − 1.77ωH 2 2 τm 2 ). 2 14 15 ξ ≤ 0.613 + 0.613 τ τm + 0.117 m . Tm1 Tm1 [ ] 2 2 0.786 1 − 0.441ωH 2 τm + 0.0569ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) Ti = 16 [ ] 2 ω τ 1.28(0.542) H 2 m 1 − 0.986 ε 2 + 0.558 ε 2 , Kc = K H2 − K m (1 + K H 2 K m ) Ti = 17 [ ( K c (1 + K H 2 K m ) ωH 2 K m 0.0172 1 + 4.62ωH 2 τm − 0.823ωH 2 2 τm 2 K c (1 + K H 2 K m ) ( ωH 2 K m 0.0476(1 + 0.996 ln[ωH 2 τm ]) 1 + 2.13 ε 2 − 1.13 ε 2 2 ] 2 2 1.14 1 − 0.466ωH 2 τm + 0.0647ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) Ti = ( K c (1 + K H 2 K m ) ωH 2 K m 0.0609 − 1 + 1.97ωH 2 τ m − 0.323ωH 2 2 τ m 2 ). ). ). b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 216 Rule Kc Ti Kc Ti 20 Kc Ti 21 Kc Ti 22 Kc Ti 18 Hwang (1995) – continued. 18 17-Mar-2009 Comment Td 19 Decay ratio = 0.1; τ 0.2 ≤ m ≤ 2.0 ; Tm1 0.471K u K m ωu 0. 6 ≤ ξ m ≤ 4. 2 . [ ] 2 2 0.794 1 − 0.541ωH 2 τm + 0.126ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) Ti = K c (1 + K H 2 K m ) ( ωH 2 K m 0.0078 1 + 8.38ωH 2 τm − 1.97ωH 2 2 τm 2 ), 2 19 20 ξ > 0.649 + 0.58 τ τm − 0.005 m . Tm1 Tm1 [ ] 2 2 0.738 1 − 0.415ωH 2 τm + 0.0575ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) Ti = 21 [ ] 1.15(0.564 )ωH 2 τ m 1 − 0.959 ε 2 + 0.773 ε 2 2 Kc = KH2 − , K m (1 + K H 2 K m ) Ti = 22 [ K c (1 + K H 2 K m ) ( ωH 2 K m 0.0124 1 + 4.05ωH 2 τm − 0.63ωH 2 2 τm 2 K c (1 + K H 2 K m ) ( ωH 2 K m 0.0335(1 + 0.947 ln[ωH 2 τm ]) 1 + 1.9 ε 2 − 1.07 ε 2 2 ] 2 2 1.07 1 − 0.466ωH 2 τm + 0.0667ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) Ti = ( K c (1 + K H 2 K m ) ωH 2 K m 0.0328 − 1 + 2.21ωH 2 τm − 0.338ωH 2 2 τm 2 ). ). ). b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Kc Ti 25 Kc Ti 26 Kc Ti 27 Kc Ti 23 Hwang (1995) – continued. 23 Ti Comment Td 24 Decay ratio = 0.1; τ 0.2 ≤ m ≤ 2.0 ; Tm1 0.471K u K m ωu 0. 6 ≤ ξ m ≤ 4. 2 . [ ] 2 2 0.789 1 − 0.527ωH 2 τm + 0.11ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) Ti = K c (1 + K H 2 K m ) ( ωH 2 K m 0.009 1 + 9.7ωH 2 τm − 2.4ωH 2 2 τm 2 2 24 25 ξ ≤ 0.649 + 0.58 2 [ ] 2 2 0.76 1 − 0.426ωH 2 τm + 0.0551ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) [ [ ( K c (1 + K H 2 K m ) ωH 2 K m 0.0153 1 + 4.37ωH 2 τ m − 0.743ωH 2 2 τ m 2 ] 1.22(0.55)ωH 2 τ m 1 − 0.978 ε 2 + 0.659 ε 2 2 Kc = KH2 − , K m (1 + K H 2 K m ) Ti = 27 ). τ τ τm τ − 0.005 m , ξ > 0.613 + 0.613 m + 0.117 m . Tm1 T T m 1 m 1 Tm1 Ti = 26 217 K c (1 + K H 2 K m ) ( ωH 2 K m 0.0421(1 + 0.969 ln[ωH 2 τm ]) 1 + 2.02 ε 2 − 1.11 ε 2 2 ] 2 2 1.111 − 0.467ωH 2 τm + 0.0657ωH 2 τm , Kc = KH2 − K m (1 + K H 2 K m ) Ti = ( K c (1 + K H 2 K m ) ωH 2 K m 0.0477 − 1 + 2.07ωH 2 τm − 0.333ωH 2 2 τm 2 ). ). ). b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 218 Rule Kc Ti Comment Td Minimum performance index: other tuning Minimum Wilton (1999). Model: Method 1 ∫ [e (t) + x K (y(t) − y ) ] dt ; T > T . ∞ 2 2 2 0 x1 Km Tm1 + Tm 2 2 ∞ m 2 m1 m2 Tm 2 1 + Tm 2 Tm1 Coefficient values (obtained from graphs): Tm 2 Tm1 = 0.25 Tm 2 Tm1 = 0.5 Tm 2 Tm1 = 0.75 x1 x2 x1 x2 x1 x2 1250.1 0.0001 22.946 0.04 20.396 0.04 22.946 0.04 4.8990 0.2 4.8990 0.2 4.8990 0.2 2.1213 1 2.1213 1 2.1213 1 180.01 0.0001 230.01 0.0001 300.01 0.0001 8.1584 0.04 9.1782 0.04 10.708 0.04 2.9394 0.2 3.1843 0.2 3.4293 0.2 1.2728 1 1.3435 1 1.4142 1 90.004 0.0001 100.00 0.0001 112.01 0.0001 4.5891 0.04 4.6911 0.04 5.354 0.04 2.0331 0.2 2.0821 0.2 2.1311 0.2 0.8768 1 1.0324 1 1.0748 1 T T 28 i d K τ m Tm1 = 0.1 τ m Tm1 = 0.5 τ m Tm1 = 1 Keane et al. c (2000). Model: Method 1 Minimum ITAE servo plus ITAE regulator, with minimum M max and with good noise attenuation; Tm1 = Tm 2 . Huang and Jeng (2003). 28 29 Kc = 29 2ξ m Tm1 Kc (ln K u )[Tu + Tm1 (1 + ln K u )] , T = d Tu K m ξ m ≤ 1.1 Model: Method 17 Td Tu Tm1 , Tu + Tm1 (1 + ln K u ) Ti = (ln K u )[Tu + Tm1 (1 + ln K u )] . (ln K u )(1 + ln K u ) + [ln K u + ln(1 + eT − K )][ln(K u + 1.33419τm ) − 1] Kc = 1.2858Tm1ξ m τm T K m τm m1 m u 0.0544 , Td = [(− 0.0349ξm + 1.0064)Tm1 + (0.4196ξm − 0.1100)τm ]2 . 2Tm1ξm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Huang and Jeng (2003) – continued. 30 Kc Ti Td Comment Ti Td ξ m > 1.1 Servo or regulator tuning – minimum IAE. x1 K m Direct synthesis: time domain criteria x 2 Tu x 3 Tu Wills (1962a). Representative coefficient values (deduced from graphs): Model: Method 1 x x2 x3 x1 x2 x3 x1 x2 x3 1 Servo response; 2.5 2.9 0.035 1.6 1.4 0.143 0.2 0.48 0.143 ξ =1. 2.75 2.9 0.073 16 1.4 0.28 8 0.48 0.36 Tm1 = Tm 2 ; 3.2 2.9 0.143 10 1.4 0.72 0.09 0.28 0.073 τ m = 0.1Tm1 . 20 2.9 0.285 0.42 0.70 0.073 0.1 0.28 0.143 12 2.9 0.57 0.4 0.70 0.143 0.11 0.28 0.28 1.4 1.4 0.035 10 0.70 0.36 1.5 1.4 0.073 2 0.70 0.70 Tm1 + Tm 2 + 31 Step disturbance. Van der Grinten Td Kc 0.5τm (1963). Stochastic Model: Method 1 32 disturbance. Ti Td Kc Pemberton (1972b). 30 2(Tm1 + Tm 2 ) 3K m τ m 0.5890 τm Kc = K m τm Tm 2 0.0030 Tm1 + Tm 2 Tm1Tm 2 Tm1 + Tm 2 Model: Method 1 2 0.0052 Tm 2 + 0.8980Tm 2 + 0.4877 τm + Tm1 , τ m 2 Ti = 0.0052 Tm 2 + 0.8980Tm 2 + 0.4877 τm + Tm1 , τm 2 0.0052 Tm 2 + 0.8980Tm 2 + 0.4877 τm τm Td = Tm1 . 2 T 0.0052 m 2 + 0.8980T + 0.4877 τ + T m2 m m1 τm 31 Kc = (T + T )τ + 2Tm1Tm 2 . 1 T +T 0.5 + m1 m 2 , Td = m1 m 2 m τm + 2(Tm1 + Tm 2 ) Km τm 32 Kc = T + Tm 2 − ωd τ m τ e − ωd τm ; Ti = − ωmτ x1 , x 1 , x1 = 1 − 0.5e− ωd τ m + m1 e Km τm e d m − ωd τ m 1 − 0.5e − ωd τ m + Tm1Tm 2e (T + T ) (Tm1 + Tm 2 )τm m1 m 2 . Td = x1 219 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 220 Rule Kc Ti Td Pemberton (1972a), (1972b). Model: Method 1 (Tm1 + Tm 2 ) K mτm Tm1 + Tm 2 Tm1Tm 2 Tm1 + Tm 2 Pemberton (1972b). Model: Method 1 2(Tm1 + Tm 2 ) 3K m τ m Tm1 + Tm 2 λTm1 + Tm 2 K m (1 + λτ m ) Tm1 + Tm 2 Chiu et al. (1973). Model: Method 1 Comment Tm1 + Tm 2 4 Tm1Tm 2 Tm1 + Tm 2 0.1 ≤ Tm1 ≤ 1.0 , Tm 2 0.2 ≤ τm ≤ 1.0 . Tm 2 0.1 ≤ Tm1 ≤ 1.0 . Tm 2 0.2 ≤ τm ≤ 1.0 . Tm 2 λ variable; suggested values are 0.2, 0.4, 0.6 and 1.0. Appropriate for “small τ m ”. Mollenkamp et 2ξ m Tm1 2ξ m Tm1 Tm1 al. (1973). K m (TCL + τ m ) 2ξ m Model: Method 1 Smith et al. Tm1 > Tm 2 λTm1 (1975). T T m 1 m 2 ( ) K m 1 + λτ m Model: Method 1 λ = pole of specified FOLPD closed loop response. Suyama (1992). Tm1 + Tm 2 Tm1Tm 2 Model: Method 1 OS = 10% T + T m1 m2 2K m τ m Tm1 + Tm 2 Rotach (1995). Model: Method 9 Wang and Clements (1995). 33 Kc = 1.06τm Damping factor for oscillations to a disturbance input = 0.75. 2ξ m Tm1 2λξ m Tm1 Tm1 K m (1 + λτ m ) 2ξ m Gorez and Klàn (2000). Model: Method 1 Vítečková (1999), Vítečková et al. (2000a). Model: Method 1 1.84τm 8.47 K m 33 Kc Model: Method 22; underdamped response. Non-dominant 2ξ m Tm1 Tm1 time delay. 2ξ m x 1 (Tm1 + Tm 2 ) Kmτm Tm1 + Tm 2 Tm1Tm 2 Tm1 + Tm 2 x1 OS (%) x1 OS (%) x1 OS (%) 0.368 0.514 0.581 0.641 0 5 10 15 0.696 0.748 0.801 0.853 20 25 30 35 0.906 0.957 1.008 40 45 50 2ξ m Tm1 . K m (2ξ mTm1 + τm ) Closed loop overshoot (OS) defined. Overdamped process; Tm1 > Tm 2 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Vítečková (1999), Vítečková et al. (2000a) – continued. Skogestad (2004b). Model: Method 1 35 Kc = Kc = Td x 1 ξ m Tm1 K mτm 2ξ m Tm1 Tm1 2ξ m Comment OS (%) x1 OS (%) x1 OS (%) 1.392 1.496 1.602 1.706 20 25 30 35 1.812 1.914 2.016 40 45 50 x1 2ξ m Tm1 Recommended TCL = τ m . Tm1 2ξ m 0.74Tm1 K mτm 2Tm1 0.5Tm1 Tm1 = Tm 2 Kc 2ξ des m Tm1 0.5Tm1 ξdes m 1.0128ξ m Tm1 K mτm 1.9747 K m τ m 0.5064Tm1 K mτm Model: Method 1 Model: Method 10 Ti Td 34 35 Kc 2 2ξdes m Tm1 K m τm Underdamped process; 0. 5 < ξ m ≤ 1 . A m = 3.14 , φ m = 61.4 0 , M max = 1.59 Chen and Seborg (2002). 34 Ti 0.736 0 1.028 5 1.162 10 1.282 15 2ξ m Tm1 K m (TCL + τ m ) Vítečková et al. (2000b). Model: Method 5 Arvanitis et al. (2000a). Bi et al. (2000). Kc ( 2ξ + 1) . des m 1 [(Tm1 + Tm 2 )τm + Tm1Tm 2 ](3TCL + τm ) − TCL3 − 3TCL 2 τm , Km (TCL + τm )3 [(Tm1 + Tm 2 )τm + Tm1Tm 2 ](3TCL + τm ) − TCL3 − 3TCL 2τm , Tm1Tm 2 + (Tm1 + Tm 2 + τm )τm 2 3T T T + Tm1Tm 2 τm (3TCL + τm ) − (Tm1 + Tm 2 + τm )TCL 3 Td = CL m1 m 2 , [(Tm1 + Tm 2 )τm + Tm1Tm 2 ](3TCL + τm ) − TCL3 − 3TCL 2τm Ti = Tise −sτ m y = . 3 d desired K c (TCLs + 1) Model: Method 1 221 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 222 Rule Kc Chen and Seborg (2002) – continued. Chidambaram (2002), p. 156. Skogestad (2004c), Manum (2005). Model: Method 1 36 Ti Td Ti Td 2ξ m Tm1 0.5Tm1 ξm 4(λ + τm ) Tm 2 Td Kc 2 1.333ξ m Tm1 K m τm Tm1 K m (λ + τm ) 37 Kc 4(λ + τm ) + Tm 2 38 Kc Ti 39 Kc Ti Comment Model: Method 1 0.5Tm1 ξ m τm x1 (Tm1 + Tm 2 ) Tm1Tm 2 ≤2 Trybus (2005). T + T m 1 m 2 Tm K m τm Tm1 + Tm 2 Model: Method 1 x1 OS x1 OS x1 OS x1 OS x1 OS x1 OS Tm1 > Tm 2 0.34 0% 0.54 5% 0.64 10% 0.74 15% 0.82 20% 0.90 25% 36 Kc = Ti = ( ) 1 2ξ mTm1τm + Tm12 (3TCL + τm ) − TCL 3 − 3TCL 2 τm , Km (TCL + τm )3 (2ξ T τ + T )(3T + τ ) − T m m1 m 2 m1 Tm12 + CL m CL 3 − 3TCL 2 τm Tise −sτ m y , = , 3 d desired K c (TCLs + 1) (2ξmTm1 + τm )τm ( ) ( ) ( )( ) T T + 4(λ + τm ) Tm 2 37 K c = m1 m 2 , Td = Tm 2 1 + . 2 4K m (λ + τm ) 4(λ + τm ) 2Tm1ξ m Tm1 [ξ m (Tm1 + 4(λ + τm )ξ m ) + 2Tm1 (1 − ξ m )] 38 K c = max , , ξm < 1 ; 2 ( ) K λ + τ m 4K m (λ + τ m ) m Ti = max{2Tm1ξ m , ξ m Tm1 + 4(λ + τ m )(2 − ξ m )} . 2 3T T 2 + Tm12 τm 3TCL + τm − 2Tm1ξm + τm TCL 3 Td = CL m1 . 2ξm Tm1τm + Tm12 3TCL + τm − TCL 3 − 3TCL 2 τm Tm1 Tm1 + 4(λ + τm ) ξ m + ξm 2 − 1 2T ξ 39 m1 m K c = max , , ξm > 1 ; 4K m (λ + τm )2 K m (λ + τm ) Ti = max{2Tm1ξm , Tm1 + 4(λ + τm )} . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc x1 Tm1 + 0.5 Trybus (2005) – Tm1 + 0.5τm K m τm continued. x1 OS x1 OS x1 OS Boudreau and McMillan (2006). Td Comment 0.5Tm1τm Tm1 + 0.5τm τm >2 Tm Ti x1 OS x1 OS x1 223 OS 0.34 0% 0.54 5% 0.64 10% 0.74 15% 0.82 20% 0.90 25% Tm1 40 Model: Method 1 Tm 2 Ti K m (λ + τm ) λ = τm (maximum load rejection capability); otherwise, λ is a multiple of Tm . Vítečková (2006). Model: Method 5 ≥ Sample results: A m = 2.0 , 2Tm1 0.5Tm1 1.0472Tm1 K mτm 2Tm1 0.5Tm1 Ti Td τ m / Tm < 2ξ m Tm 2 Tm1 > Tm 2 41 Kc ω p Tm1 42 Ti φ m = 45 o . A m = 3. 0 , φ m = 60 o . AmKm 2 Ti = Tm1 ; for an overdamped response, Ti > Kc = Tm1 = Tm 2 0.5Tm1 1.5708Tm1 K mτm Ho et al. (1994). Model: Method 1 Ho et al. (1995a). Model: Method 1 41 2Tm1 Direct synthesis: frequency domain criteria πTm1 2Tm1 0.5Tm1 Tm1 = Tm 2 AmK mτm Hang et al. (1993a). Model: Method 10; τm > 0.3 Tm1 (Yang and Clarke, 1996). 40 0.736Tm1 K m τm 4Tm1 λ + τm 1 T 1 + m1 λ + τ m 2 2 πξm 2ωp Tm1 πξm 2 + π − 2ωp τm , Ti = Tm12 + π − 2ωp τm , Tm1ωp πA m ωpTm1 π Td = 42 . 1 Ti = 2ωp − 4ωp 2 τ m π 1 + Tm1 . πTm1 πξ 2 m + πTm1 − 2Tm1ωp τm ωp . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 224 Rule Kc Ho et al. (1995a) – continued. 0.7854Tm1 K mτm Tm1 Tm 2 0.5236Tm1 K mτm Tm1 Tm 2 Ti Comment Td Sample results: A m = 2.0 , φ m = 45 o . A m = 3. 0 , φ m = 60 o . Ho et al. (1997). 43 Kc Ti Td Model: Method 1 Leva et al. (1994). 44 Kc Ti 0.25Ti Model: Method 21 43 Kc = τ 2 2 πξm + π − 2 m , Ti = Tm1 (πξm + π − 2τm ) , πA m K m Tm1 π Td = 2 τ τ πTm1 ; m ≤ 1 , 2ξ m ≥ m ; Tm 2(πξ mTm1 + πTm1 − 2τm ) Tm π + π 2 + 8πτ m ξ m A m 2 − 1 − 2 πA m (A m − 1) Tm1 φm < 4A m ( 44 Kc = ωcn Ti Km (1 + ω T ) + 4ξ ω T (1 − T T ω ) + T ω 2 cn 2 2 m1 2 i d cn 2 m 2 m1 cn 2 2 2 i 2 , cn ωcn = Ti = ) 1 2ξ m τ m Tm1 (0.5π − φ m ) 4.07 − φ m + tan −1 ; τm (0.5π − φ m )2 Tm1 2 − τ m 2 2ξ ω T 2 tan 0.5ωcn τm + φm − 0.5π − tan −1 m2 cn2 m1 , ω T − 1 ωcn cn m1 3τ m ξ m + ξ m 2 − 1 ); φ m = 70 0 (at least). ξ m ≤ 1 or ξ m > 1 (with 0.5π − φ m > Tm1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Leva et al. (1994) – continued. Wang et al. (1999). Wang and Shao (1999a). Model: Method 16 Td Kc Ti Td ξ m Tm1 K mτm 2ξ m Tm1 0.5Tm1 ξm ξ m Tm1 K mτm 2ξ m Tm1 x1 0. 5 Model: Method 16 46 Tm1 ξm Some coefficient values: Comment x1 x1 Comment 1.571 A m = 2 , φ m = 45 0 0.785 A m = 4 , φ m = 67.50 1.047 A m = 3 , φ m = 60 0 0.628 A m = 5 , φ m = 72 0 2 2 ω T 45 K c = cn m1 K m Td Comment Ti 45 ωcn + 1 2ξ m ξ m 2 − 1 − 1 2 Tm1 2 ωcn + z 2 ωcn = 225 , 2ξ m τ m Tm1 (0.5π − φ m ) 1 4.07 − φ m + tan −1 , τm (0.5π − φ m )2 Tm1 2 − τ m 2 ωcn z= , ωcn Tm1 tan φ m − 0.5π + ωcn τ m + tan −1 ξ m + ξ m 2 − 1 2 Tm1z + ξm + ξ m − 1 Tm1 , T = Ti = , ξm > 1 d z ξm + ξ m 2 − 1 Tm1z + ξ m + ξm 2 − 1 3τ m 3τ m 2 ξ m − ξ m − 1 < 0.5π − φ m ≤ ξm + ξm 2 − 1 . with Tm1 Tm1 46 ξ m > 0.7071 or 0.05 < or 0.05 < 0.7071τ m 2 Tm1 2ξ m − 1 < 0.15 , ξm τm ξ τ < 0.15 , m m > 1, ξ m < 1 . Tm1 Tm1 0.7071τ m Tm1 2ξ m 2 − 1 > 1, ξ m ≥ 1 b720_Chapter-03 Rule Kc Ti Td Wang (2000). Model: Method 1. ξ m Tm1 47 K mτm 2ξ m Tm1 0.5Tm1 ξ m Chen et al. (1999b). Model: Method 1 Kc 49 Kc 2ξ m Tm1 0.5Tm1 ξ m x 1ξ m Tm1 τm K m 2ξ m τ m τm 2ξ m x1 Am 1.00 3.14 1.22 2.58 1.34 2.34 1.40 2.24 Leva et al. (2003). Model: Method 1 0.05 < Kc = 50 51 φm φm Ms x1 Am 61.4 1.00 1.44 2.18 48.7 0 1.30 55.0 0 1.10 1.52 2.07 46.50 1.40 51.6 0 1.20 1.60 1.96 0 1.50 0 1.26 Kc 4(Tm 2 + τ m ) Kc 5Tm 2 + 4 τ m Td 0.7071τm Model: Method 16 0 50.0 16(Tm 2 + τ m ) Tm1 (K mξ m − Tm1 ) or 0.05 < Comment 48 Wang and Shao (2000b). 48 FA Handbook of PI and PID Controller Tuning Rules 226 47 17-Mar-2009 < 0.15 or 0.7071τm Tm1 (K m ξ m −Tm1 ) Ms 44.1 > 1 , for ξm > 1 ; ξm τm ξ τ < 0.15 or m m > 1 , for 0.7071 < ξ m < 1 . Tm1 Tm1 e − τ m x1 0.368 2Tm1ξm , min , x1 = 2Tm1 (K m ξm − Tm1 ) , ξm > 1 or τm Km x1 x1 = 49 Kc = 50 Kc = 51 Kc = 1.508 2ξ m Tm1 , suggested M max = 1.6. 1.451 − K m τm M max Tm1Tm 2 8K m (Tm 2 + τm )2 , Tm1 ≥ 4(Tm 2 + τ m ) . Tm1 (5Tm 2 + 4 τ m ) 8K m (Tm 2 + τ m )2 , Td = 4Tm 2 (Tm 2 + τ m ) , Tm1 ≥ 8(Tm 2 + τ m ) . 5Tm 2 + 4τ m Tm1 , ξm < 1 . ξm b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Åström and Hägglund (2006), pp. 245–246. Lee et al. (2006). Model: Method 1 52 Kc Ti K m (λ + τm ) Ti Td Comment Ti Td Model: Method 1; M s = M t = 1.4 . 53 227 Td Ti Desired closed loop response = e − sτ m (λs + 1)2 ; recommended λ = 0.5τm . Robust 54 Brambilla et al. (1990). Model: Method 1 Kc Kc = Td Representative coefficient values (deduced from graph): τm τm τm λ λ λ Tm1 + Tm 2 Tm1 + Tm 2 Tm1 + Tm 2 0.1 0.2 0.5 52 Ti 0.50 0.47 0.39 1.0 2.0 5.0 0.29 0.22 0.16 10.0 0.14 1 Tm1 T T T + 0.18 m 2 + 0.02 m1 m 2 , 0.19 + 0.37 Km τm τm τm Tm1 T T T + 0.18 m 2 + 0.02 m1 m 2 τm τm τm , Ti = 0.48 Tm1 Tm 2 T T + 0.03 2 − 0.0007 2 + 0.0012 m1 3m 2 τm τm τm τm 0.19 + 0.37 T T T T 0.29τm + 0.16Tm1 + 0.20Tm 2 + 0.28 m1 m 2 m1 m 2 τm Tm1 + Tm 2 Td = . T T T T 0.19 + 0.37 m1 + 0.18 m 2 + 0.02 m1 m 2 τm τm τm 2 2 53 Ti = 2ξm Tm1 + 54 Kc = 2 τm τm , Td = + 2(λ + τm ) 2(λ + τm ) Tm1 − Tm1 + Tm 2 + 0.5τm , Ti = Tm1 + Tm 2 + 0.5τm , K m τm (2λ + 1) Td = 3 τm 6(λ + τm ) . Ti τm Tm1Tm 2 + 0.5(Tm1 + Tm 2 )τm , 0.1 ≤ ≤ 10 . Tm1 + Tm 2 Tm1 + Tm 2 + 0.5τm b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 228 Rule Kc Lee et al. (1998). Model: Method 1 desired closed loop response = Ti K m (2λ + τm ) e − τ ms (λs + 1)2 Td 55 Ti Td 56 Ti Td 57 Ti Td 58 Ti Td Comment λ = maximum [0.25τm ,0.2Tm1 ] (Panda et al., 2004). Ti K m (λ + τm ) . Ti Huang et al. 2ξ m Tm1 Tm1 (1998), Rivera 2ξ m Tm1 K m (τ m + λ) 2ξ m and Jun (2000). Model: Method 1 λ specified graphically for different OS and TR values (Huang et al., 1998). 59 Marchetti and Model: T Td Kc i Method 1 Scali (2000). Smith (2003). Tm1 Tm1 Tm 2 λ ∈ [ τ m , Tm 2 ] , Model: Method 1 K (λ + τ ) Tm1 ≥ Tm 2 . m m Kristiansson (2003). 60 Kc 2ξ m Tm1 Tm1 2ξm Model: Method 1 2λ2 − τm 2 T 2 τm 3 , Td = Ti − 2ξ mTm1 + m1 − . 2(2λ + τm ) Ti 6Ti (2λ + τm ) 55 Ti = 2ξ m Tm1 − 56 Ti = Tm1 + Tm 2 − 2λ2 − τm 2 τm 3 T T , Td = Ti − Tm1 − Tm 2 + m1 m 2 − . 2(2λ + τm ) Ti 6Ti (2λ + τm ) τm 3 2 − T m 1 τm 2 6(λ + τm ) 57 Ti = 2ξm Tm1 + , Td = Ti − 2ξ m Tm1 . 2(λ + τm ) Ti 3 Tm1Tm 2 − τm τ ( λ + τ ) 6 58 m m Ti = Tm1 + Tm 2 + , Td = Ti − (Tm1 + Tm 2 ) . 2(λ + τm ) Ti 2ξ T + 0.5τm T (T + ξ m τm ) 59 K c = m m1 , Ti = 2ξ m Tm1 + 0.5τm , Td = m1 m1 . K m (2λ + τm ) 2ξ m Tm1 + 0.5τm 2 60 Kc = 2ξmTm1 1 1 − . Recommended values of M max : 1.4,1.7,2.0. K m τm M max b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Shamsuzzoha et al. (2006c). Model: Method 1 Shamsuzzoha and Lee (2007a). Model: Method 1 61 Kc = Ti Td Comment 61 Kc Ti Td λ , ξ not specified. 62 Kc Ti Td λ = x1Tm Ti , K m (4λξ + τm − x1 ) Ti = Tm1 + Tm 2 + x1 − Td = Kc 229 4λ2ξ 2 + 2λ2 − x 2 + x1τm − 0.5τm 2 , 4λξ + τm − x1 x 2 + x1 (Tm1 + Tm 2 ) + Tm1Tm 2 − 3 2 0.167 τm − 0.5x1τm + x 2 τm + 4λ3ξ 4λξ + τm − x1 Ti − 2 τ 4λ2ξ 2 + 2λ2 + x1τm − 0.5τm 2 − x 2 , 4λξ + τm − x1 τ 2 m 2 2 − m 2λξ − Tm1 2 λ 2 λ − 2λξ + 1 e Tm 2 + Tm 2 2 − Tm12 − + − 1 e T Tm1 m 2 T 2 T T 2 T m1 m2 m1 m2 , x1 = Tm 2 − Tm1 2 τm 2λξ − Tm1 + x1Tm1 . 1 e 1 x 2 = Tm 2 − + − Tm12 Tm1 2 62 Kc = Td = λ2 Ti 6λ2 − x 2 + x1τm − 0.5τm 2 , , Ti = Tm1 + Tm 2 + x1 − K m (4λ + τm − x1 ) 4 λ + τ m − x1 x 2 + x1 (Tm1 + Tm 2 ) + Tm1Tm 2 − 3 2 0.167 τm − 0.5x1τm + x 2 τm + 4λ3 4λ + τm − x1 Ti 2 6λ2 + x1τm − 0.5τm − x 2 , 4λ + τ m − x 1 x1 = τm τm 4 4 λ − Tm1 λ − Tm 2 2 e Tm12 1 − 1 T 1 e − − 1 − − m2 Tm1 Tm 2 Tm 2 − Tm1 , τm 4 λ − Tm 2 + x1Tm 2 . e x 2 = Tm 2 2 1 − 1 − Tm 2 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 230 Rule Shamsuzzoha and Lee (2007a) – continued. Wills (1962a). Model: Method 1 Quarter decay ratio tuning. Tm1 = Tm 2 ; τ m = 0.1Tm1 . Bohl and McAvoy (1976b). 63 17-Mar-2009 Kc Ti Comment Td Coefficients of x1 (deduced from graph): τm Tm Ms 1.8 0.27 0.29 0.31 0.33 1.9 0.18 0.20 0.22 0.24 0.25 0.27 1.8 1.9 0.2 0.34 0.28 0.3 0.35 0.29 0.4 0.38 0.32 0.5 0.41 0.34 0.6 0.46 0.38 0.7 0.51 0.41 0.8 0.55 0.43 Ultimate cycle x 2 Tu x 3Tu x1 K m x1 x2 10 8 14 33 18 2.9 1.4 2.9 2.9 1.4 63 Kc τm Tm Ms 2.0 0.12 0.15 0.17 0.19 0.21 0.22 0.23 2.0 0.25 0.26 0.28 0.30 0.33 0.36 0.38 0.9 1.0 1.25 1.5 2.0 2.5 3.0 Coefficient values (deduced from graph): x3 x1 x2 x3 x1 x2 0.036 0.036 0.068 0.143 0.068 28 6 2 0.6 30 Ti 1.4 0.67 0.48 0.28 0.67 0.143 0.068 0.068 0.068 0.143 Td 18 0.9 18 18 13 0.48 0.28 0.67 0.48 0.28 x3 0.143 0.143 0.24 0.24 0.28 Model: Method 1 τ 2.0302 + 0.9553 ln m τ Tu 2.2689 τm 1.3135+ 0.1809 ln (K u K m )+ 0.5219 ln m Kc = (K u K m ) Tu , K m Tu τ − 0.0517 − 0.0643 ln m τ τm Tu 0.5914 −0.0832 ln (K u K m )+ 0.0687 ln m Ti = 0.2693Tu (K u K m ) Tu , Tu τ − 4.2613−1.5855 ln m τ τm Tu −1.1436 −0.2658 ln (K u K m )−1.1100 ln m Td = 0.0068Tu (K u K m ) Tu . Tu b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Td Comment x 1K u x 2 Tu x 3 Tu Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm x 1 , x 2 deduced Tu Tu Tu from graphs. 0.47 0.23 0.22 0.15 0.29 0.17 0.08 0.35 0.28 0.19 0.03 0.50 ξ m = 0. 5 . 0.45 0.21 0.19 0.18 0.225 0.17 0.01 0.41 0.35 0.18 0.10 0.30 0.27 0.19 0.02 0.47 x 1K u x 2 Tu x 3 Tu Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm x 1 , x 2 deduced Tu Tu Tu from graphs. 0.45 0.30 0.14 0.20 0.34 0.20 0.07 0.35 0.28 0.18 0.03 0.50 ξm = 0.7 . 0.42 0.25 0.11 0.25 0.31 0.19 0.06 0.40 0.38 0.22 0.09 0.30 0.28 0.18 0.04 0.45 x 1K u x 2 Tu x 3 Tu Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm x 1 , x 2 deduced Tu Tu Tu from graphs. 0.45 0.47 0.12 0.20 0.37 0.24 0.07 0.35 0.28 0.18 0.03 0.50 ξm = 1 . 0.43 0.37 0.10 0.25 0.33 0.21 0.06 0.40 0.40 0.30 0.09 0.30 0.30 0.19 0.04 0.45 x 1K u x 2 Tu x 3 Tu Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm x 1 , x 2 deduced Tu Tu Tu from graphs. 0.46 0.80 0.11 0.20 0.38 0.29 0.07 0.35 0.28 0.18 0.03 0.50 ξ m = 1. 5 . 0.44 0.52 0.09 0.25 0.35 0.22 0.06 0.40 0.42 0.38 0.08 0.30 0.30 0.19 0.04 0.45 4 + x1 1 4 1 64 Landau and Voda 1 ≤ x1 ≤ 2 Kc x1 ω1350 4 + x1 ω1350 (1992). 3K u 0.51Tu 0.13Tu ωu τ m ≤ 0.16 Model: Method 1 1.9 K u 64 Kc = 4 + x1 x1 . 4 2 2 G p ( jω1350 ) 0.64Tu 0.16Tu 0.16 < ωu τ m ≤ 0.2 231 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 232 3.5.3 Ideal controller in series with a first order lag 1 1 G c (s) = K c 1 + + Td s Tf s + 1 Ti s R(s) E(s) − + U(s) 1 Tf s + 1 1 K c 1 + + Td s Ti s SOSPD model Table 22: PID controller tuning rules – SOSPD model G m (s) = G m (s) = Rule Kc Lim et al. (1985). 1 Arvanitis et al. (2000a). Model: Method 1 Panda et al. (2004). Model: Method 1 Huang et al. (2005a). Model: Method 18 Huang and Jeng (2005). 1 Kc = 2 Kc = Tm1 2 s 2 + 2ξ m Tm1s + 1 Ti Td Comment Direct synthesis: time domain criteria Model: Method 1 Tm1 + Tm 2 Tm1Tm 2 Tm1 + Tm1 Kc Kc ξ m Tm1 K mτm 2ξ m Tm1 Tm1 2ξ des m TCL 2 > τm 2 Tm1 , 2 Nξ m N not defined. Direct synthesis: frequency domain criteria ξ m Tm1 Tm1 0.025Tm1 Tf = 2ξ m Tm1 K mτm 2ξ m ξm Tm1 2ξ m Tf = A m ≈ 3 , φ m ≈ 60 0 1.1ξ m Tm1 K mτm 2ξ des m Tm1 K m 2ξ des m TCL 2 + τ m ) , Tf = 0.5Tm1 ξm 2ξ m Tm1 2 TCL 2 Tm1 + Tm 2 . , Tf = K m (2ξTCL 2 + τ m ) 2ξTCL 2 + τ m ( K m e − sτ m or (1 + sTm1 )(1 + sTm 2 ) K m e − sτ m 2ξ des m Tm1 2 Y(s) ( 2 2TCL 2 2 − τ m 2 2ξ des m TCL 2 + τ m ). Model: Method 1; Tf = 0.01τ m . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Hang et al. (1993a). Model: Method 10 Rivera and Jun (2000). Model: Method 1; λ not specified. Lee and Edgar (2002). 2Tm1 + τ m 2(λ + τ m )K m Zhang et al. (2002b). Model: Method 1; 5% overshoot: λ = 0.5τ m . 3 Ti Td Robust Tm1 + 0.5τ m Tm1 τ m 2Tm1 + τ m Model has a repeated pole (Tm1 ) . 2ξ m Tm1 λ Comment λ > 0.25τ m . Tf = λτ m . 2(λ + τ m ) 2ξ m Tm1 0.5Tm1 ξ m 2ξ m Tm1 Tm1 2ξ m Tf = Kc Ti Td Model: Method 1 Tm1 + Tm 2 K m (τ m + 2λ ) Tm1 + Tm 2 Tm1Tm 2 Tm1 + Tm 2 2ξ m Tm1 K m (2 τ m + λ ) 3 233 Tf = τ m Tf = τmλ 2τ m + λ λ2 ; 2λ + τ m H ∞ method. Kc Tm1Tm 2 Tm1 + Tm 2 Tm1 + Tm 2 Ti K m (2λ + τm ) 5 Td 4 Ti Tf = λτ m λ + 2τ m Maclaurin method. 2 Kc = 1 0.5τm − λ2 + (2ξ m Tm1 + K m (2λ + τm ) 2λ + τ m Tm 2 − τm3 0.5τ m 2 − λ2 0.5τ m 2 − λ2 + 2ξ m Tm1 + ), 6(2λ + τ m ) 2λ + τ m 2λ + τ m 2 Ti = 2ξm Tm1 + 0.5τm − λ2 + 2λ + τ m τm3 0.5τ m 2 − λ2 0.5τ m 2 − λ2 + 2ξ m Tm1 + , 6(2λ + τ m ) 2λ + τ m 2λ + τ m Tm 2 − 4 5 τm3 0.5τ m 2 − λ2 0.5τ m 2 − λ2 + 2ξ m Tm1 + . 6(2λ + τ m ) 2λ + τ m 2λ + τ m Td = Tm 2 − Kc = Tm1Tm 2 , H 2 method. K m (Tm1 + Tm 2 )(λ + 2τm ) Ti = Tm1 + Tm 2 − 2 2 2λ − τ m , Td = 2(2λ + τm ) τm 3 2λ2 − τm 2 12λ + 6τm − , Tf = 0 . Ti 2(2λ + τm ) Tm1Tm 2 − b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 234 Rule Kc Ti Td Comment 2ξ m Tm1 Tm1 2ξ m Model: Method 1 Kristiansson (2003). 6 7 Kc Panda et al. (2004). Model: Method 1 8 Kc Ti Td 9 Kc Ti Td 6 Kc = 7 Kc = 8 Kc = Kc T 2ξ m Tm1 1 T 1 , x1 = 1 + f 1 − 1 − , Tf = m1 . x1 − x12 − 1 + 2 x2 K m τm τm M max M max 2 2ξ m Tm1 T T T 2 1 + 0.019 m1 − 0.346 + 0.038 m1 + 0.00036 m12 ; K m τm τm τm τm for suggested M max = 1.7 , x 2 = 10 . 2 Td = 2 2ξ m Tm1 + 0.25Tm1 + 0.5ξ m τm 4ξ T + 0.5Tm1 + ξ m τm , Ti = m m1 , K m (λ + 2ξ m Tm1 ) 2ξ m (Tm1 + 2ξm τm )Tm1ξm 2 4ξm Tm1 + ξ m τm + 0.5Tm1 , Tf = λ (Tm1 + 2ξ m τ m ) , 4ξ m λ + 4ξ m τ m + 2Tm1 T λ = maximum 0.25 m1 + τm ,0.4ξ mTm1 . 2ξ m 1 9 Kc = Km 0.558Tm 2Tm1 Tm 2 + 0.172Tm1e − (6.9 Tm 2 / Tm1 ) + + 0.5τm Tm1 + 1.208Tm 2 Tm1 , T λ + 0.828 + 0.812 m 2 + 0.172Tm1e − (6.9Tm 2 / Tm1 ) Tm1 0.828 + 0.812 Ti = 0.828 + 0.812 0.558Tm 2Tm1 Tm 2 + 0.172Tm1e− (6.9Tm 2 / Tm1 ) + + 0.5τm , Tm1 + 1.208Tm 2 Tm1 1.116Tm 2Tm1 Tm 2 + τm + 0.172Tm1e − (6.9Tm 2 / Tm1 ) 0.828 + 0.812 T 1 . 208 T + T m1 m2 , m1 Td = Tm 2 1.116Tm 2Tm1 − (6.9 Tm 2 / Tm1 ) + 0.344Tm1e + + τm 1.656 + 1.624 Tm1 Tm1 + 1.208Tm 2 λ[1.116Tm1Tm 2 + τ m (Tm1 + 1.208Tm 2 )] , Tf = 2(λ + τ m )(Tm1 + 1.208Tm 2 ) + 2.232Tm 2 Tm1 0.279Tm 2Tm1 T λ = maximum + 0.25τm ,0.1656 + 0.1624 m 2 + 0.0344Tm1e− (6.9Tm 2 / Tm1 ) . T + 1 . 208 T T m2 m1 m1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Naşcu et al. (2006). Model: Method 19 Kc Ti Td 2ξ m Tm1 K m (2TCL 2 + τm ) 2ξ m Tm1 0.5Tm1 ξm Desired servo closed loop transfer function = 235 Comment 2 Tf = TCL 2 2TCL 2 + τm e − sτ m (TCL 2s + 1)2 ; suggested TCL 2 values: 0.4ξ m Tm1 , ξm Tm1 . Ultimate cycle Huang et al. (2005a). 10 K c = 0.080 10 Kc Ti Td Tf = 0.05Td A m ≈ 3 , φ m ≈ 60 0 ; Model: Method 18. ^ ^ ^ ^ K u Tu 6.283τm 6.283τm = sin , T 0 . 159 K T sin i u u , ^ ^ K m τm T T u u 1 + K^ cos 6.283τm u ^ ^ T Tu u . Td = 0.159 ^ Ku 6.283τm sin ^ T u b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 236 T s 1 d + 3.5.4 Controller with filtered derivative G c (s) = K c 1 + sTd Ti s 1+ N E(s) R(s) − + U(s) T s 1 d + K c 1 + Td Ti s s 1+ N SOSPD model Table 23: PID controller tuning rules – SOSPD model G m (s) = G m (s) = Rule Kc Hang et al. (1994). Model: Method 1 AmKm K m e −sτ m or (1 + sTm1 )(1 + sTm 2 ) K m e − sτ m Tm1 2 s 2 + 2ξ m Tm1s + 1 Ti ω p Tm1 Y(s) Td Comment Direct synthesis: frequency domain criteria 1 N = 20; Tm 2 Ti Tm1 > Tm 2 . 0.7854Tm1 K mτm Sample result: Tm1 Tm 2 A m = 2.0 , φ m = 45 o . Robust Hang et al. Tm1 (1994). K m (TCL + τ m ) Model: Method 1 2 Zhong and Li Kc (2002). 1 1 Ti = 2ωp − 2 2 4ωp τm π 1 + Tm1 Tm1 Tm 2 N = 20; Tm1 > Tm 2 . Ti Td Model: Method 1 . 2 Kc = 2 x1 + τm 2ξ m Tm1 (2 x1 + τm ) − x1 , Ti = , K m ( 2 x1 + τ m ) 2 2ξm Tm1 (2 x1 + τm ) − x12 2 2ξ T x 2 ( 2 x 1 + τ m ) − x 1 4 2x + τ Td = Ti Tm1 − m m1 1 , N = 1 2 m Td . (2x1 + τ m ) 2 x1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Wang (2003), pp. 50–54. Lavanya et al. (2006a). Model: Method 13; N = 10. 3 Kc Ti Td Comment Kc Ti Td Model: Method 1 2ξm Tm1 K m (λ + τm ) 2ξ m Tm1 0.5ξm Tm1 3 Ti K m (2λ + τm ) 4 λ = max [0.25τm ,0.2Tm1 ] 2 2 ξ T ( 2 x + τ ) − x1 2ξ m Tm1 (2 x1 + τm ) − x1 , Ti = m m1 1 m , 2 x1 + τ m K m ( 2 x1 + τ m ) 2 Td = 4 τm < 1; Tm1 λ not specified. τm >1 Tm1 Td Ti 2 Kc = 0.1 ≤ 237 Ti = 2ξm Tm1 − 2 2 Tm12 (2 x1 + τm ) − 2 2ξ m Tm1 (2x1 + τm ) − x1 2λ − τ m , Td = Ti − 2ξ mTm1 + 2(2λ + τm ) Tm12 − τm 3 6(2λ + τm ) . Ti 2x + τ x12 , N = 1 2 m Td . 2x1 + τm x1 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 238 1 1 + Td s 3.5.5 Classical controller G c (s) = K c 1 + Ti s 1 + Td s N E(s) R(s) + − 1 1 + Td s U(s) K c 1 + Td Ti s s 1 + N SOSPD model Table 24: PID controller tuning rules – SOSPD model G m (s) = G m (s ) = Rule Kc Y(s) K m e − sτ m or (1 + sTm1 )(1 + sTm 2 ) K m e −sτ m 2 2 Tm1 s + 2ξ m Tm1s + 1 Ti Td Comment Minimum performance index: regulator tuning 1 0.954 0.780 0.982 Approximately Tm1 > Tm 2 ; x2 x2 minimum IAE – 0.817 x1 0 . 602 x 0 . 903 x 1 1 x x x N = 8. K 1 Huang and Chao 1 m 2 (1982). Also given by Chao et al. (1989); Model: Method 1. Tm 2 0.80Tm1 = 0.25 Minimum IAE – ( ) ( ) 1 . 5 T + τ 0 . 60 T + τ m2 m m2 m K mτm Tm 2 + τ m Shinskey (1988), 0.77Tm1 Tm 2 p. 149. 1.2(Tm 2 + τ m ) 0.70(Tm 2 + τ m ) T + τ = 0.5 Model: K mτm m2 m Method 1; Tm 2 0.83Tm1 N not specified. 0.75(Tm 2 + τ m ) 0.60(Tm 2 + τ m ) T + τ = 0.75 K mτm m2 m 1 τ 1.398 x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1 m + 1.039 − . T 1 . 349 ( T T ) + m1 m2 m2 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti 239 Comment Td 2 Minimum IAE – N=8 Ti Td Kc Chao et al. 0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989). 3 Minimum IAE – Ti Td Tm 2 τm ≤ 3 Kc Shinskey (1994), 2.5Tm1 K m τm τ m + 0.2Tm 2 τ m + 0.2Tm 2 Tm 2 τm > 3 p. 159. Model: Method 1; N not specified. Minimum IAE – τm Tm1 = 0.2 , Shinskey (1996), 0.59 K u 0.36Tu 0.26Tu Tm 2 Tm1 = 0.2 . p. 121. Model: Method 1; N not specified. 0.85Tm1 K m τm 1.98τ m 0.86τ m Tm 2 Tm1 = 0.1 Minimum IAE – 0.87Tm1 K m τm 2.30τ m 1.65τ m Tm 2 Tm1 = 0.2 Shinskey (1996), 1.00Tm1 K m τm 2.50τ m 2.00τ m Tm 2 Tm1 = 0.5 p. 119. Model: Method 1 1.25T K τ 2.75τ 2.75τ T T = 1. 0 m1 m m m m m2 m1 τm Tm1 = 0.2 ; N not specified. Minimum ISE – 0.667Tm1 K m τm Model: Method 1 Tm1 Tm 2 Haalman (1965). 0 Tm1 > Tm 2 . M s = 1.9, A m = 2.36, φ m = 50 ; N = ∞ . 2 2 − 0.01133 + 0.5017ξm − 0.07813ξ m τm Kc = 2ξ T Km m m1 ( Ti = 2ξ m Tm1 1.235 − 0.4126ξ m + 0.09873ξ m ( Td = 2ξ m Tm1 1.214 − 0.6250ξ m + 0.1358ξm 3 Kc = −1.890 + 0.7902 ξ m − 0.1554 ξ m 2 ) τm 2ξ T m m1 2 ) 2 , τm 2ξ T m m1 −0.03793 + 0.3975ξ m − 0.05354 ξ m 2 , 0.5696 − 0.8484 ξ m + 0.0505ξ m 2 . 100 48 + 57 1 − e 1.2 Tm1 − τ m K m τm Tm1 Tm 2 Tm 2 + − 1 0 . 34 0 . 2 τm τm 2 , T 1.2 Tm1 T − m1 − − m2 Ti = τ m 1.5 − e 1.5τ m 1 + 0.9 1 − e τm , Td = 0.56τm 1 − e τ m + 0.6Tm 2 . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 240 Rule Minimum ISE – McAvoy and Johnson (1967). Kc Ti Td Comment x1 K m x 2τm x 3τ m Model: Method 1 ξm Representative coefficient values (deduced from graph): x1 x2 x3 ξm x1 x2 Tm1 Tm1 τm τm N = 20 1 1 1 0.5 4.0 10.0 0.7 7.6 34.3 0.97 3.33 5.00 0.75 2.03 2.7 N = 10 1 1 1 0.5 4.0 10.0 0.9 8.0 33.5 1.10 4.00 6.25 0.64 1.83 2.43 4 4 7 7 4 4 7 7 0.5 4.0 0.5 4.0 0.5 4.0 0.5 4.0 3.0 22.7 5.4 40.0 3.2 23.9 5.9 42.9 1.16 1.89 1.19 1.64 1.33 2.17 1.39 1.89 x3 0.85 1.28 0.85 1.14 0.78 1.17 0.78 1.04 0.881 4 0.883 0.881 Approximately Tm1 > Tm 2 ; x2 1.101 x1 x2 minimum ISE – 1.134x1 0.563x1 N = 8. Huang and Chao K m x 2 x1 x1 (1982). Also given by Chao et al. (1989); Model: Method 1. 5 Minimum ISE – N=8 Ti Td Kc Chao et al. 0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989). 1.036 6 0.595 1.006 Approximately Tm1 > Tm 2 ; x2 x2 0.745 x1 minimum ITAE 0 . 771 x 0 . 597 x 1 1 N = 8. Km x2 – Huang and x1 x1 Chao (1982). Also given by Chao et al. (1989); Model: Method 1. 4 5 τ 1.398 x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1 m + 1.039 − . T 1 . 349 ( T T ) + m1 m2 m2 0.2538 + 0.3875ξ m − 0.04283ξ m 2 τm Kc = 2ξ T Km m m1 ( Ti = 2ξ m Tm1 1.8283 − 0.8083ξm + 0.1852ξ m ( Td = 2ξ m Tm1 1.077 − 0.4952ξ m + 0.1062ξ m 6 −1.652 + 0.5416 ξ m − 0.09544 ξ m 2 ) , τm 2ξ T m m1 2 ) 2 τm 2ξ T m m1 0.1363+ 0.2342 ξ m − 0.01445ξ m 2 , 0.4772 + 0.0442 ξ m + 0.01694 ξ m 2 τ 1.398 x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1 m + 1.039 − . T 1 . 349 ( T T ) + m1 m2 m2 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Bohl and McAvoy (1976b). Minimum ITAE – Chao et al. (1989). 7 Kc Ti Td Comment 7 Kc Ti Td = Ti Minimum ITAE; Model: Method 1 8 Kc Ti Td 0.8 ≤ ξ m ≤ 3 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. τm Tm1 = 0.1,0.2, 0.3,0.4,0.5; Tm 2 Tm1 = 0.12,0.3 ,0.5,0.7,0.9. N = ∞ . τ τ T −1.1445 + 0.1100 ln 10 m 0.1648+ 0.1709 ln 10 m 2 − 0.2142 ln 10 m Tm1 Tm 2 Tm 1 Tm1 5.5030 τm Kc = 10 10 K m Tm1 Tm1 τ τ T 0.6212 − 0.0760 ln 10 m 0.3144 − 0.0062 ln 10 m 2 + 0.0085 ln 10 m τm Tm1 Tm 2 Tm 1 Tm1 Ti = 1.8681Tm1 10 10 Tm1 Tm1 8 2 − 0.1033 + 0.5347ξm − 0.07995ξ m τm Kc = 2ξ T Km m m1 ( Ti = 2ξ m Tm1 0.9096 − 0.1299ξ m + 0.04179ξm ( ) −1.798 + 0.7191ξ m − 0.1416 ξ m 2 ) , τm 2ξ T m m1 2 τm 2 Td = 2ξ m Tm1 1.2226 − 0.6456ξ m + 0.1373ξ m ξ 2 m Tm1 −0.1277 + 0.5220 ξ m − 0.08629 ξ m 2 , 0.4780 − 0.03653ξ m + 0.04523ξ m 2 ; N = 8. 241 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 242 Rule Kc Ti Comment Td 9 Chao et al. Ti Td 0.3 < ξ m < 0.8 Kc (1989) – N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 . continued. 10 0.884 0.907 0.869 Approximately Tm1 > Tm 2 ; x x minimum ITSE – 0.994 x1 0.570x1 2 1.032x1 2 N = 8. x1 Huang and Chao Km x2 x1 (1982). Also given by Chao et al. (1989); Model: Method 1. 11 Minimum ITSE Ti Td 0.8 ≤ ξ m ≤ 3 Kc – Chao et al. N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989). 9 1.3292 − 2.8527ξm + 2.0365ξ m 2 τm Kc = 2ξ T Km m m1 −1.6159 − 0.06616 ξ m + 0.5351ξ m 2 , ( ) τm Ti = 2ξm Tm1 exp(x1 ) + 1.4094 − 4.4581ξ m + 3.4857ξ m 2 ln 2ξ m Tm1 ( + 0.1160 − 0.8142ξ m + 0.7402ξ m 2 2 τ m , ln 2ξ m Tm1 ) ( x 1 = 3.3368 − 7.7919ξ m + 4.3556ξ m ( ) 2 ); 2 ). τm Td = 2ξ m Tm1 exp(x 2 ) + 1.8243 − 5.6458ξ m + 4.6238ξ m 2 ln 2ξ m Tm1 2 τ m 2 , + 0.2866 − 1.4727ξ m + 1.2893ξ m ln 2ξ m Tm1 ( ) ( x 2 = 2.3054 − 6.4328ξ m + 3.9743ξ m τ 1.398 10 x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1 m + 1.039 − . 1.349 + (Tm1 Tm 2 ) Tm 2 11 0.1034 + 0.4857ξ m − 0.0709ξ m 2 τm Kc = 2ξ T Km m m1 −1.713 + 0.6238ξ m − 0.1178ξ m 2 , ( 2 ) 0.1008 + 0.2625ξ m − 0.02203ξ m 2 ( ) 0.4316 + 0.08163ξ m + 0.009173ξ m 2 Ti = 2ξ m Tm1 1.5144 − 0.6060ξ m + 0.1414ξm τm 2ξ T m m1 τm 2 Td = 2ξ m Tm1 1.0783 − 0.4886ξ m + 0.1038ξm ξ 2 m Tm1 , . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Chao et al. (1989) – continued. Approximately minimum IAE – Huang and Chao (1982). 12 Kc 12 Kc Ti Td Comment Ti Td 0. 3 < ξ m < 0. 8 243 N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 . Minimum performance index: servo tuning 1.00 1.032 Tm1 > Tm 2 ; 0.673 x1 x2 13 0 . 502 x T 1 N = 8. x i Km x2 1 Also given by Chao et al. (1989); Model: Method 1. 1.8342 − 3.8984ξ m + 2.7315ξm 2 τm Kc = 2ξ T Km m m1 ( −1.5641+ 0.1628ξ m + 0.09337 ξ m 2 , ) τm Ti = 2ξm Tm1 exp(x1 ) + 0.9475 − 3.6318ξ m + 2.9969ξ m 2 ln 2ξ m Tm1 ( + − 0.06927 − 0.3875ξ m + 0.4220ξ m 2 2 τ m , ln 2ξ m Tm1 ) ( 2 ) 2 ). x 1 = 3.2026 − 7.4812ξ m + 4.3686ξ m ; ( ) τm Td = 2ξm Tm1 exp(x 2 ) + 1.0983 − 1.5794ξ m + 1.0744ξ m 2 ln 2ξ m Tm1 ( + 0.01553 − 0.01414ξ m − 0.009083ξ m ( 2 2 τ m , ln 2ξ m Tm1 ) x 2 = 1.7393 − 3.7971ξ m + 1.7608ξ m τ 1 . 398 x 13 1 . Ti = , x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1 m + 1.039 − Tm1 x Tm 2 1 . 349 + 0.148 2 + 0.979 Tm 2 x1 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 244 Rule Kc Ti Comment Td 14 Minimum IAE – N=8 Ti Td Kc Chao et al. 0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989). 0.964 15 0.971 1.028 Approximately Tm1 > Tm 2 ; x2 0.787 x1 x2 minimum ISE – 0 . 594 x 1 . 678 x 1 1 x N = 8. K m 2 Huang and Chao x1 x1 (1982). Also given by Chao et al. (1989); Model: Method 1. 16 Minimum ISE – N=8 Ti Td Kc Chao et al. 0.8 ≤ ξ m ≤ 3; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989). 14 2 − 0.1779 + 0.6763ξ m − 0.1264ξ m τm Kc = 2ξ T Km m m1 ( −0.9941+ 0.1669 ξ m − 0.04381ξ m 2 , x ) τm 1 , Ti = 2ξ m Tm1 0.2857 + 0.4671ξm − 0.08353ξ m 2 2ξ m Tm1 2 3 x 1 = 0.1665 − 0.3324ξ m + 0.1776ξ m − 0.02897ξ m ; ( ) τm Td = 2ξm Tm1 exp(x 2 ) + − 2.0552 + 2.786ξ m − 0.5687ξ m 2 ln 2ξ m Tm1 2 τ m , + − 0.5008 + 0.6232ξ m − 0.1430ξ m 2 ln 2ξ m Tm1 ( ) ( x 2 = − 0.8776 + 0.4353ξ m − 0.1237ξ m 15 16 2 ). 2 ); τ 1.398 x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1 m + 1.039 − . T 1 . 349 ( T T ) + m1 m2 m2 0.1446 + 0.4302ξ m − 0.07501ξm 2 τm Kc = 2ξ T Km m m1 −1.214 + 0.3123ξ m − 0.06889 ξ m 2 ; ( ) τm Ti = 2ξm Tm1 exp(x1 ) + − 2.2502 + 3.7015ξ m − 1.7699ξ m 2 + 0.2680ξ m 3 ln 2ξ m Tm1 ( ) + − 0.4823 + 0.9479ξm − 0.4913ξ m + 0.07792ξ m [ln (0.5τm ξTm1 )] , ( 2 ) 3 ( x 1 = − 0.1256 + 0.1434ξ m − 0.03277ξ m Td = 2ξm Tm1 0.9105 − 0.3874ξ m + 0.08662ξ m (0.5τm ξm Tm1 ) , 2 2 x2 2 x 2 = 0.0779 + 0.2855ξm − 0.01985ξm . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti 245 Comment Td 0.995 1.049 Approximately Tm1 > Tm 2 ; x2 17 minimum ITAE 0.684 x1 0 . 491 x T N = 8. 1 i K m x 2 – Huang and x1 Chao (1982). Also given by Chao et al. (1989); Model: Method 1. Minimum ITAE – Chao et al. (1989). 18 Kc Ti Td 0.8 ≤ ξ m ≤ 3 19 Kc Ti Td 0.3 < ξ m < 0.8 N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. τ 1 . 398 x 17 1 . Ti = , x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1 m + 1.039 − Tm1 x Tm 2 1 . 349 + 0.114 2 + 0.986 Tm 2 x1 18 2 − 0.2641 + 0.7736ξ m − 0.1486ξ m τm Kc = 2ξ T Km m m1 ( −0.7225 − 0.04344 ξ m − 0.001247 ξ m 2 ; x ) τm 1 , Ti = 2ξ m Tm1 0.22257 + 0.7452ξ m − 0.1451ξm 2 2ξ m Tm1 2 x 1 = 0.03138 − 0.04430ξ m + 0.01120ξ m ; ( ) τm Td = 2ξm Tm1 exp(x 2 ) + − 1.2256 + 1.8544ξ m − 0.3366ξ m 2 ln 2ξ m Tm1 2 τ m , + − 0.2315 + 0.3402ξ m − 0.06757ξ m 2 ln 2ξ m Tm1 ( ) ( x 2 = 0.05515 − 0.7031ξ m + 0.1433ξ m 19 Kc = 1.1295 − 2.8584ξ m + 2.1176ξ m 2 τm 2ξ T Km m m1 ( Ti = 2ξ m Tm1 8.6286 − 20.70ξm + 13.203ξm 2 ). 2 −0.9904 − 2.5311ξ m + 2.8008ξ m 2 ; )(0.5τ ξ T ) , m m m1 x1 2 ( ) x 1 = 0.7962 − 2.4809ξ m + 1.7611ξ m ; Td = 2ξm Tm1 exp(x 2 ) + 3.2228 − 12.034ξ m + 9.2547ξ m 2 ln (0.5τm ξm Tm1 ) ( ) + 0.6366 − 3.0208ξ m + 2.4603ξ m [ln (0.5τm ξ m Tm1 )] , 2 ( 2 2 ) x 2 = 3.9285 − 12.874ξ m + 8.6434ξ m . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 246 Rule Kc Ti Td Comment 1.00 1.028 Approximately Tm1 > Tm 2 ; x2 0.718 x1 20 minimum ITSE – 0 . 596 x T 1 N = 8. x i Huang and Chao K m x 2 1 (1982). Also given by Chao et al. (1989); Model: Method 1. 21 Minimum ITSE Ti Td 0.8 ≤ ξ m ≤ 3 Kc – Chao et al. N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 ; Model: Method 1. (1989). τ 1 . 398 x 20 1 . Ti = , x1 = Tm1 + 1.4Tm 2 , x 2 = Tm1 m + 1.039 − Tm1 x2 Tm 2 1 . 349 + 0.063 + 1.061 Tm 2 x1 21 0.0820 + 0.4471ξm − 0.07797ξ m 2 τm Kc = 2ξ T Km m m1 ( ) −0.9246 + 0.07955ξ m − 0.02436 ξ m 2 ; x τm 1 , Ti = 2ξ m Tm1 0.5930 + 0.1457ξ m − 0.02062ξ m 2 2ξ mTm1 2 x 1 = −0.2201 + 0.1576ξ m − 0.03202ξ m ; ( ) τm Td = 2ξm Tm1 exp(x 2 ) + − 1.0018 + 1.5914ξ m − 0.2831ξ m 2 ln 2ξ m Tm1 ( + − 0.2291 + 0.3064ξ m − 0.0634ξ m ( 2 2 τ m , ln 2ξ m Tm1 ) x 2 = − 0.6534 − 0.1770ξ m − 0.0240ξ m 2 ). b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Chao et al. (1989) – continued. 22 Kc Td Comment Ti Td 0.3 < ξ m < 0.8 N = 8; 0.05 ≤ 0.5τm ξm Tm1 ≤ 0.5 . Smith et al. (1975). Model: Method 1 Direct synthesis: time domain criteria Tm1 Tm 2 λ = 1 TCL ; λTm1 K m (λτ m + 1) N not specified. x 1Tm1 K mτm Smith et al. (1975). Model: Method 1; N = 10. ξm ρ 6 1.75 1.75 1.5 1.0 ≥ 0.02 0.27 0.07 0.11 0.25 Tm1 τm ρ= Tm 2 τm Tm1 + Tm 2 Coefficient values (deduced from graph): ρ ρ x1 ξm x1 ξm 0.7 0.3 0.8Tm1 Tm 2 Hougen (1979), τm pp. 348–349. Model: Method 1 0.84Tm1 0.8 Tm 2 0.2 22 Ti 247 0.51 0.46 0.36 0.33 0.46 3 1.75 1.5 1.5 1.0 0.5Tm1 + Tm 2 23 0.50 0.42 0.44 0.28 0.48 ≥ 0.04 0.13 0.33 0.08 0.17 3 τ x1 0.45 0.39 0.38 0.40 0.49 N = 10 N = 30 Td 0.9666 − 2.3853ξm + 2.4096ξm 2 τm Kc = 2ξ T Km m m1 ( ≥ 0.06 0.09 0.17 0.50 0.13 m Tm1 Tm 2 Ti Ti = 2ξ m Tm1 6.3891 − 15.773ξ m + 10.916ξm 2 1.75 1.5 1.0 1.0 −1.533 − 0.05442 ξ m +10.916 ξ m 2 ; x ) 1 τm , 2ξ T m m1 2 2 x 1 = 0.04175 − 1.3860ξ m + 1.4053ξ m ; ( ) τm Td = 2ξm Tm1 exp(x 2 ) + 2.277 − 8.9402ξ m + 7.7176ξ m 2 ln 2ξ m Tm1 ( + 0.3465 − 1.9274ξ m + 1.8030ξ m ( 2 2 τ m , ln 2ξ m Tm1 ) x 2 = 3.2620 − 10.789ξ m + 7.6420ξ m 23 Ti = 0.53Tm1 + 1.3Tm 2 , Td = 2.4(τm Tm1Tm 2 ) 0.28 . 2 ). b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 248 Rule Kc Hougen (1988). Model: Method 1 Sklaroff (1992). Ti Td Comment N = 10; τ m < Tm1 . 24 Kc 0.5Tm1 + Tm 2 Td 25 Kc Tm1 + Tm 2 Tm1Tm 2 Tm1 + Tm 2 Also given by Åström and Hägglund (1995), pp. 250–251. Model: Method 8. N = 8 – Honeywell UDC6000 controller. 26 N not defined. Ti Td K Schaedel (1997). c Model: Method 1 Skogestad (2003). 0.5Tm1 K m τm N =∞. 24 T x − x 2 log10 m 2 Tm1 1 1 Kc = 10 Km min(Tm1 ,8τ m ) Model: Method 24 Tm 2 . Coefficients of K c deduced from graphs: τ m Tm 2 0.1 0.3 0.5 1 2 x1 0.9 0.42 0.24 -0.10 -0.39 x2 0.7 0.7 0.68 0.70 0.69 Td = (Tm1 + Tm 2 )10 Td = (Tm1 + Tm 2 )10 25 Kc = 26 Kc = τ 0.33 log10 m − 0.40 Tm1 τ 0.31 log10 m − 0.63 Tm1 , Tm 2 = 0.1 ; Tm1 , 0.2 ≤ Tm 2 ≤1. Tm1 3 . 3τ m K m 1 + Tm1 + Tm 2 0.375 Km 2 2 2 2 4ξ m Tm1 + 2ξ m Tm1τm + 0.5τm − Tm1 , (2ξmTm1 + τm )2 + 1 − 1(2ξmTm1 + τm )Tm1 2(2ξm 2 − 1) − x N 2 2 2 2 x = 4ξ m Tm1 + 2ξ m Tm1 τ m + 0.5τ m − Tm1 ; 2 Ti = Td = 2 2 2 4ξ m Tm1 + 2ξ m Tm1τm + 0.5τm − Tm1 ; 2ξm Tm1 + τm (2ξmTm1 + τm )2 − 2(Tm12 + 2ξmTm1τm + 0.5τm 2 ) . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Ti Td Comment Kc 2ξ m Tm1 Tm1 2ξm Tm1 2K m τ m Tm1 Tm 2 N = 8; Model: Method 8 N not defined; Model: Method 1 Tm 2 N =∞; Tm1 > Tm 2 . Kc 27 Panda et al. (2004). Skoczowski (2004). Tm1 28 Kc Model: Method 1 Huang and Jeng T 0.495 m1 + 0.2 (2005). 0.4τ m + 0.9Tm1 τm Model: Method 1 Km Skogestad (2006b). Model: Method 1 N =∞; 4T min Tm1 , m1 Tm 2 T m1 > Tm 2 . Km Slow control; ‘acceptable’ disturbance rejection performance. Direct synthesis: frequency domain criteria Tm1 Tm1 Tm 2 K (T + τ ) m m1 m N = 5; Tm1 ≤ 5Tm 2 ; minimum φ m = 550 . ωp Tm1 KmAm x 1Tm1 K mτm Ti Tm1 Tm1 Tm 2 29 29 Tm1 = Tm 2 ; N =∞. A m = 1.571 x1 ; φ m = 0.5π − x 1 . 0 Sample result ( A m = 2.0; φ m = 45 ): 0.785Tm1 K m τm Tm1 3 . 1.5τm K m 1 + ξm Tm1 2T 28 K c ≤ m1 1 + 2 x12 − 2 x1 1 + x12 , with x1 ≈ τm 27 Tm 2 Tm 2 Kc = 1 Ti = 2ωp − Tm 2 τm 1 Poulin et al. (1996). Model: Method 1 Yang and Clarke (1996). Model: Method 10 O’Dwyer (2001a). Model: Method 1; N =∞. N = 100 4ωp 2 τm π + 1 Tm1 . ln (1 OS) π + [ln (1 OS)]2 2 . 249 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 250 Rule Kc Majhi and Litz (2003). 30 Kc Ti Td Comment Ti Tm1 N=∞ Model: Method 10; Tm1 = Tm 2 . Ultimate cycle Bohl and McAvoy (1976b). 30 Kc = 31 2φm + π(A m − 1) Tm1 , 2 K m τm 2 Am − 1 ( ) Ti = Tm1 . T Am Am [ ] ( ) ( ( ) ) 1 + m1 2 φ + π A − 1 1 − 2 φ + π A − 1 m m m m 2 τm A m 2 − 1 π A m − 1 ( 31 N =∞; Model: Method 1 Td = Ti Ti Kc ) ( ) τ 1.2264 + 0.4568 ln m τ Tu 0.5458 τm 1.2190 − 0.0958 ln (K u K m )−0.1007 ln m Kc = (K u K m ) Tu , K m Tu τ −1.9314 − 0.6221 ln m τ τm Tu −0.0703−0.1125 ln (K u K m )−0.2944 ln m Ti = 0.0393Tu (K u K m ) Tu . Tu b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 251 3.5.6 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N R(s) E(s) + − 2 T s 1 d b f 0 + b f 1s + b f 2 s K c 1 + + Ti s T 1 + a f 1s + a f 2 s 2 1+ d s N U(s) SOSPD model Table 25: PID controller tuning rules – SOSPD model G m (s) = G m (s) = Rule Huang et al. (2003). Model: Method 12 Kc 0.8ξm Tm1 K m τm Y(s) K m e − sτ m or (1 + sTm1 )(1 + sTm 2 ) K m e − sτ m 2 Tm1 s 2 + 2ξ m Tm1s + 1 Ti Comment Td Direct synthesis: time domain criteria 2ξ m Tm1 0.5Tm1 ξm b f 0 = 1 ; b f 1 not defined; b f 2 = 0 ; N = ∞ ; a f 1 = a f 2 = 0 . Robust Jahanmiri and Fallahi (1997). Model: Method 7 2ξ m Tm1 K m (τ m + λ) 2ξ m Tm1 Tm1 2ξ m b f 1 = 0.5τ m , a f1 = λτ m . 2( τ m + λ ) N = ∞ , b f 0 = 1 , b f 2 = 0 , a f 2 = 0 ; λ = 0.25τ m + 0.1ξ m Tm1 . Kristiansson (2003). Model: Method 1 1 Kc = 1 Kc 2ξ m Tm1 Tm1 2ξ m M max = 1.7 2 b f 0 = 1 , b f 1 = b f 2 = 0 , a f 1 = 0.08Tm1 , a f 2 = 0.01Tm1 , N = ∞ . 2ξ m Tm1 T T T 2 1 + 0.019 m1 − 0.346 + 0.038 m1 + 0.00036 m12 . K m τm τm τm τm b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 252 Rule Shi and Lee (2004). Kc Ti Td Comment Kc Ti Td Model: Method 1 0.5τm K m (λ + τm ) 0.5τm 0.167 τm N=∞ 2 Shamsuzzoha and Lee (2006a). Model: Method 1 af1 = 2 τmλ τm λ , af 2 = , b f 0 = 1 , b f 1 = 2ξ m Tm1 , 2(τm + λ ) 12(τm + λ ) 2 b f 2 = Tm1 . λ = x1τm ; x1 = 1.360 , M s = 1.4 ; x1 = 0.677 , M s = 1.6 ; x1 = 0.372 , M s = 1.8 ; x1 = 0.337 , M s = 2.0 . 2 Kc = 1 + ω−6dB τ m ω− 6dB 0.5 Tm1 + Tm 2 − , bf 0 = 1 , b f 1 = , 2ω−6dB ω− 6 dB K m (2 + ω− 6dBτm ) bf 2 = Ti = Tm1 + Tm 2 − τm τm 1 , a f1 = , af 2 = ; 4ω −6dB 2ω−6dB 2ω−6dB (1 + ω −6dB τ m ) 2 0.5 T T ω − 0.5[(Tm1 + Tm 2 )ω− 6dB − 0.5] , Td = m1 m 2 − 6dB , ω− 6dB ω− 6 dB [(Tm1 + Tm 2 )ω− 6dB − 0.5] N= 0.5[(Tm1 + Tm 2 )ω−6dB − 0.5] Tm1Tm 2 ω−6dB 2 − 0.5[(Tm1 + Tm 2 )ω −6dB − 0.5] . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 253 3.5.7 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1 s + N E(s) R(s) − + T s 1 d K c 1 + + Ti s Td 1+ s N − U(s) + SOSPD model Table 26: PID controller tuning rules – SOSPD model G m (s) = Y(s) K m e − sτ m or (1 + sTm ) 2 K m e − sτ K m e − sτ m or (1 + sTm1 )(1 + sTm 2 ) Tm1 2 s 2 + 2ξ m Tm1s + 1 m Rule Kc Ti Td Comment Minimum performance index: regulator tuning 1.18Tm1 K m τm 2.20τ m 0.72τ m Tm 2 Tm1 = 0.1 Minimum IAE – 1.25Tm1 K m τm Shinskey (1996), 1.67Tm1 K m τm p. 119. Model: Method 1 2.5T K τ m1 m m 2.20τ m 1.10τ m Tm 2 Tm1 = 0.2 2.40τ m 1.65τ m Tm 2 Tm1 = 0.5 2.15τ m 2.15τ m Tm 2 Tm1 = 1.0 τm Tm1 = 0.2 ; α = 0 ; β = 1 ; N = ∞ . b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 254 Rule Minimum IAE – Shinskey (1996), p. 121. Model: Method 1 Minimum IE – Shen (1999). Model: Method 1 Td Comment Ti Td Tm 2 τm ≤ 3 τ m + 0.2Tm 2 τ m + 0.2Tm 2 Tm 2 τm > 3 α = 0 ;β =1; N = ∞ . 0.85K u 0.35Tu τm Tm1 = 0.2 , 0.17Tu Tm 2 Tm1 = 0.2 . α = 0 ;β =1; N = ∞ . 2 Ti Kc Km = 1 Td 100 Kc = )] (38 + 40[1 − e ( )(K τ T )(1 + 0.34[T τ ] − 0.2[T ( ) ]), T = τ (0.5 + 1.4[1 − e ]) (1 + 0.48[1 − e ( ) ) + 0.6T . T = 0.42τ (1 − e i d 2 Ti Kc 1 Minimum IAE – Kc Shinskey (1994), 3.33Tm1 K m τm p. 159. Model: Method 1 1 FA − 1.5 Tm1 τ m m m m1 − Tm1 1.5 τ m m m m2 m m2 τm ]2 ), − Tm 2 τ m − 1.2 Tm1 τ m m2 x1 , x 2 , x 3 and x 4 are given by the following equation: 2 3 τ τ τ ξ τ exp[a + b m + c m + d m + e m m + fξ m + gξ m 2 Tm1 T T T m1 m1 m1 3 2 3 2 τ τ τ ξm τm 2 2 + i m ξ m + j m ξ m + k m ξ m ] ; T T T Tm1 m1 m1 m1 K c = Tm1x1 τm , Ti = τm x 2 , Td = τm x 3 , α = 1 − x 4 ; coefficients of x1 , x 2 , x 3 and x 4 +h are given below. 1.4 ≤ M s ≤ 2 ; β = 0 ; N = ∞ . a b c d e f g h i j k x1 x2 x3 x4 1.40 -11.60 14.17 -6.44 8.28 -0.07 0.13 -4.66 -0.99 -3.03 2.89 2.43 -2.50 -4.16 2.85 -0.30 0.1 -1.72 15.51 -7.98 -8.45 4.66 1.32 -0.51 0.70 -1.06 0.02 -0.09 -7.43 1.77 2.64 4.77 -3.74 -1.44 1.27 1.34 -0.37 1 -5.97 -0.78 -10.63 18.85 -10.97 6.44 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum IE – Shen (2000). Model: Method 1 3 3 Kc Ti Td Comment Kc Ti Td Km = 1 255 x1 , x 2 , x 3 and x 4 are given by the following equation: 2 3 τm τ m τm + c + eξ m + f ξ m 2 exp[a + b τ + T + d τ + T τ + T m1 m1 m1 m m m 2 3 τm τm τm ξ m + h ξ m + i ξ m + g τ + T τ + T τ + T m1 m1 m1 m m m 2 3 τm 2 τm τm ξ m + k ξ m 2 + l ξ m 2 ] ; + j τ + T τ + T τ + T m1 m1 m1 m m m K c = Tm1x1 τm , Ti = τm x 2 , Td = τm x 3 , α = 1 − x 4 ; coefficients of x1 , x 2 , x 3 and x 4 are given below. M s = 2 ; β = 0 ; N = ∞ ; 0 < ξ m < 1 ; 0 ≤ τm Tm1 ≤ 1 . a b c d e f g h i j k l x1 x2 x3 x4 1.73 -17.51 30.73 -24.69 0.15 -0.12 3.30 33.07 -37.64 2.63 -34.57 36.88 2.28 0.87 -24.00 15.09 -0.01 -0.02 -8.03 45.21 -22.48 3.89 -22.39 8.32 1.43 -1.94 14.82 -21.44 -0.25 0.16 -1.06 -43.39 51.16 -3.25 36.04 -34.95 -1.60 4.30 -12.16 26.54 1.33 -1.08 -17.56 38.13 -56.45 14.80 -39.33 51.18 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 256 Rule Minimum IE – Shen (2000). Model: Method 1 Madhuranthakam et al. (2008). Model: Method 1 4 17-Mar-2009 4 5 Kc Ti Td Comment Kc Ti Td Km = 1 Kc Ti Td Tm1 > Tm 2 ; τm < Tm1 + Tm 2 . x1 , x 2 , x 3 and x 4 are given by the following equation: 2 3 τm τ m τm + c + d + eξ m + f ξ m 2 exp[a + b τ +T m1 τm + Tm1 τm + Tm1 m 2 3 τm τm τm ξ m + h ξ m + i ξ m + g τ + T τ + T τ + T m1 m1 m1 m m m 2 3 τm 2 τm τm ξ m + k ξ m 2 + l ξ m 2 ] ; + j τ + T τ + T τ + T m1 m1 m1 m m m K c = Tm1x1 τm , Ti = τm x 2 , Td = τm x 3 , α = 1 − x 4 ; coefficients of x1 , x 2 , x 3 and x 4 are given below. M s = 2 ; β = 0 ; N = ∞ ; 0.4 ≤ ξ m ≤ 1 ; 1 < τm Tm1 ≤ 15 . 5 x1 x2 x3 x4 a b c d e f g h i j k l 128.9 -560.3 769.2 -338.9 -324.8 194.3 1421 -1981.6 895.0 -848.1 1186.1 -538.1 92.8 -391 521.6 -225.5 -224.5 137.5 963.9 -1310.6 574.4 -590.8 808.0 -356.7 -3.1 46.3 -107.4 63.1 5.2 -2.5 -79.6 176.0 -103.6 36.2 -78.4 45.8 -119.9 539.5 -785.1 371.8 233.9 -115.7 -1068 1560.5 -741.5 529.2 -774.6 369.2 Kc = 0.6202 Tm1 + Tm 2 1 + K m τm 0.9931 , 2 τm τm τ , + 13.81 Ti = τm 1 + m 4.566 − 14.906 τ +T +T m1 m2 m τm + Tm1 + Tm 2 Tm1 Td = 0.0921(τ m + Tm 2 ) 1.4849 τm T + Tm 2 1 + m1 Tm1 τm , N = 10, α = 0 , β = 1 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Madhuranthakam et al. (2008). Model: Method 1 6 Ti Kc Comment Td Minimum performance index: servo tuning Tm1 > Tm 2 ; Ti Td τm < Tm1 + Tm 2 . Minimum performance index: servo/regulator tuning x1 K m x 2Tm1 x 3Tm1 N=∞ Taguchi et al. τm τm (1987). α α x1 x 2 x3 β T x1 x 2 x3 β T m1 m1 Model: Method 1 11.47 0.74 0.40 0.64 0.84 0.2 0.41 1.32 1.28 -0.15 0.37 1.6 3.39 1.17 0.67 0.57 0.80 0.4 0.38 1.34 1.25 -0.24 0.27 1.8 1.70 1.35 0.89 0.47 0.75 0.6 0.35 1.38 1.21 -0.30 0.18 2.0 1.06 1.39 1.06 0.35 0.69 0.8 0.33 1.53 1.12 -0.37 -0.04 2.5 0.75 1.37 1.18 0.21 0.62 1.0 0.33 1.75 1.06 -0.37 -0.19 3.0 0.58 1.34 1.25 0.08 0.55 1.2 0.35 2.26 1.08 -0.32 -0.30 4.0 0.48 1.32 1.28 -0.05 0.46 1.4 0.37 2.82 1.19 -0.24 -0.18 5.0 0 x1K u x 2 Tu Minimum ITAE Coefficient values: – Pecharromán α α x1 x2 φc x1 x2 φc (2000a), 0 0.280 0.512 0.281 − 730 Pecharromán and 0.147 1.150 0.411 − 146 Pagola (2000). 0.170 1.013 0.401 − 140 0 0.291 0.503 0.270 − 630 Model: 0.1713 1.0059 0.4002 − 139.7 0 0.297 0.483 0.260 − 52 0 Method 15; Km = 1; 0.195 0.880 0.386 − 1330 0.303 0.462 0.246 − 410 Tm1 = 1 ; 0.210 0.720 0.342 − 1250 0.307 0.431 0.229 − 30 0 ξm = 1 . 0.234 0.672 0.345 − 115 0 0.317 0.386 0.171 − 19 0 0.249 0.610 0.323 − 1050 0.324 0.302 0.004 − 10 0 0.262 0.568 0.308 − 94 0 0.320 0.223 -0.204 − 60 0.274 0.545 0.291 − 84 0 1.0409 6 Kc = 0.5723 Tm1 + Tm 2 1 + τm K m , 1.6501 Ti = 0.2476 τm (τm + Tm1 + Tm 2 )1 + Tm1 + Tm 2 τm Tm1 Td = 0.0943(τ m + Tm 2 ) , 1.4636 T + Tm 2 τm 1 + m1 Tm1 τm , N = 10, α = 0 , β = 1 . 257 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 258 Rule Kc Ti Td Comment Minimum ITAE – Pecharromán and Pagola (2000). 0.7236K u 0.5247Tu 0.1650Tu N = 10; Model: Method 15 Minimum ITAE – Pecharromán (2000a). Model: Method 15; Km = 1; Tm1 = 1 ; β = 1 ; N = 10; ξm = 1 ; 0.1 < τ m < 10 . Minimum ITAE – Pecharromán (2000b), p. 145. Model: Method 15; Km = 1; Tm1 = 1 ; β = 1 ; N = 10; ξm = 1 . α = 0.5840 , β = 1 , φc = −139.650 , K m = 1 ; Tm1 = 1 ; ξ m = 1 . x1K u x 2 Tu x 3Tu x1 Coefficient values and other data: α x2 x3 0.803 0.509 0.167 0.585 − 146 0 0.727 0.524 0.165 0.584 − 140 0 0.672 0.532 0.161 0.577 − 134 0 0.669 0.486 0.170 0.550 − 1250 0.600 0.498 0.157 0.543 − 1150 0.578 0.481 0.154 0.528 − 1050 0.557 0.467 0.149 0.504 − 930 0.544 0.466 0.141 0.495 − 84 0 0.537 0.444 0.144 0.484 − 730 0.527 0.450 0.131 0.477 − 630 0.521 0.440 0.129 0.454 − 52 0 0.515 0.429 0.126 0.445 0.509 0.399 0.132 0.433 − 410 0.496 0.374 0.123 0.385 − 19 0 0.480 0.315 0.112 0.286 − 10 0 0.430 0.242 0.084 0.158 − 60 Coefficient values and other data: α x2 x3 φc x1K u x1 x 2 Tu φc − 30 0 x 3Tu 0.7965 0.5139 0.1638 0.5854 − 146.10 0.7139 0.5275 0.1625 0.5844 − 140.00 0.6675 0.5109 0.1600 0.5771 − 134.00 0.6540 0.4240 0.1704 0.5498 − 125.00 0.5996 0.4736 0.1629 0.5428 − 115.00 0.5801 0.4660 0.1573 0.5276 − 105.00 0.5604 0.4545 0.1512 0.5042 − 93.00 0.5500 0.4520 0.1470 0.4951 − 84.00 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Minimum ITAE – Pecharromán (2000b), p. 145 – continued. Kuwata (1987). Model: Method 1 7 Kc = Ti Td x1K u x 2 Tu x 3Tu Coefficient values and other data: α x2 x3 x1 Kuwata (1987). Model: Method 1 φc 0.5451 0.4321 0.1491 0.4844 − 73.00 0.5312 0.4431 0.1361 0.4769 − 63.00 0.5247 0.4356 0.1325 0.4544 − 52.0 0 0.5189 0.4260 0.1281 0.4452 − 41.00 0.5153 0.3971 0.1324 0.4327 − 30.00 0.4987 0.3680 0.1227 0.3855 − 19.00 0.4846 0.3131 0.1146 0.2860 − 10.00 0.4610 0.2471 0.0966 0.1577 − 6.00 Direct synthesis: time domain criteria Ti 0 Ti 7 Kc 8 Kc 9 Kc Ti Td 10 Kc Ti Td 0.8(2Tm + τm ) 1 − , x1 = K m (2Tm + τm − x1 ) K m [ 8 Comment Kc α =1 α = 1; β = 1; N =∞. (2Tm + τm )2 − 0.267τm [6Tm 2 + 6Tm τm + τm 2 ] ; ] [ ] 0.833 2Tm 2 + 4Tm τm + τm 2 1.667 2Tm 2 + 4Tm τm + τm 2 . Ti = 1 − 2Tm + τm (2Tm + τm )2 0.8(2ξ m Tm1 + τm ) 1 Kc = − , K m (2ξ m Tm1 + τm − x1 ) K m x1 = (2ξmTm1 + τm )2 − 0.267τm [6Tm12 + 6ξmTm1τm + τm 2 ] ; [ ] [ ] 0.833 2Tm12 + 4ξm Tm1τm + τm 2 1.667 2Tm12 + 4ξ mTm1τm + τm 2 . Ti = 1 − 2ξm Tm1 + τm (2ξmTm1 + τm )2 9 2 2 1.20 x1 1 x x x − (2Tm + τm )x 2 − ; Ti = 1.667 2 − 4.167 23 ; Td = 0.833x 2 1 3 , 2 Km x2 Km x1 x1 x1 − 0.833x 2 2 3 Kc = 2 2 2 2 3 x1 = 2Tm + 4Tm τm + τm , x 2 = 6Tm τm + 6Tm τm + τm . 10 ( 3 Kc = ) 2 2 1.20 x1 1 2.778 x1 − 2.5x 2 x 2 = − ; ; x1 = 2Tm12 + 4ξm Tm1τm + τm 2 ; T i K m x 22 K m x14 x1 − (2ξm Tm1 + τm )x 2 2 Td = 0.833x 2 x13 − 0.833x 2 2 , x 2 = 6Tm12 τm + 6ξ m Tm1τm 2 + τm 3 . 259 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 260 Rule Taguchi and Araki (2000). Model: Method 1 Kc Ti Td Comment 11 Kc Ti 0 ξm = 1 12 Kc Ti 0 ξ m = 0.5 τ m Tm ≤ 1.0 ; OS ≤ 20% . Taguchi and Araki (2000). Model: Method 1 14 Kc Ti Td ξm = 1 15 Kc Ti Td ξ m = 0.5 τ m Tm1 ≤ 1.0 ; OS ≤ 20% ; N not specified. 11 Kc = 1 0.5316 , 0.3717 + [τm Tm ] + 0.0003414 Km ( ) Ti = Tm 2.069 − 0.3692[τm Tm ] + 1.081[τm Tm ] − 0.5524[τm Tm ] , 2 3 α = 0.6438 − 0.5056(τm Tm ) + 0.3087(τm Tm ) − 0.1201(τm Tm ) . 2 12 Kc = 3 0.05627 1 0.1000 + , 2 Km [(τm Tm ) + 0.06041] ( ) Ti = Tm 4.340 − 16.39[τm Tm ] + 30.04[τm Tm ] − 25.85[τm Tm ] + 8.567[τm Tm ] , 2 3 4 α = 0.6178 − 0.4439(τm Tm ) − 7.575(τm Tm ) + 9.317(τm Tm ) − 3.182(τm Tm ) . 2 14 Kc = 3 4 0.6978 1 1.389 + , K m [(τm Tm1 ) + 0.02295]2 ( ) Ti = Tm1 0.02453 + 4.104[τm Tm1 ] − 3.434[τm Tm1 ] + 1.231[τm Tm1 ] , 2 3 4 τ τ τ τ Td = Tm1 0.03459 + 1.852 m − 2.741 m + 2.359 m − 0.7962 m , Tm1 Tm1 Tm1 Tm1 2 3 α = 0.6726 − 0.1285(τm Tm1 ) − 0.1371(τm Tm1 ) + 0.07345(τm Tm1 ) , 2 3 β = 0.8665 − 0.2679(τm Tm1 ) + 0.02724(τm Tm1 ) . 2 15 Kc = 0.5013 1 0.3363 + , K m [(τm Tm1 ) + 0.01147]2 ( ) T = T (0.03392 + 2.023[τ T ] − 1.161[τ T ] + 0.2826[τ T ] ) , Ti = Tm1 − 0.02337 + 4.858[τm Tm1 ] − 5.522[τm Tm1 ] + 2.054[τm Tm1 ] , 2 3 2 d m1 m m1 m m1 3 m m1 α = 0.6678 − 0.05413(τm Tm1 ) − 0.5680(τm Tm1 ) + 0.1699(τm Tm1 ) , 2 β = 0.8646 − 0.1205(τm Tm1 ) − 0.1212(τm Tm1 ) . 2 3 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Arvanitis et al. (2000a). Model: Method 1 Huang et al. (2000). 16 Direct synthesis: frequency domain criteria 0 < x1 < 1 ; Tm1 16 Ti Kc suggested T x1 2ξm + m1 x1 = 0.5 . τm α = 1, β = 1, N = ∞ . Model: Method 1 17 K c Ti Td τm ( 2 Tm1 Km ) T T 8(1 − x1 ) m1 (2ξ m τm + Tm1 ) 2ξ2 (1 − x1 ) + x1 m1 (2ξm τm + Tm1 ) + τm , τm τm ξ Kc = λ α= Comment ( where 17 Td x1Tm1 T (x1Tm1 τm )(2ξm + (Tm1 τm )) 2ξm + m1 , Ti = , K m τm τm 1 Tm1 (2ξ m τm + Tm1 ) 4ξ 2 (1 − x1 ) + x1 − x2 1 + τm τm 2 Kc = x2 = Ti ) 2 (2ξ m τ m + Tm1 ) + 1 2 x1 + 2ξ2 (1 − x1 )2 Tm1 (2ξ m τm + Tm1 )2 ≥ 0 . 2 τm Km 2Tm1ξ m + 0.4τm , Ti = 2Tm1ξ m + 0.4τm , Td = K m τm 2 Tm1 + 0.8Tm1ξ m τm 0.8Tm1ξm τm , 2Tm1ξ m 0.5Tm1 2 −1, β = − 1 , N = min[20,20Td ] ; 2 2Tm1ξ m + 0.4τ m Tm1 + 0.8Tm1ξ m τ m λ = 0.6 , τmξ m Tm1 ≤ 1 , A m = 2.83 ; λ = 0.7 , 1 < τmξ m Tm1 ≤ 2 , A m = 2.43 ; λ = 0.8 , τmξ m Tm1 > 2 , A m = 2.13 . 261 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 262 Rule Kc 18 Shamsuzzoha and Lee (2007a). Model: Method 1 Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 0. 5 . Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 0. 7 . Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 1. 0 . Kc = Td = Td Comment Ti Td λ = x1Tm τm Tm Ms 1.9 0.18 0.20 0.22 0.24 0.25 0.27 τm Tm Ms 2.0 0.12 0.15 0.17 0.19 0.21 0.22 0.23 1.8 1.9 0.2 0.34 0.28 0.3 0.35 0.29 0.4 0.38 0.32 0.5 0.41 0.34 0.6 0.46 0.38 0.7 0.51 0.41 0.8 0.55 0.43 Ultimate cycle 0 x 2 Tu x 1K u 2.0 0.25 0.26 0.28 0.30 0.33 0.36 0.38 0.9 1.0 1.25 1.5 2.0 2.5 3.0 α =1 x1 x2 τm Tu x1 x2 τm Tu x1 x2 0.05 0.11 0.83 0.39 x 1K u 0.25 0.30 0.16 0.18 x 2 Tu 0.23 0.18 0.35 0.40 0 0.20 0.20 0.15 0.45 0.14 0.50 α =1 τm Tu x1 x2 τm Tu x1 x2 τm Tu x1 x2 τm Tu 0.01 0.03 0.07 0.11 0.27 0.17 0.14 0.14 x 1K u 0.05 0.10 0.15 0.20 0.14 0.17 0.19 0.20 x 2 Tu 0.14 0.14 0.14 0.14 0.25 0.30 0.35 0.40 0 0.20 0.20 0.14 0.14 0.45 0.50 x1 x2 τm Tu x1 x2 τm Tu x1 x2 τm Tu 0.04 0.11 0.17 0.20 0.95 0.53 0.38 0.30 0.05 0.10 0.15 0.20 0.21 0.21 0.21 0.20 0.23 0.20 0.17 0.15 0.25 0.30 0.35 0.40 0.20 0.20 0.14 0.14 0.45 0.50 α =1 2 Ti 6λ2 − x 3 + x1τm − 0.5τm , Ti = Tm1 + Tm 2 + x1 − x 2 , x 2 = , K m (4λ + τm − x1 ) 4 λ + τ m − x1 x 3 + x1 (Tm1 + Tm 2 ) + Tm1Tm 2 − [ 4 [ 4 ] 3 2 0.167 τm − 0.5x1τm + x 3τm + 4λ3 4λ + τm − x1 − x2 , Ti [ ] Tm1 (1 − (λ Tm1 )) e − τ m Tm1 − 1 − Tm 2 (1 − (λ Tm 2 )) e − τ m Tm 2 − 1 , Tm 2 − Tm1 2 x1 = Kc Ti N = ∞ , α = 0.6 , β = 0 . Coefficients of x1 (deduced from graph): 1.8 0.27 0.29 0.31 0.33 18 17-Mar-2009 2 ] x 3 = Tm 2 (1 − (λ Tm 2 )) e − (τ m Tm 2 ) − 1 + x1Tm 2 . 2 4 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Td Comment Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 1. 5 . x 1K u x 2 Tu 0 α =1 Kuwata (1987); x 1 , x 2 deduced from graphs. ξ m = 0. 5 ; x1 x2 τm Tu x1 x2 τm Tu x1 x2 τm Tu 0.15 0.23 0.27 0.80 0.60 0.47 x 1K u 0.10 0.15 0.20 0.27 0.24 0.22 x 2 Tu 0.38 0.27 0.20 0.25 0.30 0.35 0.20 0.20 0.20 0.17 0.15 0.14 0.40 0.45 0.50 x2 τm Tu x1 x3 x1 x 3 Tu x2 x3 τm Tu x1 x2 x3 τm Tu 0.42 0.24 0.19 0.18 0.19 0.11 0.07 0.30 α = 1; β = 1; 0.27 0.15 0.17 0.23 0.17 0.10 0 0.36 N =∞. Kuwata (1987); x 1K u x 2 Tu x 3 Tu x 1 , x 2 deduced x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm from graphs. Tu Tu Tu ξm = 0.7 ; 0.54 0.32 0.14 0.18 0.22 0.16 0.03 0.36 0.20 0.14 0 0.50 α = 1; β = 1; 0.37 0.25 0.12 0.23 0.20 0.14 0.01 0.42 N =∞. 0.29 0.19 0.08 0.30 0.20 0.14 0 0.47 Kuwata (1987); x 1K u x 2 Tu x 3 Tu x 1 , x 2 deduced x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm from graphs. Tu Tu Tu ξm = 1 ; α = 1 ; 0.61 0.42 0.12 0.18 0.30 0.20 0.06 0.36 0.20 0.14 0 0.50 β =1; N = ∞ . 0.47 0.34 0.10 0.23 0.21 0.17 0.025 0.42 0.38 0.27 0.08 0.30 0.20 0.14 0.01 0.47 Kuwata (1987); x 1K u x 2 Tu x 3 Tu x 1 , x 2 deduced x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm from graphs. Tu Tu Tu ξ m = 1. 5 ; α = 1 ; 0.67 0.48 0.11 0.18 0.33 0.25 0.07 0.36 0.20 0.14 0 0.50 β =1; N = ∞ . 0.54 0.43 0.09 0.23 0.25 0.18 0.03 0.42 0.43 0.34 0.08 0.30 0.20 0.15 0.02 0.47 Oubrahim and Tm1 = Tm 2 ; 19 Leonard (1998). 0.5Tu 0.125Tu 0. 6 K u 0.1 < τm Tm1 < 3 Model: Method 1 2 19 α = 1 − 1.3 α = 1− 16 − K m K u 16 − K m K u , 10% overshoot. , β = 1 − 1.69 17 + K m K u 17 + K m K u 38 1.717 , 20% overshoot. , β =1− 29 + 3.5K m K u (1 + 0.121K m K u )2 263 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 264 3.5.8 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) Y(s) Model − F2 (s) Table 27: PID controller tuning rules – SOSPD model G m (s) = G m (s) = Rule Huang et al. (1996). 1 Kc 1 Kc K m e − sτ m or (1 + sTm1 )(1 + sTm 2 ) K m e − sτ m 2 2 Tm1 s + 2ξ m Tm1s + 1 Ti Td Comment Minimum performance index: regulator tuning 0 Model: Method 1 Ti Minimum IAE. Equations continued into the footnote on pp. 265 and 266; 0 < Tm 2 Tm1 ≤ 1 ; 0.1 ≤ τ m Tm1 ≤ 1 . Kc = τm τ T 1 T + 76.6760 m 2 − 3.3397 m m2 2 0.1098 − 8.6290 K m Tm1 Tm1 Tm1 τ 1 1.1863 m + T Km m1 −0.8058 T 1 − 52.0778 m 2 + T Km m1 τ + 23.1098 m Tm1 0.8405 0.6642 T − 12.1033 m 2 Tm1 τ + 20.3519 m Tm1 2.1123 2.1482 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models + τ T 1 9.4709 m m 2 Km Tm1 Tm1 + T τ 1 − 19.4944 m 2 m Km Tm1 Tm1 0.5306 + 13.6581 Tm 2 τ m Tm1 Tm1 −0.4500 − 28.2766 −1.0781 τ m Tm 2 Tm1 Tm1 1.1427 τ m Tm 2 τm Tm 2 2 T 1 Tm1 Tm 1 + − 19.1463e + 8.8420e + 7.4298e Tm1 − 11.4753 m 2 ; Km τm 2 τ T τ T τ Ti = Tm1 − 0.0145 + 2.0555 m + 0.7435 m 2 − 4.4805 m + 1.2069 m m2 2 Tm1 Tm1 Tm1 Tm1 2 3 2 T τ T τ + Tm1 0.2584 m 2 + 7.7916 m − 6.0330 m 2 m Tm1 Tm1 Tm1 Tm1 2 3 4 τ T τ T + Tm1 3.9585 m m 2 − 3.0626 m 2 − 7.0263 m Tm1 Tm1 Tm1 Tm1 3 2 2 3 τ T T τ T τ + Tm1 7.0004 m 2 m − 2.7755 m 2 m − 1.5769 m m 2 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 4 5 5 T τ T + Tm1 3.1663 m 2 + 2.4311 m − 0.9439 m 2 Tm1 Tm1 Tm1 4 2 3 T τ T τ + Tm1 − 2.4506 m 2 m − 0.2227 m 2 m Tm1 Tm1 Tm1 Tm1 2 3 4 τ T τ T + Tm1 1.9228 m m 2 − 0.5494 m m 2 ; Tm1 Tm1 Tm1 Tm1 2 τ τ T τ T x1 = Tm1 − 0.0206 + 0.9385 m + 0.7759 m 2 − 2.3820 m + 2.9230 m m2 2 Tm1 Tm1 Tm1 Tm1 2 3 2 T τ T τ + Tm1 − 3.2336 m 2 + 7.2774 m − 9.9017 m 2 m Tm1 Tm1 Tm1 Tm1 2 3 4 τ T τ T + Tm1 2.7095 m m 2 + 6.1539 m 2 − 11.1018 m Tm1 Tm1 Tm1 Tm1 265 b720_Chapter-03 266 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 3 2 2 6 τm τ m Tm 2 Tm 2 τ m − 2.274 + 5.7105 + Tm1 10.6303 Tm1 Tm1 Tm1 Tm1 Tm1 3 4 5 τm Tm 2 τ m Tm 2 + 8.0849 − 6.6597 + Tm1 − 7.9490 Tm1 Tm1 Tm1 Tm1 4 2 3 Tm 2 τ m Tm 2 τ m − 7.6469 + Tm1 − 4.4897 Tm1 Tm1 Tm1 Tm1 2 3 4 5 Tm 2 τ m Tm 2 τ m Tm 2 + 4.1225 + 5.0694 + Tm1 2.2155 Tm1 Tm1 Tm1 Tm1 Tm1 5 6 2 4 τ m Tm 2 Tm 2 Tm 2 τ m − 1.6307 − 1.1295 + Tm1 0.519 Tm1 Tm1 Tm1 Tm1 Tm1 4 2 3 3 5 τ m Tm 2 τ m Tm 2 τ m Tm 2 ; − 0.9321 + 0.9524 + Tm1 2.2875 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Huang et al. (1996). 2 2 Kc Ti Td Comment Kc Ti 0 Model: Method 1 267 Minimum IAE. Equations continued into the footnote on p. 268. 0.4 ≤ ξ m ≤ 1 , 0.05 ≤ τ m Tm1 ≤ 1 . Kc = τ 1 τ − 35.7307 + 14.19 m + 1.4023ξ m + 6.8618ξ m m K m Tm1 Tm1 + τ τ 1 − 0.9773 m + 55.5898ξ m m T T Km m1 m + τ τ 1 53.8651 m ξ m 2 + 11.4911ξ m 3 + 0.8778 m T Km Tm1 m1 + τ 1 − 29.8822 m T Km m1 + τ 1 − 25.4293ξ m 1.4622 − 0.1671ξ m 58981 + 0.0034ξ m m T Km m1 3 −0.6951 0.086 τ + 53.535 m Tm1 2 τ − 3.3093ξ m m Tm1 −0.4762 −1.6624 − 16.9807ξ m 1.1197 −2.1208 τm τm τm 1 3.0836 1.2103 Tm1 + − 25.0355 ξm − 54.9617 ξm − 0.1398e Km Tm1 Tm1 + τ m ξm ξ T 1 − 8.2721e ξm + 6.3542e Tm1 + 1.0479 m m1 ; τm Km 2 τ τ τ Ti = Tm1 0.2563 + 11.8737 m − 1.6547ξ m − 16.1913 m − 9.7061ξ m m Tm1 Tm1 Tm1 3 2 τ τ τ + Tm1 3.5927ξ m 2 + 19.5201 m − 14.5581ξ m m + 2.939ξ m 2 m Tm1 Tm1 Tm1 4 3 τm τm 3 + 50.5163ξ m + Tm1 − 0.4592ξ m − 34.6273 Tm1 Tm1 2 τm 2 τm 3 4 + ξ − ξ 8 . 6966 6 . 9436 + Tm1 8.9259ξ m m m Tm1 Tm1 b720_Chapter-03 268 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 5 4 3 τ τ τ + Tm1 27.2386 m − 20.0697ξ m m − 42.2833ξ m 2 m Tm1 Tm1 Tm1 2 τ τ + Tm1 8.5019 m ξ m 3 − 12.2957 m ξ m 4 + 8.0694ξ m 5 − 2.7691ξ m 6 Tm1 Tm1 6 5 τ τ τ + Tm1 − 7.7887 m + 2.3012ξ m m + 4.6594 m ξ m 5 Tm1 Tm1 Tm1 4 3 2 τ τ τ + Tm1 8.8984 m ξ m 2 + 10.2494 m ξ m 3 − 5.4906 m ξ m 4 ; Tm1 Tm1 Tm1 2 τ τ τ x1 = Tm1 − 0.021 + 3.3385 m + 0.185ξ m − 0.5164 m − 0.9643ξ m m Tm1 Tm1 Tm1 3 2 τ τ τ + Tm1 − 0.8815ξ m 2 + 0.584 m − 12.513ξ m m + 1.3468ξ m 2 m Tm1 Tm1 Tm1 4 3 τm τm 3 4 − ξ 15 . 3014 3 . 1988 + ξ + Tm1 2.3181ξ m − 5.2368 m m T Tm1 m1 2 5 τm τm 2 τm 3 − 2.0411 ξ m + 3.4675 + Tm1 11.9607ξ m Tm1 Tm1 Tm1 4 3 2 τm τm 2 τm 3 + Tm1 − 0.8219ξ m − ξ − ξ 15 . 0718 1 . 8859 m m T T Tm1 m1 m1 6 5 τm τm τm 4 5 + 0.529ξ m ξ m + 2.2821ξ m − 0.9315 + Tm1 0.4841 Tm1 Tm1 Tm1 4 3 τm τm 6 2 3 ξ 7 . 1176 ξ + + Tm1 − 0.6772ξ m − 1.4212 m m T Tm1 m1 2 τm τm 4 5 ; ξ + ξ 0 . 5497 + Tm1 − 2.3636 m m Tm1 Tm1 α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 . FA b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Madhuranthakam et al. (2008). Model: Method 1 3 Kc Ti Td Kc Ti 0 269 Comment Tm1 > Tm 2 ; τm < Tm1 + Tm 2 . α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = K c ; b f 3 = 0 , b f 4 = x1 , a f 5 = 0.1x1 . Madhuranthakam et al. (2008). Model: Method 1 4 Minimum performance index: servo tuning Tm1 > Tm 2 ; 0 Ti τm < Tm1 + Tm 2 . Kc α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = K c ; b f 3 = 0 , b f 4 = x1 , a f 5 = 0.1x1 . 3 0.6202 Tm1 + Tm 2 1 + Kc = K m τm 0.9931 1.4849 τ T + Tm 2 , x1 = 0.0921(τ m + Tm 2 ) m 1 + m1 Tm1 τm , 2 τm τm τ . + 13.81 Ti = τm 1 + m 4.566 − 14.906 τ +T +T m1 m2 m τm + Tm1 + Tm 2 Tm1 1.0409 4 Kc = 0.5723 Tm1 + Tm 2 1 + K m τm , x1 = 0.0943(τm + Tm 2 ) 1.6501 Ti = 0.2476 τm (τm + Tm1 + Tm 2 )1 + Tm1 + Tm 2 Tm1 τ m . 1.4636 τm Tm1 + Tm 2 1 + Tm1 τm , b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 270 Rule Huang et al. (1996). 5 17-Mar-2009 5 Kc Ti Td Comment Kc Ti 0 Model: Method 1 Minimum IAE. Equations continued into the footnote on p. 271; 0 < Tm 2 Tm1 ≤ 1 , 0.1 ≤ τ m Tm1 ≤ 1 . Kc = 1 T τm τ T − 137.8937 m 2 − 122.7832 m m2 2 7.0636 + 66.6512 K m Tm1 Tm1 Tm1 + 0.0865 2.6405 1.0309 Tm 2 τm τ 1 26.1928 m 33 . 6578 3 . 0098 + + T T T Km m1 m1 m1 + 2.345 1.0570 Tm 2 T T 1 − 10.9347 m 2 + 29.4068 m 2 141 . 511 + τm Km Tm1 Tm1 + −0.9450 −0.9282 T τ Tm 2 τ m 1 34.3156 m 2 m 70 . 1035 − Km Tm1 Tm1 Tm1 Tm1 + 0.8866 0.8148 τ m Tm 2 τ T 1 152.6392 m m 2 47 . 9791 − Km Tm1 Tm1 Tm1 Tm1 + τ m Tm 2 τm Tm 2 −0.4062 2 τ 1 ; − 57.9370e Tm1 + 10.4002e Tm1 + 6.7646e Tm1 + 7.3453 m Km Tm 2 τ τ T τ T Ti = Tm1 0.9923 + 0.2819 m − 0.2679 m 2 − 1.4510 m − 0.6712 m m2 2 Tm1 Tm1 Tm1 Tm1 2 3 2 T τ T τ + Tm1 0.6424 m 2 + 2.504 m + 2.5324 m 2 m Tm1 Tm1 Tm1 Tm1 2 3 4 τm Tm 2 τ m Tm 2 − 1.8759 + 2.0500 + Tm1 2.3641 Tm1 Tm1 Tm1 Tm1 4 3 2 2 τ m Tm 2 Tm 2 τ m τ m Tm 2 − 3.4972 − 1.3496 + Tm1 0.8519 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 3 4 5 τm Tm 2 τ m Tm 2 + 0.5862 − 3.1142 + Tm1 − 2.4216 Tm1 Tm1 Tm1 Tm1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 271 4 2 3 T τ T τ + Tm1 0.0797 m 2 m + 0.985 m 2 m Tm1 Tm1 Tm1 Tm1 2 3 5 T τ T + Tm1 1.2892 m m 2 + 1.2108 m 2 ; Tm1 Tm1 Tm1 2 τ τ T τ T x1 = Tm1 0.0075 + 0.3449 m + 0.3924 m 2 − 0.0793 m + 2.7495 m m2 2 Tm1 Tm1 Tm1 Tm1 2 3 2 T τ T τ + Tm1 0.6485 m 2 + 0.8089 m − 9.7483 m 2 m Tm1 Tm1 Tm1 Tm1 2 3 4 τm Tm 2 τ m Tm 2 − 1.0884 − 5.8194 + Tm1 3.4679 Tm1 Tm1 Tm1 Tm1 3 2 2 3 τ m Tm 2 Tm 2 τ m τ m Tm 2 − 3.7055 − 1.4056 + Tm1 12.0049 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 4 5 4 τm Tm 2 Tm 2 τ m − 6.3603 + 0.3520 + Tm1 10.0045 Tm1 Tm1 Tm1 Tm1 2 3 2 3 τ m Tm 2 Tm 2 τ m + 7.0404 + Tm1 − 3.2980 Tm1 Tm1 Tm1 Tm1 4 5 6 τ m Tm 2 Tm 2 τm + Tm1 1.4294 − 6.9064 + 0.0471 Tm1 Tm1 Tm1 Tm1 5 6 4 2 Tm 2 τ m Tm 2 Tm 2 τ m + Tm1 1.1839 + 1.7087 + 1.7444 Tm1 Tm1 Tm1 Tm1 Tm1 3 3 2 4 5 τ m Tm 2 τ m Tm 2 τ m Tm 2 ; + Tm1 − 1.2817 − 2.1281 + 1.5121 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 . b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 272 Rule Huang et al. (1996). 6 17-Mar-2009 6 Kc Ti Td Comment Kc Ti 0 Model: Method 1 Minimum IAE. Equations continued into the footnote on p. 273. 0.4 ≤ ξ m ≤ 1 ; 0.05 ≤ τ m Tm1 ≤ 1 . Kc = τ τ 1 − 8.1727 − 32.9042 m + 31.9179ξm + 38.3405ξm m K m Tm1 Tm1 τ τ τ 1 + 0.2079 m + 35.9456 m + 29.3215 m T Km T Tm1 m1 m1 1 + − 21.4045ξ m 0.1311 + 5.1159ξ m1.9814 − 21.9381ξ m1.737 Km −1.9009 0.1571 + [ 1.2234 ] + −0.1303 1.2008 τm τ 1 − 17.7448ξ m m 26 . 8655 + ξ m T Km Tm1 m1 + τm τ τ 1 − 52.9156 m ξ m 1.1207 − 22.4297 m ξ m 0.3626 − 3.3331e Tm1 Km Tm1 Tm1 τ m ξm ξ T 1 ξm + 8.5175e − 1.5312e Tm1 + 0.8906 m m1 [Huang (2002)]; τm Km 2 τ τ τ Ti = Tm1 1.1731 + 6.3082 m − 0.6937ξ m + 8.5271 m − 24.7291ξ m m Tm1 Tm1 Tm1 3 2 τ τ + Tm1 − 6.7123ξ m m + 7.9559ξ m 2 − 32.3937 m + 5.5268ξ m 6 Tm1 Tm1 4 3 τm τm 2 τm 166 . 9272 36 . 3954 + ξ + ξ + Tm1 − 27.1372 m m T T Tm1 m1 m1 2 τm 2 τm 3 3 4 − ξ − ξ + ξ 22 . 6065 1 . 6084 29 . 9159 + Tm1 − 94.8879ξ m m m m Tm1 Tm1 5 4 3 τm τm 2 τm − 93.8912ξ m − 84.3776ξ m + Tm1 49.6314 Tm1 Tm1 Tm1 2 τm 4 τm 3 5 ξ − ξ 25 . 1896 19 . 7569 ξ − + Tm1 110.1706 m m m T Tm1 m1 b720_Chapter-03 17-Mar-2009 Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 6 5 4 τ τ τ + Tm1 − 12.4348 m − 11.7589ξ m m + 68.3097 m ξ m 2 Tm1 Tm1 Tm1 3 2 τ τ τ + Tm1 − 17.8663 m ξ m 3 − 22.5926 m ξ m 4 + 9.5061 m ξ m 5 ; Tm1 Tm1 Tm1 2 τ τ τ x1 = Tm1 0.0904 + 0.8637 m − 0.1301ξ m + 4.9601 m + 14.3899ξ m m Tm1 Tm1 Tm1 3 2 τ τ + Tm1 0.7170ξ m 2 − 12.5311 m − 42.5012ξ m m + 17.0952ξ m 4 Tm1 Tm1 4 τm 2 τm 3 − 6.9555ξ m − 12.3016 + Tm1 − 21.4907ξ m Tm1 Tm1 3 2 τm τm 2 τm 3 + ξ 7 . 5855 19 . 1257 + ξ + Tm1 102.9447ξ m m m T Tm1 Tm1 m1 5 4 3 τm τm 2 τm − 110.0342ξ m − 17.2130ξ m + Tm1 10.8688 Tm1 Tm1 Tm1 2 τm 4 τm 3 5 6 ξ − ξ + ξ 16 . 7073 16 . 2013 5 . 4409 ξ − + Tm1 50.6455 m m m m T Tm1 m1 6 5 4 τm τm τm 2 ξ 10 . 9260 29 . 4445 + − ξ + Tm1 − 0.0979 m m T T Tm1 m1 m1 3 2 τm τm τm 3 4 5 ; ξ + ξ 24 . 1917 6 . 2798 ξ − + Tm1 21.6061 m m m T Tm1 Tm1 m1 α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 10 . FA 273 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 274 Rule Majhi and Atherton (1999), Atherton and Boz (1998). Model: Method 1 Kc 7 Ti Comment Td Direct synthesis: time domain criteria Coefficient x3 0 values deduced 1 1 1 + + from graphs. τm Tm 2 Tm1 Kc α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = x1 ; b f 3 = 1 , bf 4 = x 2 , a f 5 = 0 . x3 x4 x5 x6 x3 x4 x5 x6 1.2 2.6 4.5 6.4 1.8 4.2 6.9 10.0 Minimum ITSE – 1.8 servo – Majhi 4.2 and Atherton 6.9 (1999). 10.0 Minimum ISTSE 2.3 – servo – 5.2 Atherton and Boz 8.4 (1998). 12.4 Minimum ISTSE 2.3 – servo – Majhi 5.2 and Atherton 8.7 (1999). 12.4 0.58 0.91 0.89 0.98 0.17 0.22 0.25 0.26 0.17 0.22 0.25 0.26 0.08 0.11 0.14 0.13 0.08 0.11 0.12 0.13 -0.81 -0.55 -0.30 -0.22 -0.31 -0.16 -0.11 -0.08 -0.29 -0.16 -0.11 -0.08 -0.16 -0.10 -0.07 -0.04 -0.16 -0.10 -0.06 -0.04 2.36 2.49 2.18 2.23 1.08 0.96 0.95 0.92 1.05 0.96 0.95 0.92 0.71 0.65 0.68 0.63 0.69 0.65 0.62 0.64 8.5 10.8 13.3 16.0 13.0 16.2 20.3 24.0 13.0 16.8 20.3 1.02 1.03 1.02 1.00 0.28 0.31 0.29 0.30 0.28 0.27 0.29 -0.16 -0.14 -0.12 -0.09 -0.07 -0.06 -0.04 -0.04 -0.06 -0.05 -0.04 2.18 2.16 2.13 2.10 0.94 0.99 0.93 0.95 0.94 0.90 0.97 16.5 20.4 25.2 31.2 16.5 20.4 25.2 0.14 0.15 0.15 0.14 0.14 0.15 0.15 -0.04 -0.03 -0.02 -0.02 -0.03 -0.03 -0.02 0.64 0.65 0.64 0.61 0.64 0.67 0.65 Minimum ISE – servo – Atherton and Boz (1998). 7 Equations developed from information provided by Majhi and Atherton (1999). (T T + τmTm1 + τmTm 2 )3 , x = 1 1 − (Tm1Tm 2 + τmTm1 + τmTm 2 )3 x , K c = x 4 m1 m 2 1 5 2 2 2 K m τm Tm1 Tm 2 K m τm 2Tm12Tm 2 2 (T T + τ T + τ T )2 m m1 m m2 m1 m 2 x 6 − (Tm1 + Tm 2 − τ m ) τ m Tm1Tm 2 τm + Tm 2 ≤ 0. 8 . x2 = , 3 Tm1 (T T + τm Tm1 + τ m Tm 2 ) x 5 1 − m1 m 2 τ m 2 Tm12 Tm 2 2 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule 8 Hansen (1998). Model: Method 1 Kc Ti Td Kc Ti 0 275 Comment χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = K c − (1 K m ) , b f 4 = x 2 , a f 5 = 0 . 9 Ti Td Kc Arvanitis et al. 10 Ti Td Kc (2000a). Model: Method 1 α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b = 0 , a f1 f 1 = Ti , a f 2 = Ti Td ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 0 . 8 Kc = [ [ 2 Tm1Tm 2 + (Tm1 + Tm 2 )τm + 0.5τm 2 ] 3 9K m Tm1Tm 2 τm + 0.5(Tm1 + Tm 2 )τm 2 + 0.167 τm 3 ] 2 , Ti = 1.54 x1 , α = 0.802 , nearly zero overshoot; Ti = 1.27 x1 , α = 0.711 , nearly minimum IAE; Ti = 1.75x1 , α = 0.857 , conservative tuning; ] [T T + (T + T )τ + 0.5τ ] , 2[T T + (T + T )τ + 0.5τ ] T +T +τ . x = − K 3K [T T τ + 0.5(T + T )τ + 0.167 τ ] x1 = [ 3 Tm1Tm 2 τm + 0.5(Tm1 + Tm 2 )τm + 0.167 τm 2 3 2 m1 m 2 m1 m2 m m 2 2 m1 m 2 2 m1 m2 m m m1 2 m m1 m 2 m m1 m2 m 3 m m2 m m xT T Tm1τm 9 ; K c = 1 m1 2ξm + m1 , 0 < x1 < 1 ; suggested x1 = 0.4 ; Td = τm K m τm x1 ( 2ξ m τm + Tm1 ) Ti = 10 Kc = 1 1 1 τm x + 2ξ 2 − 1 + 2ξ − 1 x 2 + ξ2 − 1 , with 2 2 x2 x1 x1 x1 2 1 τm τm 2 1+ + ξ2 − 1 ≥ 0 , x 2 = 1 + . x1Tm1 (2ξ m τm + Tm1 ) x1Tm1 (2ξm τm + Tm1 ) x1 1 (2ξTCL 2 + τm )(2ξm τm + Tm1 )Tm1 − τmTCL 2 2 , K m τm TCL 2 2 + τm (2ξTCL 2 + τm ) (2ξTCL 2 + τm )(2ξm τm + Tm1 )Tm1 − τmTCL 2 2 , τm 2 + τm 2Tm1 (2ξ m τm + Tm1 ) 2 2 Tm1 [TCL 2 + τm (2ξTCL 2 + τm )] Td = . (2ξTCL 2 + τm )(2ξm τm + Tm1 )Tm1 − τmTCL 22 Ti = b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 276 Rule Huang et al. (2000). Model: Method 1 Kc Ti Td 11 K c Ti 0 Comment α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 x 2 , x 2 not specified. Åström and Hägglund (2006), pp. 252, 265. Model: Method 24 11 Kc = λ 12 Kc Direct synthesis: frequency domain criteria Ti Td M s = M t = 1.4 α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 ; a f 3 = x1 , Td 30 ≤ x1 ≤ Td 3 ; 2 a f 4 = 0 . 5 x1 ; K 0 = 0 . 2 2Tm1ξ m + 0.4τm T + 0.8Tm1ξ m τm , Ti = 2Tm1ξ m + 0.4τm , x1 = m1 ; K m τm 0.8Tm1ξm τm λ = 0.5, τm ξ m Tm1 ≤ 1 , λ = 0.6,1 < τm ξm Tm1 ≤ 2 , λ = 0.7, τmξ m Tm1 > 2 , servo; λ = 0.6, τm ξ m Tm1 ≤ 1 , λ = 0.7,1 < τm ξ m Tm1 ≤ 2 , λ = 0.8, τm ξm Tm1 > 2 , regulator. 12 Kc = 1 Tm1 T + 0.5x1 T (T + 0.5x1 ) + 0.18 m 2 + 0.02 m1 m 2 0.19 + 0.37 , Km τm + 0.5x1 τm + 0.5x1 τm + 0.5x1 Tm1 T + 0.5x1 T (T + 0.5x1 ) + 0.18 m 2 + 0.02 m1 m 2 τm + 0.5x1 τm + 0.5x1 τm + 0.5x1 Ti = , Tm1 (Tm 2 + 0.5x1 ) Tm 2 + 0.5x1 0.48 Tm1 0 . 0012 0 . 0007 + + 0.03 − τm + 0.5x1 (τm + 0.5x1 )3 (τm + 0.5x1 )2 (τm + 0.5x1 )2 0.19 + 0.37 Td = Tm1 (Tm 2 + 0.5x1 ) τm + 0.5x1 Tm1 Tm 2 + 0.5x1 Tm1 (Tm 2 + 0.5x1 ) 0.19 + 0.37 + 0.18 + 0.02 τ m + 0. 5 x 1 τm + 0.5x1 τ m + 0. 5 x 1 0.29(τm + 0.5x1 ) + 0.16Tm1 + 0.2(Tm 2 + 0.5x1 ) + 0.28 Tm1 + Tm 2 + 0.5x1 T +T +τ +x . m2 m 1 m1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 277 3.6 SOSPD Model with a Zero K (1 + sTm 3 )e −sτ m K m (1 + Tm 3 s )e − sτ m G m (s) = m or G m (s) = 2 (1 + sTm1 )(1 + sTm 2 ) Tm1 s 2 + 2ξ m Tm1s + 1 or G m (s) = K m (1 − Tm 4 s )e − sτ m K m (1 − Tm 4 s )e −sτ m or G m (s) = 2 (1 + sTm1 )(1 + sTm 2 ) Tm1 s 2 + 2ξ m Tm1s + 1 1 3.6.1 Ideal PI controller G c (s) = K c 1 + Ti s R(s) + 1 K c 1 + Ti s E(s) − U(s) Model Y(s) Table 28: PI controller tuning rules – SOSPD model with a zero K m (1 − Tm 4s )e −sτ m K m (1 + Tm3s )e −sτ m K m (1 − Tm 4s )e −sτ m G m (s ) = or or 2 2 (1 + sTm1 )(1 + sTm 2 ) Tm12s 2 + 2ξmTm1s + 1 Tm1 s + 2ξ m Tm1s + 1 Rule Chien et al. (2003). Model: Method 1 Kc 1 Kc 2 Kc Ti Direct synthesis: time domain criteria Tm1 Tm1 > Tm 2 Ti Pomerleau and Poulin 1 Tm1 (2004). K m Tm1 + Tm 4 + τm Model: Method 3 1 Kc = Comment 1.5Tm1 OS < 10%; Tm1 = Tm 2 . Tm1 , K m (1.414x1 + Tm 4 + τm ) x1 = 0.707Tm 2 + 0.5 4(Tm 4 τm + Tm 2Tm 4 + Tm 2 τm ) + 2Tm 2 . 2 2 Kc = Ti , K m (2.414x1 + Tm 4 + τm − Ti ) 4.828ξ m Tm1x1 + 2ξ mTm1 (Tm 4 + τm ) + Tm 4 τm − 2.414 x1 , 2ξ m Tm1 + Tm 4 + τm 2 Ti = x1 obtained numerically from the solution of a cubic equation. b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 278 Rule Kc Comment Ti Khodabakhshian and Golbon (2004). Direct synthesis: frequency domain criteria Ti Tm1 > Tm 2 3 Kc Marchetti and Scali (2000). Model: Method 1 Also Desbiens (2007), p. 90. Model: Method 1 Robust Negative zero. 2ξ m Tm1 + 0.5τm 2ξ mTm1 + 0.5τm K m (2λ + τm ) 2ξm Tm1 + 0.5τm K m (2λ + τm + 2Tm 4 ) Positive zero. 2ξ m Tm1 + 0.5τm Ultimate cycle Luyben (2000). Tm 4 Tm1 values. x 1 values. 3 Kc = Ti Km K u x1 K m 4 Model: Method 1 Tu x 2 0.2 0.4 0.8 1.6 3.9 3.3 2.8 3.0 τm Tm 4 = 0.2 3.8 3.4 3.0 2.9 τm Tm 4 = 0.4 4.1 3.6 3.2 3.1 τm Tm 4 = 0.6 4.3 3.8 3.4 3.2 τm Tm 4 = 0.8 4.8 4.3 3.7 3.3 τm Tm 4 = 1.0 5.2 4.6 3.9 3.4 τm Tm 4 = 1.2 5.5 5.0 4.4 3.7 τm Tm 4 = 1.4 6.0 5.4 4.7 3.8 τm Tm 4 = 1.6 (Tm1Tm 2 )2 ω6 + (Tm12 + Tm 22 )ω4 + ω2 , with ω given by solving (TiTm 4 )2 ω4 + (Ti 2 + Tm 4 2 )ω2 + 1 1 − 10−0.1M max = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 4ω) − tan −1 (Tm 2ω) cos −1 2 − tan −1 (Tm1 ω) − τ m ω with recommended M max = 0dB (Khodabakhshian and Golbon, 2004); ω given by solving, for φm = 58.13 (Desbiens, 2007): 1.015 = 0.5π + tan −1 (Ti ω) − tan −1 (Tm 4ω) − tan −1 (Tm 2ω) − tan −1 (Tm1 ω) − τ m ω ; 2 T T τ τ Ti = Tm1 1 + 0.175 m + 0.3 m 2 + 0.2 m 2 , m < 2 ; Tm1 Tm1 Tm1 Tm1 2 Tm 2 T τ τm Ti = Tm1 0.65 + 0.35 + 0.2 m 2 , m > 2 . + 0.3 Tm1 Tm1 Tm1 Tm1 4 x 2 = (0.5 + 1.56[Tm 4 Tm1 ]) + (3.44 − 1.56[Tm 4 Tm1 ])(τm Tm 4 ) ; Tm1 = Tm 2 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 1 + Td s 3.6.2 Ideal PID controller G c (s) = K c 1 + Ti s R(s) E(s) − + 1 K c 1 + + Td s T s i U(s) Y(s) Model Table 29: PID controller tuning rules – SOSPD model with a zero K m (1 + sTm3 )e −sτ m K m (1 − sTm 4 )e −sτ m K m (1 − sTm 4 )e −sτ m G m (s ) = or or (1 + sTm1 )(1 + sTm 2 ) Tm12s 2 + 2ξmTm1s + 1 Tm1 2 s 2 + 2ξ m Tm1s + 1 Rule Kc Wang et al. (1995a). 1 Ti Td Comment Minimum performance index: servo tuning p 0 q 2 − p 2 p1 Model: Method 1 p 1 p 0 q 1 − p1 q1 − 1 − 1 r p0 p 0 q 1 − p1 p 0 Km p0 2 p 0 = Tm1 + Tm 2 + Tm 4 + τm , p1 = Tm1Tm 2 + 0.5τm (Tm1 + Tm 2 ) − 0.5τm Tm 4 , p 2 = 0.5Tm1Tm 2 τ m , q 1 = Tm1 + Tm 2 + 0.5τ m , q 2 = Tm1Tm 2 + 0.5τ m (Tm1 + Tm 2 ) , r = δ0 + δ1 T T τ T T τ τm + δ 2 m 2 + δ3 m 4 + δ4 m + δ7 m 2 + δ9 m 4 m Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 τ τ T T T T T T + δ7 m + δ5 m 2 + δ8 m 4 m 2 + δ9 m + δ8 m 2 + δ6 m 4 m 4 ; Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Tm1 Minimum IAE: δ 0 = 3.5550 , δ1 = −3.6167 , δ 2 = 2.1781 , δ 3 = −5.5203 , δ 4 = 1.4704 , δ 5 = −0.4918 , δ 6 = 2.5356 , δ 7 = −0.4685 , δ 8 = −0.3318 , δ 9 = 1.4746 . Minimum ISE: δ 0 = 3.9395 , δ1 = −3.2164 , δ 2 = 1.6185 , δ 3 = −5.8240 , δ 4 = 1.0933 , δ 5 = −0.6679 , δ 6 = 2.5648 , δ 7 = −0.2383 , δ 8 = −0.0564 , δ 9 = 1.3508 . Minimum ITAE: δ 0 = 3.2950 , δ1 = −3.4779 , δ 2 = 2.5336 , δ 3 = −5.5929 , δ 4 = 1.4407 , δ 5 = −0.3268 , δ 6 = 2.7129 , δ 7 = −0.5712 , δ 8 = −0.5790 , δ 9 = 1.5340 . 279 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 280 Rule Kc Chidambaram (2002), p. 157. Model: Method 1 2 Kc 3 Kc 4 Kc 5 Ti Comment Td Direct synthesis: time domain criteria ‘Smaller’ Tm1Tm 2 Tm1 + Tm 2 τm Tm 4 . Tm1 + Tm 2 ‘Larger’ τm Tm 4 . 0.5 2ξ m Tm1 Tm1 ξm Kc ‘Smaller’ τm Tm 4 . ‘Larger’ τm Tm 4 . Direct synthesis: frequency domain criteria Wang et al. (2000b), (2001a). Model: Method 2 6 2ξ m Tm1 Kc Tm1 2 M s = 1.5 Robust Marchetti and 2ξ mTm1 + 0.5τm Scali (2000). K m (2λ + τm ) Model: Method 1 7 Kc 2 3 4 5 6 Kc = Kc = 1.571(Tm1 + Tm 2 ) 2ξ m Tm1 + 0.5τm Tm1 (Tm1 + ξm τm ) 2ξm Tm1 + 0.5τm ( ) . ( ) . ) . ) . K m τm 1 + 3.142 Tm 4 τm 2 2.358(Tm1 + Tm 2 ) K m τm 1 + 4.712 Tm 4 τm 2 Negative zero. Positive zero. 2 Kc = 3.142ξ mTm1 ( K m τm 1 + 3.142 Tm 4 τm 2 2 Kc = Kc = 4.716ξ m Tm1 ( K m τm 1 + 4.712 Tm 4 τm 2 1 cos θ 2ξ m Tm1ω ; ω is determined by solving the M s cos ωτ m − tan −1 (x1ω) K x 2ω2 + 1 [ ] m 1 cos θ = tan −1 (x1ω) − ωτ m − 0.5π , with equation: tan −1 M s − sin θ ( ( ) ) x τ ω2 − 1 cos(ωτ m ) − ωτ m sin (ωτ m ) θ = tan −1 1 m 2 ; x1 = Tm3 or −Tm 4 , as appropriate. x1τm ω − 1 sin (ωτ m ) + ωτ m sin (ωτ m ) 7 Kc = 2ξm Tm1 + 0.5τm . K m (2λ + τm + 2Tm 4 ) b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Lee et al. (2006). K (T + τ ) m CL m Model: Method 1 9 Kc 8 Comment 8 Ti Td Recommended τm ≤ TCL ≤ 2τm . Ti Td Recommended τ m ≤ λ ≤ 2τm . 2 Desired closed loop transfer function = 9 Td 0 . 5τ m T 0. 5 τ m τ (0.1667 τm − 0.5Tm 3 ) , Td = + m1 + m − Tm3 ; TCL + τm Ti Ti (TCL + τm ) TCL + τm 2 Ti = 2ξm Tm1 − Tm3 + Ti 281 2 2 e − sτ m . TCLs + 1 T (τ − λ ) + 0.5τm Ti , Ti = 2ξm Tm1 + m 4 m , K m (λ + τm + 2Tm 4 ) λ + τm + 2Tm 4 2 Kc = Tm 4 (τm − λ ) + 0.5τm T τ (0.1667 τm + 0.5Tm 4 ) + m1 − m ; λ + τm + 2Tm 4 Ti Ti (λ + τm + 2Tm 4 ) 2 Td = Desired closed loop transfer function = (1 − sTm 4 )e −sτ . (1 + sTm 4 )(1 + sλ ) m 2 2 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 282 3.6.3 Ideal controller in series with a first order lag 1 1 + Td s G c (s) = K c 1 + Ti s Tf s + 1 R(s) + E(s) U(s) 1 1 K c 1 + + Td s Ti s Tf s + 1 − Y(s) Model Table 30: PID controller tuning rules – SOSPD model with a zero K m (1 + sTm3 )e −sτ m K m (1 + Tm3s )e −sτ m K m (1 − sTm 4 )e −sτ m G m (s ) = or or (1 + sTm1 )(1 + sTm 2 ) (1 + sTm1 )(1 + sTm 2 ) Tm1 2 s 2 + 2ξ m Tm1s + 1 Rule Chien et al. (2003). Model: Method 1; Tf = Tm3 . Kc Ti Comment Td Direct synthesis: time domain criteria 2ξ m Tm1 0.829ξm Tm1 Tm1 K m τm 2ξ m 1 Kc 2 Kc Pomerleau and Tm1 1 Poulin (2004). K m Tm1 + τ m Model: Method 3 Tm1 Tm1 0 Ti 0 1.5Tm1 0 ( Tm1 = Tm 2 , Tf = Tm 3 ; OS < 10%. ) 2 2 , x 1 = 0.707T m 2 + 0.5 2 T m 2 + τ m + 4T m 2 τ m . 1 Kc = 2 Kc = Ti , K m (2.414 x1 + τm − Ti ) Ti = 4.828ξm Tm1x1 + 2ξ m Tm1τm + 0.5τm − 2.414 x1 , 2ξm Tm1 + τm K m (1.414 x1 + Tm3 + τm ) Tm1 > Tm 2 2 2 x1 is obtained numerically from the solution of a cubic equation. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Li et al. (2004). Model: Method 1 Kc Ti Td Kc Tm1 + Tm 2 Tm1Tm 2 Tm1 + Tm 2 2ξm Tm1 K m (λ + τm ) 2ξ m Tm1 3 283 Comment Robust Bialkowski (1996). Model: Method 1 Chen et al. (2006). Model: Method 1 Chen et al. (2006). 3 Kc = 4 Kc 0.6K u Tm1 2ξ m Tf = Tm3 or Tm 4 ; λ not defined. Tm1Tm 2 Tm1 + Tm 2 τm ≤ λ ≤ 10τm Ultimate cycle 0.5Tu 0.125Tu Tf = 0.0125Tu Tm1 + Tm 2 2 T − Tm 4 τm Tm1 + Tm 2 ; , Tf = CL 2 K m (2TCL 2 + Tm 4 + τm ) 2TCL 2 + Tm 4 + τm desired closed loop transfer function = − 4 Kc = (λ − τ m )Tm 4 Tm1 + Tm 2 . , Tf = K m (λ + 2Tm 4 + τ m ) λ + 2Tm 4 + τ m (1 − sTm 4 )e−sτ . (TCL 2s + 1)2 m b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 284 Td s 1 + 3.6.4 Controller with filtered derivative G c (s) = K c 1 + Ti s 1 + Td s N R(s) + U(s) T s 1 d + K c 1 + Td Ti s s 1+ N E(s) − Y(s) Model Table 31: PID controller tuning rules – SOSPD model with a zero K (1 + sTm 3 )e − sτ m K (1 − sTm 4 )e −sτ m K m (1 + sTm3 )e −sτ m or or m or G m (s) = m 2 2 (1 + sTm1 )(1 + sTm 2 ) Tm1 s + 2ξmTm1s + 1 (1 + sTm1 )(1 + sTm 2 ) K m (1 − sTm 4 )e −sτ m Tm12s 2 + 2ξ mTm1s + 1 Rule Kc Chien et al. (2003). Model: Method 1 0.829ξm Tm1 K m τm 2 Kc Ti Td Comment Direct synthesis: time domain criteria 1 2ξ m Tm1 − Tm 3 Td 2ξ m Tm1 − 0.1Td Td N = 10 2 1 Td = Tm1 Tm1 − Tm3 , N = −1 . 2ξ m Tm1 − Tm3 Tm 3 (2ξ mTm1 − Tm3 ) 2 Kc = Tm1 2 , x1 = 0.1Td + 0.5 4Tm 4 τm + 0.4Td Tm 4 + 0.4Td τm + 0.04Td ; K m (2 x1 + Tm 4 + τm ) 2 Td = Tm1 10ξ m − 3.02 11ξm − 1 , ξm > 0.302 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Td Comment Td Tm1 > Tm 2 > Tm 3 285 Robust 3 Kc Chien (1988). ξ 2 T m m1 − Tm 3 Model: Method 1 K m (λ + τ m ) Ti 2ξ m Tm1 − Tm 3 4 Td 5 Kc Ti Td 6 Kc Ti Td Tm1 > Tm 2 > Tm 4 N = 10; λ = [Tm1 , τ m ] (Chien and Fruehauf, 1990). 3 Kc = 4 Td = 5 Tm1 + Tm 2 − Tm3 T T − (Tm1 + Tm 2 − Tm3 )Tm3 . , Ti = Tm1 + Tm 2 − Tm3 , Td = m1 m 2 K m (λ + τm ) Tm1 + Tm 2 − Tm 3 Tm2 1 − (2ξ m Tm1 − Tm 3 )Tm3 . 2ξ m Tm1 − Tm3 Tm 4 τm Tm 4 τm λ + Tm 4 + τm , Ti = Tm1 + Tm 2 + , K m (λ + Tm 4 + τm ) λ + Tm 4 + τm Tm1 + Tm 2 + Kc = Td = 6 Tm 4 τm Tm1Tm 2 + . λ + Tm 4 + τm Tm1 + Tm 2 + (Tm 4 τm (λ + Tm 4 + τm )) Tm 4 τm Tm 4 τm λ + Tm 4 + τm , Ti = 2ξm Tm1 + , K m (λ + Tm 4 + τm ) λ + Tm 4 + τm 2ξ m Tm1 + Kc = 2 Td = Tm 4 τm Tm1 + . λ + Tm 4 + τm 2ξ m Tm1 + (Tm 4 τm (λ + Tm 4 + τm )) b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 286 1 1 + Td s 3.6.5 Classical controller G c (s) = K c 1 + Ti s 1 + Td s N R(s) + 1 1 + Td s U(s) K c 1 + Td Ti s s 1 + N E(s) − Y(s) Model Table 32: PID controller tuning rules – SOSPD model with a zero K (1 + sTm 3 )e − sτ m K (1 − sTm 4 )e −sτ m or m G m (s) = m (1 + sTm1 )(1 + sTm 2 ) (1 + sTm1 )(1 + sTm 2 ) Rule Kc Zhang (1994). Model: Method 1 Tm1 K m τm Chien et al. (2003). Model: Method 1; Tm1 > Tm 2 . 0.414Tm1 K m τm Poulin et al. (1996). Model: Method 1 Kc = Comment Td Direct synthesis: time domain criteria Tm1 Tm 2 ‘Small’ τ m ; ‘good’ servo response. N = Tm 2 Tm 3 or N = ∞ . 1 Tm1 Tm 2 N = Tm 2 Tm3 N = 10 Kc Direct synthesis: frequency domain criteria Tm1 Tm1 Tm 2 N = Tm 2 Tm 3 K m (Tm1 + τ m ) Minimum φ m = 90 0 ; φ m = 60 0 , τ m = Tm1 . 2 1 Ti Kc Tm1 Tm 2 Minimum φ m = 650 . Tm1 , K m (2 x1 + Tm 4 + τm ) 2 x1 = 0.1Tm 2 + 0.5 4Tm 4 τm + 0.4Tm 2Tm 4 + 0.4Tm 2 τm + 0.04Tm 2 . 2 Kc = T (T + 2Tm 4 ) Tm1 , φ m = 56 0 when Tm 4 = Tm1 = τ m . , N = m 2 m1 K m (Tm1 + 2Tm 4 + τ m ) Tm1Tm 4 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule O’Dwyer (2001a). Model: Method 1 Kc Ti Td x1Tm1 K m τm Tm1 Tm 2 Comment A m = 0.5π x 1 ; φ m = 0.5π − x 1 ; N = Tm 2 Tm 4 ; sample result: 0.785Tm1 K mτm Tm1 Tm 2 A m = 2.0; φ m = 450 . Robust Chien (1988). Model: Method 1 Chien (1988). Model: Method 1 Chien (1988). Model: Method 1; Tm1 > Tm 2 > Tm 4 . Chien (1988). Model: Method 1 Bialkowski (1996). Model: Method 1 Tm1 > Tm 2 ; N = 10; [Tm1 , τ m ] λ ∈ Tm1 (Chien and Tm1 Tm 2 K m (λ + τ m ) Fruehauf, 1990). Tm 2 1.1(Tm1 − Tm 3 ) Tm1 > Tm 2 > Tm 3 Tm 2 K m (λ + τ m ) N = 11; [Tm1 , τ m ] λ ∈ Tm1 1.1(Tm 2 − Tm 3 ) Tm1 (Chien and K m (λ + τ m ) Fruehauf, 1990). Tuning rules converted to this equivalent controller architecture. N = 10; Tm 2 Tm1 Tm 2 [Tm1 , τ m ] λ ∈ K m (λ + τ m ) (Chien and Tm1 Tm 2 Tm1 Fruehauf, 1990). K m (λ + τ m ) Tm 2 K m (λ + τ m ) Tm 2 Tm1 Tm 2 K m (λ + τ m ) Tm 2 3 Td Tm1 K m (λ + τ m ) Tm1 4 Td N = 11; λ ∈ [T, τ m ] , T = dominant time constant (Chien and Fruehauf, 1990). Tuning rules converted to this equivalent controller architecture. Tm1 Tm 2 Tm1 > Tm 2 Tm1 K m (λ + τm ) 3 Td = 1.1Tm1 + 1.1Tm 4 τm . λ + Tm 4 + τm 4 Td = 1.1Tm 2 + 1.1Tm 4 τ m . λ + Tm 4 + τ m N = Tm 2 Tm3 or N = Tm 2 Tm 4 . 287 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 288 3.6.6 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N R(s) + E(s) − 2 T s 1 d b f 0 + b f 1s + b f 2 s K c 1 + + Ti s T 1 + a f 1s + a f 2 s 2 1+ d s N U(s) Model Y(s) Table 33: PID controller tuning rules – SOSPD model with a zero G m (s) = Rule K m (1 + sTm 3 )e −sτ m 2 Tm1 s 2 + 2ξ m Tm1s + 1 Kc or K m (1 − sTm 4 )e−sτ m Tm12s 2 + 2ξ mTm1s + 1 Ti Td Comment 0.167 τm Negative zero. Robust Shamsuzzoha and Lee (2006a). Model: Method 1 0.5τm K m (λ + τm ) 1 0.5τm Positive zero. Kc af1 = 0.5(λ + Tm3 )τm + Tm3λ − 0.083τm + 0.5Tm3τm or λ + τm af1 = 0.5(λ + Tm 4 )τm + Tm 4λ − 0.083τm − 0.5Tm 4 τm ; λ + τm + 2Tm 4 2 2 af 2 = 0.5τm Tm3 (λ − 0.167 τm ) 0.5τm Tm 4 (λ + 0.167 τm ) or a f 2 = ; λ + τm λ + τm + 2Tm 4 2 b f 0 = 1 , b f 1 = 2ξ m Tm1 , b f 2 = Tm1 , N = ∞ . λ = x1τm ; x1 = 1.360 , M s = 1.4 ; x1 = 0.677 , M s = 1.6 ; x1 = 0.372 , M s = 1.8 ; x1 = 0.337 , M s = 2.0 . 1 Kc = 0.5τm . K m (λ + τm + 2Tm 4 ) b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 289 3.6.7 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N E(s) R(s) + − T s 1 d K c 1+ + Ti s Td 1+ s N − U(s) Y(s) Model + Table 34: PID controller tuning rules – SOSPD model with a zero K m (1 + sTm 3 )e −sτ m K m (1 − sTm 4 )e −sτ m K m (1 + sTm3 )e−sτ m G m (s) = or or or (1 + sTm1 )(1 + sTm 2 ) (1 + sTm1 )(1 + sTm 2 ) Tm12s 2 + 2ξmTm1s + 1 K m (1 − sTm 4 )e −sτ m Tm12s 2 + 2ξ m Tm1s + 1 Rule Kc Madhuranthakam et al. (2008). Model: Method 1 1 Ti Kc = c α = 0 , β = 1 , N = 10. 0.4938 Tm1 + Tm 2 − Tm 3 1 + K m τm Ti = 12.585 Comment Minimum performance index: regulator tuning Ti Td Tm1 > Tm3 K 1.3594 1 Td , 0.1 ≤ τm ≤ 0.66 ; τm + Tm1 + Tm 2 − Tm3 Tm 2 (Tm1 + Tm 2 − Tm3 ) Tm1 + Tm 2 − Tm3 1 + τm τm 2.4858 , T T + Tm 2 − Tm3 Td = 6.465(Tm1 + Tm 2 − Tm 3 ) m 2 1 + m1 τm τm 2.9185 . b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 290 Rule Madhuranthakam et al. (2008). Model: Method 1 Chien et al. (2003). Model: Method 1 Huang et al. (2000). Model: Method 1 Kc 2 Kc Ti Kc = Comment Td Minimum performance index: servo tuning Ti Td Tm1 > Tm3 α = 0 , β = 1 , N = 10. Direct synthesis: time domain criteria 3 4 Ti Kc Kc α = 0 ; β =1; N = 10. Td Direct synthesis: frequency domain criteria N = min[20, 2Tm1 ξ m + 0.4τ m Td 20Td ] 1.2206 2 FA 0.5254 Tm1 + Tm 2 − Tm 3 1 + K m τm , 0.1 ≤ τm ≤ 0.66 ; τm + Tm1 + Tm 2 − Tm3 1.4569 T + Tm 2 − Tm3 Ti = 0.5657 τm 1 + m1 τm , Td = 6.7572(Tm1 + Tm 2 − Tm3 ) 3 Kc = T + Tm 2 − Tm 3 Tm 2 1 + m1 τm τm Tm1 , x1 = 0.1Td + 0.5 4Tm 4 τm + 0.4Td Tm 4 + 0.4Td τm + 0.04Td 2 ; K m (2 x1 + Tm 4 + τm ) 2 Ti = 2ξ m Tm1 − 0.1Td ; Td = Tm1 10ξ m − 3.02 11ξm − 1 , ξm > 0.302 . 4 Kc = λ α= 2 T + 0.8Tm1ξ m τm 2Tm1ξ m + 0.4τm 2T ξ + 0.4τm ; or K c = λ m1 m , Td = m1 K m (τm + 2Tm 3 ) K m (τm + 2Tm 4 ) 0.8Tm1ξm τm 2Tm1ξ m 0.5Tm1 2 −1 ; β = −1 ; 2Tm1ξ m + 0.4 τ m Tm1 2 + 0.8Tm1ξ m τ m λ = 0.6, τm τ ξ m ≤ 1 , A m = 2.83 ; λ = 0.7,1 < m ξ m ≤ 2 , A m = 2.43 ; Tm1 Tm1 λ = 0.8, τm ξ m > 2 , A m = 2.13 . Tm1 2.9057 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Jyothi et al. (2001). Model: Method 1; α = 0 , β =1, N =∞. Kc Ti Td Comment 5 Kc Tm1 + Tm 2 τm Tm3 < 1 6 Kc Tm1Tm 2 Tm1 + Tm 2 7 Kc τm Tm3 < 1 8 Kc Tm1 2ξ m 0.5(Tm1 + Tm 2 ) 0.5Tm1Tm 2 Tm1 + Tm 2 τm Tm 4 < 1 Tm1 + Tm 2 Tm1Tm 2 Tm1 + Tm 2 2ξ m Tm1 Tm1 2ξ m 2ξ m Tm1 Tm1 + Tm 2 K m Tm 4 9 0.5 Kc Tm1 + Tm 2 K m Tm 4 10 Kc ξm Tm1 K m Tm 4 11 5 6 7 8 9 10 11 Kc = Kc = Kc = Kc = Kc = Kc = Kc = Kc 1.57(Tm1 + Tm 2 ) K m τm 1 + 9.870(Tm3 τm )2 0.79(Tm1 + Tm 2 ) K m τm 1 + 2.467(Tm3 τm )2 3.14ξ m Tm1 K m τm 1 + 9.870(Tm3 τm )2 1.57ξ m Tm1 K m τm 1 + 2.467(Tm3 τm )2 1.57(Tm1 + Tm 2 ) K m τm 1 + 2.467(Tm 4 τm )2 0.79(Tm1 + Tm 2 ) K m τm 1 + 2.467(Tm 4 τm )2 1.57ξ m Tm1 K m τm 1 + 2.467(Tm 4 τm )2 . . . . . . . τm Tm3 > 1 τm Tm3 > 1 τm Tm 4 > 1 τm Tm 4 < 1 τm Tm 4 > 1 τm Tm 4 < 1 τm Tm 4 > 1 291 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 292 3.6.8 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) − Y(s) Model F2 (s) Table 35: PID controller tuning rules – SOSPD model with a zero K m (1 + sTm 3 )e −sτ m K m (1 − sTm 4 )e −sτ m G ( s ) = or G m (s) = m Tm12s 2 + 2ξ mTm1s + 1 Tm1 2 s 2 + 2ξ m Tm1s + 1 Rule Kc 1 Huang et al. (2000). Model: Method 1 Kc Ti Td Comment Direct synthesis: time domain criteria 2Tm1ξ m + 0.4τ m 0 α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = x 1 ; a f 5 = x1 x 2 , x 2 not specified. Chien et al. (2003). Model: Method 1 2 Kc Tm1 0 Tm1 > Tm 2 α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 ; b f 3 = 1 , b f 4 = Tm 2 , a f 4 = 0.1Tm 2 . 1 Kc = λ 2 2Tm1ξ m + 0.4τm 2T ξ + 0.4τm T + 0.8Tm1ξ m τm ; or K c = λ m1 m , x1 = m1 K m ( τm + 2Tm3 ) K m (τm + 2Tm 4 ) 0.8Tm1ξ m τm λ = 0.5, τ m ξ m Tm1 ≤ 1 ; λ = 0.6,1 < τm ξ m Tm1 ≤ 2 ; λ = 0.7, τmξ m Tm1 > 2 ; servo. λ = 0.6, τmξ m Tm1 ≤ 1 ; λ = 0.7,1 < τm ξ m Tm1 ≤ 2 ; λ = 0.8, τm ξm Tm1 > 2 ; regulator. 2 Kc = Tm1 , x1 = 0.1Tm 2 + 0.5 4Tm 4 τm + 0.4Tm 2 (Tm 4 + τm ) + 0.04Tm 2 2 . K m (2 x1 + Tm 4 + τm ) b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 3.7 TOSPD Model G m (s) = K m G m (s) = G m (s) = 1 + b 1s + b 2 s 2 + b 3 s 3 e −sτ m or 1 + a 1s + a 2 s 3 + a 3 s 3 K m e −sτ m or (1 + sTm1 )(1 + sTm 2 )(1 + sTm3 ) K m (Tm 3 s + 1)e −sτ m (T s + 2ξ T s + 1)(T s + 1) m1 2 2 m m1 m2 1 3.7.1 Ideal PI controller G c (s) = K c 1 + Ti s E(s) R(s) + − 1 K c 1 + T is U(s) TOSPD model Y(s) Table 36: PI controller tuning rules – third order system plus time delay model K m e − sτ m 1 + b1s + b 2s 2 + b3s3 −sτ m G m (s ) = K m e or or 3 3 (1 + sTm1 )(1 + sTm 2 )(1 + sTm3 ) 1 + a1s + a 2s + a 3s K m (Tm3s + 1)e −sτ m (T s + 2ξ T s + 1)(T s + 1) 2 2 m1 Rule Minimum IAE – Murata and Sagara (1977). Model: Method 2; K c , Ti deduced from graphs. Minimum IAE – Murata and Sagara (1977). Model: Method 2; K c , Ti deduced from graphs. m m1 Kc m2 Comment Ti Minimum performance index: regulator tuning x1 K m x 2Tm1 Tm1 = Tm 2 = Tm3 x1 1.3 0.9 0.7 x2 τm Tm1 x1 x2 τm Tm1 3.4 0.2 0.6 3.6 0.8 3.4 0.4 0.6 3.8 1.0 3.5 0.6 Minimum performance index: servo tuning x1 K m x 2Tm1 Tm1 = Tm 2 = Tm3 x1 x2 τm Tm1 x1 x2 τm Tm1 0.9 0.7 0.6 3.0 3.0 3.1 0.2 0.4 0.6 0.5 0.5 3.3 3.5 0.8 1.0 293 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 294 Rule Kc Kuwata (1987). Model: Method 1 1 Comment Ti Direct synthesis: time domain criteria Ti Tm1 = Tm 2 = Tm3 Kc Direct synthesis: frequency domain criteria Hougen (1979), pp. 350–351. Model: Method 1; Tm1 ≥ Tm 2 ≥ Tm 3 . 0.7 Tm1 K m τ m 2 Vrančić et al. (1996), Vrančić (1996), p. 121. Model: Method 1 Vrančić et al. (2004a). Model: Method 1 Vrančić et al. (2004b). 1 Kc = 3 0.333 τm Tm1 ≤ 0.04 Kc 0.5A 3 ( A1A 2 − K m A 3 ) A3 A2 0.5A 3 ( A1A 2 − K m A 3 ) 4 undefined 5 τ m Tm1 > 0.04 Ti A m ≥ 2, φ m ≥ 60 0 (Vrančić et al., 1996). Ti A1 A 3 A1A 2 − K m A 3 G c = 1 (Tis ) Ti Model: Method 1 Kc 0.4(3Tm1 + τm ) 0.5 − , Ti = 1.25x1 − 0.25(3Tm1 + τm ) , with K m (3Tm1 + τm − x1 ) K m x1 = (3Tm1 + τm )2 − 0.267[6Tm13 + 18Tm12 τm + 9Tm1τm 2 + τm3 ] . 0.333 1 Tm1 T + Tm 2 + Tm 3 + 0.8 m1 0.7 , Ti = 1.5τm 0.08 Tm1 (Tm 2 + Tm3 ) . 2K m τm ( Tm1Tm 2Tm 3 )0.333 2 Kc = 3 A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a 1 − b1 τ m + 0.5τ m ( ( 2 ), 2 3 ) A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m . A1A3 (A1A 2 − K m A 3 ) ) . (A1A 2 − K m A3 )2 + K m A3 (A1A 2 − K m A3 ) + 0.25K m 2A32 4 Ti = 5 Kc = A1A 2 − A3K m − [sign (A1A 2 − A 3K m )]A1 A 2 − A1A 3 2 A 3K m 2 − 2A1A 2 K m + A13 , A1A 2 − A3K m − [sign (A1A 2 − A 3K m )]A1 A 2 − A1A 3 2 Ti = 2A1 ( A 3K m 2 − 2A1A 2 K m + A13 )(1 + K c K m ) 2 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Comment Robust Marchetti and Scali (2000). Model: Method 1 a1 + 0.5τm (3λ + τm )K m a1 + 0.5τm (3λ + τm + 2Tm3 )K m b1 = b 2 = b3 = 0 or a 1 + 0. 5 τ m b 2 = b3 = 0; b1 = Tm 3 . b 2 = b3 = 0 ; b1 = −Tm3 . Ultimate cycle x 2 Tu Non-oscillatory x 1K u TOSPD model – x1 x 2 τm Tu x 1 x 2 τm Tu x 1 x 2 τm Tu Kuwata (1987). x 1 , x 2 deduced from 0.13 0.40 0.05 0.22 0.21 0.25 0.225 0.155 0.45 0.17 0.33 0.10 0.225 0.19 0.30 0.225 0.155 0.50 graphs. 0.19 0.30 0.15 0.225 0.17 0.35 0.21 0.24 0.20 0.225 0.16 0.40 Yu (2006), Ziegler–Nichols. 0.45K u 0.83Tu pp. 107–112. 0.33K u 2Tu Model: Method 3 Oscillatory TOSPD model. 295 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 296 1 + Td s 3.7.2 Ideal PID controller G c (s) = K c 1 + Ti s R(s) + E(s) − U(s) 1 K c 1 + + Td s Ti s Table 37: PID controller tuning rules – TOSPD model G m (s) = G m (s ) = Rule Kc 1 Polonyi (1989). Model: Method 1 Jones and Tham (2006). Marchetti and Scali (2000). Kuwata (1987); x 1 , x 2 deduced from graphs. 1 Kc K m e − sτ m or (1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 ) K m e − sτ m 1 + a1s + a 2s3 + a 3s3 Ti Comment Td Minimum performance index: other tuning Ti τm Tm1 > 10(Tm 2 + τ m ) Standard form optimisation – binomial. 2 Ti τm Kc Standard form optimisation – minimum ITAE. Robust 3 Tm1 + Tm 2 + Tm3 Td Kc a1 + 0.5τm (3λ + τm )K m a 1 + 0. 5 τ m x2 a 2 + 0.5a1τm a1 + 0.5τm Model: Method 1 Model: Method 1 Ultimate cycle x 2 Tu x 3 Tu x 1K u x1 Y(s) TOSPD model x3 τm Tu x1 x2 x3 τm Tu x1 x2 τm Tu 0.45 0.57 0.16 0.05 0.41 0.29 0.08 0.25 0.29 0.19 0.04 0.45 0.46 0.47 0.13 0.10 0.38 0.25 0.08 0.30 0.28 0.18 0.03 0.50 0.45 0.40 0.11 0.15 0.35 0.22 0.07 0.35 0.43 0.33 0.10 0.20 0.32 0.20 0.06 0.40 T T T K c = 1 − 4 m 2 + m 2 m1 , Ti = 4 6Tm 2 τm + τm . 6τm Tm3 τm Tm 2 T T + m 2 m1 , Ti = 2.7 3.4Tm 2 τm + τm . K c = 1 − 2.1 3.4τm Tm3 τm T + Tm 2 + Tm3 T T + Tm 2Tm 3 + Tm1Tm3 3 K c = m1 , Td = m1 m 2 , λ not specified. K m (λ + τm ) Tm1 + Tm 2 + Tm3 2 x3 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 297 3.7.3 Ideal controller in series with a first order lag 1 1 + Td s G c (s) = K c 1 + Ti s Tf s + 1 R(s) E(s) − + U(s) Y(s) 1 1 K m e − sτ K c 1 + + Td s 2 3 1 + a 1s + a 2 s + a 3 s Ti s Tf s + 1 m Table 38: PID controller tuning rules – TOSPD model G m (s) = Rule Schaedel (1997). Model: Method 1 1 Kc = Kc 1 Td 1 + a 1s + a 2 s 2 + a 3 s 3 Comment Direct synthesis: frequency domain criteria Ti Td Tf = Td N ; 5 ≤ N ≤ 20 . 2 2 0.375Ti a − a 2 + a1τm + 0.5τm , Ti = 1 , T a1 + τm − Td K m a1 + τm + d − Ti N 2 Td = Kc Ti K m e − sτ m 2 3 a 2 + a1τm + 0.5τm a + a 2τm + 0.5a1τm + 0.167 τm . − 3 a1 + τ m a 2 + a1τm + 0.5τm 2 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 298 Td s 1 + 3.7.4 Controller with filtered derivative G c (s) = K c 1 + Ti s 1 + Td s N R(s) E(s) − + T s 1 d K c 1 + + Ti s Td 1+ s N U(s) K m e − sτ 2 1 + a 1s + a 2 s + a 3 s Table 39: PID controller tuning rules – TOSPD model G m (s) = Rule Schaedel (1997). 1 Kc = Td = N= Kc 1 Kc Ti Y(s) m Td 3 K m e − sτ m 1 + a 1s + a 2 s 2 + a 3 s 3 Comment Direct synthesis: frequency domain criteria Model: Method 1 Ti Td 2 2 0.375Ti a − a 2 + a1τm + 0.5τm , Ti = 1 , T a1 + τm − Td K m a1 + τm + d − Ti N a 2 + a1τm + 0.5τm 2 a 3 + a 2 τm + 0.5a1τm 2 + 0.167 τm 3 , − a1 + τ m a 2 + a1τm + 0.5τm 2 Td , 2 ( x1 + τm − Ti ) Ti Td x1 + τm − Ti + 0.375 − + 2 4 Kv 0.05 ≤ x1 ≤ 0.2 , K v = prescribed overshoot in the manipulated variable for a step response in the command variable. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 299 3.7.5 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N E(s) R(s) + − T s 1 d K c 1 + + Ti s Td 1+ s N − U(s) + TOSPD model Y(s) Table 40: PID controller tuning rules – third order system plus time delay model 1 + b 1s + b 2 s 2 + b 3 s 3 −sτ m K m e − sτ m G m (s) = K m e or G m (s) = 3 3 (1 + sTm1 )(1 + sTm 2 )(1 + sTm3 ) 1 + a 1s + a 2 s + a 3 s Rule Taguchi and Araki (2000). 1 Kc = Kc 1 Ti Td Comment Minimum performance index: servo/regulator tuning 0 Model: Method 1 Ti K c OS ≤ 20% ; Tm1 = Tm 2 = Tm3 ; τ m Tm1 ≤ 1.0 . 1 0.7399 0.2713 + , Km (τm Tm1 ) + 0.5009 ( Ti = Tm1 2.759 − 0.003899[τm Tm1 ] + 0.1354[τm Tm1 ] 2 α = 0.4908 − 0.2648(τm Tm1 ) + 0.05159(τm Tm1 ) . 2 ), b720_Chapter-03 Rule Taguchi and Araki (2000). 2 Kc Ti Td Comment Kc Ti Td Model: Method 1 OS ≤ 20% ; N not specified; Tm1 = Tm 2 = Tm3 ; τ m Tm1 ≤ 1.0 . Kuwata (1987). Tm1 = Tm 2 = Tm3 . Kuwata (1987). Model: Method 1 Kc = FA Handbook of PI and PID Controller Tuning Rules 300 2 17-Mar-2009 3 Kc 4 Kc Direct synthesis: time domain criteria 0 Ti α = 1; Model: Method 1 Ti Tm1 = Tm 2 = Tm3 Td α = 1, β = 1, N = ∞ . 1 1.275 0.4020 + , [τm Tm1 ] + 0.003273 Km ( T = T (0.8335 + 0.2910[τ T ] − 0.04000[τ T ] ) , ) Ti = Tm1 0.3572 + 7.467[τm Tm1 ] − 12.86[τm Tm1 ] + 11.77[τm Tm1 ] − 4.146[τm Tm1 ] , 2 3 4 2 d m1 m m1 m m1 α = 0.6661 − 0.2509[τm Tm1 ] + 0.04773[τm Tm1 ] , 2 β = 0.8131 − 0.2303[τm Tm1 ] + 0.03621[τm Tm1 ] . 2 3 Kc = x1 = 0.8(3Tm1 + τm ) 1 − , with K m (3Tm1 + τm − x1 ) K m (3Tm1 + τm )2 − 0.267[6Tm13 + 18Tm12τm + 9Tm1τm 2 + τm3 ] ; [ ] [ ] 0.833 6Tm12 + 6Tm1τm + τm 2 1.667 6Tm12 + 6Tm1τm + τm 2 Ti = 1 − . 3Tm1 + τm (3Tm1 + τm )2 4 Kc = x 1.20 x13 1 x 2 x 2 − (3T + τ )x − , Ti = 1.667 2 − 4.167 23 , Td = 0.833x 2 1 3 m1 m 2 2 ; 2 x1 Km x2 Km x1 x1 − 0.833x 2 2 2 3 2 2 3 x1 = 6Tm1 + 6Tm1τm + τm , x 2 = 6Tm1 + 18Tm1 τm + 9Tm1τm + τm . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Vrančić (1996), pp. 136–137. 5 Ti Comment Td Direct synthesis: frequency domain criteria Model: Method 1 0 Ti Kc Ultimate cycle 0 x 2 Tu x 1K u Kuwata (1987); x 1 , x 2 deduced from graphs. α = 1. 301 Tm1 = Tm 2 = Tm3 x1 x2 τm Tu x1 x2 τm Tu x1 0.14 0.17 0.19 0.33 0.29 0.26 x 1K u 0.05 0.10 0.15 0.20 0.21 0.21 x 2 Tu 0.22 0.20 0.18 0.20 0.25 0.30 0.21 0.20 0.20 x 3 Tu x2 τm Tu 0.16 0.35 0.15 0.40 0.14 0.45 Tm1 = Tm 2 = Tm3 Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm x 1 , x 2 deduced T T Tu u u from graphs. α = 1 , β = 1 , 0.69 0.41 0.13 0.05 0.38 0.26 0.08 0.25 0.21 0.15 0.01 0.45 0.60 0.38 0.12 0.10 0.32 0.22 0.07 0.30 0.20 0.14 0 0.50 N =∞. 0.51 0.36 0.11 0.15 0.28 0.18 0.04 0.35 0.43 0.31 0.10 0.20 0.23 0.16 0.02 0.40 5 Kc = A1A 2 − K m A 3 − x1 (1 − [1 − α] )(K A + A − 2K A A ) , K A − A A < 0 ; x = (K A − A A ) − (1 − [1 − α ] )A (K A + A − 2K A A ) ; 2 2 m 3 1 3 m m 1 2 m 3 3 1 2 2 1 Kc = A1A 2 − K m A 3 + x1 1 2 2 2 3 m 3 3 1 m 1 2 (1 − [1 − α] )(K A + A − 2K A A ) , K A − A A > 0 . 2 2 m A1 Ti = 2 Km + m 3 1 3 m ( 1 KK 2 + c m 1 − [1 − α ] 2K c 2 ( ) 1 3 1 2 2 with ) A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a1 − b1τm + 0.5τm , 2 ( 2 3 ) A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m ; 0.2 ≤ α ≤ 0.5 – good servo and regulator response (p. 139). b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 302 3.7.6 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) − Y(s) Model F2 (s) Table 41: PID controller tuning rules – third order system plus time delay model K m e − sτ m G m (s ) = (1 + sTm1 )(1 + sTm 2 )(1 + sTm3 ) Rule Kc Ti Kc Ti Td Comment Td 0.5τm ≤ λ ≤ 3τm Robust 1 Jones and Tham (2006). Model: Method 1 α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K0 = 1 Tm1 + Tm 2 + Tm3 − Kc = 0.4 ; bf 3 = 1 , bf 4 = 0 , a f 5 = 0 . K m (τm + 0.75) 0.4τm τm + 0.75 K m (λ + τm ) Tm1 + Tm 2 + Tm3 − , Ti = 1+ 0.4τm τm + 0.75 0.4τm τm + 0.75 Td = , Tm1Tm 2 + Tm1Tm 3 + Tm 2Tm3 . 0.4τm Tm1 + Tm 2 + Tm 3 − τm + 0.75 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 303 3.8 Fifth Order System Plus Delay Model K (1 + b1s + b 2 s 2 + b 3s 3 + b 4 s 4 + b s s 5 )e − sτm G m (s ) = m 1 + a 1s + a 2 s 2 + a 3s 3 + a 4 s 4 + a 5s 5 ( ) 1 + Td s 3.8.1 Ideal PID controller G c (s) = K c 1 + Ti s U(s) E(s) R(s) − + ( Y(s) ) ) K m 1 + b1s + b 2s 2 + b 3s 3 + b 4s 4 + b 5s 5 e − sτ 1 + a 1s + a 2s 2 + a 3s 3 + a 4s 4 + a 5s 5 1 K c 1 + + Td s Ti s ( m Table 42: PID controller tuning rules – fifth order model with delay K (1 + b 1s + b 2 s 2 + b 3 s 3 + b 4 s 4 + b s s 5 )e − sτ m G m (s) = m 1 + a 1s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5 ( Rule Kc Vrančić et al. (1999). 1 1 Kc ) Ti Comment Td Direct synthesis: frequency domain criteria Model: Method 1 Ti Td Note: equations continued into the footnote on p. 304. Kc = a13 − a12 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1 [ 2K m − a12 b1 + a1a 2 + a1b12 − a 3 − b1b 2 + b3 + y 2 ( ) ] , with y1 = τ m a 1 − a 1 b1 − a 2 + b 2 + 0.5(a 1 − b1 )τ m + 0.167τ m and 2 2 3 y 2 = (a1 − b1 ) τ m + (a1 − b1 )τm + 0.333τm − (a1 − b1 + τm ) Td ; 2 Ti = 2 3 2 a13 − a12 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1 [a − a b − a + b + (a − b )τ + 0.5τ − (a − b + τ )T ] ; 2 1 2 1 1 2 2 1 1 m m 1 1 m d Td is found by solving the following equation: 2 3 4 Td x1 + 360 x 2 − 360τm x 3 + 180τm x 4 − 60τm x 5 − 15τm x 6 − 3τ m x 7 + 7τ m (b1 − a 1 ) − τ m = 0 , with 5 4 6 3 x 1 = 360[a 1 b 2 − a 1 (a 2 b1 + a 3 + b1b 2 + b 3 ) 2 2 2 2 + a 1 (a 2 + a 2 ( b1 − 2 b 2 ) + 3a 3 b1 + 2a 4 + b1b 3 + b 2 − b 4 ) 2 − a 1 (a 2 ( 2a 3 − 3b 3 ) + a 3 ( 2 b1 − b 2 ) + 3a 4 b1 + a 5 − b1 b 4 + 2 b 2 b 3 − b 5 ) 2 2 2 − a 2 b1 b 3 + a 3 + a 3 ( b1 b 2 − 2 b 3 ) + a 4 b1 + a 5 b1 − b1 b 5 + b 3 ] 4 3 2 2 − 360τ m [a 1 b1 − a 1 (a 2 + b1 + 2 b 2 ) + 3a 1 (a 3 + b1 b 2 ) 2 + a 1 (3a 2 b 2 − 3a 3 b1 − 3a 4 − b1 b 3 − 2 b 2 + 2 b 4 ) 7 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 304 2 − a 2 ( b 1 b 2 + b 3 ) + a 3 ( b 1 − b 2 ) + 2a 4 b 1 + a 5 − b 1 b 4 + 2 b 2 b 3 − b 5 ] 2 4 3 2 2 + 180τ m [a 1 − 4a 1 b1 + 3a 1 ( b1 + b 2 ) + a 1 (3a 2 b1 − 3a 3 − 5b1 b 2 + b 3 ) 2 2 − a 2 ( b1 + 2 b 2 ) + a 3 b1 + 2a 4 + b1 b 3 + 2( b 2 − b 4 )] 3 3 2 2 + 60τ m [4a 1 − 9a 1 b1 − a 1 (3a 2 − 5b1 − 4b 2 ) +4a 2 b1 − a 3 − 5b1 b 2 + b 3 ] 4 2 5 2 6 + 15τ m [ 9a 1 −14a 1 b1 − 4a 2 + 5b1 + 4 b 2 ] − 42 τ m ( b1 − a 1 ) + 7τ m , 4 3 x 2 = a 1 b 3 − a 1 ( a 2 b 2 + a 1 b 3 + a 4 + b1 b 3 ) 2 2 2 + a 1 (a 2 b1 + a 2 ( 2a 3 + b1 b 2 − 3b 3 ) + a 3 ( b1 + b 2 ) + 2a 4 b1 + a 5 + b 2 b 3 − b 5 ) 3 2 2 2 − a 1 ( a 2 + a 2 ( b 1 − 2 b 2 ) + a 2 ( 2a 3 b 1 − 2 b 1 b 3 + b 2 − b 4 ) 2 2 2 + 2a 3 + a 3 ( 2 b1 b 2 − 3b 3 ) + a 4 ( b1 + b 2 ) + a 5 b1 − b1 b 5 + b 3 ) 2 2 2 + a 2 a 3 + a 2 [a 3 ( b1 − 2 b 2 ) − a 5 − b1 b 4 + b 5 ] + a 3 b1 2 + a 3 ( a 4 − b1 b 3 + b 2 − b 4 ) + a 4 ( b1 b 2 − b 3 ) + a 5 b 2 − b 2 b 5 + b 3 b 4 , 4 3 x 3 = a 1 b 2 − a 1 ( a 2 b1 + a 3 + b1 b 2 + b 3 ) 2 2 2 2 + a 1 (a 2 + a 2 ( b1 − 2 b 2 ) + 3a 3 b1 + 2a 4 + b1b 3 + b 2 − b 4 ) 2 −a 1 (a 2 ( 2a 3 − 3b 3 ) + a 3 ( 2 b1 − b 2 ) + 3a 4 b1 + a 5 − b1 b 4 + 2 b 2 b 3 − b 5 ) 2 2 2 − a 2 b1 b 3 + a 3 + a 3 ( b1 b 2 − 2 b 3 ) + a 4 b1 + a 5 b1 − b1 b 5 + b 3 , 4 3 2 x 4 = a 1 b1 − a 1 ( a 2 + b1 + 2 b 2 ) 2 2 + 3a 1 (a 3 + b1 b 2 ) + a 1 (3a 2 b 2 − 3a 3 b1 − 3a 4 − b1 b 3 − 2 b 2 + 2b 4 ) 2 − a 2 ( b1 b 2 + b 3 ) + a 3 ( b 1 − b 2 ) + 2a 4 b 1 + a 5 − b 1 b 4 + 2 b 2 b 3 − b 5 ) , 4 3 2 2 x 5 = a 1 − 4a 1 b1 + 3a 1 ( b1 + b 2 ) + a 1 (3a 2 b1 − 3a 3 − 5b1b 2 + b 3 ) 2 2 − a 2 ( b 1 + 2 b 2 ) + a 3 b 1 + 2a 4 + b 1 b 3 + 2( b 2 − b 4 ) , 3 2 2 x 6 = 4a 1 − 9a 1 b1 − a 1 (3a 2 − 5b1 − 4 b 2 ) +4a 2 b1 −a 3 − 5b1 b 2 + b 3 , 2 2 x 7 = 9a 1 − 14a 1 b1 − 4a 2 + 5b1 + 4b 2 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 305 3.8.2 Controller with filtered derivative E(s) R(s) + − Td s 1 G c (s) = K c 1 + + Ti s 1 + (Td N )s Y(s) U(s) Td s K m 1 + b1s + b 2s 2 + b 3s 3 + b 4s 4 + b 5s 5 e − sτ m 1 + K c 1 + T Ti s 1 + a 1s + a 2s 2 + a 3s 3 + a 4s 4 + a 5s 5 1+ d s N ( ) ) ( Table 43: PID controller tuning rules – fifth order model with delay K (1 + b 1s + b 2 s 2 + b 3 s 3 + b 4 s 4 + b s s 5 )e − sτ m G m (s) = m 1 + a 1s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5 ( Rule ) Kc Ti Comment Td Direct synthesis: frequency domain criteria Vrančić (1996), pp. 118–119. Model: Method 1 1 A3 Ti = 2 A 2 − Td A1 − Td = Ti 2(A1 − Ti ) Td N 1 N < 10 Td Ti , − (A 32 − A 5A1 ) + (A − A A ) − N4 (A A − A )(A A − A A ) 2 2 3 5 1 3 2 5 5 2 4 2 (A3A 2 − A5 ) N ( A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a 1 − b1 τ m + 0.5τ m 2 3 , ), ( ) A = K (b − a + A a − A a + A a − b τ + 0.5b τ + 0.167 b τ + 0.042 τ ) A = K (a − b + A a − A a + A a − A a + b τ − 0.5b τ ) + K (0.167 b τ − 0.042b τ + 0.008τ ) . 2 3 A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m , 2 4 m 4 4 3 1 5 m 5 5 4 1 2 2 1 3 3 m 3 2 m 4 1 m m 2 3 2 2 3 1 4 4 m 3 m 3 m 2 m 4 1 m 5 m b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 306 Rule Vrančić et al. (1999). 2 17-Mar-2009 Kc = [ 2 Kc Ti Td Comment Kc Ti Td 8 ≤ N ≤ 20 ; Model: Method 1 a13 − a12 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1 2K m − a12 b1 + a1a 2 + a1b12 − a 3 − b1b 2 + b3 + (a1 − b1 ) 2 τ m + y 2 ( ) ] , with y1 = τ m a 1 − a 1 b1 − a 2 + b 2 + 0.5(a 1 − b1 )τ m + 0.167τ m and 2 2 3 y 2 = (a1 − b1 )τm + 0.333τm − (a1 − b1 + τm ) Td − 2 3 3 Ti = 2 Td 2 (a1 − b1 + τm ) ; N 2 a1 − a1 b1 + a1b 2 − 2a1a 2 + a 2 b1 + a 3 − b3 + y1 2 2 T 2 a1 − a1b1 − a 2 + b 2 + (a1 − b1 )τ m + 0.5τ m − (a1 − b1 + τ m )Td − d N Td is found by solving the following equation: 4 3 2 Td N −3z1 + Td N −2 z 2 + Td N −1z3 + Td x1 + 360 x 2 − 360τm x 3 + 180τm x 4 − 60τm x 5 − 15τm x 6 − 3τ m x 7 + 7τ m (b1 − a 1 ) − τ m = 0 2 2 5 4 6 7 with x 1 , x 2 , x 3 , x 4 , x 5 , x 6 and x 7 as defined in Table 42; z 1 , z 2 and z 3 are defined as follows: 3 2 z 1 = 360[a 1 − a 1 b1 + a 1 ( b 2 − 2a 2 ) + a 2 b1 + a 3 − b 3 ] 2 2 3 + 360τ m [a 1 − a 1 b1 − a 2 + b 2 ] + 180τ m (a 1 − b1 ) + 60τ m , 3 2 z 2 = 360(a 1 − b1 )[a 1 − a 1 b1 + a 1 ( b 2 − 2a 2 ) + a 2 b1 + a 3 − b 3 ] 3 2 2 + 360τ m [2a 1 − 3a 1 b1 − a 1 (3a 2 − b1 − 2b 2 ) + 2a 2 b1 + a 3 − b1 b 2 − b 3 ] 2 2 2 3 4 + 180τ m (3a 1 − 4a 1 b1 − 2a 2 + b1 + 2 b 2 ) + 240τ m (a 1 − b1 ) + 60τ m , 2 3 4 5 z 3 = z 4 + z 5 τ m + z 6 τ m + z 7 τ m + 135(a 1 − b1 ) τ m + 27τ m , with 4 3 2 2 z 4 = −360[a 1 b1 − a 1 (a 2 + b1 + b 2 ) + a 1 ( 2(a 3 + b1 b 2 ) − a 2 b1 ) 2 2 2 + a 1 ( a 2 + a 2 ( b 1 + b 2 ) − a 3 b 1 − 2a 4 − b 1 b 3 − b 2 + b 4 ) − a 2 ( a 3 + b1 b 2 ) + a 4 b1 + a 5 + b 2 b 3 − b 5 ] , 4 3 2 2 z 5 = 360[a 1 − 3a 1 b1 + a 1 ( 2( b1 + b 2 ) − a 2 ) + a 1 (3a 2 b1 − a 3 − 3b1 b 2 ) 2 2 − a 2 ( b1 + b 2 ) + a 4 + b1 b 3 + b 2 − b 4 ] , 3 2 2 z 6 = 540[a 1 − 2a 1 b1 − a 1 (a 2 − b1 − b 2 ) + b1 (a 2 − b 2 )] and 2 2 z 7 = 180[ 2a 1 − 3a 1 b1 − a 2 + b1 + b 2 ] . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Vrančić et al. (2004a), Vrančić and Lumbar (2004). 3 Kc Ti Td Comment Kc Ti Td 8 ≤ N ≤ 20 ; Model: Method 1 3 307 K c , Ti and Td are determined by solving the following equations: 0.5 K c . Td = A2 (A 4 A1 − A 5 K m ) − 0.5 A 4 (A1A 2 − A 3 K m ) A1 Km 2 2 K c 1 + 2K m K c + K m K c = Ti 2 A1 + K m 2 K cTd ( ) 2 and 2 Kc = , 2 A 5 A 1 K m + A 1 A 2 A 3 − A 4 A1 − A 3 K m 2 A 3K m − A1 K m K cTd − A 2 A1 + A 2 K m K c Td + x 2A1A 2 K m − A13 − A 3K m 2 [ ( , with ][ ) ] x 2 = (2A1K m K cTd + A 2 )A 2 − A1A 3 − A 3 K m + A1 K c Td A1 + K m K cTd , and with 2 2 2 ( A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a 1 − b1 τ m + 0.5τ m 2 ), ( ) A = K (b − a + A a − A a + A a − b τ + 0.5b τ + 0.167 b τ + 0.042τ ) A = K (a − b + A a − A a + A a − A a + b τ − 0.5b τ ) + K (0.167 b τ − 0.042b τ + 0.008τ ) . 2 3 A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m , 2 4 m 4 4 3 1 2 2 1 3 5 m 5 5 4 1 3 2 2 3 3 m 3 2 m 4 1 m m 2 1 4 4 m 3 m 3 m 2 m 4 1 m 5 m b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 308 3.8.3 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N E(s) R(s) − + T s 1 d K c 1 + + Ti s Td 1+ s N − U(s) Y(s) Model + Table 44: PID controller tuning rules – fifth order model with delay K (1 + b 1s + b 2 s 2 + b 3 s 3 + b 4 s 4 + b s s 5 )e − sτ m G m (s) = m 1 + a 1s + a 2 s 2 + a 3 s 3 + a 4 s 4 + a 5 s 5 ( Rule ) Kc Ti Comment Td Direct synthesis: frequency domain criteria Vrančić et al. (2004a), Vrančić and Lumbar (2004). 1 1 Ti Kc Model: Method 1; 8 ≤ N ≤ 20 . Td Note: equations continued into the footnote on the next page. K c , Ti and Td are determined by solving the following equations: A A 0.5 2 (A 4 A1 − A 5 K m ) − 0.5 4 (A1A 2 − A 3 K m ) A1 Km K c . Td = , 2 2 A 5 A 1 K m + A 1 A 2 A 3 − A 4 A1 − A 3 K m K c 1 + 2K m K c + K m 2 K c 2 = and Ti 2 A1 + K m 2 K cTd ( ) A K − A12 K m K cTd − A 2 A1 + A 2 K m 2 K c Td + x Kc = 3 m , with 2A1A 2 K m − A13 − A 3K m 2 [ ( ) ][ ] x 2 = (2A1K m K cTd + A 2 )A 2 − A1A 3 − A 3 K m + A1 K c Td A1 + K m K cTd , and with 2 2 2 ( A1 = K m (a 1 − b1 + τ m ) , A 2 = K m b 2 − a 2 + A1a 1 − b1 τ m + 0.5τ m ( 2 3 ) A 3 = K m a 3 − b 3 + A 2 a 1 − A1a 2 + b 2 τ m − 0.5b1 τ m + 0.167 τ m , 2 ), b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models ( A = K (a − b + A a − A a + A a − A a + b τ − 0.5b τ ) + K (0.167 b τ A 4 = K m b 4 − a 4 + A 3 a 1 − A 2 a 2 + A 1a 3 − b 3 τ m + 0.5b 2 τ m 2 + 0.167b1 τ m 3 + 0.042 τ m 4 309 ) 2 5 m 5 5 4 1 3 2 2 3 1 4 4 m 3 m m α = 1, β = 1. 2 m 3 4 5 ) − 0.042b1 τ m + 0.008τ m ; b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 310 3.9 General Model G m (s) = K m e − sτ m (1 + sTm1 ) n 1 3.9.1 Ideal PI controller G c (s) = K c 1 + Ti s E(s) R(s) − + 1 K c 1 + Ti s U(s) K m e − sτ Y(s) m (1 + sTm1 ) n Table 45: PI controller tuning rules – general model G m (s) = Rule Minimum IAE – Murata and Sagara (1977). n = 4. Model: Method 3; K c , Ti deduced from graphs. Minimum IAE – Murata and Sagara (1977). n = 5. Model: Method 3; K c , Ti deduced from graphs. Minimum IAE – Murata and Sagara (1977). n = 4. Model: Method 3; K c , Ti deduced from graphs. Minimum IAE – Murata and Sagara (1977). n = 5. Model: Method 3; K c , Ti deduced from graphs. Kc K m e − sτ m (1 + sTm1 ) n Comment Ti Minimum performance index: regulator tuning x1 K m x 2Tm1 x1 x2 τm Tm1 x1 x2 τm Tm1 0.9 0.7 0.6 4.0 4.0 4.2 0.2 0.4 0.6 0.5 0.5 4.4 4.7 0.8 1.0 x1 K m x 2Tm1 x1 x2 τm Tm1 x1 x2 τm Tm1 0.7 0.6 0.5 4.5 4.7 4.9 0.2 0.4 0.6 0.5 0.4 5.2 5.5 0.8 1.0 Minimum performance index: servo tuning x1 K m x 2Tm1 x1 x2 τm Tm1 x1 x2 τm Tm1 0.7 0.6 0.5 4.5 4.7 4.9 0.2 0.4 0.6 0.5 0.4 5.2 5.5 0.8 1.0 x1 K m x 2Tm1 x1 x2 τm Tm1 x1 x2 τm Tm1 0.6 0.5 0.5 4.1 4.3 4.6 0.2 0.4 0.6 0.5 0.4 5.0 5.3 0.8 1.0 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Kuwata (1987). 1 Davydov et al. (1995). Kc = x1 = 2 Kc = Comment Direct synthesis: time domain criteria Model: Method 1 Ti K c Model: Method 1; ξ = 0.9. Direct synthesis: frequency domain criteria τ m + Tar 0.5 ‘Normal’ design. 0.5 Km n −1 n ( n + 1)( τ m + Tar ) 0.375 n + 1 ‘Sharp’ design. 2n K m n − 1 2 [ Schaedel (1997). Model: Method 2 1 Ti 311 Ti Kc ] 0.4(nTm1 + τm ) 0.5 − , Ti = 1.25x1 - 0.25(nTm1 + τm ) , with K m (nTm1 + τm − x1 ) K m (nTm1 + τm )2 − 0.267[n (n − 1)(n − 2)Tm13 + 3n (n − 1)Tm12τm + 3nTm1τm 2 + τm3 ] . τ*m , Ti = (nTm1 + τ m )0.624 − 0.467 nTm1 + τ m K m 4.345 − 0.151 nTm1 + τm 1 τ*m n (n − 1)! 1 ( ) n 1 ! with τ*m = τ m + Tm1 (n − 1) − . + − n −1 − (n −1) n −i (n − 1) e ( ) (i − 1)! i =1 n − 1 ∑ b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 312 1 + Td s 3.9.2 Ideal PID controller G c (s) = K c 1 + Ti s E(s) R(s) + − U(s) Y(s) Model Table 46: PID controller tuning rules – general model Comment Kc Ti Td Rule Gorez and Klàn (2000), Klán (2000). Model: Method 4 1 K c 1 + + Td s T s i Direct synthesis: time domain criteria Td 1 Ti K m (Ti + τm ) Ti Ti τm τ 1 − m Tar Tar Alternative tuning. 0.25Ti Alternative tuning. Klán (2000). < 0.25Ti 2 τ τ 1 + 1 + 2 m − 2 m Tar Tar 1 General delayed model. Ti = Tar , 2 2 T T K 2τ τ Td = ar (Ti + Tcr ) cr + Ti − Taa − m 1 − c 1 + m , Ti Tar 2Tar K m 3Ti Tar = average residence time of the process (which equals Tm + τ m for a FOLPD process, for example); Taa = additional apparent time constant; K τ Tcr = 1 − c 1 + m τm . K m 2Ti b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Gorez and Klàn (2000). Lee and Teng (2003). 313 Kc Ti Td Comment 2 Kc Ti Td Model: Method 4 3 Kc Ti Td Model: Method 1 2 x1 K m 2Tm1 0.5Tm1 0.2 ≤ τm Tm1 ≤ 2 Trybus (2005). Model: Method 1 τm n Tm1 0.20 0.22 0.25 0.29 0.33 0.40 0.50 0.67 1.0 2.0 2 x1 0.23 0.23 0.22 0.22 0.21 0.21 0.20 0.18 0.16 0.12 3 x1 0.14 0.14 0.14 0.14 0.14 0.13 0.13 0.12 0.11 0.09 4 x1 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.09 0.07 5 General delayed model has a positive zero. K c = Ti , K m [Ti + (1 + x1 )τm ] 2 τ τ τ 1 + 1 + 2 m − 2x1 m 1 − (3 + 2.5x1 ) m Tar Tar τm (1 + 1.5x1 ) Tar Ti = Tar − , 2 Tar Tar Tcr 1 + Ti − Taa − Tca , (Ti + Tcr ) Ti Tar 1 + x1 K c τ m K m Tar with Tar = average residence time of the process, Taa = additional apparent time Td = constant and x1 is a normalised area parameter characterising the inverse response of the process. K K 2τ τ 2 τ Tcr = 1 − (1 + x1 ) c 1 + m τm ; Tca = 1 − (1 + x1 ) c 1 + m m . K m 2Ti Km 3Ti 2Tar 3 General delayed model. The three dominant desired poles of the closed loop response are − ξωCL ± jωCL 1 − ξ 2 and − x1ωCL . G p ( jω) evaluated at − ξωCL + jωCL 1 − ξ 2 equals x 2∠φ2 , with G p ( jω) evaluated at − x1ωCL equalling − x 3 . Then, [ 2 Kc = − ] [ ] [ x1 x 3 sin cos −1 ξ + φ2 + x 3 sin cos −1 ξ − φ2 + x1x 2 sin 2 cos −1 ξ [ ] [ x 2 x 3 x12 − 2 x1ξ + 1 sin cos −1 ξ ]+ x sin[cos ξ − φ ]+ x x sin[2 cos ξ] ; x ω {x sin [cos ξ] + x [sin (cos ξ − φ ) + x sin φ ]} x x sin [cos ξ] + x [x sin (cos ξ + φ ) − sin φ ] T = . ω [x x sin (cos ξ + φ ) + x sin (cos ξ − φ ) + x x sin (2 cos ξ )] Ti = [ ] ]; 2 x1 x 3 sin cos −1 ξ + φ2 1 CL 2 2 CL 1 3 3 −1 −1 3 −1 1 2 d −1 −1 3 2 3 2 2 1 −1 1 −1 −1 1 2 2 2 2 2 1 2 −1 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 314 Rule Trybus (2005) – continued. Kc Ti Td Comment (1 + τm ) x1 Km Tm1 + 0.5τm 0.5Tm1τm Tm1 + 0.5τm τm >2 Tm1 τm Tm1 2.2 20 n x1 0.12 0.13 0.15 0.16 0.19 0.21 0.26 0.32 0.43 3 x1 0.09 0.10 0.11 0.13 0.15 0.17 0.21 0.26 0.37 4 x1 0.07 0.08 0.09 0.10 0.12 0.14 0.17 0.22 0.33 5 2.5 2.9 3.3 4.0 5.0 6.7 10 G m (s) = K m e −sτ m (1 + sTm1 ) . n Direct synthesis: frequency domain criteria Skoczowski and Tarasiejski (1996). 4 Kc Td Model: Method 1 0.5(n − 1)Tm1 λ ≥ nTm1 Tm1 Robust Larionescu (2002). Model: Method 1 4 nTm1 K m (λ + τm ) ωg Tm1 K m e − sτ m G m (s ) = . Kc ≤ n Km (1 + sTm1 ) nTm1 G m (s) = K m e −sτ m (1 + sTm1 ) . n (1 + ω T ) 2 n −1 m1 2 2 1 + ωg Td 2 g , Td = n −1 Tm1 , n ≥ 2 with n+2 2n + 4 τm 4n + 2 − Tm1 n 2 − 2n − 2 + + φm ± b π Tm1 π ωg = , and with + τ − 4 n 2 2 n 2 2 2 m + φm 2Tm1 n − 4n + 3 + π Tm1 π 2 b = Tm1 2 2n + 4 τ m 4 n + 2 + φm + c , with n − 2n − 2 + π Tm1 π 2 4n + 2 τ m 2n − 2 + φm , c = 4(n + 2)1 + φm n 2 − 4n + 3 + π π Tm1 π φ m = specified phase margin. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 315 3.9.3 Ideal controller in series with a first order lag 1 1 G c (s) = K c 1 + + Td s Tf s + 1 Ti s E(s) R(s) U(s) 1 1 K m e − sτ K c 1 + + Td s (1 + sTm1 ) n Ti s Tf s + 1 m + − Y(s) Table 47: PID controller tuning rules – general model G m (s) = Rule Schaedel (1997). Model: Method 2 1 Ti = Kc Ti Td K m e − sτ m (1 + sTm1 ) n Comment Direct synthesis: frequency domain criteria 1 Td Tf = Td N , 0.563 n + 1 Ti 5 ≤ N ≤ 20 . Km n − 2 3(n + 1) (τm + Tar ) , Td = n + 1 (τm + Tar ) . 5n − 1 6n b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 316 Td s 1 + 3.9.4 Controller with filtered derivative G c (s) = K c 1 + Ti s 1 + s Td N R(s) T s 1 d + K c 1 + Td Ti s 1+ s N E(s) − + U(s) K m e − sτ (1 + sTm1 ) n m Table 48: PID controller tuning rules – general model G m (s) = Rule Kc Ti Y(s) K m e − sτ m (1 + sTm1 ) n Comment Td Direct synthesis: time domain criteria Davydov et al. (1995). ξ = 0.9; N = Km ; Model: Method 1 1 2 Kc Ti 0.25Ti τ*m > 0.5 nTm1 + τm Kc Ti Td τ*m < 0. 5 nTm1 + τm 0.5(n − 1)Tm1 λ ≥ nTm1 ; N = 5. Robust Larionescu (2002). Model: Method 1 1 Kc = nTm1 K m (λ + τm ) nTm1 τ*m , , Ti = (nTm1 + τm )0.911 − 0.716 nTm1 + τm K m 3.540 − 0.718 nTm1 + τm 1 τ*m (n − 1)! + (n − 1)! n 1 with τ*m = τm + Tm1 (n − 1) − . n −1 − (n −1) n −i (n − 1) e (n − 1) (i − 1)! i =1 ∑ 2 Kc = 0.150τ*m . , Ti = (nTm1 + τ m )0.559 − nT + τ * τ m1 m m K m 2.766 − 0.521 nTm1 + τ m 1 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 317 3.9.5 Two degree of freedom structure 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N E(s) R(s) − + T s 1 d K c 1 + + Ti s Td 1+ s N − U(s) + Y(s) K m e − sτ (1 + sTm1 )n Table 49: PID controller tuning rules – general model G m (s) = Rule Kc Kuwata (1987). Model: Method 1 1 Kc = x1 = 1 Kc 2 Kc Ti Td m K m e − sτ m (1 + sTm1 ) n Comment Direct synthesis: time domain criteria 0 Ti Ti Td 0.8(nTm1 + τm ) 1 − , α = 1, K m (nTm1 + τm − x1 ) K m (nTm1 + τm )2 − 0.267[n (n − 1)(n − 2)Tm13 + 3n (n − 1)Tm12τm + 3nTm1τm 2 + τm3 ] , [ ] [ ] 0.833 n ( n − 1)Tm12 + 2nTm1τm + τm 2 1.667 n ( n − 1)Tm12 + 2nTm1τm + τm 2 Ti = 1 − . nTm1 + τm (nTm1 + τm )2 2 Kc = x 2 − (nT + τ m )x 2 x x 1.20 x13 1 − , Ti = 1.6671 − 2.5 22 2 , Td = 0.833x 2 1 3 m1 , 2 Km x2 Km x1 x1 x 1 − 0.833x 2 2 α = 1 , β = 1 , N = ∞ ; x1 = n (n − 1)Tm12 + 2nTm1τm + τm 2 , x 2 = n (n − 1)(n − 2 )Tm1 + 3n (n − 1)Tm1 τm + 3nTm1τm + τm . 3 2 2 3 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 318 3.10 Non-Model Specific 1 3.10.1 Ideal PI controller G c (s) = K c 1 + Ti s R(s) E(s) + − 1 K c 1 + Ti s U(s) Non-model specific Y(s) Table 50: PI controller tuning rules – non-model specific Rule Kc Ti Comment Ziegler and Nichols (1942). Idzerda et al. (1955). 0.45K u Ultimate cycle 0.83Tu Quarter decay ratio. Johnson (1956). 0.5K u Tu Hwang and Chang (1987). 0.45K u 1 5.22 5.22 Tu − p1 Tu T1 0.83K u 1 3TuK c p1 , T1 = decay rate, period measured under proportional control when K c = 0.5K u . 1 Parr (1989). 0. 5K u 0.43Tu p. 191. p. 192. Parr (1989). 0.33K u 2Tu Chen and Yang (1993). Hang et al. (1993b), p. 59. Pessen (1994). 0.36K u 5.25Tu 0.25K u 0.25Tu 0.25K u 0.167Tu McMillan (1994). 0.3571K u Tu Dominant time delay process. Dominant time delay process. p. 90. TuK c = period of the continuous oscillation signal (of the measured variable) generated when K c = 0.83K u , with Ti being decreased until continuous oscillations exist. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Åström and Hägglund (1995), pp. 141–142. Kc Ti Comment 0.4698K u 0.4373Tu A m = 2, φ m = 20 0 . 0.1988K u 0.0882Tu A m = 2.44, φ m = 610 . 0.2015K u 0.1537Tu A m = 3.45, φ m = 46 0 . Calcev and Gorez (1995). Edgar et al. (1997), p. 8-15. Klán and Gorez (2000). ABB (2001). 0.3536 K u 0.1592Tu 0.59 K u 0.81Tu 0.5 K m 0.5Tu 0. 5K u 0.8Tu 0.3K u 2 Ti 0.3K u 3 Ti 0.5 K m 0.4Tu Robbins (2002). Prokop et al. (2005). Wang et al. (1995b). Vrančić et al. (1996), Vrančić (1996), p. 121. Vrančić (1996), p. 140. Friman and Waller (1997). 4 φm = 450 , small τ m φm = 150 , large τ m . Minimum IAE – regulator. K m assumed known. Minimum IAE – servo. Good response – regulator. K m assumed known. Direct synthesis: frequency domain criteria Ti Kc 0.5A 3 ( A1A 2 − K m A 3 ) A3 A2 A m ≥ 2, φ m ≥ 60 0 (Vrančić et al., 1996). 0.45K u A3 A2 Modified Ziegler– Nichols method. 0.4830 3.7321 ω1500 A m > 2, G p ( jω1500 ) 2 Ti = (0.24 + 0.111K u K m )Tu . 3 Ti = (0.27 + 0.054K u K m )Tu . 1 jωCL G CL ( jωCL ) 1 4 Kc = Im , Ti = ωCL ωCL G p ( jωCL ){1 − G CL ( jωCL )} φ m > 150 . jωCLG CL ( jωCL ) Im ( ) { ( ) } ω − ω G j 1 G j CL CL p CL . jωCLG CL ( jωCL ) Rl G p ( jωCL ){1 − G CL ( jωCL )} 319 b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 320 Rule Kc Kristiansson and Lennartson (2000). KuKm K 1350 K m K 1350 K m 6 Ti = 0.50K u K m − 0.36 . (0.33K u K m − 0.17)ωu 7 Ti = 9 10 Ti = K c = K135 0 K u K m > 0. 5 7 Ti K u K m < 0.1 ; K 1350 K m ≤ 0.1 . K u K m < 0.1 ; Ti K 1350 K m > 0.1 . KuKm ω u (0.18K u K m + 1) K u K m ≤ 10 Kc Ti K u K m > 10 10 Kc Ti 1.6 ≤ M s ≤ 1.9 (0.315K1350 K m − 0.175)ωu 5.4K1350 K m − 13.6 Ti 9 20K135 0 K m − 160 (1.32K1350 K m − 3.2)ωu 6 8 2 Kristiansson and 0.33K u K m − 0.15 Lennartson (2002a). K (0.18K K + 1) m u m 1.18K u K m − 1.72 . (0.33K u K m − 0.17)ωu 0.1 ≤ K u K m ≤ 0.5 2 5.4 K 1350 K m − 13.6 Ti = Ti 2 20 K 1350 K m − 160 5 5 2 0.50K u K m − 0.36 Kristiansson (2003). Comment Ti 1.18K u K m − 1.72 KuKm 8 FA . . 0.09K1350 K m + 0.25 0.09K1350 K m + 1.2 , Ti = ( K1350 K m ω1350 0.09K1350 K m + 1.2 ). 2 K150 0 ω150 0 K150 0 0.075 Kc = 0 . 2 + = ω + − , 0 . 06 T 0 . 06 1 . 6 0 150 i K . K150 0 Km Km m + 0 . 05 Km b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Kristiansson and Lennartson (2006). 11 Kristiansson and Lennartson (2003). 12 Kristiansson and Lennartson (2006). 13 0.05 ≤ Ti K 1500 Km ≤ 0. 9 ; 1.65 ≤ M max ≤ 1.85 . 0.05 ≤ Ti Kc K 1500 Km ≤ 0. 9 ; 1.60 ≤ M max ≤ 1.85 . Kc 1.6(Tar − τm ) 0.15 Ku 0.8Tu 1 + 3.7(K u K m ) 0.05 ≤ K 1500 Km ≤ 0. 9 ; 1.60 ≤ M max ≤ 1.85 . Hägglund and Åström (2004). K u ≥ 0.2 K m (robust); K u ≥ 0.5K m (close to optimal tuning). Åström and Hägglund (2006), p. 239. Aikman (1950). Comment Ti Kc 321 0.16 Ku Tu 1 + 4.5(K u K m ) 0.81K 37% Other tuning rules 1.29T37% K u > 0.2K m (close to optimal tuning); M s = M t = 1.4 . 37% decay ratio. [0.63K 33% ,0.97 K 33% ] [1.1T33% ,1.3T33% ] Rutherford (1950). MacLellan (1950). 33% decay ratio. 0.90K 37% T37% 37% decay ratio. 1.19 K 37% 0.92T37% 37% decay ratio. 1.02 K 37% 0.92T37% 37% decay ratio. 0.075 2 0.2 + K150 0 K m + 0.05 11 Kc = , K m 0.06 + 1.6 K150 0 K m − 0.06 K150 0 K m 2 [ ( ( ) ) )] ( Ti = [ ( 2 ) ( ω150 0 0.06 + 1.6 K150 0 K m − 0.06 K150 0 K m K ω K 0 0 0.075 12 , T = 2T 0.7 − 0.45 150 0 . K c = 2Tar 0.7 − 0.45 150 150 0.2 + i ar K150 0 K m Km Km + 0.05 Km 13 K c = 1.6(Tar − τm ) ω150 0 0.075 0.2 + . K m K150 0 K m + 0.05 ( ) )2 ] . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 322 Rule Kc Ti Comment Young (1955). K 37% < 1.1Tu 37% decay ratio. K x% 0.5 + 0.361 ln x Tx % 1.2 1 + 0.025 ln 2 x 0.999 K 25% 0.814T25% Parr (1989), p. 191. 0.667 K 25% T25% Example: x = 25 i.e. quarter decay ratio. Quarter decay ratio. McMillan (1994), pp. 42–43. 0.42 K 25% T25% ‘Fast’ tuning. 0.33K 25% T25% ‘Slow’ tuning. Wade (1994), p. 114. 0.7515K 25% 0.7470T25% ECOSSE team (1996b). Hay (1998), p. 189. 0.65K 50% 0.7917T50% K 25% T25% Bateson (2002), p. 616. K 25% Tu McMillan (2005), p. 62. 0.25K 25% Chesmond (1982), p. 400. sin φ m Åström (1982). G p ( jω900 ) Leva (1993). Åström and Hägglund (1995), p. 248. τ 0.1 + 65.31 m T25% tan φ m ω900 Ti 0.5 4 ω1350 G p ( jω1350 ) 0.25 1. 6 ω1350 G p ( jω1350 ) Hägglund and Åström (1991). 14 Kc = ^ ^ 0.25 K u tan (φm − φω − 0.5π ) G p ( jω) 1 + tan (φm − φω − 0.5π ) 2 ‘Modified’ ultimate cycle method. T25% Kc 14 x = 100(decay ratio). 0.2546 Tu , Ti = 2 tan( φ m − φ ω − 0.5π) >0 Alfa-Laval Automation ECA400 controller. Alfa-Laval Automation ECA400 controller – large delay. Model: Method 2 tan (φm − φω − 0.5π ) . ω b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc NI Labview (2001). Model: Method 2 0.4 K u ^ 0.18 K u ^ ‘Some’ overshoot. ^ ‘Little’ overshoot. ∧ Model: Method 5 0.8 Tu ^ 0.3 K u 1.6 Tu ∧ 0.333 A p 2 Tu Cox et al. (1994). 0.2 hTu sin φ m Ap Jones et al. (1997). Model: Method 2 0.5787 0.16Tu tan φ m ^ h Ap x1 Tu Majhi (1999), pp. 132–137. 0.4614 K m h K m A p Majhi (1999), pp. 132–137. 0.5380 K m h K m A p 2.5465h sin φ m Lee et al. (2005). Quarter decay ratio. 0.8 Tu ^ Tang et al. (2002). Model: Method 4 ^ 0.8 Tu 0.13 K u Pagola and Pecharromán (2002). Parr (1989). Comment Ti ^ 323 0.7958 ≤ x1 ≤ 1.5915. ^ 0.5468 Minimum ISTE – servo. ^ 0.5752 Minimum ISTE – regulator. 0.5554 Tu K m h 3.7736 A p 0.9395 Tu K m h 3.2541 A p ^ tan φ m ^ A p x 1 ω900 A m ω900 0.45 1 + Tm 2 x12 Km 5.215 x1 0.81 ≤ x 1 ≤ 1 ; x 1 = 1 , ‘small’ τ m ; x 1 = 0.91 , ‘large’ τ m . Equivalent to Ziegler and Nichols (1942). 15 15 Representative result for FOLPD process model. Other formulae may be deduced, equivalent to other ultimate cycle tuning rules, for this process model. In addition, tuning rules may also be deduced for process models in SOSPD form, FOLIPD form, FOLIPD form (with a negative zero) and SOSIPD form, as well as processes of 4th, 5th and 6th order without an explicit time delay term. x1 = 2 τm 16(τm Tm ) + 66 − 181(τm Tm )2 + 762(τm Tm ) + 3081 (τm Tm ) + 17 , 0.1 ≤ τm ≤ 2.0 . Tm b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 324 1 3.10.2 Ideal PID controller G c (s) = K c 1 + + Td s Ti s E(s) R(s) + − U(s) 1 K c 1 + + Td s Ti s Non-model specific Y(s) Table 51: PID controller tuning rules – non-model specific Rule Kc Ziegler and Nichols (1942). [ 0. 6 K u , K u ] Ti Td Ultimate cycle 0.5Tu 0.125Tu Farrington Tu [ 0.1Tu , 0.25Tu ] [ 0.33K u ,0.5K u ] (1950). McAvoy and 0.54 K u Tu 0.2Tu Johnson (1967). Atkinson and 0.25K u 0.75Tu 0.25Tu Davey (1968). 20% overshoot – servo response. Ku 0.5Tu 0.125Tu Carr (1986); 0.6667 K u Tu 0.167Tu Pettit and Carr (1987). 0. 5K u 1.5Tu 0.167Tu Tu 0.2Tu 0. 5K u Tu 0.25Tu 0. 5K u 0.34Tu 0.08Tu Tinham (1989). 0.4444 K u 0.6Tu 0.19Tu Blickley (1990). 0. 5K u Tu Corripio (1990), p. 27. 0.75K u 0.63Tu Parr (1989), pp. 190,191,193. 0. 5K u [0.125Tu , 0.167Tu ] 0.1Tu Comment Quarter decay ratio. Underdamped. Critically damped. Overdamped. ξ ≈ 0.45 Less than quarter decay ratio response. Quarter decay ratio. Quarter decay ratio. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Åström (1982). Kc K u cos φ m Ti Td 1 x1Ti Ti Comment 0.87 K u Representative results: 0.55Tu 0.14Tu φ m = 30 0 0.71K u 0.77Tu 0.19Tu φ m = 450 0.50K u 1.19Tu 0.30Tu φ m = 60 0 Ku Am Arbitrary 4 π 2 Ti Specify A m . K u cos φ m x1Td 2 Specify φ m . Hang and Åström (1988b). K u sin φ m Tu (1 − cos φ m ) π sin φ m Tu (1 − cos φ m ) 4π sin φ m Åström and Hägglund (1988), p. 60. 0.35K u 0.77Tu 0.19Tu 0.4698K u 0.4546Tu 0.1136Tu 0.1988K u 1.2308Tu 0.3077Tu 0.2015K u 0.7878Tu 0.1970Tu 0.27 K u 2.40Tu 1.32Tu 0.906K u 0.5Tu 0.125Tu φ m = 250 0.866K u 0.5Tu 0.125Tu φ m = 30 0 0. 5K u 0.5Tu 0.125Tu p. 90. 0.5Tu 0.125Tu p. 21. 0.1592Tu 0.0398Tu Åström and Hägglund (1984). Åström and Hägglund (1995), pp. 141–142. Chen and Yang (1993). De Paor (1993). McMillan (1994) McMillan (2005) [0.3K u ,0.5K u ] 1 2 Calcev and Gorez (1995). 0.3536K u ABB (1996). 0.625K u Tu Td 325 Am ≥ 2 , φ m ≥ 450 . A m = 2, φ m = 20 0 . A m = 2.44, φ m = 610 . A m = 3.45, φ m = 46 0 . 0 φ m = 45 , ‘small’ τ m ; φ m = 150 , ‘large’ τ m . 0.5Tu 0.083Tu ‘P+D’ tuning rule. T Ti = tan φm + 4 x1 + tan 2 φm u . x1 = 0.25 (Åström, 1982); 4απ x1 = 0.5 (Seki et al., 2000); 0.1 ≤ x1 ≤ 0.3 (Seki et al., 2000). T Td = tan φ m + 1 + tan 2 φ m u , x1 not specified. π b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 326 Rule Kc Ti Td Luo et al. (1996). 0.48K u 0.5Tu 0.125Tu [0.213Tu , [0.133Tu , 1.406Tu ] 0.469Tu ] 0.46K u 2.20Tu 0.16Tu 0. 4 K u 0.5Tu 0.125Tu Based on work by Tyreus and Luyben (1992). p. 26. 0. 5K u Tu 0.125Tu p. 27. [0.32 K u ,0.6K u ] Karaboga and Kalinli (1996). Luyben and Luyben (1997), p. 97. Tan et al. (1999a). Tan et al. (1999a). Tighter damping than quarter decay ratio tuning. 0. 4 K u 0.333Tu 0.083Tu Wojsznis et al. (1999). Yu (1999), p. 11. Tan et al. (2001). Robbins (2002). Smith (2003). 0.125Tu Some overshoot. 0. 2 K u 0.5Tu 0.125Tu No overshoot. Kc Ti 0.25Ti 0.33K u 0.5Tu 0.333Tu 0. 2 K u 0.5Tu 0.333Tu 0.45K u 4 Ti 0.25Tu 0.45K u 5 Ti 0.25Tu 0.625Tu 0.1Tu 0.625Tu 0.1Tu 0.6Tu 1 + 2(K u K m ) Td 0.75K u Åström and Hägglund (2006), pp. 240–241. Ti ωu G 'c ( jωu ) 2 0.5Tu [K u ,1.67K u ] Alfaro Ruiz (2005a). Kc = 0.33K u 3 Chau (2002), p. 115. 3 Comment 2 0.25Ti ωu + 1 6 Kc , Ti = 0.3183Tu tan ‘Just a bit of’ overshoot. No overshoot. Minimum IAE – servo. ‘Good’ response – regulator. Quarter decay ratio. K u > 0.2K m (close to optimal tuning); M s = M t = 1. 4 . 0.5π + ∠G 'c ( jωu ) , 2 G 'c ( jωu ) = desired frequency response of controller G c ( jω) at ωu . 4 Ti = (0.24 + 0.111K m K u )Tu . 5 Ti = (0.27 + 0.054K m K u )Tu . K 1 1 − (K u K m ) Kc = 0.3 − 0.1 u , Td = 0.15Tu . Ku K 1 − 0.95(K u K m ) m 4 6 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti Rutherford (1950). 1.25K 37% T37% Other rules 0.125T37% 1.12 K 37% T37% 0.25T37% 1.77 K 37% 1.29T37% 0.16T37% 1.67 K 37% 1.29T37% 0.16T37% MacLellan (1950). Harriott (1964), pp. 179–180. Lipták (1970), pp. 846–847. Comment Td 1.49 K 37% 0.98T37% 0.24T37% K 25% 0.167T25% 0.667T25% K 25% 0.667T25% 0.167T25% Kc Ti 0.25Ti 0.999 K 25% 0.488T25% 0.122T25% K 25% 0.67T25% 0.17T25% 37% decay ratio. 37% decay ratio. 0.83K 25% 0.5T25% 0.1T25% Quarter decay ratio. Quarter decay ratio. x = 100(decay ratio). Example: x = 25 i.e. quarter decay ratio. Quarter decay ratio. ‘Fast’ tuning. 0.67 K 25% 0.5T25% 0.1T25% ‘Slow’ tuning. Wade (1994), [ 1.002 K 25% , p. 114. 1.67 K 0.45T25% 0.1125T25% 0.8497 K 50% 0.475T50% 0.1188T50% K 25% 0.667T25% 0.167T25% K 25% 0.6667Tu 0.1667Tu 7 Chesmond (1982), p. 400. Parr (1989), p. 191. McMillan (1994), p. 43. ECOSSE team (1996b). Hay (1998), p. 189. Bateson (2002), p. 616. Rotach (1994). 7 Kc = 8 Kc = 8 25% ] Kc Ti − 0.5 Tx % K x% , Ti = . 0.5 + 0.361ln x 2 1 + 0.025 ln 2 x M max 2 M max − 1 G p ( jωM max ) , Ti = − 2 . dφωM 2 max ωM max dωM max dφ ωM max dω M max ‘Modified’ ultimate cycle. 327 b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 328 Rule Rotach (1994) – continued. García and Castelo (2000). Ti Td 9 Kc Ti Td 10 Kc Ti 0.25Ti ^ 0. 4 K u Parr (1989). 0.5 A p Zhang et al. (1996a). 11 Kc = Kc = ^ Td = ^ 0.159 Tu ∧ ∧ Tu 0.25 Tu Ti Td G p ( jωr ) , dφωr 0.5 1 1 −1 2 [ ωr + φ ωr ]. + φωr + tan π − sin −1 1 + tan π − sin M dωr M ωr max max ( cos 1800 + φm − ∠G p ( jω1 ) ) , T = 2[sin(180 + φ − ∠G ( jω )) + 1] . ω cos(180 + φ − ∠G ( jω ) ) 0 i G p ( jω1 ) m 0 1 1 p m 1 p 1 π 2 2 , with x1 = , x2 = Ap − ε4 , 2 4h 1+ x x3 1 + x12 x 2 2 + sin 2 φm 3 x3 = Ti = Self-regulating processes. Non selfregulating processes. ^ 0.083 Tu Tu Kc ω1 < ωu 2 , dφω 1 1 −1 −1 2 r + φω r ωr ωr + φω r − tan π − sin 1 + tan π − sin M max M max dωr Td = − 11 ^ M max 2 − 1 Ti = − 10 ^ 0.3333 Tu 0. 4 K u M max Kc = Comment Kc Lloyd (1994). Model: Method 3 9 FA x 2 − cos φm 2 2 x 2 + sin φm sin φm or x 3 = cos φm − x 2 x 2 2 + sin 2 φm A x −x Am2 − 1 Am2 − 1 or Ti = ; Td = m 4 2 1 or A mωg (x 4 − A m x1 ) A mωg (x 4 + A m x1 ) ωg A m − 1 ( A m x 4 + x1 ( 2 ) ) , with x ∈ [A x + 1.0, A x + 1.2] or ωg A m − 1 4 m 1 m 1 ∧ ∧ ∧ ∧ ∧ x 4 ∈ ωpc − ωu ωu − 0.4, ωpc − ωu ωu + 0.4 , ωu obtained from an ideal ∧ ∧ relay experiment. x 4 = A m x1 + 1.2 or x 4 = ωpc − ωu ωu + 0.4 (minimum IAE, regulator). sin φm ; b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Jones et al. (1997). Shin et al. (1997). Majhi (1999), pp. 132–137. Majhi (1999), pp. 132–137. NI Labview (2001). Model: Method 2 12 Kc = Kc 0.7490h A p Comment Td Ti ^ Model: Method 2 ^ x1 Tu 329 0.125 Tu 0.2526 ≤ x1 ≤ 0.5053 . 12 Kc Ti x 3Ti Model: Method 2 13 Kc Ti Td 14 Kc Ti Td Minimum ISTE – servo. Minimum ISTE – regulator. Quarter decay ratio. ‘Some’ overshoot. ∧ ∧ 0.6 K u ∧ 0.5 Tu ∧ 0.12 Tu ∧ 0.25 K u ∧ 0.5 Tu 0.12 Tu ρ ∧ 1 1 − x 2 x 3 ωu Ti − ∧ x1 x 3ωu Ti − − y 2 − ρx 2 ωu Ti T ω u i , ωu ωu (x 2 − x1 ) + (x1 − x 2 )2 + 4 x 3ρx1ωu + x 3 y2 ωu ρx1 + y∧ 2 ∧ Ti = , ∧ 2 x 3ρx1ωu + x 3 y 2 ωu ∧ ωu − ωu 2 1 1 1− ξ , x1 = − , x 2 = ∧ cos ∠G p ( j ωu ) , ρ = Ku ωu ξ Ku ∧ 1 y 2 = ∧ sin ∠G p ( j ωu ) . Typical x 3 : 0.1; typically 0.3 ≤ ξ ≤ 0.7 . Ku ∧ 13 Kc = ∧ K h K h 1 0.5585 m + 0.03835 , Ti = 0.259 Tu m Ap Ap Km 0.516 , ∧ ( ) ∧ ( ) Td = 0.124 Tu K m h A p 0.403 . 14 Kc = ∧ K h K h 1 0.7837 m − 0.0526 , Ti = 0.257 Tu m Ap Ap Km 0.197 , Td = 0.131Tu K m h A p 1.109 . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 330 Rule Kc NI Labview (2001) – continued. Chen and Yang (1993). Ti ∧ 0.5 Tu 0.12 Tu ‘Little’ overshoot. 4Td Td Model: Method 6 ∧ 0.15 K u 15 0.25 + 0.87 x1 0.25 + x12 Comment Td ∧ 2.5465h sin φ m 16 ^ Tang et al. x 2Td 0.81 ≤ x 1 ≤ 1 Td A p x 1 ω900 A m (2002). Model: Method 4 x 1 = 1 , ‘small’ τ m ; x 1 = 0.91 , ‘large’ τ m ; 1.5 ≤ x 2 ≤ 4 . Dutton et al. (1997). 17 Tue 0.5K eu 0.25Tue pp. 576–578. Direct synthesis: time domain criteria Ramasamy and Ti 18 Sundaramoorthy K (τ + T ) Td Ti m CL CL (2007). 2 15 x1 = π Ap − εr Td = 16 Td = 2 , 2ε r = relay deadband, d r = relay output amplitude; 4d r − x 2 ± x 22 + 1 ^ 2 ωu , x2 = 0.43 − 0.5x1 . 0.25 + 0.87 x1 − cot φ m + cot 2 φ m + 4 x 2 ^ . 2 ω90 0 17 K *c K eu = (8 − r )2 1− , Tue = 6.283 (8 − r )3.5 1 − ωd 1110 0.286 with r = ratio of the height of the first peak 55 to the height of the first trough of the closed loop response, when controller gain K *c is applied; ωd = damped natural frequency. 18 Desired closed loop transfer function = e −sτ CL 1 + sTCL . 2 − τCL − t − σ 2 _ − τCL 3 τCL Ti = t − τCL + , Td = Ti + τCL − t + , − 2(τCL + TCL ) 2Ti 6Ti (τCL + TCL ) 2 − with t = mean and σ 2 = variance of the process impulse response. b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ramasamy and Sundaramoorthy (2007) – continued. 19 Kc Ti Td Ti Td Direct synthesis: frequency domain criteria 0 < KmKu < ∞ ; 21 Ti Td Kc M = 1.4. 20 Åström and Hägglund (1995), p. 217. max 22 Åström and Hägglund (1995), p. 217. Streeter et al. (2003), Keane et al. (2005). 19 Comment 23 24 Kc Ti Td Kc Ti Td ( 0 < KmKu < ∞ ; M max = 2.0. 0 < KmKu < ∞ ; M max = 2.0. ) Desired closed loop transfer function = e −sτ CL 1 + 2ξCL TCL1s + TCL12s 2 . Kc = 2 2 − 2TCL1 τCL Ti , Ti = t − τCL + , K m (τCL + 2ξCL TCL1 ) 2(τCL + 2ξCL TCL1 ) − 2 − τCL − t − σ 2 _ − τCL 3 , with t = mean Td = Ti + τCL − t + − 2Ti 6Ti (τCL + 2ξCL TCL1 ) and σ 2 = variance of the process impulse response. 20 21 22 23 24 1 Tuning rules converted from formulae developed for G c (s) = K c b + + Td s . Ti s ( 2 ) ( ) ( 2 ) ( 2 ) K c = 0.19e −1.61κ + 2.5 κ K u , Ti = 0.44e −2.9 κ + 3.14 κ Tu , Td = 0.29e0.84 κ −5.6 κ Tu 1 Tuning rules converted from formulae developed for G c (s) = K c b + + Td s . T s i ( 2 2 ) ( 2 ) K c = 0.18e −1.04 κ +1.08 κ K u , Ti = 0.15e −0.74 κ + 0.26 κ Tu , Td = 0.60e −1.96 κ + 0.68 κ Tu . ( K c x1 = 0.72e ( Ti 0.59e = x1 Td x1 = )K − 0.0012340T − 6.1173.10 , )([ 0.72e )K − 0.0012340T − 6.1173.10 ]T , [0.72e ]K − 0.040430e −1.6 κ +1.2 κ 2 u −1.3 κ + 0.38 κ 2 0.108e u −1.6 κ +1.2 κ 2 −1.6 κ +1.2 κ − 3 κ +1.76 κ 2 u u −1.3 κ + 0.38 κ 2 2 ( ) K u Tu − 0.0026640 e 2 −6 ( Tu log 1.6342 log K u 0.72K u e −1.6 κ +1.2 κ − 0.0012340Tu − 6.1173.10− 6 ) −6 u u 2 , x1 = 0.25e0.56 κ − 0.12 κ + Ku . eK u 331 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 332 3.10.3 Ideal controller in series with a first order lag 1 1 G c (s) = K c 1 + + Td s Tf s + 1 Ti s R(s) E(s) − + Rule Kristiansson (2003). 1 Kc = 1 K c 1 + + Td s Ti s 1 Tf s + 1 U(s) Non-model specific Y(s) Table 52: PID controller tuning rules – non-model specific Comment Kc Ti Td 1 Kc Minimum performance index: other tuning Ti Td ‘Almost’ optimal tuning rule; K 1500 ≥ 0.03 ; 1.6 ≤ M s ≤ 1.9 . 0.60 , 2 K150 0 K150 0 K150 0 − 0.8 Km + 0.07 0.32 + 1.6 Km K m Km 1.5 , 2 K150 0 K150 0 ω150 0 0.32 + 1.6 − 0.8 Km K m 0.6667 , Td = 2 K150 0 K150 0 ω150 0 0.32 + 1.6 − 0.8 Km K m 0.4 or Tf = 2 2 K K K 0 0 0 K ω150 0 150 + 0.07 0.32 + 1.6 150 − 0.8 150 min 6 + m ,25 K K K K 0 m m m 150 Ti = Tf = 1 . 2 K150 0 K150 0 K150 0 ω150 0 20 + 0.5 0.32 + 1.6 − 0.8 Km Km K m b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ramasamy and Sundaramoorthy (2007). 2 2 Kc 3 Kc Ti 333 Comment Td Direct synthesis: time domain criteria Applies to x1 + Tf Td processes with x 3 + Tf dominant Tf numerator x 2 + Tf 1+ dynamics. x2 Desired closed loop transfer function = e −sτ CL 1 + sTCL . Kc = 2 − Tf τCL , Tf not specified; 1 + , x1 = t − τCL + K m (τCL + TCL ) x1 2(τCL + TCL ) Ti 2 − τ − t − σ 2 3 − CL τCL x1 + τCL − t + + Tf − 2 x1 6 x1 (τCL + TCL ) Td = , Tf 1+ 2 _ τCL t − τCL + 2(τCL + TCL ) − with t = mean and σ 2 = variance of the process impulse response (treated as a statistical distribution). 3 ( ) Desired closed loop transfer function = e −sτ CL 1 + 2ξCL TCL1s + TCL12s 2 . Kc = Tf τCL x2 , 1 + , x 2 = t − τCL + K m (τCL + 2ξCLTCL1 ) x 2 2(τCL + 2ξCL TCL1 ) − 2 2 − 2TCL1 2 − τCL − t − σ 2 _ − τCL3 , with t = mean x 3 = x 2 + τCL − t + − 2x 2 6 x 2 (τCL + 2ξCLTCL1 ) and σ 2 = variance of the process impulse response (treated as a statistical distribution). b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 334 Rule Kc Wang et al (1995b). Kristiansson and Lennartson (2002b). 4 17-Mar-2009 Ti 4 Kc 5 Kc Td Comment Direct synthesis: frequency domain criteria Ti Td Ti 0.4444Ti Plant does not contain an integrator; 1.6 ≤ M max ≤ 1.9 . For a stable process, K m is assumed known; for an integrating process, K m and τ m are assumed known. K c = 1 jωCL G CL ( jωCL ) Im , ωCL G p ( jωCL ){1 − G CL ( jωCL )} jωCL G CL ( jωCL ) Im G j 1 G j ω − ω ( ) { ( ) } 1 CL CL p CL Ti = ωCL jωCL G CL ( jωCL ) Rl G p ( jωCL ){1 − G CL ( jωCL )} + jωCL G CL ( jωCL ) j2ωCL G CL ( j2ωCL ) 1 Rl − Rl , 3 G p ( jωCL )[1 − G CL ( jωCL )] G p ( j2ωCL )[1 − G CL ( j2ωCL )] j2ωCLG CL ( j2ωCL ) jωCLG CL ( jωCL ) Rl − Rl G p ( j2ωCL )[1 − G CL ( j2ωCL )] 1 G p ( jωCL )[1 − G CL ( jωCL )] Td = . 3 jωCL G CL ( jωCL ) Im G p ( jωCL )[1 − G CL ( jωCL )] 5 Kc = Ti = Tf = 0.48 1.5 − 0.15 K150 0 K m + 0.07 ( [ ) ( ) ( )] K m 0.34 + 1.5 K150 0 K m − 0.7 K150 0 K m 2 [ 1.5 ( ) ( ω150 0 0.34 + 1.5 K150 0 K m − 0.7 K150 0 K m [ ( , )2 ] , 0.48 − 0.15 K150 0 K m + 0.07 ( ) ( ) ) ]2 min{3 + 2(K m K150 ),25} ω150 0 0.34 + 1.5 K 150 0 K m − 0.7 K 150 0 K m 2 0 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule 6 Kristiansson and Lennartson (2006). Kc Ti Td Kc Ti Td Comment Plant does not contain an integrator; 1.65 ≤ M max ≤ 1.85 . 7 Kc Ti Td 8 Kc Ti Td Plant contains an integrator. Ultimate cycle 0.5Tu 0.125Tu [0.3K u ,0.5K u ] McMillan (2005). M max ≈ 1.7 0.45 1. 5 − 0.1 1.5 6 K150 0 K m + 0.07 , T = , Kc = i K m 0.86 K150 0 K m + 0.44 ω150 0 0.86 K150 0 K m + 0.44 ( [ ( Td = Tf = 7 Kc = Ti = 8 Kc = Td = ω1500 Km ) [ ( ] ) ] ) 0.6667 1 , Tf = or K150 0 K 1500 K 1500 ω150 0 0.86 + 0.44 ω1500 0.86 + 0.44 2 + 14 Km Km Km 0.45 − 0.1 K K 0 . 07 + 150 0 m ( [ ( ) ) ] { ( ) } ω150 0 0.86 K150 0 K m + 0.44 2 min 3 + 2 K m K150 0 ,25 3.333ω150 0 e Km 3.333 ω150 0 3.5 − 3.5 − 2.94ω150 0 Km 6.4 − 9ω e 4.4 − 3.5 − G p jω 3ω150 0 150 0 ( ( G p jω150 0 6.5ω 150 0 Km 3ω150 0 Km ) G p jω150 0 Km 1 3ω150 0 Km ( 150 0 Km ( ) , Tf = , Td = ) G p jω 150 0 ( ) ( ) Km ≤ 0.91 . 0.083 ω1500 , G p jω1500 ≥ 0.7 ; ω1500 Km ( ) , Ti = ( 2.94 ω150 0 3.5 − 3ω150 0 2.94 0.11 4.4− G p jω150 0 , Tf = e 3.5 − ω1500 ω150 0 Km ( K 1500 3ω150 0 0.834 G p jω150 0 . 3.5 − ω150 0 Km ) G p jω150 0 ; 0.04 ≤ ) G p jω1500 < 0.7 , 1.7 ≤ M max ≤ 1.8 . 1 3ω150 0 Km 6.5ω 150 0 Km ( ) ( G p jω150 0 G p jω 1500 ) , ) , 335 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 336 Td s 1 + 3.10.4 Controller with filtered derivative G c (s) = K c 1 + Ti s 1 + Td s N U(s) T s 1 d Non-model + K c 1 + Td Ti s specific s 1+ N E(s) R(s) − + Y(s) Table 53: PID controller tuning rules – non-model specific Comment Td Kc Ti Rule Direct synthesis: frequency domain criteria Ti 6.283 x1ω φω > φm − π Leva (1993). N not specified. 1 Kc 2 Kc x1Td Td 6 ≤ x1 ≤ 10 Yi and De Moor (1994). Vrančić (1996), p. 130. 3 Kc 4Td Td N not defined. 4 Td N < 10 1 Ti 2(A1 − Ti ) ωTi Kc = 2 2π + ω2Ti 2 G p ( jω) 1 − ωTi x1 ωTi 2 Kc = , 2 G p ( jω) 1 − ω2Ti Td + ω2Ti 2 ( 4 Kc = Ti = 0.5 cos θ G p ( jωφ ) , Td = A3 , Ti = tan (φm − φω − 0.5π) 2π ω 1+ tan (φm − φω − 0.5π ) x1 , x1 = 10 . ) Td = 3 Ti ( A 2 − Td A1 − Td 2 − x1ω + x12ω2 + 4x1ω2 tan 2 (φm − φω − 0.5π ) 2 x1ω2 tan (φm − φω − 0.5π ) , φω < φm − π . tan θ + 4 + tan 2 θ , θ = 2250 − ∠G p ( jωφ ) . 2ωφ N ), 2 Td = − (A 3 − A 5A1 ) + (A − A A ) − (4 N )(A A − A )(A A − A A ) . 2 2 3 5 1 3 (2 N )(A3A 2 − A5 ) 2 5 5 2 4 3 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Ti 5 Kc Ti Vrančić (1996), p. 134. 6 Kc Vrančić et al. (1999). Lennartson and Kristiansson (1997). Kristiansson and Lennartson (1998a). 7 Vrančić (1996), pp. 120–121. 5 Kc = 337 Comment Td A3A4 − A 2 A5 A 3 2 − A1 A 5 N ≥ 10 Ti x1Ti N = 10 ; 0.2 ≤ x1 ≤ 0.5 . Kc Ti Td 8 ≤ N ≤ 20 8 Kc Ti 0.4Ti K u K m ≥ 1.67 9 Kc Ti 0.4Ti ( ) (A − A A )(A A − A K ) − (A A − A A )A , 0.5A3 A32 − A1A5 2 3 1 5 1 2 3 m 3 4 2 2 1 5 Ti = ( A3 A 32 − A1A 5 ) (A − A A )A − (A A − A A )A . 2 3 1 5 2 3 4 2 6 Kc = 7 Kc = A − A 2 − 4x1A1A 3 0.5Ti . , Ti = 2 A1 − K m Ti 2 x1A1 A3 2 T 2 2 A1A 2 − A 0 A 3 − Td A1 − d A 0 A1 N Td is obtained by solving: A3 , Ti = 2 A 2 − Td A1 − ( Td Km N , ) A 0 A 3 4 A1A 3 3 A 0 A 5 − A 2 A 3 2 2 Td + Td − Td + A 3 − A1A 5 Td + (A 2 A 5 − A 3A 4 ) = 0 . N N3 N2 8 Kc = KuKm Ti = 9 N= [ K u K m + 2.5 12K u 2 K m 2 − 35K u K m + 30 Kc 3 2 − 0.053ωu + 0.47ωu − 0.14ωu + 0.11 Kc = KuKm Ti = 12K u 2 K m 2 − 35K u K m + 30 ,N= 2.5 35 30 + 12 − . K m K u K m K u K m 2 K u 2 12K u 2 K m 2 − 35K u K m + 30 [ K u K m + x1 12K u 2 K m 2 − 35K u K m + 30 Kc − 0.525ωu 3 + 0.473ωu 2 − 0.143ωu + 0.113 , ], ]; ωu ≤ 0.4 ; KuKm x1 35 30 + 12 − ; x 1 = 3 , K u K m > 10 ; x 1 = 2.5 , K u K m < 10 . K m K u K m K u K m 2 K u 2 2 5 1 b720_Chapter-03 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 338 Rule Kc Ti Kristiansson and Lennartson (1998a) – continued. Kristiansson and Lennartson (1998b). Kristiansson and Lennartson (2000). Kc 10 10 FA Comment Td Ti 0.4Ti 11 Kc Ti 0.4Ti 12 Kc Ti 0.333Ti 13 Kc Ti Td K c as in footnote 9; Ti = N = 2.5 0. 1 ≤ K u K m ≤ 0.5 Kc 3 − 0.185ωu + 1.052ωu 2 ωu ≥ 0.4 . − 0.854ωu + 0.309 K u K m , ωu 9.14 11 > 0.45; − + 3.14 , K u K m > 1.67 , Kc = 2 2 KuKm K mK u Km Ku 7.71 Ti = 12 Kc = 12K u 3K m 2 − 35K u 2 K m + 30K u [ 3 K m K u 3 + 2.5 12K u 2 K m 2 − 35K u K m + 30 3 Ti = 13 2 Kc = [ KcKm Ku 2 2 2 (1.1K u K m − 2.3K u K m + 1.6) − 0.63ωu + 0.39ωu 2 + 0.15ωu + 0.0082 . ], ωu 0.95K m K u − 2K m K u + 1.4 2 Kc 3 ] ,N = 2.5 35 30 + 12 − . K m K u K m K u K m 2 K u 2 2 2 K m K u (−20 + 13K m K u )(1 + 0.37 K m K u ) 2 1.6( −20 + 13K m K u )(1 + 0.37 K m K u ) − 1 , 2 2 1.1K m K u − 2.3K m K u + 1.6 Ti = 2 2 K m K u (1.1K u K m − 2.3K u K m + 1.6) 1.6( −20 + 13K m K u )(1 + 0.37 K m K u ) − 1 , ωu (−20 + 13K m K u )(1 + 0.37 K m K u ) 2 1.1K m 2 K u 2 − 2.3K m K u + 1.6 Td = K m K u (1.1K u K m − 2.3K u K m + 1.6) ωu (−20 + 13K m K u )(1 + 0.37 K m K u ) 2 2 2 ( −20 + 13K m K u ) 2 (1 + 0.37 K m K u ) 2 2 2 2 (1.1K m K u − 2.3K m K u + 1.6) − 1 , 1.6( −20 + 13K m K u )(1 + 0.37 K m K u ) −1 2 2 1.1K m K u − 2.3K m K u + 1.6 N= 2 2 K K (1.1K u K m − 2.3K u K m + 1.6) Td . , Tf = m u Tf ωu (−20 + 13K m K u )(1 + 0.37 K m K u ) 2 b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Kristiansson and Lennartson (2000) – continued. 14 Ti Td Comment 14 Kc Ti Td K u K m > 0.5 15 Kc Ti Td K u K m < 0.1 Kc = 2 2 (1.1K u K m − 2.3K u K m + 1.6) 2 4.8K m K u (1 + 0.37 K m K u ) − 1 , 2 2 2 2 2 3ωu K m K u (1 + 0.37 K m K u ) 1.1K m K u − 2.3K m K u + 1.6 Ti = 2 2 (1.1K u K m − 2.3K u K m + 1.6) 4.8K m K u (1 + 0.37 K m K u ) − 1 , 2 2 2 3ωu (1 + 0.37 K m K u ) 1.1K m K u − 2.3K m K u + 1.6 3K m 2 K u 2 (1 + 0.37 K m K u ) 2 2 2 2 2 2 (1.1K u K m − 2.3K u K m + 1.6) (1.1K m K u − 2.3K m K u + 1.6) , Td = 1 − 4.8K m K u (1 + 0.37 K m K u ) 3ωu (1 + 0.37 K m K u ) 2 −1 2 2 1.1K m K u − 2.3K m K u + 1.6 N= 15 Kc = Ti = 2 2 Td 1.1K u K m − 2.3K u K m + 1.6 . , Tf = Tf 3ωu (1 + 0.37 K m K u ) 2 (−6 + 3.7 K1350 K m ) 2 20.8(1.8 + 0.3K m K135 0 ) 2 13K m (1.8 + 0.3K m K1350 ) ( −6 + 3.7 K m K1350 ) − 1 , (−6 + 3.7 K1350 K m )K1350 K m 20.8(1.8 + 0.3K m K135 0 ) − 1 , 13ω1350 (1.8 + 0.3K m K135 0 ) 2 (−6 + 3.7 K m K1350 ) 169(1.8 + 0.3K m K 0 ) 2 135 2 (−6 + 3.7 K m K135 0 )K m K1350 (−6 + 3.7 K m K1350 ) Td = − 1 , 13ω1350 (1.8 + 0.3K m K135 0 ) 2 20.8(1.8 + 0.3K m K1350 ) −1 (−6 + 3.7 K m K1350 ) (−6 + 3.7 K m K1350 ) K m K1350 T . N = d , Tf = Tf 13ω1350 (1.8 + 0.3K m K1350 ) 2 339 b720_Chapter-03 Rule Kc Kristiansson and Lennartson (2002a). Ti Td Comment 16 Kc Ti Td K u K m ≤ 10 17 Kc Ti Td K u K m > 10 K u [1.5(K m K u + 4)(0.4K m K u + 0.75) − K m K u x1 ]x1 , Km (0.4K m K u + 0.75) 2 ( 4 + K m K u ) Kc = Ti = K m K u [1.5(K m K u + 4)(0.4K m K u + 0.75) − K m K u x1 ] , ωu (0.4K m K u + 0.75) 2 (4 + K m K u ) Td = K mK u (4 + K m K u ) , ωu [1.5(4 + K m K u )(0.4K m K u + 0.75) − K m K u x1 ] 1 + N −1 ( 2 N= K K x1 Td , , Tf = m u Tf ωu (0.4K m K u + 0.75) 2 (4 + K m K u ) 2 2 [ Km (0.44K m K1350 + 1.4) 2 x 2 [ K m K135 0 1.5(0.44K m K135 0 + 1.4) x 2 − K m K u x 3 Td = (0.44K m K135 0 + 1.4) 2 x 2 ω135 0 K m K1350 ω1350 2 m ], x2 , + 1 .4) − K m K1350 x 3 1 + N −1 135 [1.5x (0.44K K 2 N= ] , K1350 1.5(0.44K m K135 0 + 1.4) x 2 − K m K1350 x 3 x 3 Kc = Ti = ) 2 x 1 = 0.13 + 0.16K m K u − 0.007 K m K u . 17 FA Handbook of PI and PID Controller Tuning Rules 340 16 17-Mar-2009 0 ]( ) 2 K m K1350 x3 Td , , Tf = ω135 0 Tf (0.44K m K1350 + 1.4) 2 x 2 x 2 = min(14 + 0.5K m K 1350 ,25) , x 3 = 1.35 + 0.35K m K 1350 − 0.006K m 2 K 1350 2 . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 1 1 + sTd 3.10.5 Classical controller G c (s) = K c 1 + Ti s 1 + s Td N R(s) + E(s) − Rule U(s) 1 1 + sTd Non-model K c 1 + Td Ti s specific 1 + s N Y(s) Table 54: PID controller tuning rules – non-model specific Comment Td Kc Ti Minimum performance index: regulator tuning Minimum IAE – Edgar et al. (1997), p. 8-15. Pessen (1953). N =∞. Pessen (1954). N =∞. Atkinson (1968). N =∞. 0.56K u 0.39Tu 0.25K u Ultimate cycle 0.33Tu 0.5Tu 0. 2 K u 0.25Tu N = 10 0.14Tu ‘Optimum’ servo response. 0.5Tu ‘Optimum’ regulator response – step changes. 0.33K u 0.5Tu 0.33Tu 0.750K u 0.75Tu ‘Some’ overshoot. 0.25Tu 0.375K u ‘Rather underdamped response’ – p. 396. 0.75Tu 0.25Tu Kinney (1983). 0.25K u ‘Good results’ – p. 396. 0.5Tu 0.12Tu N not specified. Corripio (1990), p. 27. Hang et al. (1993b), p. 58. 0. 6 K u 0.5Tu 0.125Tu 10 ≤ N ≤ 20 0.35K u 1.13Tu 0.20Tu N not specified. 0. 5K u 1.2Tu 0.125Tu 0.25K u 1.2Tu 0.125Tu ‘Aggressive’ tuning. ‘Conservative’ tuning. St. Clair (1997), p. 17. N ≥ 1 . 341 b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 342 Rule Kc Ti Td Comment Harrold (1999). 0.67 K u 0.5Tu 0.125Tu N not specified. Tan et al. (1999a), p. 26. 0.3K u 0.15Tu 0.25Tu N=∞ O’Dwyer (2001b) – representative results – Åström and Hägglund (1995) equivalent, p. 142. N =∞. Tan et al. (2001). 1 17-Mar-2009 Kc = Ti = Ziegler–Nichols ultimate cycle equivalent. A m = 2, 0.2349 K u 0.2273Tu 0.2273Tu 0.0944 K u 0.6154Tu 0.6154Tu 0.1008K u 0.3939Tu 0.3939Tu φ m = 46 0 . Ti 0.25Ti N=∞ 1 Kc Ti ωu G 'c ( jωu ) Ti 2ωu 2 + 1 0.0625Ti 2ωu 2 + 1 , 2∠G 'c ( jωu ) + π 2 2 6.25ωu + 4ωu tan 2 − 2.5ωu 2 2∠G 'c ( jωu ) + π ωu 2 tan 2 , G 'c ( jωu ) = desired controller frequency response at ω = ωu . φ m = 20 0 . A m = 2.44, φ m = 610 . A m = 3.45, b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 343 3.10.6 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N R(s) E(s) + − U(s) 2 T s b + b s + b s 1 Non-model d f1 f2 f0 K c 1 + + Ti s specific Td 1 + a f 1 s + a f 2 s 2 s 1+ N Y(s) Table 55: PID controller tuning rules – non-model specific Comment Kc Ti Td Rule Kristiansson et al. (2000). 1 Kc Minimum performance index: other tuning Ti Td M s ≈ 1.7 ‘Almost’ optimal tuning rule. 2 G p ( jωu ) G p ( j ωu ) − + 1.6 1.6 2 . 3 1 . 1 2 Km Km 1.6 1 , Kc = , Ti = G p ( jωu ) G p ( jωu ) ωu 0.37 + K m 0.37 + Km Km Td = 2 G (j ) 1.6 G p ( jωu ) − 2.3 p ωu + 1.1 2 Km Km 0.625 , x1 = , 2 G p ( jωu ) G ( j ) G ( j ) ω ω p u u ωu 0.37 + 13 − 20 p ωu 0.37 + Km K K m m Tf = x1 , G p ( jω u ) Km ≤ 0.5 ; Tf = x1 G p ( j ω u ) , > 0. 5 ; Km 3 2 b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , a f 1 = Tf , a f 2 = Tf , N = ∞ . b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 344 Rule Kristiansson (2003). 2 Kc Ti Td Kc Ti Td Comment ‘Almost’ optimal tuning rule; K 1500 ≥ 0.03 ; 1.6 ≤ M s ≤ 1.9 . Direct synthesis: time domain criteria Ramasamy and Sundaramoorthy (2007). 2 3 x1 + Tf Kc 0.60 Kc = 2 K 0 K 0 K 0 K m 150 + 0.07 0.32 + 1.6 150 − 0.8 150 Km K m Km 1.5 , Ti = 2 K150 0 K150 0 ω150 0 0.32 + 1.6 − 0.8 Km K m Td = Td , 0.6667 , bf 0 = 1 , bf 1 = 0 , bf 2 = 0 , N = ∞ , 2 K150 0 K150 0 ω150 0 0.32 + 1.6 − 0.8 Km K m 0.8 , af1 = 2 K150 0 K150 0 K150 0 ω150 0 20 + 0.5 0.32 + 1.6 − 0.8 Km Km K m 1 . af 2 = 2 2 2 K K K 0 0 0 2 ω150 0 20 150 + 0.5 0.32 + 1.6 150 − 0.8 150 Km Km K m 2 − af1 a f 2 + (a f 1 + x 2 )x1 τCL x1 3 Kc = , , x1 = t − τCL + , Td = 1 + K m (τCL + TCL ) x1 x1 + a f 1 2(τCL + TCL ) 2 − 2 3 τ CL − t − σ _ τ CL with x 2 = x 1 + τ CL − t + , − 2x 1 6x 1 (τ CL + TCL ) − a f 1 , a f 2 not specified; b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , N = ∞ ; t = mean and σ2 = variance of the process impulse response; desired closed loop transfer function = e −sτ CL 1 + sTCL . b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Ramasamy and Sundaramoorthy (2007). 4 4 Kc Ti Td Kc x3 + af1 Td Comment Applies to processes with dominant numerator dynamics. ( ) Desired closed loop transfer function = e −sτ CL 1 + 2ξCL TCL 2s + TCL 2 2s 2 . Kc = af1 a f 2 + (a f 1 + x 4 )x 3 x3 with 1 + , Td = K m (τCL + 2ξCL TCL 2 ) x3 x3 + a f1 − x 3 = t − τCL + 2 2 τCL − 2TCL 2 , 2(τCL + 2ξCL TCL 2 ) 2 − τCL − t − σ 2 _ τCL 3 , x 4 = x 3 + τCL − t + − 2x 3 6x 3 (τCL + 2ξCL TCL 2 ) − a f 1 , a f 2 not specified; b f 0 = 1 , b f 1 = 0 , b f 2 = 0 , N = ∞ ; t = mean and σ 2 = variance of the process impulse response. 345 b720_Chapter-03 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 346 3.10.7 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d K α + Td c 1+ s N E(s) R(s) + − Rule Edgar et al. (1997), p. 8–15. Majhi (1999), pp. 132–137. 1 Kc = T s 1 d + K c 1 + Td Ti s s 1+ N − + U(s) Non-model specific Y(s) Table 56: PID controller tuning rules – non-model specific Comment Td Kc Ti Minimum performance index: regulator tuning 0.77 K u 0.48Tu 0.11Tu 1 Kc Minimum IAE; α = 0 , β = 1 , N = ∞ . Minimum ISTE; Ti Td N = 10. ∧ K h + 4.267 A 17.08 K m h − 0.4079A p m p , Ti = 0.654 Tu , K m K m h + 14.9A p K m h + 12.15A p ∧ K h − 0.6498A m p Td = 2.891Tu , α = 0 , β =1. K m h + 10.52A p b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models Rule Kc Majhi (1999), pp. 132–137. Shen (2002). 2 Kc 3 Kc Minimum performance index: servo tuning Minimum ISTE; Ti Td N = 10. Minimum performance index: other tuning ∧ Ti Td Ku > 1 Km Kc Direct synthesis: frequency domain criteria 2 0 M s = 1.4 0.90T e −4.4 κ + 2.7 κ 4 Åström and Hägglund (1995), p. 215. Comment Td Ti 347 u 2 α = 1 − 1.1e −0.0061κ +1.8κ ; 0 < K m K u < ∞ . 2 0.13K u e1.9 κ −1.3κ 0.90Tu e −4.4 κ + 2.7 κ 0 2 M s = 2.0 2 α = 1 − 0.48e 0.40 κ −0.17 κ ; 0 < K m K u < ∞ . 2 Kc = ∧ K h + 0.1205A 15.75 K m h − 0.3287A p m p , Ti = 4.602 Tu , K m K m h + 18.04A p K m h + 16.07 A p ∧ K h − 0.5917 A m p Td = 2.93 Tu , α = 0 , β =1. K m h + 18.22A p 3 Minimise ∞ ∞ 0 0 ∫ r( t) − y(t) dt + ∫ d( t) − y(t) dt ; N = ∞ , β = 0 , M ≤ 2 ; s ∧ ^ ^ 2 K c = K u exp 0.17 − 2.62 K m K u + 1.79 K m 2 K u , ∧ ^ Ti = T u exp − 0.02 − 2.62 K m K u + 1.34 2 ^ 2 K m K u , ∧ ^ ^ 2 Td = T u exp − 1.70 − 0.59 K m K u − 0.25 K m 2 K u , ^ ^ 2 α = 1 − exp − 0.30 − 0.48 K m K u + 0.93 K m 2 K u . 4 2 K c = 0.053K u e 2.9 κ − 2.6 κ . b720_Chapter-03 FA Handbook of PI and PID Controller Tuning Rules 348 Rule Vrančić (1996), pp. 136–137. 5 17-Mar-2009 5 Kc Ti Td Kc Ti 0 Ultimate cycle 0.5Tu 0.125Tu Mantz and Tacconi (1989). 0.6K u VanDoren (1998). 0.75K u 0.625Tu 0.1Tu OMEGA Books (2005). 0.61K u 0.625Tu 0.1Tu Kc = Comment Quarter decay ratio; α = 0.83 , β = 0.346 , N = ∞ . A1A 2 − K m A 3 − x1 α = 0 ,β =1, N =∞. α = 0 ,β =1, N =∞. (1 − [1 − α] )(K A + A − 2K A A ) , K A − A A < 0 ; 2 2 m m 3 1 3 m 1 A1A 2 − K m A 3 + x1 Kc = 3 1 2 2 (1 − [1 − α] )(K A + A − 2K A A ) , K A − A A > 0 ; x = (K A − A A ) − (1 − [1 − α ] )A (K A + A − 2K A A ) ; 2 2 m 2 1 Ti = m 3 1 m 3 1 3 m 1 2 3 m 2 2 2 A1 2 ( 1 K K 2 Km + + c m 1 − [1 − α ] 2K c 2 ) 3 3 3 1 1 2 m 1 2 . 0.2 ≤ α ≤ 0.5 – good servo & regulator action (Vrančić, 1996, p. 139). b720_Chapter-03 17-Mar-2009 FA Chapter 3: Controller Tuning Rules for Self-Regulating Process Models 3.10.8 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) Model − Y(s) F2 (s) Table 57: PID controller tuning rules – non-model specific Comment Td Kc Ti Rule 1 Bateson (2002), pp. 629–637. Direct synthesis: frequency domain criteria Model: Method 1 Ti 2 ω180 0 Kc α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 ; b f 3 = 0 ; b f 4 = 0.1Td ; a f 5 = 0 . 1 K c = 10 { ( min − G p jω 0 140 [ ) } or {− G (jω ) − 6} p ] [ 180 0 ( , G p jω1400 ] Ti = min ω82 0 ,0.2ω170 0 . min ω82 0 ,0.2ω170 0 . ) and G p (jω180 ) are given in dB; 0 349 b720_Chapter-04 04-Apr-2009 FA Chapter 4 Controller Tuning Rules for Non-Self-Regulating Process Models 4.1 IPD Model G m (s) = K m e −sτ m s 1 4.1.1 Ideal PI controller G c (s) = K c 1 + Ti s R(s) E(s) + − 1 K c 1 + Ti s U(s) K m e − sτ s m Y(s) Table 58: PI controller tuning rules – IPD model G m (s) = K m e − sτ m s Rule Kc Ti Comment Ziegler and Nichols (1942). Model: Method 2 0.9 K m τm Process reaction 3.33τ m Quarter decay ratio. 1.0 K m τm ∞ Quarter decay ratio. Two constraints method – Wolfe (1951). Model: Method 2 Also given by Coon (1956), (1964) and NI Labview (2001). Decay ratio = 0.4. 0.6 K m τm 2.78τ m Decay ratio is as small as possible. Minimum error integral (regulator mode). 0.87 K m τm 4.35τ m 350 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Comment Åström and Hägglund (1995), p. 13. Model: Method 1 Hay (1998), p. 188. 0.63 K mτm 3.2 τ m 0.42 K m τm 5. 8τ m Ultimate cycle Ziegler–Nichols equivalent. Model: Method 3 Hay (1998), p. 199. Model: Method 1; K c and Ti deduced from graphs. Kp s(1 + sTp ) n NI Labview (2001). Model: Method 12 Minimum IAE – Shinskey (1988), p. 123. Minimum IAE – Shinskey (1994), p. 74. Minimum IAE – Shinskey (1988), p. 148. Model: Method 1 K m τ m = 0. 1 3.5 K m τ m = 0. 2 2.5 2.2 K m τ m = 0. 4 2.0 K m τ m = 0.5 1.7 K m τ m = 0.6 0% overshoot (servo). x 2τm x 3 K m τm x1 K m τ m = 0.3 4.5K m τ m x1 K m τ m Bunzemeier (1998). Model: Method 5 Gp = 7.5 x2 0.18 1.350 0.24 3.050 0.22 1.910 0.21 1.690 0.20 1.584 0.20 1.557 0.44 K m τm x3 0.23 0.40 0.30 0.28 0.26 0.26 10% overshoot (servo). Coefficient values: n x1 x2 0 1 2 3 4 5 0.20 0.19 0.19 0.18 0.18 1.477 1.463 1.453 1.385 1.378 x3 n 0.25 0.24 0.24 0.24 0.24 6 7 8 9 10 ‘Some’ overshoot. ∞ ‘No’ overshoot. 0.9 K m τm 3.33τm Quarter decay ratio. 0.4 K m τm 5.33τm ‘Some’ overshoot. 0.24 K m τm 5.33τm ‘No’ overshoot. 0.26 K m τm Minimum performance index: regulator tuning 0.9524 Model: Method 1 4τ m K τ m m 0.9259 K m τm 4τ m Model: Method 1 0.61K u Tu Also specified by Shinskey (1996), p. 121. 351 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 352 Rule Kc Ti Comment Minimum ISE – Hazebroek and Van der Waerden (1950). Minimum ISE – Haalman (1965). Model: Method 1 Minimum ITAE – Poulin and Pomerleau (1996). Model: Method 1 1.5 K mτm 5.56τ m Model: Method 2 0.6667 K m τm ∞ Skogestad (2001). Model: Method 1 Skogestad (2003). Model: Method 1 Skogestad (2001). Model: Method 1 Process output step load disturbance. Process input step 0.5327 K m τm 3.8853τ m load disturbance. Minimum performance index: other tuning 0.28 K m τm 7τ m M max = 1.4 0.5264 K m τm 4.5804 τ m 0.404 K m τm 7τ m M max = 1.7 0.49 K m τm 3.77τ m M max = 2.0 Kuwata (1987). Model: Method 1 Direct synthesis: time domain criteria Also given by Hay ∞ 0.5 K m τm (1998), p. 188. Regulator – Gorecki et al. (1989). Model: Method 1 e m K m τm ∞ 0.828 K m τm 5.828τm 0.487 K mτm 8.75τ m 0.31K u 2.2Tu 0.5 K m τm 5τ m 0.75 K m τm 2.41τ m Tyreus and Luyben (1992). Model: Method 1 or Method 10. Fruehauf et al. (1993). Rotach (1995). Model: Method 6 Wang and Cluett (1997). 1 M s = 1.9 , A m = 2.36 , φ m = 50 0 . Kc = τ +T − m m T Pole is real and has maximum attainable multiplicity. Maximum closed loop log modulus = 2dB ; TCL = τ m 10 . Model: Method 2 Damping factor for oscillations to a disturbance input = 0.75. 1 Model: Method 1; Ti Kc ξ = 0.707 . 1 , Ti = (1.4020TCL + 1.2076)τm , τm ≤ TCL ≤ 16τm . K m τm (0.7138TCL + 0.3904) b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Wang and Cluett (1997) – continued. Cluett and Wang (1997). Model: Method 1 Kc Ti Comment Kc Ti Model: Method 1; ξ =1 . 0.9588 K m τ m 3.0425τ m TCL = τ m 0.6232 K m τ m 5.2586τ m TCL = 2τ m 2 0.4668 K m τ m 7.2291τ m TCL = 3τ m 0.3752 K m τ m 9.1925τ m TCL = 4τ m 0.3144 K m τ m 11.1637 τ m TCL = 5τ m 0.2709 K m τ m 13.1416τ m TCL = 6τ m 1.04Tu M max = 5 dB Poulin and Pomerleau 2.13 0.34 K u or (1999). K m Tu Model: Method 1 x1 K m τ m Vítečková (1999), Vítečková et al. OS x1 (2000a). 0.368 0% Model: Method 1 0.514 5% 0.581 10% 0.641 15% Chidambaram and 1.1111 K m τm Sree (2003). Huba and Žáková (2003). Skogestad (2003), (2004b). Model: Method 1 0.696 20% 0.748 25% 0.801 30% 0.853 35% 4.5τ m 0.23 K m τm 2.914 τ m 3.555τ m 1 K m (TCL + τ m ) 4ξ 2 (TCL + τ m ) 0.5 K mτm 0.4689 K m τm Sree and Chidambaram (2006). Model: Method 1 2x1 (1 + x1 )K m τm Kc = Coefficient values: OS x1 0.281 K m τ m Benjanarasuth et al. (2005). Model: Method 1 2 ∞ 0.30K u x1 OS 0.906 0.957 1.008 40% 45% 50% Model: Method 1 Model: Method 1 Suggested ξ = 0.7 or 1. 8τ m ‘Good’ robustness – TCL = τ m , ξ = 1 . ∞ ‘Stability index’ = 2.5. 0.5(1 + x1 )τm x1 − 1 p. 47. x1 > 1 ; x1 = 1.25 (Chidambaram and Sree, 2003). 1 , Ti = (1.9885TCL + 1.2235)τm , τm ≤ TCL ≤ 16τm . K m τm (0.5080TCL + 0.6208) 353 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 354 Rule Kc Comment Ti Direct synthesis: frequency domain criteria Maximise crossover 3 frequency. ∞ Kc Hougen (1979), p. 333. Model: Method 1 Buckley et al. (1985), pp. 389–390. Model: Method 1 Chidambaram (1994), Srividya and Chidambaram (1997). ≤ 0.67075 Kmτm ωp Gain and phase margin – Kookos et al. (1999). Model: Method 1 ≥ 20τm ‘Well-damped’ response. 3.6547τ m Model: Method 7; recommended A m = 2. 0.45 K m τm AmKm 1 ω p 0.5π − ω p τ m 0.942 K m τ m Representative results: 4.510τ m ( ) A m = 1.5; φ m = 22.5 0 . Chen et al. (1999a). Model: Method 11 0.698 K m τ m 4.098τ m A m = 2; φ m = 30 0 . 0.491 K m τ m 6.942 τ m A m = 3; φ m = 450 . 0.384 K m τ m 18.710τ m A m = 4; φ m = 600 . π φ m + 2 (A m − 1) K mτm Am2 − 1 ( 3 ) Ti 0.5236 K m τ m 8τ m Values recorded deduced from a graph: 0.1 0.2 0.3 0.4 K mτm 7.0 Kc 4 Ti = 5 Kc = φm ≥ Ti Kc 5 Cheng and Yu (2000). Model: Method 1 4 3.5 2.2 1.8 A m = 2.83; φ m = 46.10 . 0.5 0.6 0.7 0.8 0.9 1.0 1.4 1.1 1.0 0.8 0.75 0.7 2 π τm Am − 1 . 4 φm + 0.5π(A m − 1) φm + 0.5π(A m − 1) 0.5π − φm − 2 Am − 1 r1π(2r1A m − 1) ( ) 4K m τm r12 A m 2 − 1 , Ti = ( 2 2 ) 4τm r1 A m − 1 2 πr1 A m (r1A m − 2 )(2r1A m − 1) 2 π 1.0607 1 − 2 A m . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule O’Dwyer (2001a). Model: Method 1 Kc Ti Comment x1 K m τ m x2τm 6 Representative coefficient values: Am φm x1 x2 Am x2 0.558 1.4 1.5 46.2 0 0.357 4.3 4.0 60.0 0 0.484 1.55 2.0 45.50 0.305 12.15 5.0 75.0 0 0.458 3.35 3.0 0 O’Dwyer (2001a). Model: Method 1 59.9 x1K u x 2Tu 7 2.01 − 0.80φ 6.283(1.72φ − 2.55) ωφ 1.920 < φ < 2.356 ; M s = 1.2 0.33 6.61 ω 2.09 φ = 2.09 radians (‘good’ choice). 2.50 − 0.92φ 6.283(1.72φ − 3.05) ωφ 2.094 < φ < 2.443 ; M s = 1.4 . 5.36 ω 2.27 φ = 2.27 radians (‘reasonable’ choice). ( ) Hägglund and Åström (2004). Model: Method 1 G p jωφ G p ( jω2.09 ) ( ) G p jωφ 0.41 G p ( jω2.27 ) 6 Am = x2 π + 4x 1 2 1+ π2 π − 4 x2 2 2 π π − 4 x2 2 x2 π + 4 2 2 , x 2 > 1.273 ; φ m = tan −1 0.5x 1 2 x 2 2 + 0.5x 1 x 2 x 1 2 x 2 2 + 4 − 0.5x 1 2 + 0.5 7 Am = 2x 2 π + πx1 2 1+ π2 π − 4 4x 2 π + x12 π2 2 1 φm x1 x12 x 2 2 + 4 . 2 π2 π − 4 4x 2 φ m = tan −1 x 3 − 0.125π2 x12 + x1 x2 2 ; x 3 = 2π2 x12 x 2 2 + 2πx1x 2 4π2 x12 x 2 2 + 4 ; πx1 4π2 x12 x 2 2 + 4 . 16 x 2 355 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 356 Rule Kc Ti Comment 6.283(0.86φ − 1.50) ωφ 2.356 < φ < 2.705 ; M s = 2. 0 . G p ( jω2.53 ) 0.52 4.25 ω 2.53 φ = 2.53 radians (‘good’ choice). Hägglund and Åström (2004). 0.35 K m τ m 7τm Model: Method 2 Åström and Hägglund (2006), pp. 228–229. 0.35 K m τ m 13.4τm Model: Method 2; ‘AMIGO’ tuning rule; M s = M t = 1. 4 . Clarke (2006). Model: Method 4 0.222 0.444 , K m τm K m τm [1.592τm ,3.183τm ] Hägglund and Åström (2004) – continued. Model: Method 1 Litrico and Fromion (2006). Litrico et al. (2006). 8 Kc = 3.43 − 1.15φ ( ) G p jωφ Kc Ti 0.333 K m τm 6τm 8 A m = 12.7dB , φm = 44.50 . Model: Method 1 Model: Method 1 A − Am 1.571 − 20m 10 sin 0.0175φm + 1.57110 20 , K m τm Am − Am Ti = (0.637 τm )10 20 tan 0.0175φm + 1.57110 20 ; A m in dB, φm in degrees. b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Chien (1988). Model: Method 1 1 2λ + τ m K m [λ + τ m ]2 Boudreau and McMillan (2006). Model: Method 1 1 2λ + τ m K m [λ + τ m ]2 357 Comment Ti Robust Ogawa (1995). Model: Method 1; coefficients of K c x1 K m τ m x1 x2 and Ti deduced from graphs. 0.45 0.39 0.34 0.30 0.27 11 12 13 14 15 Alvarez-Ramirez et al. (1998). Model: Method 1 1 1 1 + K m TCL Tm 9 2λ + τ m 9 4(λ + τm ) 2λ + τ m 2 > Overdamped response. x 2τm 20% uncertainty in process parameters. 30% uncertainty in process parameters. 40% uncertainty in process parameters. 50% uncertainty in process parameters. 60% uncertainty in process parameters. Tm + τ m TCL > τ m λ ∈ [1 K m , τm ] (Chien and Fruehauf, 1990); λ = 3τ m (Bialkowski, 1996); λ > τ m + Tm (Thomasson, 1997); 1.5τm ≤ λ ≤ 4.5τm (Zhang et al., 1999b). From a graph, λ = 1.5τ m ….overshoot = 58%, settling time = 6τ m λ = 2.5τ m ….overshoot = 35%, settling time = 11τ m λ = 3.5τ m ….overshoot = 26%, settling time = 16τ m λ = 4.5τ m ….overshoot = 22%, settling time = 20τ m . λ = e max / 100 K m ; e max = maximum output error after a step load disturbance (Andersson, 2000, p. 12); Ti se − τ ms y = (Chen and Seborg, 2002); 2 d desired K c (λs + 1) λ = τ m (Smith, 2002); λ = 1.65τm , minimum IAE, servo (Panda et al., 2006); λ = 1.15τm , minimum IAE, regulator (Panda et al., 2006); λ = 2.65τm , overshoot (servo) ≈ 20% (Panda et al., 2006); λ = τm , Maximum load rejection capability (Boudreau and McMillan, 2006); λ > τm (Yu, 2006, p. 19). b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 358 Rule Kc Zhong and Li (2002). Model: Method 2 2 x1 + τm Smith (2002). Skogestad (2004a). Model: Method 1 Ti Comment K m (x1 + τm )2 2 x1 + τ m x1 not specified. 1 K m τm τm Model: Method 1 10 4 K cK m Kc Ultimate cycle Litrico et al. (2007). 1.234 K u 10 ∧ 11 Kc ^ 0.37 K u − Am 20 A − m φm = 900 1 − 10 20 Model: Method 10 ∞ Ti ^ 1.5 Tu Representative result; A m = 10 dB; φm = 430 . Other methods Penner (1988). Model: Method 1 10 0.50 K m τm ∞ Maximum closed loop gain = 1.0. 0.58 K mτm 10τ m Maximum closed loop gain = 1.26. 0.8 K mτm 5.9 τ m Maximum closed loop gain = 2.0. K c ≥ ∆d 0 ∆y max , ∆d 0 = maximum magnitude of a sinusoidal disturbance over all frequencies, ∆y max = maximum controlled variable change, corresponding to the sinusoidal disturbance. 11 A − Am ∧ − m K c = 1.234 K u 10 20 sin 0.0175φm + 1.57110 20 , Am − Am ∧ Ti = 0.159 Tu 10 20 tan 0.0175φ m + 1.57110 20 ; A m in dB, φm in degrees. b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 1 4.1.2 Ideal PID controller G c (s) = K c 1 + + Td s Ti s E(s) R(s) + − U(s) 1 K c 1 + + Td s Ti s K m e − sτ s Y(s) m K m e − sτ m s Table 59: PID controller tuning rules – IPD model G m (s) = Rule Kc Callender (1934). τm = 1 ; Model: Method 1 Ziegler and Nichols (1942). 0.2155 K m Ti 0.1 K m Process reaction 2.529 ∞ 5.0 x1 K m τ m 2τm 0.5τm Ford (1953). Model: Method 3 1.48 K m τm 2τ m 0.37τ m 1.36 K m τm ∞ 0.265τm Åström and Hägglund (1995), p. 139. Hay (1998), p. 188. 0.94 K m τm 2τ m 0.5τ m Hay (1998), p. 199. Model: Method 1; K c , Td deduced from graphs. 10.0 NI Labview (2001). Model: Method 12 1.1 K m τm 2τm 0.53 K m τm 4τm 0.8τm 0.32 K m τm 4τm 0.8τm x1 K m τ m 2. 5τ m 0.4τm Alfaro Ruiz (2005a). Comment Td Decay ratio = 0.11. Decay ratio = 0. 1.2 ≤ x1 ≤ 2.0 ; Model: Method 2 Decay ratio 2.7:1. Model: Method 1 Ultimate cycle Ziegler–Nichols equivalent. 0.4 K m τm 3.2 τ m 0.8τ m Model: Method 3 0.5 K m τm ∞ 0.5τm Model: Method 3 0.55τ m K m τ m = 0. 1 0.30τ m K m τ m = 0. 2 0.25τ m K m τ m = 0.3 0.25τ m K m τ m = 0. 4 0.25τ m K m τm = 0.5, 0.6 0.5τm Quarter decay ratio. ‘Some’ overshoot. ‘No’ overshoot. 4.0 2.5 2.0 3. 2 K m τ m 1.8 2 2.0 ≤ x1 ≤ 3.33 ; Model: Method 1 359 b720_Chapter-04 FA Handbook of PI and PID Controller Tuning Rules 360 Rule Kc Visioli (2001). Model: Method 1 Paz Ramos et al. (2005). Åström and Hägglund (2004). Model: Method 1 Alenany et al. (2005). Model: Method 1 Regulator – Gorecki et al. (1989). Model: Method 1 Leonard (1994). Model: Method 1 Ti Comment Td Minimum performance index: regulator tuning x1 K m τ m x 2τm x 3τ m Visioli (2001). Model: Method 1 1 04-Apr-2009 Coefficient values: 1.49 0.59 Minimum ISE. 1.66 0.53 Minimum ITSE. 1.83 0.49 Minimum ISTSE Minimum performance index: servo tuning ∞ x1 K m τ m x 3τ m 1.37 1.36 1.34 Coefficient values: 0.49 0.45 0.45 ∞ 2.1883τm 1.03 0.96 0.90 0.5992 K m τm Minimum ISE. Minimum ITSE. Minimum ISTSE Minimum IAE; Model: Method 1 Minimum performance index: other tuning x1 K m τ m x 2τm x 3τ m x1 x2 x3 0.139 76.9 0.261 23.3 0.367 12.2 0.460 7.85 0.543 5.78 0.645 K m τm 0.346 0.365 0.378 0.389 0.400 1.104 K m τm Coefficient values: M max x1 1.1 1.2 1.3 1.4 1.5 x2 x3 M max ∞ 0.616 4.58 0.681 3.82 0.740 3.28 0.793 2.89 0.841 2.61 0.357 τm 0.410 1.6 0.418 1.7 0.426 1.8 0.434 1.9 0.440 2.0 Optimum servo. 2.405τm 0.417 τm Optimum regulator. Direct synthesis: time domain criteria Pole is real and Tm has maximum ∞ 1 1 + 4(Tm τm ) Kc attainable 2.785 K m τm 3.731τm 0.263τm multiplicity. OS (step input) < 0.74 K m τm 12.2 τ m 0.41τ m 10%; Minimum IAE (disturbance 0.47 K u 3.05Tu 0.10Tu ramp). τ + 4Tm − Kc = m e K m Tm τm τ m + 2 Tm Tm . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Td 2 Kc Ti Td 3 Kc Ti Td x1 K m τ m x 2τm x 3τ m Wang and Cluett (1997). Cluett and Wang (1997). Model: Method 1 Rotach (1995). Model: Method 6 x1 x2 x3 Coefficient values: x4 x1 0.9588 3.0425 0.3912 1 0.6232 5.2586 0.2632 2 0.4668 7.2291 0.2058 3 1.21 K m τm 1.60τ m 361 Comment Model: Method 1 TCL = x 4 τ m x2 x3 x4 0.3752 9.1925 0.1702 0.3144 11.1637 0.1453 0.2709 13.1416 0.1269 0.48τ m 4 5 6 Damping factor for oscillations to a disturbance input = 0.75. Chidambaram Model: Method 1 1.2346 K m τm 4.5τ m 0.45τm and Sree (2003). Sree and Chidambaram Model: Method 1 0.896 K m τm 2.5τ m 0.55τm (2005b). 4 Chen and Seborg Model: Method 1 Ti Td Kc (2002). 2 Kc = 1 , Ti = (1.4020TCL + 1.2076)τm , K m τm (0.7138TCL + 0.3904) Td = 3 Kc = τm ; τm ≤ TCL ≤ 16τm , ξ = 0.707 . 1.4167TCL + 1.6999 1 , Ti = (1.9885TCL + 1.2235)τm , K m τm (0.5080TCL + 0.6208) Td = 4 y = d desired Tis(1 + 0.5τms )e −sτ m K c (TCLs + 1) 3 , Kc = τm ; τm ≤ TCL ≤ 16τm , ξ = 1 . 1.0043TCL + 1.8194 τm (3TCL + 0.5τm ) , Ti = 3TCL + 0.5τm , K m (TCL + 0.5τm )3 2 Td = 2 3 3 1.5TCL τm + 0.75TCL τm + 0.125τm − TCL . τm (3TCL + 0.5τm ) b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 362 Rule Sree and Chidambaram (2006). Model: Method 1 Regulator – Gorecki et al. (1989). Chen et al. (1999a). Kc Ti Td Comment 1 K m τm ∞ 0.5τm p. 46. 4x12 1 + x1 2 K m τm ) 0.5(1 + x1 ) τm x1 − 1 0.25(1 + x1 ) τm x1 0.8956 K m τm 2.5τm 0.5τm ( p. 47. x1 > 1 ; x1 = 1.25 (Chidambaram and Sree, 2003). p. 48. Direct synthesis: frequency domain criteria Model: Method 1 ∞ 0.75 K m τm 0.333τm Low frequency part of the magnitude Bode diagram is flat. 1.661 AmK mτm ∞ 0.4603 K m τm 7.880τm 0.390τm M s = M t = 1.4 ; Model: Method 1 0.45 K m τm 8τm 0.5τm M s = M t = 1.4 ; Model: Method 2 Åström and Hägglund (2006), p. 244. Åström and Hägglund (2006), pp. 263–264. 5 Td 6 Td Model: Method 1 Robust λ + 0.25τ m 2 τm Zou et al. (1997), K (λ + 0.5τ ) 2λ + τ m 2λ + τ m m m Zou and Brigham (1998). Model: Method 8 or Method 9; 0.5τ m ≤ λ ≤ 3τ m . Rice (2004). 2λ + τm Model: Method 1 K (λ + 0.5τ )2 m 2λ + τ m m 0.25τm + λ τm 2λ + τm 7 2 Recommended 1 0.5τm Lee et al. (2006). K (T + τ ) τm ≤ TCL ≤ 2τm . ∞ TCL + τm m CL m Model: Method 1 Desired closed loop response = e −sτ m (1 + sTCL ) . 5 6 7 Td = 1.273τm [1 − 0.6002A m (1.571 − φm )] . rA Td = τm 1.2732 − 1 m . 1.6661 λ = 2.149τm , ‘very aggressive’; λ = 2.444τm , ‘aggressive’; λ = 2.780τm , ‘moderately aggressive’; λ = 3.162τm , ‘moderate’; λ = 4.091τm , ‘moderately conservative’; λ = 5.293τm , ‘conservative’; λ = 6.847 τm , ‘very conservative’. b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Shamsuzzoha and Lee (2006a). Shamsuzzoha and Lee (2007a). Kc Ti Td Comment Kc Ti Td Model: Method 1 8 Also given by Shamsuzzoha et al. (2006c); λ , ξ not specified. 9 Model: Method 1 Ti Td K c Ultimate cycle Litrico and Fromion (2006). 8 Kc = 10 Kc Ti , Td = x1K m (2λξ + τm − x 2 ) Ti = x1 + x 2 − Ti x1x 2 − Model: Method 10 Td 0.167 τm 3 − 0.5x 2 τm 2 2 λ2 + x 2 τm − 0.5τm 2λξ + τm − x 2 , Ti 2λξ + τm − x 2 ( ) 2 2 λ2 − 2λξx1 + x1 e− τ m x1 λ2 + x 2 τm − 0.5τm , x 2 = x1 1 − , 2 2λξ + τm − x 2 x1 x1 is a constant of a ‘sufficiently large value’. 9 Kc = 2 2 3λ2 + 2x 2 τm − 0.5τm − x 2 Ti , , Ti = x1 + 2 x 2 − x1K m (3λ + τm − 2x 2 ) 3λ + τm − 2 x 2 2 x1x 2 + x 2 2 − Td = λ3 + 0.167 τm 3 − x 2 τm 2 + x 2 2 τm 2 2 3λ2 + 2x 2 τm − 0.5τm − x 2 3λ + τm − 2 x 2 , Ti 3λ + τm − 2 x 2 3 λ 1 − e − τ m Tm , x1 is a constant of a ‘sufficiently large value’; x 1 From graphs: λ = 2.3τm , M s = 1.6 ; λ = 1.7 τm , M s = 1.8 ; λ = 1.4τm , M s = 2.0 . x 2 = x1 1 − 10 A − Am ∧ − m K c = 1.234 K u 10 20 sin θ , θ = 0.0175φm + 1.57110 20 ; A Am m ∧ x ∧ 10 20 , Ti = 0.159 Tu 1 10 20 tan θ , Td = 0.159 Tu x1 tan θ 1 + x1 A − m with 2 ≤ x1 ≤ 6 , φm < 901 − 10 20 , A m in dB, φm in degrees. 363 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 364 4.1.3 Ideal controller in series with a first order lag 1 1 G c (s) = K c 1 + + Td s Tf s + 1 Ti s R(s) E(s) − + 1 K c 1 + + Td s Ti s U(s) 1 1 + Tf s K m e − sτ s m Table 60: PID controller tuning rules – IPD model G m (s) = Rule Kc Ti Td Y(s) K m e − sτ m s Comment Robust Tan et al. τm 0.463λ + 0.277 (1998b). 0 λ = 0.5 K mτm 0.238λ + 0.123 Model: Method 1 Tf = τm (5.750λ + 0.590) ; labelled H ∞ optimal. Zhang et al. (1999a). Rivera and Jun (2000). Rice and Cooper (2002). Model: Method 1 Ou et al. (2005a). Model: Method 1 1 2 3 4 Kc = Kc = Kc = Kc = ( 1 Kc 2λ + τ m 0 2 Kc 2(τ m + λ ) 0 3 Kc 2λ + 1.5τ m 0.5τ m 2 + λτ m 2λ + 1.5τ m 4 Kc 3λ + τm 2λ + τm K m λ2 + 2λτ m + 0.5τm 2(τm + λ ) 2 2 , Tf = 2 ) K m ( 2τ m + 4 τ m λ + λ ) ( 2λ + 1.5τm K m λ2 + 2λτ m + 0.5τm , Tf = ) λ2 τm . 2λ2 + 4λτ m + τm 2 τ m λ2 2 2 τ m + 4 τ m λ + λ2 , Tf = 2 (6λ + τm )τm 4(3λ + τm ) . 0.5λ2 τ m λ2 + 2λτ m + 0.5τ m 2 1 4(3λ + τm ) 4λ3 , Tf = . 2 2 2 K m 12λ + 6λτ m + τm 12λ + 6λτ m + τm 2 . λ not specified; Model: Method 1 λ not specified; Model: Method 1 Suggested λ = 3.162τ m . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Ou et al. (2005b). Model: Method 1 Wang et al. (2005). Arbogast and Cooper (2007). Model: Method 1 Kc Ti 5 Kc 2λ + 1.5τm 6 Kc 7 365 Comment Td (2λ + τm )τm λ > 0.3034τm 4λ + 3τm (for stability). 2λ + 1.3866τm Td Kc 2TCL + 1.5τm (TCL + 0.5τm )τm λ not specified; Model: Method 1 0.465 K m τm 7.82τm 2TCL + 1.5τm 0.468τm Tf = 0.297 τm Recommended TCL = 3.16τm . 5 Kc = 6 Kc = λ2 τ m 1 2λ + 1.5τm , Tf = . 2 2 2 K m λ + 2λτ m + 0.5τm 2λ + 4λτ m + τ m 2 1 2λ + 1.3866τm 0.3866τm (2λ + τm ) , Td = , K m λ2 + 3.2286λτ m + 0.4896τm 2 2λ + 1.3866τm Tf = τm 7 Kc = ( 2(2TCL + 1.5τm ) 2 K m 2TCL + 4τm TCL + τm 2 ) , Tf = 0.3866λ2 − 0.2494λτ m − 0.1247 τm 2 TCL 2 τm 2 2TCL + 4TCL τm + τm 2 λ2 + 3.2286λτ m + 0.4896τm 2 . . b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 366 Td s 1 + 4.1.4 Controller with filtered derivative G c (s) = K c 1 + Ti s 1 + sTd N R(s) T s 1 d + K c 1 + Td Ti s s 1+ N E(s) + − U(s) K m e − sτ s Table 61: PID controller tuning rules – IPD model G m (s) = Rule Kc Ti m Y(s) K m e − sτ m s Comment Td Direct synthesis: frequency domain criteria Kristiansson and Lennartson (2002a). 1 Chien (1988). Model: Method 1 2 Kc Ti Kc 2λ + τ m Yu (2006), p. 19. 2 Model: Method 1 K (λ + 0.5τ ) m m 1 2λ + τ m Model: Method 1 Td Robust τ m (λ + 0.25τ m ) 2λ + τ m τ m (λ + 0.25τ m ) 2λ + τ m ( ) N = 10; λ > τm . Kc = 0.059 K 1 2 + (0.055 K 1 ) − 0.08 2.5ωu 0.055 0.059 2 − 0.08 + + − , 2 0.4(10 + 1 K 1 ) K m K1 K1 2.3 − 2 K 1 Ti = 2 2.5 2 0.059 K1 + (0.055 K1 ) − 0.08 − , ωu 2.3 − 2K1 0.4(10 + 1 K1 ) ( ) ( ) −1 2.5 2 0.059 K12 + (0.055 K1 ) − 0.08 1 Td = − + 1 , 0.4(10 + 1 K1 ) ωu 2.3 − 2K1 N N= 2 N = 10 Kc = ( −1 ) 2 Td 0.059 K1 + (0.055 K1 ) − 0.08 , Tf = , 0.5 ≤ K1 ≤ 1 . Tf 0.16ωu (10 + 1 K1 ) 2λ + τm K m (0.5λ + τm )2 ; λ ∈ [1 K m , τ m ] (Chien and Fruehauf, 1990). b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Td Kc Ti Td 0.447 K m τm 7.528τm 0.189τm 3 Arbogast and Cooper (2007). Model: Method 1 Comment N = 0.636 Recommended TCL = 3.16τm . 3 Kc = (2TCL + 1.5τm )(2TCL 2 + 4τmTCL + τm 2 ) − TCL 2τm , ( 2 0.5K m 2TCL + 4τmTCL + τm ) 2 2 2 Ti = 2TCL + 1.5τm − Td = τm (TCL + 0.5τm ) 2 TCL τm 2 2TCL + 4τm TCL + τm Td TCL 2 τm . , Tf = 2 Tf 2TCL + 4τm TCL + τm 2 , − 2 2TCL + 1.5τm − N= TCL τm 2TCL 2 + 4τm TCL + τm 2 2 TCL τm 2 2TCL + 4τm TCL + τm 2 , 367 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 368 1 1 + Td s 4.1.5 Classical controller G c (s) = K c 1 + Ti s 1 + Td s N R(s) 1 1 + sTd K c 1 + Td Ti s 1 + s N E(s) + U(s) K m e − sτ s m Y(s) Table 62: PID controller tuning rules – IPD model G m (s) = K m e − sτ m s − Rule Kc Bunzemeier (1998). Model: Method 5; Gp = x1 K m τ m Ti Comment Td Process reaction Kp s(1 + sTp ) . n Minimum IAE – Shinskey (1988), p. 143. Minimum IAE – Shinskey (1994), p. 74. Model: Method 1 Minimum IAE – Shinskey (1996), p. 117. x 2τm x 4 K m τm x1 x2 0% overshoot (servo). 10% overshoot (servo). x 3τ m Coefficient values: x3 x4 N n 0.21 0.68 0.47 0.35 0.30 0.27 0.25 0.23 0.22 0.22 0.730 0.280 0.24 5.60 0 1.570 0.796 0.94 5.99 2 1.200 0.669 0.62 6.03 3 0.981 0.587 0.46 5.99 4 0.866 0.530 0.38 6.02 5 0.828 0.487 0.34 6.01 6 0.801 0.452 0.32 6.03 7 0.782 0.424 0.30 5.97 8 0.767 0.401 0.29 5.99 9 0.756 0.381 0.27 5.95 10 Minimum performance index: regulator tuning Model: 0.93 Method 2; 1 . 57 τ 0 . 58 τ m m K mτm N not specified. 0.93 K mτm 0.93 K mτm 1.60τ m 0.58τ m N = 10 1.48τ m 0.63τ m N = 20 1.57 τ m 0.56τ m Model: Method 2; N not specified. b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Minimum IAE – Shinskey (1996), p. 121. 0.56K u Ti Comment Td Model: Method 1; N not specified. Direct synthesis – frequency domain criteria 1 ∞ 0.45τm Kc Hougen (1979), pp. 333–334. 0.39Tu 0.15Tu Model: Method 1; maximise crossover frequency; 10 ≤ N ≤ 30 . 2 ∞ 10[log10 τ m +1.65] Five criteria are Kc fulfilled; N = 10. Robust λ ∈ [1 K m , τ m ] 0.5τm Chien (1988). K (λ + 0.5τ )2 0.5τ m 2λ + 0.5τ m (Chien and m Model: Method 1 m Fruehauf, 1990); 2λ + 0.5τm N = 10. 2λ + 0.5τ m 0.5τ m K m (λ + 0.5τm )2 Hougen (1988). Model: Method 1 2λ + 0.5τm Chien (1988). K (λ + 0.5τ )2 m Model: Method 1 m 0.5τm 2λ + 0.5τ m 0.55τm 0.5τ m 2.2λ + 0.55τm K m (λ + 0.5τm )2 Tuning rules converted to this equivalent controller architecture. 3 0.5τ m 3λ + 0.5τm λ not specified; Kc Model: Method 1 Suggested 4 2λ + τ m 0.5τ m λ = 3.162τ m . Kc Zhang et al. (1999a). Rice and Cooper (2002). Model: Method 1 1 2 3 4 Values deduced from graph: 0.1 0.2 K mτm 0.3 0.4 0.5 Kc 9 4.2 2.8 2.1 1.7 K mτm 0.6 0.7 0.8 0.9 1.0 Kc 1.45 1.2 1.1 0.95 Equations deduced from graph; K c ≈ Kc = Kc = 0.5τm ( K m 3λ2 + 1.5λτ m + 0.25τm 2 ( 2λ + τm 2 λ ∈ [1 K m , τ m ] (Chien and Fruehauf, 1990); N = 11. K m λ + 2λτ m + 0.5τm 2 ) ) 1 10 K 'm , N= , N= 0.85 τ − log10 m' T m , Km = K 'm Tm' . (3λ + 0.5τm )(3λ2 + 1.5λτm + 0.25τm 2 ) . λ3 λ2 + 2λτ m + 0.5τ m 2 λ2 . 369 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 370 Rule Kc Ti Td 2λ + 0.5τm 0.5τm Kc 2TCL + τm 0.5τm 0.435 K m τm 7.324τm 0.5τm Rice (2004). 2λ + 0.5τm Model: Method 1 K (λ + 0.5τ )2 m Arbogast and Cooper (2007). Model: Method 1 m 6 Comment N=∞ 5 N = 1.684 Recommended TCL = 3.16τm . Ultimate cycle Luyben (1996). Model: Method 10 0.46K u 2.2Tu 0.16Tu Maximum closed loop log modulus of +2dB ; N = 10. Belanger and Luyben (1997). Model: Method 1 3.11K u 2.2Tu 3.64Tu N = 0.1 5 λ = 2.149τm , ‘very aggressive’; λ = 2.444τm , ‘aggressive’; λ = 2.780τm , ‘moderately aggressive’; λ = 3.162τm , ‘moderate’; λ = 4.091τm , ‘moderately conservative’; λ = 5.293τm , ‘conservative’; λ = 6.847 τm , ‘very conservative’. 6 Kc = ( 2(2TCL + τm ) 2 K m 2TCL + 4τm TCL + τm 2 ) , N= TCL 2 + 2TCL τm + 0.5τm 2 TCL 2 . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 371 4.1.6 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N R(s) E(s) + − 2 U(s) T s 1 K m e − sτ m d b f 0 + b f 1s + b f 2 s K c 1 + + 2 Ti s T 1 + a f 1s + a f 2 s s 1+ d s N Table 63: PID controller tuning rules – IPD model G m (s) = Rule Kc Ti Kc 0.667 τm Y(s) K m e − sτ m s Td Comment 0.25τm Model: Method 1 Robust Shamsuzzoha et al. (2007). Yu (2006), p. 18. Model: Method 1 1 Kc = − 1 0.46K u Ultimate cycle 2.22Tu 0.16Tu b f 0 = 0 , b f 1 = 0.16Tu , b f 2 = 0 , a f 1 = 0.016Tu , a f 2 = 0 . 3 4τm λ , b f 0 = 1 , b f 1 = x1 1 + e τ m x1 − 1 , b f 2 = 0 , K m x1 (6τm + 18λ − 6b f 1 ) x1 2 N = ∞ , af1 = af 2 = 0 . 2b f 1τm + τm + 12λτ m + 18λ2 + x1 ; x1 , λ not specified explicitly; 6τm + 18λ − 6 x1 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 372 4.1.7 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N E(s) − R(s) + T s 1 d + K c 1 + Td Ti s s 1+ N U(s) − + K m e − sτ s Table 64: PID controller tuning rules – IPD model G m (s) = Rule Minimum IAE – Shinskey (1988), p. 143. Minimum IAE – Shinskey (1994), p. 74. Minimum IAE – Shinskey (1988), p. 148. Minimum IAE – Shinskey (1996), p. 121. Kc Ti Td m Y(s) K m e − sτ m s Comment Minimum performance index: regulator tuning α = 0 , β =1, 1.28 1.90τ m 0.48τ m N =∞. K mτm Model: Method 2 α = 0 , β =1, 1.28 1.90τ m 0.46τ m N =∞. K mτm Model: Method 1 α = 0 , β =1, 0.82 K u 0.48Tu 0.12Tu N =∞. Model: Method 1 α = 0 , β =1, 0.77 K u 0.48Tu 0.12Tu N =∞. Model: Method 1 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Minimum ISE – 0.6330 K m τm Arvanitis et al. (2003a). 1.4394 K m τm Model: Method 1 0.6114 K m τm Arvanitis et al. 1.2986 K m τm (2003a). Model: Method 1 Ti Td Comment 2.3800τ m 0 α =1 2.4569 τ m 0.3982τ m α = 1, β = 1, N =∞. 2.7442 τ m 0 α =1 3.2616τ m 0.4234τ m α = 1, β = 1, N =∞. ∞ ∫ [e ( t) + K u ( t)]dt . 2 2 Minimise performance index m 2 0 0.6146 K m τm Arvanitis et al. 1.1259 K m τm (2003a). Model: Method 1 2.6816τ m 0 α =1 6.7092τ m 0.4627τ m α = 1, β = 1, N =∞. ∞ 2 du Minimise performance index e 2 ( t ) + K m 2 dt . dt 0 Minimum performance index: servo tuning 0 0.6270 K m τm 2.4688τ m α =1 ∫ Minimum ISE – Arvanitis et al. (2003a). 1.5221 K m τm Model: Method 1 0.6064 K m τm Arvanitis et al. 1.4057 K m τm (2003a). Model: Method 1 2.1084 τ m 0.3814τ m α = 1, β = 1, N =∞. 2.8489 τ m 0 α =1 2.5986τ m 0.4038τ m α = 1, β = 1, N =∞. ∞ ∫ [e ( t) + K u ( t)]dt . 2 2 Minimise performance index m 2 0 0.6130 K m τm Arvanitis et al. 1.3332 K m τm (2003a). Model: Method 1 2.7126τ m 0 α =1 3.0010τ m 0.4167τ m α = 1, β = 1, N =∞. ∞ Minimise performance index 2 2 2 du e ( t ) + K m dt . dt 0 ∫ 373 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 374 Rule Minimum ITAE – Pecharromán (2000a). Model: Method 11 Minimum ITAE – Pecharromán and Pagola (2000). Model: Method 11 Kc Ti x1K u Comment Td Minimum performance index: other tuning 0 x 2 Tu Km = 1 Coefficient values: φc x1 x1 x2 α x2 α φc 0.049 2.826 0.506 − 164 0 0.066 2.402 0.512 − 160 0 0.218 1.279 0.564 − 1350 0.250 1.216 0.573 − 130 0 0.099 1.962 0.522 − 1550 0.286 1.127 0.578 − 1250 0.129 1.716 0.532 − 150 0 0.330 1.114 0.579 − 120 0 1.506 0.544 0 0.159 0.351 1.093 0.577 − 118 0 0.189 1.392 0.555 − 145 − 140 0 x 2 Tu x1K u x 3Tu x1 x2 Coefficient values: x3 α φc 1.672 0.366 0.136 0.601 − 164 0 1.236 0.427 0.149 0.607 − 160 0 0.994 0.486 0.155 0.610 − 1550 0.842 0.538 0.154 0.616 − 150 0 0.752 0.567 0.157 0.605 − 1450 0.679 0.610 0.149 0.610 − 140 0 0.635 0.637 0.142 0.612 − 1350 0.590 0.669 0.133 0.610 − 130 0 0.551 0.690 0.114 0.616 − 1250 0.520 0.776 0.087 0.609 − 120 0 0.509 0.810 0.068 0.611 − 118 0 Km = 1 0 x1K u x 2 Tu Minimum ITAE Coefficient values: – Pecharromán α α x1 x2 φc x1 x2 φc (2000b), p. 142. 0 0.0469 2.8052 0.5061 − 164.1 0.2174 1.2566 0.5642 − 135.00 Model: Method 11 0.0662 2.3876 0.5119 − 160.50 0.2511 1.1948 0.5730 − 130.00 0.0988 1.9581 0.5220 − 155.00 0.2870 1.1281 0.5777 − 125.00 0.1271 1.6757 0.5319 − 150.00 0.3280 1.0860 0.5794 − 120.00 0.1589 1.5052 0.5439 − 145.00 0.3501 1.0822 0.5767 − 118.00 0.1866 1.3619 0.5546 − 140.00 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Minimum ITAE – Pecharromán (2000b), p. 148. Model: Method 11 Comment Kc Ti Td x1K u x 2 Tu x 3Tu x1 x2 Coefficient values: x3 α φc 1.6695 0.3684 0.1323 0.6007 − 164.10 1.2434 0.4389 0.1507 0.6073 − 160.00 0.9901 0.4846 0.1587 0.6102 − 155.00 0.8378 0.5393 0.1562 0.6163 − 150.00 0.7383 0.5789 0.1548 0.6052 − 145.00 0.6728 0.6283 0.1488 0.6105 − 140.00 0.6196 0.6221 0.1376 0.6120 − 135.00 0.5948 0.6939 0.1326 0.6096 − 130.00 0.5513 0.7050 0.1190 0.6157 − 125.00 0.5271 0.7471 0.0992 0.6088 − 120.00 0.5229 0.8886 0.0862 0.6115 − 118.0 0 α = 0.6810 Taguchi and 0 0.7662 K m τm 4.091τ m Araki (2000). 1.253 K m τm 2.388τ m 0.4137τ m OS ≤ 20% ; α = 0.6642 , β = 0.6797 , N fixed but not specified. Model: Method 1 ∞ 0.82 K m τm 0.40τm β = 0.41 ; Taguchi et al. N = 10. (2002). N = 10; ∞ 0 . 4046 τ 1 m 0.8203 K m τm Model: 0.1 ≤ τm ≤ 1.0 . Method 1; ∞ 0.87 K m τm 0.44τm β = 0.44 ; α=0. N= ∞. Minimum IAE – 0.571 K m τm 0 6.139τm Tavakoli et al. M max = 2 ; α = 0.417 ; Model: Method 1. (2005). Direct synthesis: time domain criteria Kuwata (1987). 0 1 K m τm 3.33τm α =1 Model: Method 1 2 1.067 K m τm 0 . 313 τ α = 1, β = 1, m 6.25τm N =∞. 1 β = 0.3623 + 0.02488 . τm + 0.1035 375 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 376 Rule Chien et al. (1999). Model: Method 2 Hansen (2000). Model: Method 1 Chidambaram (2000b). Model: Method 1 Kc Ti Td Comment 2 Kc 1.414TCL 2 + τ m 0 α =1 3 Kc α = 1, β = 1, N =∞. Underdamped system response: ξ = 0.707 . 1.414TCL 2 + τ m Td 0.938K m τ m 2.7 τ m 0.313τ m 0.64 K m τ m 3.333τ m 0.487 K m τ m 8.75τ m 0.9425 K m τm 2τ m α = 0.833 , β =1, N = ∞ . α = 0.65 0 α = 0.557 In general, α = 0.6 (on average) or α = 0.5[1 − τ m Ti ] . 0.5τ m N = ∞ ; β = 1 ; α = 0.707 or α = 0.65 . Hägglund and Åström (2002). Model: Method 2 0.35 K mτm Arvanitis et al. (2003a). Model: Method 1 (8 − x1 )K m τm 4 ( ) 16 2ξ2 + 1 32ξ2 + 4 K m τ m Chidambaram and Sree (2003). Model: Method 1 Sree and Chidambaram (2006). Model: Method 1 Klán (2001). Model: Method 1 2 3 Kc = Kc = ( ( ) 4τ m x1 0 α = 1, 4 . x1 = 2 4ξ + 1 16ξ 2 + 4 τm 16 2ξ 2 + 1 α = 1, β = 1, N =∞. (2ξ + 1)τ 2 m ( 0.3 < α < 0.5 . ) 4.5τ m 0.45τ m 0.9 K m τm 3.33τm 0 1.2 ≤ x1 ≤ 2 2τ m 0.5τ m 0.8956 K m τm 2.5τm 0.5τm x1 K m τ m , α = 0.6 , β = 0 , N =∞. α = 0.65 . pp. 37–38. α = 0.65 , β = 1 , N =∞. p. 48. N = ∞ , β = 0 , α = 0.6 . Direct synthesis: frequency domain criteria 0 α = 0.19 0.29 K m τm 8.9τm K m TCL 2 2 + 1.414TCL 2τm + τm 2 ( 0 1.2346 K mτm 1.414TCL 2 + τm ). 2 1.414TCL 2 + τm 2 M s = 1.4; 7τ m K m TCL 2 + 0.707TCL 2 τm + 0.25τm 2 ) , Td = 0.25τm + 0.707TCL 2 τm . 1.414TCL 2 + τm b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Td Comment 0 α =1 0.25τm + λ τm 2λ + τm α = 0 ,β =1; N =∞, λ = 3τm . Ti 377 Robust Arvanitis et al. (2003a). Model: Method 1 2 4 4π τ m x 1 π 2 − x1 ( Kc 2λ + τm Rice (2006). K m (λ + 0.5τm )2 Model: Method 1 Kc = 2λ + τ m α = 0.6 , β = 0 , N =∞. λ = 2.3τm , M s = 1.6 ; λ = 1.7 τm , M s = 1.8 ; λ = 1.4τm , M s = 2.0 ; 5 Shamsuzzoha and Lee (2007a). Model: Method 1 4 ) Ti Kc Td equations for λ deduced from graphs. 4π2 2 [(8 − x )π + x ]K τ with 0 < x < π ; 1 2 1 2 1 m m recommended x1 = 5 Kc = 2 π2 16 , ξ > 0.3941 . 1− 1− 2 2 2 π 4ξ + 1 ( 2 ) 2 Ti 3λ + 2x 2 τm − 0.5τm − x 2 , , Ti = x1 + 2 x 2 − x1K m (3λ + τm − 2x 2 ) 3λ + τm − 2 x 2 2 2 x1x 2 + x 2 − Td = x 2 = x1 1 − 3 2 2 λ3 + 0.167 τm − x 2 τm + x 2 τm 2 2 3λ2 + 2x 2 τm − 0.5τm − x 2 3λ + τm − 2 x 2 − , Ti 3λ + τm − 2 x 2 3 λ 1 − e − τ m Tm , x1 is a constant with a ‘sufficiently large value’. x 1 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 378 4.1.8 Two degree of freedom controller 2 T s 1 d 1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) , U(s) = K c 1 + + 0 Ti s T 1 + a f 1s 1+ d s N 1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4 F(s) = 1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4 R(s) F(s) + − T s 1 d 1 + b f 1s K c 1 + + Ti s Td 1 + a f 1 s 1+ s N + U(s) − K m e − sτ s K0 Table 65: PID controller tuning rules – IPD model G m (s) = Rule Alenany et al. (2005). Model: Method 1 Kc Ti Y(s) m Td K m e − sτ m s Comment Minimum performance index: servo tuning 1.104 K m τm 2.405τm 0.417 τm N=∞ a f 1 = 0 , bf 1 = 0 ; bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . a f 2 = 1.866τm : OS ≈ 7% ; a f 2 = 2.38τm : OS ≈ 0% . Normey-Rico et al. (2000). Model: Method 1 0.563 K m τm Direct synthesis: time domain criteria 1.5τ m 0.667 τ m N=∞ a f 1 = 0.13τm , b f 1 = 0 ; a f 2 = 0.75τm , b f 2 = 0.30τm ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0. 5 K m τ m . Normey-Rico et al. (2001). Model: Method 1 1 Kc (1.5 − x1 )τm Td a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0. 5 K m τ m . 1 Kc = 1.5 − x1 τm 1 − x1 (1.5 − x1 ) 4 x1 + 1 − 4 x12 + 0.5x1 , Td = − x1τm , N = ; (1.5 − x1 )x1 K m τm 1.5 − x1 Recommended x1 = 0.5 ; non-oscillatory response. b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Benjanarasuth et al. (2005). Model: Method 1 Kc Ti Td 0.37 K u Tu 0 Comment a f 1 = 0 , b f 1 = 0 ; a f 2 = Tu , b f 2 = 0 ; a f 3 = 0 , b f 3 = 0 ; a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . 0.63K u 0.76Tu 0.078Tu N=∞ a f 1 = 0 , b f 1 = 0 ; a f 2 = 0.76Tu , b f 2 = 0 ; 2 a f 3 = 0.06Tu , b f 3 = 0 ; a f 4 = 0 , a f 5 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0 . 2 Huang et al. (2005b). Model: Method 1 Kc Ti [ 0.6325 τm ] N=∞ a f 1 = max 0.0063 τm ,0.01 , b f 1 = 0 ; a f 2 = Ti , b f 2 = 1.0630τm ; 1.5 a f 3 = 0.6723τm + 0.6988K m τm 2.5 , b f 3 = 0.2652τm 2 ; a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . Robust Lee and Edgar (2002). Model: Method 1 3 Ti Kc 0 a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u . 4 Zhang et al. (2002a). Model: Method 1 2λ 2 + 1.5τm Kc Td N=∞ 2 a f1 = 0.5λ 2 τ m 2 λ 2 + 2λ 2 τ m + 0.5τ m 2 , b f 1 = 0 ; a f 2 = 2λ 2 + τ m , b f 2 = 2λ1 + τm ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , bf 5 = 0 , K 0 = 0 . 2 K c = 0.3221 + (0.3099 K m τm ) , Ti = τm (1.0630 + 1.1048K m τm ) . 3 Kc = 4 Kc = 2 0.25K u Ti 4 τm − τm + . , Ti = (λ + τm ) KuKm 2(λ + τm ) 1 2λ 2 + 1.5τm (2λ 2 + τm )0.5τm , λ ,λ not specified. , Td = 1 2 K m λ 2 2 + 2λ 2 τm + 0.5τm 2 2λ 2 + 1.5τm 379 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 380 Rule Kc Ti Td Comment 5 2λ 2 + (1 + δ)τm Td N=∞ Kc Zhang et al. λ 2 2δτm − r2 τm 2 (2λ + τm ) (2006). af1 = 2 , b f 1 = 0 ; a f 2 = 2λ 2 + τ m , Model: Method 1 λ + 2λ δτ − r τ 2 − r τ (2λ + τ ) 2 2 m 2 m 1 m m b f 2 = 2λ1 + τm ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , bf 5 = 0 , K 0 = 0 . 5 Kc = 2 1 2λ 2 + τm + δτm 2λ 2 + δτm T = , , d K m λ 2 2 + 2λ 2δτm − r2 τm 2 − r1τm (2λ + τm ) 2λ 2 + (1 + δ)τm λ1 , λ 2 , λ , r1 , r2 , δ not specified. b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 4.1.9 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) − Y(s) Model F2 (s) Table 66: PID controller tuning rules – IPD model G m (s) = Rule Kc Ti Td K m e − sτ m s Comment Minimum performance index: other tuning 0.45 K m 8τ m 0.5τ m Åström and α = 1 , τm Tm ≤ 1 ; α = 0 , τm Tm > 1 ; β = 1 ; χ = 0 ; δ = 0 ; N = ∞ ; Hägglund (2004). bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , Model: Method 1 2 a f 3 = 0.2τm , a f 4 = 0.01τm ; K 0 = 0 . Kaya and Atherton (1999), Kaya (2003). Model: Method 10 1 Direct synthesis: frequency domain criteria ∧ 0 1 0.313 K u 0.733 T u α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ∧ 0.567 K m K u K0 = ∧ K m τm T u ∧ bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; bf 3 = 1 ; a f 5 = 0 . 0.667 0.333 ∧ K m K u 1 τm Tm − − 0.313 K u , b f 4 = 0.753 . ∧ τm K m τm T u ∧ 381 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 382 Rule Kc Ti Kc Ti Td Comment Robust 2 Lee and Edgar (2002). Model: Method 1 Td α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25K u ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 . 3 Kc Ti x1 α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b f 1 = 0 , a f 1 = Td , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = Td , a f 4 = 0 ; K 0 = 0.25K u ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 . 2 3 Kc = 2 2 0.25K u 4 4 τm τm − τm + , − τm + , Ti = (λ + τm ) K u K m 2(λ + τm ) KuKm 2(λ + τm ) Td = 2 4 0.25K u τm 3 τm τ (1 − 0.25K u K m τm ) 2 + + m . 0.25τm − 2 (λ + τm ) 6(λ + τm ) 4(λ + τm ) 0.5K u K m (λ + τm ) Kc = 2 2 0.25K u 4 τm 4 τm − τm + + x1 , Ti = − τm + + x1 (λ + τm ) K u K m 2(λ + τm ) KuKm 2(λ + τm ) 2 τm τm (1 − 0.25K u K m τm ) x1 = τm 0.5 − + + 2 6( λ + τ m ) 4( λ + τ m ) 0.5K u K m (λ + τm ) 0.5 . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 4.2 IPD Model with a Zero G m (s) = K m (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτ m or G m (s) = m s s 1 4.2.1 Ideal PI controller G c (s) = K c 1 + Ti s E(s) R(s) − + 1 K c 1 + Ti s U(s) Model Y(s) Table 67: PI controller tuning rules – IPD model with a negative zero K (1 + sTm3 )e −sτ m K (1 − sTm 4 )e −sτ m G m (s) = m or m s s Rule Kc 2 3 4 Kc = Kc = Kc = Kc = 1.571 3 Kc ( ) . ( ) . K m τm 1 + 3.142 Tm 4 τm 2 2.358 Comment Direct synthesis: frequency domain criteria τm Tm3 < 1 ∞ 4 τm Tm3 > 1 Kc Jyothi et al. (2001). Model: Method 1 1 Ti Direct synthesis: time domain criteria 1 ‘Smaller’ τm Tm 4 . Kc ∞ 2 ‘Larger’ τm Tm 4 . Kc Chidambaram (2002), p. 157. Model: Method 1 K m τm 1 + 4.712 Tm 4 τm 2 1.57 K m τm 1 + 9.870(Tm3 τm )2 0.79 K m τm 1 + 2.467(Tm3 τm )2 . . 383 b720_Chapter-04 Rule Kc Jyothi et al. (2001). Model: Method 1 0.5 K m Tm 4 Kc = FA Handbook of PI and PID Controller Tuning Rules 384 5 04-Apr-2009 5 Ti Kc 0.79 K m τm 1 + 2.467(Tm 4 τm )2 . ∞ Comment τm Tm 4 < 1 τm Tm 4 > 1 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 4.3 FOLIPD Model G m (s) = K m e − sτm s(1 + sTm ) 1 4.3.1 Ideal PI controller G c (s) = K c 1 + Ti s R(s) E(s) + − 1 K c 1 + Ti s U(s) K m e − sτ s(1 + sTm ) m Y(s) Table 68: PI controller tuning rules – FOLIPD model G m (s) = Rule Kc Ti K m e − sτ m s(1 + sTm ) Comment Process reaction Quarter decay ratio Coon (1956), (1964). criterion. ∞ K m (τ m + Tm ) Model: Method 1; Coefficient values: coefficients of K c τ m Tm x1 τ m Tm x1 τ m Tm x1 deduced from graphs. 0.020 5.0 0.25 2.2 4.0 1.1 0.053 4.0 0.43 1.7 0.11 3.0 1.0 1.3 Minimum performance index: regulator tuning Minimum IAE – 0.556 Shinskey (1994), Model: Method 1 3.7(τ m + Tm ) K m (τ m + Tm ) p. 75. Minimum IAE – 0.952 Shinskey (1994), Model: Method 1 4(Tm + τ m ) K m (Tm + τ m ) p. 158. x1 385 b720_Chapter-04 Rule Minimum ITAE – Poulin and Pomerleau (1996). Model: Method 1 Process output step load disturbance. Process input step load disturbance. Velázquez-Figueroa (1997). Model: Method 1 Velázquez-Figueroa (1997). Model: Method 1 Kuwata (1987). Model: Method 1 Kc = FA Handbook of PI and PID Controller Tuning Rules 386 1 04-Apr-2009 1 Kc Ti Comment Kc x 1 (τ m + Tm ) K c and Ti coefficients deduced from graph. τ m Tm x1 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 5.0728 4.9688 4.8983 4.8218 4.7839 3.9465 3.9981 4.0397 4.0397 4.0397 0.029 Tm K m τm Coefficient values: x2 τ m Tm 0.5231 0.5237 0.5241 0.5245 0.5249 0.5320 0.5315 0.5311 0.5311 0.5311 1.2 1.4 1.6 1.8 2.0 1.2 1.4 1.6 1.8 2.0 0.157 ∞ x1 x2 4.7565 4.7293 4.7107 4.6837 4.6669 4.0337 4.0278 4.0278 4.0218 4.0099 0.5250 0.5252 0.5254 0.5256 0.5257 0.5312 0.5312 0.5312 0.5313 0.5314 0.1 ≤ τm ≤ 0.33 Tm 0.195 τ τ 0.1 ≤ m ≤ 0.5 6.824Tm m Tm Tm Minimum performance index: servo tuning 0.528 τ 0.031 Tm 0.1 ≤ m ≤ 0.33 ∞ Tm K m τm 0.406 0.119 τ τm 0.042 τm 0.1 ≤ m ≤ 0.5 8 . 807 T T m T m K m Tm m Direct synthesis: time domain criteria 0.5 ∞ K (τ + T ) 0.044 τm K m Tm m m 0.122 m x2 Tm 2 +1 . K m (τm + Tm ) x1 (τm + Tm )2 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule 2 Huba and Žáková (2003). Model: Method 1 Kc Ti Kc ∞ 387 Comment Representative tuning rules; tuning rules for other τ m Tm values may be obtained from a graph. 0.193 K m τ m 3.717 τ m τ m Tm = 0.5 0.207 K m τ m 3.386τ m τ m Tm = 1 0.219 K m τ m 3.127 τ m τ m Tm = 2 Direct synthesis: frequency domain criteria Maximise crossover 3 1.26x 1 frequency. ∞ Kmτm Hougen (1979), p. 338. Model: Method 1 Poulin and Pomerleau 2.13 0.34 K u or (1999). K T m u Robust ∞ Velázquez-Figueroa 0.833(2λ + Tm + τm ) (1997). K m (λ + τm )2 McMillan (1984). Model: Method 1 4 Kc Tuning rules developed from K u , Tu . ^ K u tan 2 φ m Perić et al. (1997). Model: Method 8 − 2Tm + τm + 4Tm 3 4 2 2 2 Tm + τ m − τ m + 4 Tm K m τ m 2e 0.1592 Tu tan φ m 1 + tan 2 φ m 2 Kc = Model: Method 1; λ not specified. Ultimate cycle Ti ^ 2 Model: Method 1; M max = 5 dB . 1.04Tu 2 . 2 Tm x 1 values recorded deduced from a graph: τ m Tm 1 2 3 4 5 6 7 8 9 10 x1 0.31 0.19 0.14 0.11 0.09 0.078 0.068 0.06 0.053 0.05 2 T 0.65 1.477 Tm 1 m Kc = , Ti = 3.33τm 1 + . τ K m τ m 2 1 + (Tm τ m )0.65 m b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 388 1 + Td s 4.3.2 Ideal PID controller G c (s) = K c 1 + Ti s E(s) R(s) + − U(s) 1 K c 1 + + Td s Ti s K m e − sτ s(1 + sTm ) m Y(s) Table 69: PID controller tuning rules – FOLIPD model G m (s) = Rule Kc Ti Td K m e − sτ m s(1 + sTm ) Comment Minimum performance index: regulator tuning Minimum ISE – 0.6667 K m τm Model: Method 1 ∞ Tm Haalman (1965). 0 A m = 2.36 ; φ m = 50 ; M s = 1.9 . 0.717 Velázquez 1 Figueroa (1997). 0.062 Tm Td Ti Model: Method 1 K m τm Minimum performance index: servo tuning 0.75 0.73 Velázquez τm 2 Figueroa (1997). 0.064 Tm 1 . 2 T T m i T Model: Method 1 K m τm m Direct synthesis: time domain criteria Van der Grinten ∞ 1 K m τm Tm + 0.5τ m Step disturbance. (1963). −2 ω d τ m Stochastic e Model: Method 1 3 disturbance. ∞ Td K τ m m Tachibana (1984). 1 2 λ K m (1 + λτ m ) τ Ti = 3.537Tm m Tm 0.951 ∞ τ , Td = 1.458Tm m Tm Tm 0.754 . Ti = 8.071τm , 0 < τm Tm ≤ 0.0333 ; Ti = 22.183τm − 0.4701Tm , 0.0333 < τm Tm ≤ 0.2121 ; Ti = 8.2076Tm − 18.724τm , 0.2121 < τm Tm ≤ 0.3256 ; Ti = 1.7571Tm + 1.089τm , τm Tm > 0.3256 . 3 T Td = 1 − 0.5e − ωd τ m + m e − ωd τ m τme − ωd τ m . τ m Model: Method 5 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti 389 Comment Td Vítečková ∞ x1 K m τ m Tm (1999), Coefficient values: Vítečková et al. OS OS OS OS x x x1 x1 1 1 (2000a). 0% 0.641 15% 0.801 30% 0.957 45% Model: Method 1 0.368 0.514 5% 0.696 20% 0.853 35% 1.008 50% 0.581 10% 0.748 25% 0.906 40% 4 Chen and Seborg Model: Method 1 3TCL + τ m Td Kc (2002). Sree and Model: Chidambaram Method 1; T T 5 i d Kc (2006). p. 58. Direct synthesis: frequency domain criteria 6 2 Åström and 0.67e−1.9 τ + 0.2 τ Hägglund (1995), K (T + τ ) m m m pp. 212–213. −9.3τ + 6.6 τ 2 Model: Method 1 4.8e or Method 3. K m (Tm + τm ) = d desired 0.56τm e1.1τ −1.1τ ωpTm Chen et al. (1999a). 4 y 2 0.13τm e13.6 τ −11.0 τ 14.2τm e −13.3τ + 8.6 τ K m A mTd r1ωx Tm K mTd 2 0.68τm e 2.26 τ −1.8τ 2 2 M max = 1.4 M max = 2.0 ∞ 7 Td Model: Method 1 ∞ 8 Td Model: Method 1 (3TCL + τm )se−sτ , K = (3TCL + τm )(Tm + τm ) , c (TCL + τm )3 K c (TCLs + 1)3 m 2 Td = 0.895 − 0.100 5 Kc = K m τm Tm τm Tm T 0.450 + 0.960 m τm τm τ , T = τ . , Ti = Tm m d Tm m 0.358 − 0.040 0.895 − 0.100 τm τm 0.895 − 0.100 1 Tuning rules converted from formulae developed for G c (s) = K c [1 − α ] + + Tds . Tis 1 7 . Td = 2 −1 2ωp − 1.273ωp τm + Tm 6 8 Td = 1 ωx − 1.273ωx 2 τm + Tm −1 , ωx = 0.25π(2r1A m − 1) (r A − 1)τ 2 1 2 m m . 3 2 3TCL Tm + 3TCL Tm τm − TCL + Tm τm . (3TCL + τm )(Tm + τm ) b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 390 Rule Kc Klán (2001). 0.67e−1.9 τ + 0.2 τ Model: Method 1 K m (Tm + τm ) Ti Td 9 Ti Td ∞ Tm Comment 2 O’Dwyer (2001a). Model: Method 1 Åström and Hägglund (2006), p. 246. x 1Tm K mτm A m = 0. 5 x 1 ; φ m = 0.5π − x 1 . x1 Representative coefficient values: Am φm x1 Am 0.785 2 10 45 0 0.524 Ti Kc Rivera and Jun τ m + Tm + 2λ (2000). K m (τ m + λ )2 Model: Method 1 Td Robust Tm (τ m + 2λ ) τ m + Tm + 2λ τ m + Tm + 2λ φm 3 60 0 M s = M t = 1. 4 ; Model: Method 1 λ not specified. 2 Recommended 1 0.5τm Tm + Lee et al. (2006). K (T + τ ) τ m ≤ TCL ≤ 2τ m . ∞ TCL + τm m CL m Model: Method 1 Desired closed loop response = e −sτ m (1 + sTCL ) . Shamsuzzoha et al. (2006c). Model: Method 1 9 11 Ti Kc Td λ , ξ not specified. 1 Tuning rules converted from formulae developed for G c (s) = K c [1 − α ] + + Tds . Tis Ti = 0.13(τm + Tm )e13.6 τ −11.0 τ ; Td = 14.2(τm + Tm )e−13.3τ +8.6 τ . 2 10 11 2 T T 0.37 + 0.02 m 0.16 + 0.28 m 1 Tm τm τm Kc = τ , T = τ . 0.37 + 0.02 , Ti = Tm m d Tm m K m τm τm 0.03 + 0.0012 0.37 + 0.02 τm τm Kc = Td = Ti , Ti = x1 + Tm + x 2 − x 3 , x1K m (4λξ + τm − x 2 ) x 4 + x 2 (Tm + x1 ) + x1Tm − 0.167 τm 3 − 0.5x 2 τm 2 + x 4 τm + 4λ3ξ 4λξ + τm − x 2 − x3 ; Ti x1 is a constant with ‘a sufficiently large value’, 2 x2 = 2 2 2 2 Tm x 5e − τ m x 1 − x1 x 5e − τ m x1 + x1 − Tm 4λ2ξ 2 + 2λ2 + x 2 τm − 0.5τm − x 4 , x3 = , x1 − Tm 4λξ + τm − x 2 2 λ2 2λξ 2 x 4 = Tm x 5e − τ m Tm − 1 + x 2Tm , x 5 = 2 − +1 . T Tm m [ ] b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Shamsuzzoha and Lee (2007a). 12 Kc Ti Td Comment Ti Td Model: Method 1 0.25Ti Tuning rules developed from K u , Tu . Ultimate cycle McMillan (1984). Model: Method 1 Perić et al. (1997). Model: Method 8 12 Kc = Td = Kc Ti K u cos φ m x1Td 13 ∧ 14 Td Recommended x1 = 4 . Ti , Ti = x1 + Tm + x 2 − x 3 , x1K m (4λ + τm − x 2 ) x 4 + x 2 (Tm + x1 ) + x1Tm − 3 2 0.167 τm − 0.5x 2 τm + x 4 τm + 4λ3 4λ + τm − x 2 − x3 ; Ti x1 is a constant with ‘a sufficiently large value’, 2 x2 = [ ] 2 [ ] 2 x1 x 5e − τ m x1 − 1 − Tm x 5e − τ m Tm − 1 6λ2 − x 4 + x 2 τm − 0.5τm , x3 = , Tm − x1 4λ + τ m − x 2 [ 4 ] λ x 4 = Tm x 5e − 1 + x 2Tm ; x 5 = 1 − ; x 1 From graphs: λ = 0.26τm , M s = 1.6 , 0.05 ≤ τm Tm ≤ 1.1 ; 2 − τ m Tm λ = 0.21τm , M s = 1.8 , 0.05 ≤ τm Tm ≤ 1.75 ; λ = 0.19τ m , M s = 2.0 , 0.05 ≤ τm Tm ≤ 2.5 . 13 14 2 T 0.65 1.111 Tm 1 m Kc = , Ti = 2τm 1 + . τ K m τ m 2 1 + (Tm τ m )0.65 m Td = tan φ m + ^ T 4 + tan 2 φ m u . 4π x1 391 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 392 4.3.3 Ideal controller in series with a first order lag 1 1 G c (s) = K c 1 + + Td s 1 + Tf s Ti s R(s) E(s) + − U(s) K e −sτ m 1 m Tf s + 1 s(1 + sTm ) 1 K c 1 + + Td s Ti s Y(s) Table 70: PID controller tuning rules – FOLIPD model G m (s) = Rule Kc Kristiansson and Lennartson (2006). Model: Method 1; 1.7 < M max < 1.8 . 1 Kc = 10e −8 τ m T m K m Tm 2 Ti Kc 2 Kc Ti Kc Ti (1 − 0.35e Comment Td Direct synthesis: frequency domain criteria Ti Td τ 0 ≤ m < 0.25 Tm 1 3 1.7 τ m + 1.3Tm ( 2 1.7 − e − τ m 1.3Tm 1.7 τ m + 1.3Tm ( 2 1.7 − e − τ m 1.3Tm − 4 τ m Tm m ( −4.5 τ m T m m m d ) 4e K m Tm 2 − τ m 1.3Tm m ) m − 4 τ m Tm −8 τ Tm 0.25e m Tm (2.4τm + 0.9Tm )2 . m − τ m 1.3Tm i 3 m τ 1 ≤ m < 50 Tm )(1.7τ + 1.3T ) , )(1.7τ + 1.3T ) , T = 0.85τ + 0.65T . T = 2(1.7 − e τ 2 0.06 + 0.004 (1.7 − e )(1.7τ + 1.3T ) , K = T K T )(1.7τ + 1.3T ) , T = 0.85τ + 0.65T . T = 2(1.7 − e Kc = (1.7 − e τm ) 0.25 ≤ T < 1 )(2.4τ + 0.9T ) , T = 2(12−.40τ.35+e0.9T ) , Ti = 2 1 − 0.35e −4 τ m Tm (2.4τm + 0.9Tm ) , Tf = 2 K m e − sτ m s(1 + sTm ) − τ m 1.3Tm m c 2 m m m m m f m m m m f m m m m i − τ m 1.3Tm b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Kc Ti Td Comment Td Model: Method 1 Robust H ∞ optimal – Tan et al. (1998b). 4 λ = 0.5 (performance), λ = 0.1 (robustness), λ = 0.25 (acceptable). 5 3λ + Tm + τ m 1.5τ m ≤ λ (3λ + τ m )Tm Kc Zhang et al. 3λ + τ m + Tm ≤ 4.5τ m (1999b). Model: Method 1 λ = 1.5τ m , OS = 58%, TS = 6τ m ; λ = 2.5τ m , OS = 35%, TS = 11τ m ; λ = 3.5τ m , OS = 26%, TS = 16τ m ; λ = 4.5τ m , OS = 22%, TS = 20τ m ; λ > 0.2291τm for stability (Ou et al., 2005b). Tan et al. (1998a). Model: Method 1; λ not specified. Rivera and Jun (2000). Model: Method 1 4 Kc = Kc = Kc Tm + 8.1633τ m Tm τ m 0.1225Tm + τ m Tf = 0.5549τ m 7 Kc Tm + 5.3677 τ m Tm τ m 0.1863Tm + τ m Tf = 0.4482τ m 8 Kc Tm + 3.9635τ m Tm τ m 0.2523Tm + τ m Tf = 0.2863τ m 9 Kc 2( τ m + λ ) + Tm (0.463λ + 0.277)([0.238λ + 0.123]Tm + τm ) K m τm 2 Ti = Tm + 5 6 , Tf = K m 3λ2 + 3λτ m + τm 2 ) , Tf = 6 Kc = 0.0337Tm τm 1 + . 2 K m τm 0.1225Tm 7 Kc = 0.0754Tm τm 1 + . 2 0 . 1863 T K m τm m λ3 3λ2 + 3λτ m + τ m 2 . 0.1344Tm τm 1 + . 2 K m τm 0.2523Tm 2(τ m + λ ) + Tm τ m λ2 9 Kc = , Tf = . 2 2 2 K m 2τ m + 4τ m λ + λ 2τ m + 4 τ m λ + λ2 8 Kc = ( ) λ not specified. τm , 5.750λ + 0.590 Tm τm τm , Td = . (0.238λ + 0.123)Tm + τm 0.238λ + 0.123 3λ + Tm + τm ( 2Tm (τ m + λ ) 2( τ m + λ ) + Tm 393 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 394 Td s 1 + 4.3.4 Controller with filtered derivative G c (s) = K c 1 + Ti s 1 + s Td N R(s) T s 1 d + K c 1 + Td Ti s s 1+ N E(s) + − U(s) K m e − sτ s(1 + sTm ) m Table 71: PID controller tuning rules – FOLIPD model G m (s) = Rule Kc Ti Y(s) K m e − sτ m s(1 + sTm ) Comment Td Direct synthesis: frequency domain criteria Kristiansson and Lennartson (2002a). 1 Td Model: Method 1; 0.5 ≤ K1 ≤ 1 . Tm (2λ + τ m ) 2λ + Tm + τ m λ ∈ [τ m , Tm ] (Chien and Fruehauf, 1990); N = 10. Ti Kc Robust Chien (1988). 2λ + τ m + Tm Model: Method 1 K m (λ + τ m )2 1 2λ + Tm + τ m ( ) Kc = 0.059 K 1 2 + (0.055 K 1 ) − 0.08 2.5ωu 0.055 0.059 2 − 0.08 + + − , 2 0.4(10 + 1 K 1 ) K m K1 K1 2.3 − 2 K 1 Ti = 2 2.5 2 0.059 K1 + (0.055 K1 ) − 0.08 − , ωu 2.3 − 2K1 0.4(10 + 1 K1 ) ( ) ( ) −1 2.5 2 0.059 K12 + (0.055 K1 ) − 0.08 1 Td = − + 1 , 0.4(10 + 1 K1 ) ωu 2.3 − 2K1 N N= ( ) 2 Td 0.059 K1 + (0.055 K1 ) − 0.08 , Tf = . Tf 0.16ωu (10 + 1 K1 ) −1 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 1 1 + Td s 4.3.5 Classical controller G c (s) = K c 1 + Td s Ti s 1 + N R(s) 1 1 + sTd K c 1 + Td Ti s 1 + s N E(s) + − U(s) Y(s) K m e − sτ s(1 + sTm ) m Table 72: PID controller tuning rules – FOLIPD model G m (s) = Rule Kc Ti K m e − sτ m s(1 + sTm ) Comment Td Minimum performance index: regulator tuning Minimum IAE – τ m = Tm ; 0.78 Shinskey (1994), K ( τ + T ) 1.38( τ m + Tm ) 0.66( τ m + Tm ) N = 10. m m m p. 75. Model: Method 1 Minimum IAE – Shinskey (1994), pp. 158–159. 1 Minimum ITAE – Poulin and Pomerleau (1996), (1997). Model: Method 1. Process output step load disturbance. 3 2 Kc Kc Kc τm Tm < 2 0.56τ m + 0.75Tm Ti Model: Method 1; N not specified. x 2 (τ m + Tm ) Tm τm Tm ≥ 2 3.33 ≤ N ≤ 10 Tm N Coefficient values (deduced from graphs): τm x1 x2 x1 T N 0.2 0.4 0.6 0.8 1.0 5.0728 4.9688 4.8983 4.8218 4.7839 τm m 0.5231 0.5237 0.5241 0.5245 0.5249 ( 1.2 1.4 1.6 1.8 2.0 [ 4.7565 4.7293 4.7107 4.6837 4.6669 ]) 1 Kc = 0.926 , Ti = 1.57 τm 1 + 1.2 1 − e − Tm τ m . K m τm (1.22 − 0.03(Tm τm )) 2 Kc = 0.926 . K m τm (1 + 0.4(Tm τm )) 3 Kc = τm x2 Tm 2 +1 ; 0 ≤ ≤ 2. (Tm N ) K m (τm + Tm ) x1 (τm + Tm )2 x2 0.5250 0.5252 0.5254 0.5256 0.5257 395 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 396 Rule Kc Ti Poulin and Pomerleau (1996), (1997) – continued. Process input step load disturbance. 0.2 0.4 0.6 0.8 1.0 Skogestad (2003). Model: Method 1 0.5 K mτm Comment Td Coefficient values (continued): 3.9465 0.5320 1.2 4.0337 3.9981 0.5315 1.4 4.0278 4.0397 0.5311 1.6 4.0278 4.0397 0.5311 1.8 4.0218 4.0397 0.5311 2.0 4.0099 0.5312 0.5312 0.5312 0.5313 0.5314 Direct synthesis: time domain criteria Hougen (1979), p. 338. 8τ m N=∞ Tm Direct synthesis: frequency domain criteria 0.7 0.3 ∞ 0 . 106 x 4 1 0.135 NTm τm Kmτm Model: Method 1; maximise crossover frequency; 10 ≤ N ≤ 30 . N = 5. Poulin et al. 0.5455 (1996). Tm M max = 1 dB K m (τm + 0.2Tm ) 21(0.2Tm + τ m ) Model: Method 1 Robust Tm λ ∈ [τ m , Tm ] Chien (1988). Tm 2λ + τ m (Chien and K m (λ + τ m )2 Model: Method 1 Fruehauf, 1990); 2λ + τ m N = 10. 2λ + τ m Tm K m (λ + τ m )2 Chien (1988). Model: Method 1 2λ + τ m K m (λ + τ m ) 2 Tm K m (λ + τ m )2 2λ + τ m 1.1Tm Tm 2.2λ + 1.1τm λ ∈ [τ m , Tm ] (Chien and Fruehauf, 1990); N = 11. Tuning rules converted to this equivalent controller architecture. 4 x 1 values deduced from graph: τ m Tm 1 2 3 4 5 6 x1 6 3 2.1 1.8 1.6 1.3 τ m Tm 7 8 9 10 11 12 x1 1.1 1.0 0.9 0.8 0.7 0.7 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 397 4.3.6 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N R(s) E(s) + 2 U(s) T s 1 K m e − sτ d b f 0 + b f 1s + b f 2 s K c 1 + + Ti s T 1 + a f 1s + a f 2 s 2 s(1 + sTm ) 1+ d s N m − K m e − sτ m s(1 + sTm ) Table 73: PID controller tuning rules – FOLIPD model G m (s) = Rule Tsang et al. (1993). Model: Method 2 Kc x1 K m τ m Direct synthesis: time domain criteria 0 ∞ bf 0 = 1 2 ξ 1.6818 0.0 1.3829 0.1 1.1610 0.2 x1 K m τ m x1 Coefficient values: ξ x1 0.9916 0.8594 0.7542 0.3 0.4 0.5 ξ 0.6693 0.6 0.6000 0.7 0.5429 0.8 0.35τm ∞ x1 ξ 0.4957 0.4569 0.9 1.0 bf 0 = 1 2 2 b f 1 = 0.5τm , b f 2 = 0.0833τm , a f 1 = 0.2τm , a f 2 = 0.01τm . x1 Tsang and Rad (1995). Model: Method 2 Comment Td b f 1 = Tm + 0.25τm , b f 2 = 0.25Tm τm , a f 1 = 0.2τm , a f 2 = 0.01τm . x1 Tsang et al. (1993). Model: Method 2 Ti Y(s) ξ 1.8512 0.0 1.5520 0.1 1.3293 0.2 0.809 K m τm x1 Coefficient values: ξ x1 1.1595 1.0280 0.9246 0.3 0.4 0.5 0.8411 0.7680 0.6953 ∞ 0 ξ x1 ξ 0.6 0.7 0.8 0.6219 0.5527 0.9 1.0 Maximum overshoot = 16%. b f 0 = 1 , b f 1 = Tm + 0.5τm , b f 2 = 0.5Tm τm , a f 1 = 0.12τm , 2 a f 2 = 0.0036τm . b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 398 Rule Kc Shamsuzzoha et al. (2006a). Model: Method 1 b Ti Td Comment Direct synthesis: frequency domain criteria τ 1 0 ≤ m ≤ 2.5 ∞ 0.333τm Tm K m (λ + τm ) f 0 = 1 , b f 1 = Tm , b f 2 = 0 , a f 1 = 0.167 τ m 2λ − τm , af 2 = 0 , N = ∞ . λ + τm λ values deduced from graph: λ = 1.4τm , M s = 1.4 ; λ = 0.7 τm , M s = 1.6 ; λ = 0.4τm , M s = 1.8 ; λ = 0.35τm , M s = 2.0 . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 399 4.3.7 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s α + β d K c T 1+ d s N E(s) − R(s) + T s 1 d K c 1 + + Ti s Td 1+ s N − U(s) + m Table 74: PID controller tuning rules – FOLIPD model G m (s) = Rule Kc Ti Y(s) K m e − sτ s(1 + sTm ) K m e − sτ m s(1 + sTm ) Comment Td Minimum performance index: regulator tuning Minimum IAE – α = 0 ,β =1, 1.16 Shinskey (1994), K ( τ + T ) 1.38( τ m + Tm ) 0.55( τ m + Tm ) N =∞. m m m p. 75. Model: Method 1 Minimum IAE – Shinskey (1994), p. 159. 1 1 1.28 Kc = ( Kc 0.48τ m + 0.7Tm Ti K m τm 1 + 0.24(Tm τm ) − 0.14(Tm τm )2 α = 0 ,β =1, N =∞. Model: Method 1 ) , T = 1.9τ (1 + 0.75[1 − e i m − (T m τ m ) ) ] . b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 400 Rule Kc 4 (8 − x1 )K m τm Arvanitis et al. (2003b). Model: Method 1 Ti Td Comment 4τm x1 0 α =1 Minimum ISE. 2 Minimum performance index. 3 2 2 3 τ τ τ τ x1 = 2.6302 − 2.1248 m + 2.5776 m − 1.8511 m + 0.82557 m T T T m m m Tm 5 6 4 7 τ τ τ τ − 0.23641 m + 0.044128 m − 0.0053335 m + 0.00040201 m T T T m m m Tm 9 10 τ τ τ − 1.7164.10 −5 m + 3.1685.10 −7 m , 0 < m < 4.1 ; T T T m m m τ τ x1 = 1.7110073 − 0.003554 m − 4.1 , m ≥ 4.1 . Tm Tm ∞ 3 Performance index = ∫ [e ( t) + K u ( t)]dt . 2 2 2 m 0 2 3 τ τ τ τ x1 = 2.1054 − 0.76462 m + 0.79445 m − 0.51826 m + 0.21771 m Tm Tm Tm Tm 5 6 7 τ τ τ τ − 0.06 m + 0.010927 m − 0.0013006 m + 9.717.10 −5 m Tm Tm Tm Tm 9 10 τ τ τ − 4.132.10 −6 m + 7.6235.10 −8 m , 0 < m < 4.6 ; Tm Tm Tm τ τ x1 = 1.69667833406 − 0.00148768 m − 4.6 , m ≥ 4.6 . T m Tm 4 8 8 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Arvanitis et al. (2003b) – continued. Arvanitis et al. (2003b). Model: Method 1 Ti 401 Comment Td Minimum performance index. 4 5 0 Ti Kc α =1 Minimum ISE. 6 ∞ 4 Performance index = 2 2 2 du e ( t ) + K m dt . dt 0 ∫ τ τ x1 = 1.6505623 − 0.0034985 m − 8 , m ≥ 2 ; T m Tm 2 3 τ τ τ τ x1 = 2.1806 − 0.60273 m + 0.3875 m − 0.14955 m + 0.034535 m Tm Tm Tm Tm 5 6 4 7 τ τ τ τ − 0.0042243 m + 8.2395.10 −5 m + 5.0912.10 −5 m − 7.283.10 −6 m Tm Tm Tm Tm 9 8 10 τ τ τ + 4.2603.10 −7 m − 9.5824.10 −9 m , m range not defined. Tm Tm Tm 12 . 5664 τ + T − 4 ( ) 5 m m Kc = K m 2(τm + Tm )(12.5664(τm + Tm ) − 4 ) − 1.5τm 2 + 3τm Tm + 2Tm 2 x1 [ ) ] ( 3.4674τm + 3.1416Tm − 1 12.5664(τm + Tm ) − 4 ]x 2 ; Ti = . x1 τm with x1 = [0.5708 + 2 6 3 τ τ τ τ x 2 = 0.99315 + 1.9536 m − 2.5157 m + 1.9163 m − 0.8893 m Tm Tm Tm Tm 5 6 7 4 τ τ τ τ + 0.26167 m − 0.049809 m + 0.0061098 m − 0.00046593 m Tm Tm Tm Tm 9 10 τ τ τ + 2.0083.10 −5 m − 3.7368.10 −7 m , 0 < m < 6.0 ; Tm Tm Tm τ τ x 2 = 1.96705757 + 0.01308189 m − 6 , m ≥ 6.0 . T m Tm 8 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 402 Rule Kc Ti Arvanitis et al. (2003b) – continued. Td Comment Minimum performance index. 7 Minimum performance index. 8 ∞ 7 Performance index = ∫ [e ( t) + K u ( t)]dt . 2 2 m 2 0 2 3 τ τ τ τ x 2 = 0.73114 + 2.3461 m − 2.8509 m + 2.0965 m − 0.95252 m T T T m m m Tm 5 6 4 7 τ τ τ τ + 0.27639 m − 0.052089 m + 0.0063413 m − 0.00048064 m T T T m m m Tm 9 8 10 τ τ τ + 2.0611.10 −5 m − 3.8178.10 −7 m , 0 < m < 4.5 ; T T T m m m τ τ x 2 = 1.916423116 + 0.018175439 m − 4.5 , m ≥ 4.5 . Tm Tm ∞ 8 Performance index = 2 2 2 du e ( t ) + K m dt . dt 0 ∫ 2 3 τ τ τ τ x 2 = 0.60896 + 3.0593 m − 3.9339 m + 2.933 m − 1.3353 m Tm Tm Tm Tm 5 6 7 4 τ τ τ τ + 0.38696 m − 0.072771 m + 0.0088395 m − 0.00066864 m Tm Tm Tm Tm 9 10 τ τ τ + 2.8623.10 −5 m − 5.2941.10 −7 m , 0 < m < 4.5 ; Tm Tm Tm τ τ x 2 = 1.9283751 + 0.016846129 m − 4.5 , m ≥ 4.5 . T m Tm 8 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Ti Td Comment 4 τm x1 (8 − x1 ) τm 16 α = 1, β = 1, N =∞. Kc 16 Arvanitis et al. (16 − 3x )K τ 1 m m (2003b). Model: Method 1 Minimum ISE. 9 Minimum performance index. 10 2 9 3 τ τ τ τ x1 = 1.5841 + 0.3101 m + 0.38841 m − 0.4622 m + 0.21918 m Tm Tm Tm Tm 7 6 5 4 τ τ τ τ − 0.060691 m + 0.010771 m − 0.0012466 m + 9.1201.10 −5 m Tm Tm Tm Tm 10 9 τ τ − 3.8302.10 −6 m + 7.0316.10 −8 m . Tm Tm ∞ 10 Performance index = ∫ [e ( t) + K u ( t)]dt . 2 2 m 2 0 2 3 τ τ τ τ x1 = 0.066485 + 2.0757 m + 1.0972 m − 2.7957 m + 1.9019 m Tm Tm Tm Tm 7 6 5 4 τ τ τ τ − 0.68418 m + 0.14763 m − 0.019733 m + 0.0016016 m Tm Tm Tm Tm 9 10 τ τ τ − 7.2368.10 −5 m + 1.397.10 −6 m , 0 < m < 4.0 ; Tm Tm Tm τ τ x1 = 2.128638 − 0.00560812 m − 4.0 , m ≥ 4.0 . T m Tm 8 8 403 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 404 Rule Kc 11 Arvanitis et al. (2003b). Model: Method 1 Kc Ti Td Comment Ti Td α = 1, β = 1, N =∞. Minimum ISE. 12 Minimum performance index. 13 11 Kc = 12.5664(τm + Tm ) − 4 K m τm [25.1328(τm + Tm ) − 8 − (τm + 2Tm )x1 ] with x1 = [0.5708 + Ti = 12.5664(τm + Tm ) − 4 x1 2 . , Td = Tm − Tm x1 12.5664(τm + Tm ) − 4 2 12 3.4674τm + 3.1416Tm − 1 ]x 2 , τm 3 τ τ τ τ x 2 = 1.178 + 4.5663 m − 7.0371 m + 5.5784 m − 2.6103 m Tm Tm Tm Tm 7 6 5 4 τ τ τ τ + 0.7679 m − 0.1458 m + 0.017833 m − 0.0013561 m Tm Tm Tm Tm 8 10 9 τ τ τ + 5.83.10 −5 m − 1.0823.10 −6 m , 0 < m < 3.5 , Tm Tm Tm τ τ x 2 = 2.2036 − 0.0033077 m − 6.5 , m ≥ 3.5 . T m Tm ∞ 13 Performance index = ∫ [e ( t) + K u ( t)]dt . 2 2 m 2 0 2 3 τ τ τ τ x 2 = −0.1185 + 4.781 m − 3.9977 m + 1.733 m − 0.42245 m T T T m m m Tm 5 7 6 4 τ τ τ τ + 0.057708 m − 0.0038421 m + 2.6386.10 −5 m + 1.078.10 −5 m T T T m m m Tm 9 τ − 4.2486.10 −7 m . Tm 8 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Arvanitis et al. (2003b). Model: Method 1 Ti Comment Td Minimum performance index: servo tuning 4 4τm (8 − x1 )K m τm 0 α =1 x 1 Minimum ISE. 14 Minimum performance index. 15 Minimum performance index. 16 2 14 3 τ τ τ τ x1 = 1.997 − 0.82781 m + 1.0002 m − 0.72106 m + 0.32269 m Tm Tm Tm Tm 5 6 4 7 τ τ τ τ − 0.092452 m + 0.017215 m − 0.0020706 m + 0.00015505 m T T T m m m Tm 8 10 9 τ τ τ − 6.5691.10 −6 m + 1.2025.10 −7 m , 0 < m < 5.5 ; T T T m m m τ τ x1 = 1.6281745 − 0.001171 m − 5.5 , m ≥ 5.5 . T m Tm ∞ 15 Performance index = ∫ [e ( t) + K u ( t)]dt . 2 2 m 2 0 2 3 τ τ τ τ x1 = 1.5693 + 0.1066 m − 0.10101 m + 0.05217 m − 0.016147 m T T T m m m Tm 7 6 5 4 τ τ τ τ + 0.0030412 m − 0.00032187 m + 1.2181.10 −5 m + 9.9698.10 −7 m T T T m m m Tm −7 τ m 9 10 τ τ + 3.7365.10 −9 m , 0 < m < 4.6 . − 1.2274.10 T T T m m m ∞ 16 Performance index = 2 2 2 du e ( t ) + K m dt dt 0 ∫ τ τ x1 = 0.780651417 + 4.7464131 m − 0.1 , 0 < m < 0.3 ; Tm Tm τ τ x1 = 1.729934 − 0.051535 m − 0.3 , 0.3 ≤ m < 2 . T T m m 8 405 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 406 Rule Kc Arvanitis et al. (2003b). Model: Method 1 17 Kc Ti Td Comment Ti 0 α =1 Minimum ISE. 18 Minimum performance index. 19 17 Kc = 12.5664(τm + Tm ) − 4 [ 3.4674τm + 3.1416Tm − 1 12.5664(τm + Tm ) − 4 ]x 2 , Ti = . x1 τm with x1 = [0.5708 + 2 18 ) ] ( K m 2(τm + Tm )(12.5664(τm + Tm ) − 4 ) − 1.5τm 2 + 3τm Tm + 2Tm 2 x1 3 τ τ τ τ x 2 = 0.79001 + 2.114 m − 2.5601 m + 1.8971 m − 0.86871 m Tm Tm Tm Tm 7 6 5 4 τ τ τ τ + 0.25371 m − 0.048068 m + 0.0058771 m − 0.00044707 m T T T m m m Tm 8 10 9 τ τ τ + 1.9231.10 −5 m − 3.5718.10 −7 m , 0 < m < 4.5 ; T T T m m m τ τ x 2 = 1.903177576 + 0.0181584 m − 4.5 , m ≥ 4.5 . T m Tm ∞ 19 Performance index = ∫ [e ( t) + K u ( t)]dt . 2 2 2 m 0 2 3 τ τ τ τ x 2 = 0.48124 + 2.5549 m − 2.896 m + 2.0495 m − 0.91115 m T T T m m m Tm 5 6 7 4 τ τ τ τ + 0.26094 m − 0.048775 m + 0.0059064 m − 0.00044618 m T T T m m m Tm −5 τ m 9 10 τ τ − 3.5335.10 −7 m , 0 < m < 4.0 ; + 1.9095.10 T T T m m m τ τ x 2 = 1.87054524 + 0.021802 m − 4 , m ≥ 4.0 . T m Tm 8 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Arvanitis et al. (2003b) – continued. Arvanitis et al. 16 (2003b). (16 − 3x1 )K m τm Model: Method 1 Ti 407 Comment Td Minimum performance index. 20 α = 1, β = 1, N =∞. (8 − x1 ) τm 16 4 τm x1 Minimum ISE. 21 ∞ 20 Performance index = 2 2 2 du e ( t ) + K m dt . dt 0 ∫ 2 3 τ τ τ τ x 2 = 0.31373 + 3.7796 m − 5.0294 m + 3.8589 m − 1.7956 m T T T Tm m m m 7 6 5 4 τ τ τ τ + 0.52906 m − 0.10079 m + 0.012369 m − 0.00094357 m T T T m m m Tm 8 10 9 τ τ τ + 4.0678.10 −5 m − 7.5694.10 −7 m , 0 < m < 3.5 ; T T T m m m τ τ x 2 = 1.86144723 + 0.021989 m − 3.5 , m ≥ 3.5 . Tm Tm 2 21 3 τ τ τ τ x1 = 1.2171 + 0.7339 m + 0.083449 m − 0.14905 m + 0.030257 m T T T Tm m m m 5 6 7 4 τ τ τ τ + 0.0045567 m − 0.0029943 m + 0.00056821 m − 5.4957.10 −5 m T T T m m m Tm 9 10 τ τ + 2.7505.10 −6 m − 5.6642.10 −8 m . T m Tm 8 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 408 Rule Kc Arvanitis et al. (2003b) – continued. Arvanitis et al. (2003b). Model: Method 1 Ti Comment Td Minimum performance index. 22 23 Ti Kc Td α = 1, β = 1, N =∞. Minimum ISE. 24 ∞ 22 ∫ [e ( t) + K u ( t)]dt . 2 Performance index = 2 m 2 0 2 3 τ τ τ τ x1 = 0.083709 + 2.2875 m − 0.18739 m − 1.0059 m + 0.79718 m Tm Tm Tm Tm 7 6 5 τ τ τ τ − 0.3036 m + 0.067859 m − 0.0093104 m + 0.00077161 m Tm Tm Tm Tm 4 8 10 9 τ τ τ − 3.5473.10 −5 m + 6.9487.10 −7 m , 0 < m < 4.5 Tm Tm Tm τ τ x 1 = 2.444547 − 0.0146274714 m − 4.5 , m ≥ 4.5 . T m Tm 12.5664(τm + Tm ) − 4 23 Kc = K m τm [25.1328(τm + Tm ) − 8 − (τm + 2Tm )x1 ] with x1 = [0.5708 + Ti = 12.5664(τm + Tm ) − 4 x1 2 . , Td = Tm − Tm x1 12.5664(τm + Tm ) − 4 2 24 3 3.4674τm + 3.1416Tm − 1 ]x 2 τm τ τ τ τ x 2 = 1.1571 + 5.0775 m − 8.2021 m + 6.6353 m − 3.1355 m Tm Tm Tm Tm 7 6 5 4 τ τ τ τ + 0.92683 m − 0.17635 m + 0.02158 m − 0.0016402 m Tm Tm Tm Tm 9 10 τ τ τ + 7.0439.10 −5 m − 1.3056.10 −6 m , 0 < m < 4.5 Tm Tm Tm τ τ x 2 = 2.124307 − 0.004844182 m − 4.5 , m ≥ 4.5 . T Tm m 8 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Arvanitis et al. (2003b) – continued. Td Comment Minimum performance index. 25 Minimum performance index. 26 ∞ 25 Performance index = ∫ [e ( t) + K u ( t)]dt . 2 2 m 2 0 2 3 τ τ τ τ x 2 = −0.70861 + 7.4212 m − 8.1214 m + 4.8168 m − 1.7217 m T T T Tm m m m 7 6 5 4 τ τ τ τ + 0.38998 m − 0.056975 m + 0.0053141 m − 0.00030233 m T T T m m m Tm 8 10 9 τ τ τ + 9.3969.10 −6 m − 1.1861.10 −7 m , 0 < m < 4.5 T T T m m m τ τ x 2 = 2.1069751341 − 0.0017098 m − 4.5 , m ≥ 4.5 . Tm Tm ∞ 26 Performance index = 2 2 2 du e ( t ) + K m dt . dt 0 ∫ 2 3 τ τ τ τ x 2 = 0.05486 + 0.81662 m − 1.4989 m + 1.1955 m − 0.48464 m Tm Tm Tm Tm 7 6 5 4 τ τ τ τ + 0.11011 m − 0.013386 m + 0.00059665 m + 3.8633.10 −5 m Tm Tm Tm Tm 9 10 τ τ τ − 5.307.10 −6 m + 1.6437.10 −7 m , 0 < m < 6.0 ; Tm Tm Tm τ τ x 2 = 2.1111 − 0.001025 m − 6 , m ≥ 6 . T Tm m 8 409 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 410 Rule Kc Ti x1 K m Comment Td Minimum performance index: other tuning x 2Tm x 3Tm N=∞ Taguchi et al. τm τm (1987). α α x1 x 2 x3 β x1 x 2 x3 β Tm Tm Model: Method 1 12.06 0.81 0.38 0.67 0.84 0.2 0.67 5.25 1.26 0.66 0.74 1.6 3.93 1.55 0.62 0.67 0.82 0.4 0.59 5.82 1.34 0.66 0.74 1.8 2.19 2.23 0.78 0.67 0.80 0.6 0.53 6.36 1.42 0.66 0.73 2.0 1.50 2.88 0.90 0.66 0.79 0.8 0.42 7.74 1.61 0.66 0.72 2.5 1.14 3.50 1.00 0.66 0.77 1.0 0.36 9.06 1.82 0.66 0.72 3.0 0.92 4.10 1.09 0.66 0.76 1.2 0.27 11.38 2.26 0.66 0.70 4.0 0.77 4.68 1.18 0.66 0.75 1.4 0.22 14.76 2.59 0.68 0.76 5.0 x1K u x 2 Tu x 3Tu Minimum ITAE Coefficient values: – Pecharromán α x x x3 φc 1 2 and Pagola 1.672 0.366 0.136 0.601 (2000). − 164 0 Model: 1.236 0.427 0.149 0.607 − 160 0 Method 4; 0.994 0.486 0.155 0.610 − 1550 Km = 1; 0.842 0.538 0.154 0.616 T =1; − 150 0 m β = 1 , N = 10; 0.05 < τ m < 0.8 . Minimum ITAE – Pecharromán (2000a). Model: Method 4; Km = 1; Tm = 1 ; 0.05 < τ m < 0.8 . 0.752 0.567 0.157 0.605 − 1450 0.679 0.610 0.149 0.610 − 140 0 0.635 0.637 0.142 0.612 − 1350 0.590 0.669 0.133 0.610 − 130 0 0.551 0.690 0.114 0.616 − 1250 0.520 0.776 0.087 0.609 − 120 0 0.509 0.810 0.068 0.611 − 118 0 x1K u 0 x 2 Tu x1 x2 α 0.049 2.826 0.066 Coefficient values: φc x1 x2 α φc 0.218 1.279 0.564 − 1350 − 160 0 0.250 1.216 0.573 − 130 0 0.522 − 1550 0.286 1.127 0.578 − 1250 1.716 0.532 − 150 0 0.330 1.114 0.579 − 120 0 0.159 1.506 0.544 − 1450 0.351 1.093 0.577 − 118 0 0.189 1.392 0.555 − 140 0 0.506 − 164 0 2.402 0.512 0.099 1.962 0.129 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc x1K u Minimum ITAE – Pecharromán x1 x2 (2000b), p. 142. 0.0469 2.8052 Model: Method 4; 0.0662 2.3876 Km = 1; 0.0988 1.9581 Tm = 1 ; 0.05 < τ < 0.8 . 0.1271 1.6757 m Ti Td x 2 Tu 0 Comment Coefficient values: φc x1 α 411 x2 α φc 0.5061 − 164.10 0.2174 1.2566 0.5642 − 135.00 0.5119 − 160.50 0.2511 1.1948 0.5730 − 130.00 0.5220 − 155.00 0.2870 1.1281 0.5777 − 125.00 0.5319 − 150.00 0.3280 1.0860 0.5794 − 120.00 0.1589 1.5052 0.5439 − 145.00 0.3501 1.0822 0.5767 − 118.00 0.1866 1.3619 0.5546 − 140.00 Minimum ITAE – Pecharromán (2000b), p. 148. Model: Method 4; Km = 1; Tm = 1 ; β = 1 , N = 10; 0.05 < τm < 0.8 . Taguchi and Araki (2000). Model: Method 1 27 Kc = x1K u x 2 Tu x 3Tu x1 x2 Coefficient values: x3 α φc 1.6695 0.3684 0.1323 0.6007 − 164.10 1.2434 0.4389 0.1507 0.6073 − 160.00 0.9901 0.4846 0.1587 0.6102 − 155.00 0.8378 0.5393 0.1562 0.6163 − 150.00 0.7383 0.5789 0.1548 0.6052 − 145.00 0.6728 0.6283 0.1488 0.6105 − 140.00 0.6196 0.6221 0.1376 0.6120 − 135.00 0.5948 0.6939 0.1326 0.6096 − 130.00 0.5513 0.7050 0.1190 0.6157 − 125.00 0.5271 0.7471 0.0992 0.6088 − 120.00 0.5229 0.8886 0.0862 0.6115 − 118.0 0 OS ≤ 20% ; τm Tm ≤ 1.0 . 27 Kc Ti 0 0.2839 1 , 0.1787 + Km (τm Tm ) + 0.001723 Ti = 4.296 + 3.794(τm Tm ) + 0.2591(τm Tm ) , α = 0.6551 + 0.01877(τ m Tm ) . 2 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 412 Rule Kc Comment Ti Td Ti Td x1 K m x 2Tm Taguchi and α x1 x 2 x3 β Araki (2002). 8.91 1.73 0.46 0.32 0.72 Model: Method 1 11.13 1.40 0.39 0.45 0.91 12.04 0.81 0.38 0.67 0.84 x 3Tm Taguchi and Araki (2000) – continued. 28 Kc x4 x1 x2 N=∞ x3 α β x4 0.1 12.36 0.68 0.39 0.66 0.80 1.5 0.5 13.00 0.54 0.40 0.67 0.76 2.0 1.0 Load disturbance model = K m e −sτ m (1 + sx 4Tm ) ; τm Tm = 0.2 . Taguchi et al. (2002). Model: Method 1; α=0. Taguchi et al. (2002). Model: Method 1; α=0. Taguchi et al. (2002). 28 Kc = ∞ x1Tm K m x2 β τm Tm 16.54 0.32 5.33 0.52 1.96 0.76 1.17 0.91 x1Tm K m 0.89 0.85 0.78 0.73 0.1 0.2 0.4 0.6 x1 N = 10 x 2Tm ∞ x1 β x2 β τm Tm x1 x2 26.56 7.75 2.54 1.42 0.29 0.50 0.78 0.95 0.81 0.79 0.74 0.70 0.1 0.2 0.4 0.6 0.98 0.74 0.35 0.15 1.08 1.19 1.59 2.77 ∞ Kc τm Tm 0.68 0.8 0.64 1.0 0.50 2.0 0.37 5.0 N= ∞ x1 29 β x2 0.83 1.02 0.65 1.10 0.32 1.47 0.14 1.47 x 2Tm τm Tm 0.66 0.8 0.62 1.0 0.50 2.0 0.37 5.0 N = 10; Model: Method 1 Td 1 0.5184 , 0.7608 + 2 K m ((τm Tm ) + 0.01308) ( ) T = T (0.03432 + 2.058(τ T ) − 1.774(τ T ) + 0.6878(τ T ) ) , α = 0.6647 , Ti = Tm 0.03330 + 3.997(τm Tm ) − 0.5517(τm Tm ) , 2 2 d m m m m m 3 m m β = 0.8653 − 0.1277(τ m Tm ) + 0.03330(τ m Tm ) , N not specified. 2 2 29 Kc = τ τ Tm 0.2938 , β = 0.9305 − 0.4147 m + 0.1234 m , 0.4073 + 2 K m Tm ((τm Tm ) + 0.03829) Tm 2 3 τm τm τm τm Td = Tm 0.08289 + 2.682 − 2.959 + 1.309 T , α = 0 , 0.1 ≤ T ≤ 1.0 . Tm T m m m b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Kuwata (1987). Model: Method 1 Comment Td Direct synthesis: time domain criteria 3.33(Tm + τm ) 0 K m (Tm + τm ) 1 Kc Ti 4 (8 − x1 )K m τm 4τm x1 30 Arvanitis et al. (2003b). Model: Method 1 Ti α = 1, β = 1 , N =∞. α = 1; 4 x1 = 2 . 4ξ + 1 Td 0 Ti 31 Kc 32 Kc (2ξ + 1)τ 33 Kc Ti 2 α = 1, β = 1, N =∞. Td m α =1 Td Direct synthesis: frequency domain criteria 2 M s = 1.4 5.7τ m e1.7 τ−0.69 τ Kc 0 Ti M s = 2.0 35 K Åström and Hägglund (1995), pp. 210–212. 34 c 0.14 ≤ τm Tm ≤ 5.5 ; Model: Method 3. Klán (2001). 30 31 1.067(Tm + τm ) 33 6.25τm (2Tm + τm ) 2 K m τm (2Tm + τm ) 2 0 Ti Kc 3 Kc = 2 ; Ti = 2 (Tm + τm ) 2 Model: Method 1 2.5τm (Tm + 0.5τm ) 3 ; Td = (Tm + τm ) 3 (1 + 4ξ )(τ + T ) , T = (1 + 4ξ )(τ + T ) . K [τ (0.5 + 8ξ ) + τ T (1 + 16ξ ) + 8ξ T ] 16(2ξ + 1) 16ξ + 4 K = (32ξ + 4)K τ , T = 16(2ξ + 1) τ . τ T (1 + 4ξ ) τ + T + 4τ ξ . K = , T = τ + T + 4τ ξ , T = τ + T + 4τ ξ K τ (1 + 8ξ ) 2 Kc = 2 m 32 36 2 m m m m m c 2 m m 2 m m 2 2 2 i m m m 2 2 c 2 d m 2 m 2 m m 2 i m m m 2 m m d m m 2 m 2 2 34 Kc = 2 0.41e−0.23τ + 0.019 τ , α = 1 − 0.33e 2.5τ−1.9 τ . K m (Tm + τm ) 35 Kc = 2 2 0.81e−1.1τ + 0.76 τ , Ti = 3.4τm e0.28τ − 0.0089 τ , α = 1 − 0.78e −1.9 τ+1.2 τ . K m (Tm + τm ) 2 2 36 2 2 0.41e−0.23τ + 0.019 τ Kc = , Ti = 5.7(τm + Tm )e1.7 τ − 0.69 τ , α = 1 − 0.33e 2.5τ−1.9 τ . K m (Tm + τm ) . 413 b720_Chapter-04 FA Handbook of PI and PID Controller Tuning Rules 414 Rule Kc Wang and Cai 37 Kc (2001). Model: Method 1 0.6068 K m τm Wang and Cai (2002). Xu and Shao (2003a). 0.6189 K m τm Xu and Shao (2003c). 1.5τ m + 0.5Tm Xu and Shao (2004a). Model: Method 1 37 04-Apr-2009 Kc = 40 Td Ti Td 2 (2λ − 1)τm + Tm K m λ2 τm 2 Comment β = 0, N = ∞. 5.9784 τ m 38 Td 5.9112 τ m 39 Td 2(Tm + 1.5τ m ) Kc 2K m τ m Ti Tm + τ m (Tm + 0.5τm )2 (Tm + 1.5τ m ) 1.5Tm τ m 1.5τ m + 0.5Tm (2λ − 1)Tm τm (2λ − 1)τm + Tm (2λ − 1)τm + Tm N =∞; α = Model: Method 1 Model: Method 1 41 Model: Method 1 λ not defined. λτ m λ ; β= . Tm + (2λ − 1)τm 2λ − 1 τm (1.3608M s − 1.2064) 1 1.2064 , 1.3608 − , Ti = K m τm Ms 0.29M s − 0.3016 Td = (Tm + 0.1τm )(1.450Ms − 1.508) , α = 1.3608M s − 1.2064 0. 2 M s , 1.3 ≤ M s ≤ 2 . 1.3608M s − 1.2064 38 Td = 0.8363(Tm + 0.1τm ) , α = 0.3296 , M s = 1.6 ( A m > 2.66 , φ m > 36.4 0 ) . 39 Td = 0.8460(Tm + 0.1τm ) , N = ∞ , β = 0 , α = 0.3232 , A m = 3 , φ m = 60 0 . 40 Kc = 41 α= τm 0.5(Tm + 1.5τm ) , N= ∞ , β = 0 , A m > 2 , φ m > 60 0 . , α= Tm + 1.5τ m K m τm (Tm + τm ) τm , β = 0.6667 , N= ∞ , A m > 2 , φ m > 60 0 . 1.5Tm + 0.5τ m b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Td Comment Td α = 0. 6 , β = 0 . 415 Robust 42 Shamsuzzoha and Lee (2007a). Model: Method 1 Ti Kc λ = 0.26τm , M s = 1.6 , 0.05 ≤ τm Tm ≤ 1.1 ; λ = 0.21τm , M s = 1.8 , 0.05 ≤ τm Tm ≤ 1.75 ; λ = 0.19τ m , M s = 2.0 , 0.05 ≤ τm Tm ≤ 2.5 ; Kuwata (1987); x 1 , x 2 deduced from graphs. x 1K u 42 Kc = Td = α =1 x1 x2 τm Tu x1 x2 τm Tu x1 x2 τm Tu 0.12 0.19 0.25 1.14 1.09 0.91 x 1K u 0.08 0.10 0.13 0.30 0.34 0.36 x 2 Tu 0.89 0.86 0.84 0.15 0.18 0.20 0.38 0.83 0.25 x2 x3 x2 α = 1, β = 1, N =∞. τm Tu x3 0.71 0.61 0.69 0.62 0.125Tu 0.08 0.20 0.08 0.22 20% overshoot. x1 Oubrahim and Leonard (1998). Model: Method 1 equations for λ deduced from graphs. Ultimate cycle 0 x 2 Tu 0.80 0.54 0.73 0.57 0. 6 K u τm Tu 0.11 0.15 0.09 0.17 0.5Tu x 3 Tu x1 α = 0.9 ; β = 0.99 ; N = ∞ ; 0.05 < τm Tm < 0.8 . 2 Ti 6λ2 − x 4 + x 2 τm − 0.5τm , Ti = x1 + Tm + x 2 − x 3 , x 3 = , x1K m (4λ + τm − x 2 ) 4λ + τm − x 2 x 4 + x 2 (Tm + x1 ) + x1Tm − 0.167 τm 3 − 0.5x 2 τm 2 + x 4 τm + 4λ3 4λ + τ m − x 2 − x3 , Ti x1 is a constant with a ‘sufficiently large value’; x2 = λ 2 x1 1 − x1 4 e τ − m x1 τm 4 λ − Tm 2 − 1 − Tm 1 − − e 1 Tm , Tm − x1 τm 4 λ − Tm 2 e − 1 + x 2Tm , N = ∞ . x 4 = Tm 1 − Tm b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 416 4.3.8 Two degree of freedom controller 2 T s 1 d 1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) , U(s) = K c 1 + + 0 Ti s T 1 + a f 1s 1+ d s N 1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4 F(s) = 1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4 R(s) F(s) + − T s 1 d 1 + b f 1s K c 1 + + Ti s Td 1 + a f 1s 1+ s N + U(s) − K m e − sτ s(1 + sTm ) m K0 Table 75: PID controller tuning rules – FOLIPD model G m (s) = Rule Kc 1 Huang et al. (2005b). Model: Method 1 Kc Ti [ Td Direct synthesis: time domain criteria Ti T m ] Y(s) K m e − sτ m s(1 + sTm ) Comment N=∞ a f 1 = max 0.01 τm ,0.01 , b f 1 = 0 ; a f 2 = Ti , b f 2 = 1.452τm + Tm ; a f 3 = Ti Tm , b f 3 = 1.452τmTm ; a f 4 = 0 , a f 5 = 0 , b f 4 = 0 , b f 5 = 0 , K0 = 0 . Robust Lee and Edgar (2002). Model: Method 1 2 Kc Ti 0 a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , b f 3 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = 0.25K u . 1 Kc = 2.2221K m τ m 0.2328 T . 1 + 0.6887 m + 0.5173 , Ti = (1.452τ m + Tm )1 + K m τm τm 1 + 0.6887(Tm τ m ) 2 Kc = 0.25K u Ti 4 τm , T = . −τ + (λ + τm ) i K u K m m 2(λ + τm ) 2 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule 3 Liu et al. (2003). Model: Method 1 Kc Ti Td Kc Ti Td 417 Comment a f 1 = 0 , b f 1 = 0 ; a f 2 = 3λ + τm , b f 2 = λ ; a f 3 = 0 , a f 4 = 0 , a f 5 = 0 , bf 3 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . Representative λ values (deduced from graph): OS Rise time OS Rise time λ λ 0.3τ m ≈ 57% ≈ 3τ m 2τ m ≈ 0% ≈ 8τ m 0.5τ m ≈ 33% ≈ 3.5τ m 2.5τ m ≈ 0% ≈ 10.5τ m τm ≈ 7% ≈ 5τ m 3τ m ≈ 0% ≈ 12.5τ m 1.5τ m ≈ 0% ≈ 8.5τ m 3.5τ m ≈ 0% ≈ 14τ m a f 1 = 0 , b f 1 = 0 ; a f 2 = 7λ + τm , b f 2 = 3λ ; a f 3 = λ(16λ + 4τm ) , b f 3 = 3λ2 ; a f 4 = 4λ2 (3λ + τm ) , b f 4 = λ3 , a f 5 = 0 , b f 5 = 0 , K 0 = 0 . Representative λ values (deduced from graph): OS Rise time OS Rise time λ λ 0.3τ m ≈ 43% ≈ 3.5τ m 2τ m ≈ 0% ≈ 16.5τ m 0.5τ m ≈ 15% ≈ 4.5τ m 2.5τ m ≈ 0% ≈ 20τ m τm ≈ 0% ≈ 8τ m 3τ m ≈ 0% ≈ 24τ m 1.5τ m ≈ 0% ≈ 12.5τ m 3.5τ m ≈ 0% ≈ 28τ m a f 1 = 0 , b f 1 = 0 ; a f 2 = 4λ + τm , b f 2 = 3λ ; a f 3 = λ (3.25λ + τm ) , b f 3 = 3λ2 ; a f 4 = 0.25λ2 (3λ + τm ) , b f 4 = λ3 , a f 5 = 0 , b f 5 = 0 , K0 = 0 . Representative λ values (deduced from graph): OS Rise time OS Rise time λ λ 0.3τ m 3 Kc = ≈ 3τ m 2τ m ≈ 40% ≈ 3τ m 2.5τ m ≈ 4% ≈ 8τ m ≈ 15% ≈ 3.5τ m 3τ m ≈ 1% ≈ 10τ m 1.5τ m ≈ 7% ≈ 5τ m 3.5τ m ≈ 0% ≈ 12τ m Ti 2 m ≈ 6.5τ m τm K m 3λ + 3λτ m + τm m ≈ 4% 0.5τ m 2 x1 2 ) x1 , Ti = (3λ + 3λτ + τ ) T (3λ + τ )(3λ + 3λτ + τ ) λ T = − d ( ≈ 61% 2 2 m m 2 m ,N= m Tm (3λ + τm ) −1 λ3 x1 3 3λ2 + 3λτ m + Tm 2 , ( ) x1 = (3λ + τm + Tm ) 3λ2 + 3λτ m + τm − λ3 . 2 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 418 4.3.9 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) − Y(s) Model F2 (s) Table 76: PID controller tuning rules – FOLIPD model G m (s) = Rule Kc Ti Td K m e − sτ m s(1 + sTm ) Comment Direct synthesis: time domain criteria 1 2 0 Skogestad K m (TCL + τ m ) 4ξ (TCL + τ m ) (2004b). Model: Method 1 For ‘good’ robustness: TCL = τ m , ξ = 0.7 or 1. α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 ; b f 3 = 1 ; b f 4 = Tm , Xu and Shao (2004a). Model: Method 1 (λ − 1)τm + Tm 2 K m λ τm 2 a f 5 = Tm x1 , 2 ≤ x1 ≤ 10 . (λ − 1)τm + Tm (λ − 1)Tm τm (λ − 1)τm + Tm λ not defined. α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 λK m τ m ; b f 3 = 1 ; b f 4 = Tm ; a f 5 = 0 . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Majhi and Mahanta (2001). Model: Method 6 Kc 0.5236 Kmτm Ti Td 419 Comment Direct synthesis: frequency domain criteria A m = 3; 0 2τ m e τ ms φ = 60 0 . m α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; b f 1 = Tm , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; b f 2 = Tm , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.5 K m τm ; b f 3 = 1 ; b f 4 = Tm ; a f 5 = 0 . Wang et al. (2001b). Model: Method 7 0.4189 K m τm 4τ m 1.25(Tm + 0.1τ m ) A m = 3; φ m = 60 0 . α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0. 2 K m τ m ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 . Robust 1 Lee and Edgar (2002). Model: Method 1 Kc Ti Td α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 0.25K u ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 . 2 Kc Ti x1 α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b f 1 = 0 , a f 1 = Td , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = Td , a f 4 = 0 ; K 0 = 0.25K u ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 . 1 2 Kc = 2 2 0.25K u 4 4 τm τm − τm + , − τm + , Ti = (λ + τm ) K u K m 2(λ + τm ) KuKm 2(λ + τm ) Td = 2 3 4 0.25K u 4Tm τm τm τ (1 − 0.25K u K m τm ) 2 + 0.5τm − + + m . 2 (λ + τm ) K u K m 6(λ + τm ) 4(λ + τm ) 0.5K u K m (λ + τm ) Kc = 2 2 0.25K u 4 4 τm τm − τm + + x1 − τm + + x1 , Ti = 2(λ + τm ) KuKm 2(λ + τm ) λ + τm K u K m 3 4 4Tm τm τm (1 − 0.25K u K m τm )τm 2 2 x1 = + 0.5τm − + + 2 6( λ + τ m ) 4( λ + τ m ) 0.5K u K m (λ + τm ) K u K m 0.5 . b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 420 4.4 FOLIPD Model with a Zero K (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτm G m (s) = m or G m (s) = m s(1 + sTm1 ) s(1 + sTm1 ) 1 + Td s 4.4.1 Ideal PID controller G c (s) = K c 1 + Ti s E(s) R(s) − + U(s) 1 K c 1 + + Td s Ti s Model Y(s) Table 77: PID controller tuning rules – FOLIPD model with a zero K (1 − sTm 4 )e −sτ m K (1 + sTm3 )e−sτ m G m (s ) = m or m s(1 + sTm1 ) s(1 + sTm1 ) Rule Kc Ti Td Comment Direct synthesis: time domain criteria 1 Chidambaram (2002), p. 157. Model: Method 1 Jyothi et al. (2001). Model: Method 1 1 2 3 4 Kc = Kc = Kc = Kc = 1.571 Kc ∞ ‘Smaller’ τm Tm 4 . 2 Kc ‘Larger’ τm Tm 4 . 3 Kc 4 Kc Direct synthesis: frequency domain criteria τm Tm3 < 1 ∞ Tm1 τm Tm3 > 1 ( ) . ( ) . K m τm 1 + 3.142 Tm 4 τm 2 2.358 Tm1 K m τm 1 + 4.172 Tm 4 τm 2 1.57 K m τm 1 + 9.870(Tm3 τm )2 0.79 K m τm 1 + 2.467(Tm3 τm )2 . . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Jyothi et al. (2001). Model: Method 1 0.5 K m Tm 4 5 Kc Comment Ti Td ∞ Tm1 τm Tm 4 > 1 Td Recommended τm ≤ λ ≤ 2τm . τm Tm 4 < 1 Robust Lee et al. (2006). Model: Method 1 5 6 Kc = 6 ∞ Kc 0.79 K m τm 1 + 2.467(Tm 4 τm )2 . T (τ − λ ) + 0.5τm 1 ; , Td = Tm1 + m 4 m K m (λ + τm + 2Tm 4 ) λ + τm + 2Tm 4 2 Kc = (1 − sTm 4 )e −sτ . (1 + sTm 4 )(1 + sλ ) m Desired closed loop transfer function = 421 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 422 4.4.2 Ideal controller in series with a first order lag E(s) R(s) − + 1 1 G c (s) = K c 1 + + Td s Tf s + 1 Ti s U(s) 1 1 K c 1 + + Tds Tis Tf s + 1 Y(s) Model Table 78: PID controller tuning rules – FOLIPD model with a zero K (1 + sTm 3 )e − sτ m K (1 − sTm 4 )e −sτ m G m ( s) = m or m s(1 + sTm1 ) s(1 + sTm1 ) Rule Td Comment Ti Td Model: Method 1 Ti Td Model: Method 1 Kc Ti 1 Kc 2 Kc Robust Anil and Sree (2005). Shamsuzzoha et al. (2006c). 1 Kc = Tm1 + Tm 4 + τm + 3λ , T = T + T + τ + 3λ , K [(λ + T + τ ) + 2λ + λ (T + τ ) − T τ ] 2 m 2 x1 is a constant of a ‘sufficiently large value’. m4 m 2 m4 i m m1 m4 m m4 m Td = Tm1 (Tm 4 + τm + 3λ ) λ3 − Tm 4 τm (3λ + Tm 4 + τm ) . , Tf = Tm1 + Tm 4 + τm + 3λ (λ + Tm 4 + τm )2 + 2λ2 + λ(Tm 4 + τm ) − Tm 4τm Kc = Ti , λ , ξ not specified; Ti = x1 + Tm1 + x 2 − x 3 , Tf = Tm3 , x1K m (4λξ + τm − x 2 ) Td = x 4 + x 2 (Tm1 + x1 ) + x1Tm1 − 0.167 τm3 − 0.5x 2 τm 2 + x 4 τm + 4λ3ξ 4λξ + τm − x 2 − x3 , Ti x2 = Tm12 x 5e − τ m Tm1 − x12 x 5e − τ m x1 + x12 − Tm12 , x1 − Tm1 x3 = 4λ2ξ2 + 2λ2 − x 4 + x 2 τm − 0.5τm 2 2 , x 4 = Tm1 x 5e− τ m Tm1 − 1 + x 2Tm1 , 4λξ + τm − x 2 [ 2 λ2 2λξ x5 = − + 1 . T 2 T m1 m1 ] b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 423 1 1 + Td s 4.4.3 Classical controller G c (s) = K c 1 + Td s Ti s 1 + N E(s) R(s) − + 1 1 + sTd K c 1 + Ti s 1 + s Td N U(s) Y(s) Model Table 79: PID controller tuning rules – FOLIPD model with a zero K (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτ m G m (s) = m or m s(1 + sTm1 ) s(1 + sTm1 ) Rule Kc Ti Comment Td Direct synthesis: time domain criteria Anil and Sree (2005). Model: Method 1 1 0.5τ m Tm1 λ3K m 2 Kc Tm1 2.414λ + 0.5τm N= 2.414λ + 0.5τm Tm3 Tm1 Td N=∞ 1 λ is determined by solving λ3 − 1.207 τm λ2 − 0.603τm 2λ − 0.125τm 3 = 0 . 2 Kc = 0.5τm Tm1 , Td = 2.414λ + 0.5τm + Tm 4 ; K m λ3 − 0.5τmTm 4 [2.414λ + 0.5τm + Tm 4 ] ( ) 3 2 λ is determined by solving λ − 1.207τm λ − (0.6035τm + 2.414Tm 4 )τm λ 2 2 − τm (0.125τm + 0.5Tm 4 τm + Tm 4 ) = 0 . b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 424 4.5 I 2 PD Model G m (s) = K m e −sτm s2 1 + Td s 4.5.1 Ideal PID controller G c (s) = K c 1 + Ti s R(s) + E(s) − U(s) 1 K c 1 + + Td s Ti s K m e − sτ s Table 80: PID tuning rules – I 2 PD model G m (s) = Rule Åström and Hägglund (2006), pp. 244–245. Kc Ti Td m Y(s) 2 K m e − sτ m s2 Comment Direct synthesis: frequency domain criteria M s = M t = 1. 4 ; 0.02140 17.570τm 14.019τm Model: Method 1 K m τm 2 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 1 1 + Td s 4.5.2 Classical controller G c (s) = K c 1 + Ti s 1 + Td s N E(s) R(s) + − 1 1 + Td s U(s) K c 1 + Td Ti s s 1 + N K m e − sτ m s2 Table 81: PID tuning rules – I 2 PD model G m (s) = Rule Kc Skogestad (2003). Model: Method 1 0.0625 Ti Y(s) K m e − sτ m Td s2 Comment Direct synthesis: time domain criteria K mτm2 8τ m 8τ m N=∞ 425 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 426 4.5.3 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N E(s) R(s) + − T s 1 d + K c 1 + Td Ti s s 1+ N + U(s) Hansen (2000). Model: Method 1 Kc Ti K me s2 + Table 82: PID tuning rules – I 2 PD model G m (s) = Rule Y(s) − sτ m Td K m e − sτ m s2 Comment Direct synthesis: time domain criteria 2 5.4 τ m 2.5τ m N = ∞ ; β =1; 3.75K m τ m α = 0.833 . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 427 4.5.4 Two degree of freedom controller 2 T s 1 d 1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) , U(s) = K c 1 + + 0 Ti s T 1 + a f 1s 1+ d s N 1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4 F(s) = 1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4 R(s) T s 1 d 1 + b f 1s K c 1 + + Ti s Td 1 + a f 1s 1+ s N + F(s) − + U(s) − Kc Ti Kc Ti Y(s) m s2 K0 K m e − sτ m Table 83: PID tuning rules – I 2 PD model G m (s) = Rule K m e − sτ s2 Comment Td Robust 1 Liu et al. (2003). Model: Method 1 Td a f 1 = 0 , b f 1 = 0 ; a f 2 = 4λ + τm , b f 2 = 2λ ; 2 a f 3 = 6λ + 4λτ m + τm , b f 3 = λ2 ; a f 4 = 0 , a f 5 = 0 , b f 4 = 0 , 2 bf 5 = 0 , K 0 = 0 . 1 Kc = ( Ti K m 4λ + 6λ τm + 4λτ m + τm 2 d 2 2 2 m m 2 3 2 2 m m 2 m m 3 m 3 4 m 4λ + 6λ τ m + 4λτ m 2 + τ m 3 3 x1 2 N= 3 3 ) x1 , Ti = (4λ + 6λ τ + 4λτ + τ ) , (6λ + 4λτ + τ )(4λ + 6λ τ + 4λτ + τ ) − λ T = 3 ( 2 ) 6λ2 + 4λτ m + τm 2 3 − 1 , x1 = (4λ + τm ) 4λ3 + 6λ2 τm + 4λτ m + τm − λ4 . λ3x1 ; b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 428 Rule Liu et al. (2003) – continued. Kc Ti Comment Td Representative λ values (deduced from graph): OS Rise time OS Rise time λ λ 0.8τ m ≈ 33% ≈ 4τ m 2.5τ m ≈ 0% ≈ 10τ m τm ≈ 20% ≈ 5τ m 3τ m ≈ 0% ≈ 12.5τ m 1.5τ m ≈ 3% ≈ 7τ m 3.5τ m ≈ 0% ≈ 14τ m 2τ m ≈ 0% ≈ 8τ m 4τ m ≈ 0% ≈ 16τ m 2 3 a f 1 = 0 , b f 1 = 0 ; b f 2 = 4λ , b f 3 = 6λ , b f 4 = 4λ , b f 5 = λ4 ; a f 2 = 8λ + τm , a f 3 = 26λ2 + 8λτ m + τm ( 2 ) 2 ( 2 ) a f 4 = 4λ 10λ2 + 20λτ m + 4τm , a f 5 = 4λ2 6λ2 + 4λτ m + τm , K 0 = 0 . λ 0.8τ m Representative λ values (deduced from graph): OS Rise time OS Rise time λ ≈ 14% ≈ 3τ m 2.5τ m ≈ 0% ≈ 7τ m ≈ 4% ≈ 3.5τ m 3τ m ≈ 0% ≈ 8.5τ m 1.5τ m ≈ 0% ≈ 4.5τ m 3.5τ m ≈ 0% ≈ 10.5τ m 2τ m ≈ 0% ≈ 6τ m 4τ m ≈ 0% ≈ 13τ m τm a f 1 = 0 , b f 1 = 0 ; b f 2 = 4λ , b f 3 = 6λ2 , b f 4 = 4λ3 , b f 5 = λ4 ; 2 ( a f 2 = 5λ + τm , a f 3 = 10.25λ2 + 5λτ m + τm , 2 ) ( 2 ) a f 4 = λ 7λ2 + 4.25λτ m + τm , a f 5 = 0.25λ2 6λ2 + 4λτ m + τm , λ 0.8τ m K0 = 0 . Representative λ values (deduced from graph): OS Rise time OS Rise time λ ≈ 46% ≈ 6τ m 2.5τ m ≈ 4% ≈ 20.5τ m τm ≈ 34% ≈ 13.5τ m 3τ m ≈ 2% ≈ 24τ m 1.5τ m ≈ 16% ≈ 13τ m 3.5τ m ≈ 1% ≈ 28τ m 2τ m ≈ 8% ≈ 16.5τ m 4τ m ≈ 0% ≈ 32τ m b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 4.5.5 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) − Y(s) Model F2 (s) Table 84: PID tuning rules – I 2 PD model G m (s) = Rule Kc Ti K m e −sτm Td s2 Comment Direct synthesis: time domain criteria 0.25 0 Skogestad 2 4ξ 2 (TCL + τ m ) K m (TCL + τ m ) (2004b). Model: Method 1 For ‘good’ robustness: TCL = τ m , ξ = 0.7 or 1; α = 0 ; β = 1 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 1 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 1 ; b f 3 = 1 , b f 4 = 4(TCL + τm ) , a f 5 = b f 4 x 2 , 2 ≤ x 2 ≤ 10 . Huba (2005). Model: Method 1 − 0.079 K m τm 2 ∞ 5.835τm α = 1; β = 1; χ = 1 ; δ = 0 ; ε = 0 ; φ = 1; ϕ = 0 ; N = ∞ ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 0 . 429 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 430 4.6 SOSIPD Model G m (s) = K m e −sτ m s(1 + Tm1s) or G m (s) = 2 K m e − sτ m 2 s(Tm1 s 2 + 2ξ m Tm1s + 1) 1 4.6.1 Ideal PI controller G c (s) = K c 1 + Ti s E(s) R(s) + − 1 K c 1 + Ti s U(s) K m e − sτ Y(s) m s(1 + sTm1 ) 2 Table 85: PI controller tuning rules – SOSIPD model G m (s) = Rule Kc Ti s(1 + sTm1 )2 Comment Direct synthesis: time domain criteria Kuwata (1987). Model: Method 1 0.5 K m (2Tm1 + τm ) ∞ K m e − sτ m b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 431 4.6.2 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N R(s) + E(s) − T s 1 d K c 1+ + Ti s Td 1+ s N − U(s) + Table 86: PID controller tuning rules – SOSIPD model G m (s) = G m (s) = Rule Minimum ITAE – Pecharromán (2000a). Model: Method 2; Km = 1; Tm1 = 1 ; 0.1 < τ m < 10 . K m e − sτ m or s(1 + Tm1s) 2 K m e − sτ m 2 2 s(Tm1 s + 2ξm Tm1s + 1) Kc Ti x1K u Y(s) SOSIPD model Comment Td Minimum performance index: other tuning 0 x 2 Tu x1 x2 α 0.049 2.826 0.066 Coefficient values: φc x1 x2 α φc 0.218 1.279 0.564 − 1350 − 160 0 0.250 1.216 0.573 − 130 0 0.522 − 1550 0.286 1.127 0.578 − 1250 1.716 0.532 − 150 0 0.330 1.114 0.579 − 120 0 0.159 1.506 0.544 − 1450 0.351 1.093 0.577 − 118 0 0.189 1.392 0.555 − 140 0 0.506 − 164 0 2.402 0.512 0.099 1.962 0.129 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 432 Rule Kc x1K u Minimum ITAE – Pecharromán x1 x2 (2000b), p. 142. 0.0469 2.8052 Model: Method 2; 0.0662 2.3876 Km = 1; 0.0988 1.9581 Tm1 = 1 ; 0.05 < τ < 10 . 0.1271 1.6757 m Ti Td x 2 Tu 0 Comment Coefficient values: φc x1 α x2 α φc 0.5061 − 164.10 0.2174 1.2566 0.5642 − 135.00 0.5119 − 160.50 0.2511 1.1948 0.5730 − 130.00 0.5220 − 155.00 0.2870 1.1281 0.5777 − 125.00 0.5319 − 150.00 0.3280 1.0860 0.5794 − 120.00 0.1589 1.5052 0.5439 − 145.00 0.3501 1.0822 0.5767 − 118.00 0.1866 1.3619 0.5546 − 140.00 Minimum ITAE – Pecharromán and Pagola (2000). Model: Method 2; Km = 1; x1K u x 2 Tu x 3Tu x1 x2 Coefficient values: x3 α φc 1.672 0.366 0.136 0.601 − 164 0 1.236 0.427 0.149 0.607 − 160 0 0.994 0.486 0.155 0.610 − 1550 Tm1 = 1 ; 0.842 0.538 0.154 0.616 − 150 0 0.1 < τ m < 10 ; 0.752 0.567 0.157 0.605 − 1450 β = 1 , N = 10. 0.679 0.610 0.149 0.610 − 140 0 0.635 0.637 0.142 0.612 − 1350 0.590 0.669 0.133 0.610 − 130 0 0.551 0.690 0.114 0.616 − 1250 0.520 0.776 0.087 0.609 − 120 0 0.509 0.810 0.068 0.611 − 118 0 α φc Minimum ITAE – Pecharromán (2000b), p. 148. Model: Method 2; Km = 1; Tm1 = 1 ; β = 1 , N = 10; 0.05 < τm < 0.8 . x1K u x 2 Tu x 3Tu x1 x2 Coefficient values: x3 1.6695 0.3684 0.1323 0.6007 − 164.10 1.2434 0.4389 0.1507 0.6073 − 160.00 0.9901 0.4846 0.1587 0.6102 − 155.00 0.8378 0.5393 0.1562 0.6163 − 150.00 0.7383 0.5789 0.1548 0.6052 − 145.00 0.6728 0.6283 0.1488 0.6105 − 140.00 0.6196 0.6221 0.1376 0.6120 − 135.00 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Pecharromán (2000b) – continued. Ti Td x1K u x 2 Tu x 3Tu x1 x2 Coefficient values: x3 α φc 0.5948 0.6939 0.1326 0.6096 − 130.00 0.5513 0.7050 0.1190 0.6157 − 125.00 0.5271 0.7471 0.0992 0.6088 − 120.00 0.5229 0.8886 0.0862 0.6115 1 Kc Ti 0 − 118.0 0 τm Tm1 ≤ 1.0 ; 2 Kc Ti Td OS ≤ 20% . x1Tm1 K m ∞ Taguchi and Araki (2000). Model: Method 1 Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 . Kc = x 2Tm1 x1 x2 β τm Tm1 x1 x2 β τm Tm1 1.48 1.04 0.71 0.56 1.35 1.41 1.49 1.57 0.89 0.81 0.72 0.67 0.1 0.2 0.4 0.6 0.46 0.40 0.24 0.12 1.63 1.70 2.02 3.08 0.63 0.60 0.50 0.38 0.8 1.0 2.0 5.0 3 1 Comment Kc Kc ∞ Td 1 0.3840 0.07368 + , T = 8.549 + 4.029[τm Tm1 ] , [τm Tm1 ] + 0.7640 i Km α = 0.6691 + 0.006606[τm Tm1 ] . 2 Kc = 1 0.5667 0.1778 + , T = T (1.262 + 0.3620[τm Tm1 ]) , [τm Tm1 ] + 0.002325 d m1 Km ( ) Ti = Tm1 0.2011 + 11.16[τm Tm1 ] − 14.98[τm Tm1 ] + 13.70[τm Tm1 ] − 4.835[τm Tm1 ] , 2 3 α = 0.6666 , β = 0.8206 − 0.09750[τm Tm1 ] + 0.03845[τm Tm1 ] ; N not specified. 2 3 Kc = Tm1 0.3084 , 0.1309 + Km ([τm Tm1 ] + 0.1315) [ ] Td = Tm 1.308 + 0.5000[τm Tm1 ] − 0.1119[τm Tm1 ] , 2 β = 0.9231 − 0.6049[τm Tm1 ] + 0.2906[τm Tm1 ] ; 0.1 ≤ τm Tm1 ≤ 1.0 . 2 4 433 b720_Chapter-04 Rule Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 . Kc Ti Td Comment x1Tm1 K m ∞ x 2Tm1 ξm = 0.5 x1 x2 β τm Tm1 x1 x2 β τm Tm1 0.52 0.43 0.34 0.30 1.11 0.89 0.61 0.50 0.82 0.67 0.26 -0.13 0.1 0.2 0.4 0.6 0.28 0.26 0.21 0.13 0.49 0.53 0.91 2.26 -0.34 -0.39 -0.08 0.23 0.8 1.0 2.0 5.0 Kc ∞ Td x1Tm1 K m ∞ x 2Tm1 4 Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 . Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 . Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 . Taguchi et al. (2002). Model: Method 1; N = 10; α = 0 . Kc = FA Handbook of PI and PID Controller Tuning Rules 434 4 04-Apr-2009 x2 β τm Tm1 1.06 1.34 0.80 1.32 0.57 1.30 0.46 1.31 x1Tm1 K m 0.89 0.81 0.70 0.62 0.1 0.2 0.4 0.6 x1 x2 2.18 1.34 1.48 1.46 0.93 1.64 0.69 1.77 x1Tm1 K m x1 x2 β τm Tm1 ∞ 0.40 1.33 0.35 1.37 0.23 1.64 0.12 2.74 x 2Tm1 0.56 0.8 0.52 1.0 0.40 2.0 0.33 5.0 ξ m = 1. 2 β τm Tm1 x1 β 0.92 0.87 0.81 0.77 0.1 0.2 0.4 0.6 ∞ x2 β τm Tm1 2.93 1.29 1.90 1.46 1.13 1.71 0.81 1.90 x1Tm1 K m 0.93 0.89 0.83 0.80 0.1 0.2 0.4 0.6 x1 x1 ξm = 0.8 ∞ x2 0.55 1.88 0.46 1.98 0.26 2.37 0.12 3.42 x 2Tm1 x1 x2 0.63 2.06 0.51 2.19 0.27 2.68 0.12 3.75 x 2Tm1 τm Tm1 0.73 0.8 0.70 1.0 0.59 2.0 0.43 5.0 ξ m = 1. 4 β τm Tm1 0.77 0.8 0.74 1.0 0.64 2.0 0.47 5.0 ξ m = 1. 6 x1 x2 β τm Tm1 x1 x2 β τm Tm1 3.80 2.37 1.34 0.93 1.24 1.45 1.76 2.00 0.94 0.90 0.85 0.82 0.1 0.2 0.4 0.6 0.70 0.56 0.28 0.12 2.20 2.36 2.95 4.08 0.79 0.77 0.68 0.51 0.8 1.0 2.0 5.0 Tm1 0.08343 , 0.1904 + Km ((τm Tm1 ) + 0.1496) [ ] Td = Tm1 1.237 − 2.031(τm Tm1 ) + 1.344(τm Tm1 ) , 2 β = 1.048 − 1.637(τm Tm1 ) − 1.676(τm Tm1 ) + 1.898(τm Tm1 ) ; 0.1 ≤ τm Tm1 ≤ 1.0 . 2 3 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Kuwata (1987). Model: Method 1 Comment Td Direct synthesis: time domain criteria 1 0 K m (2Tm1 + τm ) 3.33(2Tm1 + τm ) 5 Kuwata (1987); x 1 , x 2 deduced from graphs. Ti Ti Kc α = 1, β =1, N = ∞ . Td Ultimate cycle 0 x 2 Tu x 1K u α =1 α =1 x1 x2 τm Tu x1 x2 τm Tu x1 x2 0.16 0.19 0.23 1.05 0.97 0.95 x 1K u 0.01 0.03 0.05 0.26 0.30 0.32 x 2 Tu 0.89 0.88 0.86 0.08 0.10 0.13 0.35 0.36 0.38 0.85 0.15 0.84 0.18 0.83 0.25 α =1 x 3 Tu τm Tu Kuwata (1987); x 1 x 2 x 3 τm x 1 x 2 x 3 τm x 1 x 2 x 3 τm x 1 , x 2 deduced Tu Tu Tu from graphs; 0.98 0.45 0.12 0.03 0.79 0.54 0.11 0.10 0.70 0.60 0.09 0.17 β =1, N = ∞ . 0.90 0.47 0.12 0.05 0.76 0.56 0.10 0.12 0.70 0.62 0.08 0.20 0.82 0.51 0.11 0.08 0.72 0.58 0.09 0.15 5 Kc = 1.067 K m 2T Td = 1.25 [ (2Tm1 + τm )3 2 2 m1 + 4Tm1τ m + τ m (2.5T ; Ti = 2 ] 2 2 m1 + Tm1τ m + 0.25τm )(2T [ 6.25 2Tm12 + 4Tm1τm + τm 2 (2Tm1 + τm )2 2 2 m1 + 4Tm1τ m + τm 3 (2Tm1 + τm ) ). ]; 2 435 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 436 4.7 SOSIPD Model with a Zero G m (s) = K m (1 + sTm 3 )e − sτ m s(1 + sTm1 )(1 + sTm 2 ) 1 1 + Td s 4.7.1 Classical controller G c (s) = K c 1 + Ti s 1 + Td s N R(s) E(s) + − 1 1 + Td s K c 1 + Td Ti s s 1 + N Y(s) U(s) G m (s) Table 87: PID controller tuning rules – SOSIPD model with a zero K (1 + sTm3 )e −sτ m G m (s ) = m s(1 + sTm1 )(1 + sTm 2 ) Rule Kc Bialkowski (1996). Model: Method 1 Tm1 Ti Td Comment Robust K m (λ + τm )2 Tm1 2λ + τ m N= 2λ + τm ; Tm3 Tm1 > Tm 2 . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 437 4.8 TOSIPD Model G m (s) = K m e − sτ m K m e −sτ m or G m (s) = 3 s(1 + sTm1 )(1 + sTm 2 )(1 + sTm 3 ) s(1 + sTm1 ) 4.8.1 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N E(s) R(s) + − Rule Kuwata (1987); x 1 , x 2 deduced from graphs. T s 1 d K c 1 + + Ti s Td 1+ s N − Y(s) U(s) Model + Table 88: PID controller tuning rules – TOSIPD model Comment Kc Ti Td Ultimate cycle 0 x 2 Tu x 1K u α =1 x1 x2 τm Tu x1 x2 τm Tu x1 x2 τm Tu 0.23 0.26 0.28 0.91 0.89 0.88 x 1K u 0.01 0.03 0.05 0.31 0.33 0.35 x 2 Tu 0.86 0.86 0.85 0.08 0.10 0.13 0.36 0.37 0.38 0.84 0.84 0.83 0.15 0.18 0.25 x 3 Tu Kuwata (1987); x1 x 2 x 3 τm x1 x 2 x 3 τm x1 x 2 x 3 τm x 1 , x 2 deduced Tu Tu Tu from graphs. 0.85 0.50 0.11 0.03 0.74 0.56 0.09 0.10 0.70 0.62 0.08 0.18 α = 1, β = 1, 0.80 0.52 0.11 0.05 0.72 0.58 0.09 0.12 0.70 0.63 0.075 0.22 N =∞. 0.77 0.55 0.10 0.08 0.70 0.61 0.08 0.16 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 438 4.9 General Model with Integrator G m (s) = K m e − sτ m s(1 + sTm1 ) ∏ (T s + 1)∏ (T K G (s) = s ∏ (T s + 1)∏ (T m1i m m m 2i i m 3i m 4i or ) e s + 2ξ T s + 1) 2 2 s + 2ξ m n i Tm 2i s + 1 i i n 2 2 i m di m 4i 1 4.9.1 Ideal PI controller G c (s) = K c 1 + Ti s R(s) + U(s) 1 K c 1 + T is E(s) − G m (s) Y(s) Table 89: PI controller tuning rules – general model with integrator Comment Kc Ti Rule Direct synthesis: time domain criteria 0.5 ∞ K m (nTm1 + τm ) Kuwata (1987). Model: Method 1 Direct synthesis: frequency domain criteria (1) Symmetrical Kc optimum method. α1TQ ( 2) Non-symmetrical Kc optimum method. 1 Loron (1997). Model: Method 1 1 Kc (1) 1 + sin φ m , α1 = α1 TQ cos φ m 1 = Km 2 ξ md i Tm 4i − ξ m n i Tm2i + τ m . TQ = Tm3i + 2 Tm1i + 2 i i i i ∑ Kc ( 2) = ∑ λ K m α 2 TQ , λ= ∑ Mc2 ∑ 2 1 + sin ∆φ −1 , α 2 = , ∆φ = cos (1 λ ) , 2 Mc −1 cos ∆φ M c = desired closed loop resonant peak. −sτ m b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 439 4.9.2 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N T s 1 d K c 1 + + Ti s Td 1+ s N E(s) R(s) + − U(s) − K m e − sτ n s(1 + sTm1 ) m + Table 90: PID controller tuning rules – general model with integrator Comment Kc Ti Td Rule Direct synthesis: time domain criteria Kuwata (1987). 1 Model: Method 1 K (nT + τ ) 3.33(nTm1 + τm ) 0 m m1 m Kuwata (1987). Model: Method 1 1 Kc = 1 Ti Kc Td α =1 α = 1; β = 1; N =∞. [ 1.067 (nTm1 + τm )3 6.25 n (n − 1)Tm12 + 2nTm1τm + τm 2 , Ti = 2 2 2 K m n (n − 1)T + 2nT τ + τ (nTm1 + τm )2 m1 m1 m m [ ] [(nT + τ ) − 0.75(n(n − 1)T + 2nT τ + τ )] , T = 1.25x 2 d Y(s) 1 m1 m 2 m1 3 nTm1 + τm ( ) ], 2 2 m1 m m x1 = [n (n − 1)Tm1 + 2nTm1τm + τm ] . 2 2 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 440 4.10 Unstable FOLPD Model G m (s) = K m e −sτ m Tm s − 1 1 4.10.1 Ideal PI controller G c (s) = K c 1 + Ti s R(s) E(s) + − 1 K c 1 + Ti s U(s) Y(s) K m e − sτ Tm s − 1 m Table 91: PI controller tuning rules – unstable FOLPD model G m (s) = Rule Minimum ITSE – Majhi and Atherton (2000). 0 < τm Tm < 0.693 . Kc 1 2 K m e − sτ m Tm s − 1 Comment Ti Minimum performance index: servo tuning Model: Method 1 Ti K c Model: Method 3 Ti Kc Direct synthesis: time domain criteria TS is the ±1% 0.4ξ 2TS settling time. Jacob and Chidambaram (1996). Model: Method 1 10 TS Chidambaram (1997). 1.678 Tm ln K m τ m 0.4015Tm e 5.8τ m Tm Valentine and Chidambaram (1998). Model: Method 1 2.695 −1.277 τ m Tm e Km 0.4015Tm e 5.8τ m Tm Model: Method 1 3 ( ξ = 0.333 ; τm Tm < 0.7 . ) 1 Kc = e − τ m Tm − 0.064 2.6316Tm e τ m Tm − 0.966 1 , Ti = . 0.889 + τ T m m Km e e − τ m Tm − 0.377 − 0.990 2 Kc = 1 1 Tm K (1 + K ) 0.889 + , Ti = , K = − Ap Kmh . K m K (1 + K ) 0.5(1 − 0.6382K ) tanh −1 K 3 ±1% Ts = 7.5Tm , τ m Tm = 0.1 ; ±1% Ts = 8.75Tm , τ m Tm = 0.2 ; ( ) ±1% Ts = 11.25Tm , τ m Tm = 0.3 ; ±1% Ts = 15.0Tm , τ m Tm = 0.4 ; ±1% Ts = 17.5Tm , τ m Tm = 0.5 ; ±1% Ts = 20.0Tm , τ m Tm = 0.6 . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Model: Method 1; ‘small’ τm Tm . Chandrashekar et al. (2002). 5 Kc Ti Model: Method 1 Chidambaram (2002), p. 156. Model: Method 1 Kc = Tm Ti 2 2 0.04TS ξ + Ti τm 49.535 − 21.644 τm Tm e Km 0.8668 Tm K m τm 0.8288 0.1523Tm e τm < 0.1 Tm 7.9425τ m Tm 0. 1 ≤ τm ≤ 0. 7 Tm 6.0000 K m Alternative tuning rules: 0.6000Tm τ m Tm = 0.1 3.2222 K m 1.4500Tm τ m Tm = 0.2 2.2963 K m 2.6571Tm τ m Tm = 0.3 1.8333 K m 4.4000Tm τ m Tm = 0.4 1.5556 K m 7.0000Tm τ m Tm = 0.5 1.3265 K m 14.6250Tm τ m Tm = 0.6 1.1875 K m 31.0330Tm τ m Tm = 0.7 τm 2.695 −1.277 Tm e K m, 0.4015Tme τ 5.8 m Tm 0< τm ≤ 0.6 Tm 2 , Ti = 0.04TS ξ2 + Tm τm + 0.4TSξ2 . Tm − τm Desired closed loop transfer function = Kc = 6 Comment 4 Chandrashekar et al. (2002). Model: Method 1 5 Ti Chidambaram (2000c). 6 4 Kc 441 ( ( (Tis + 1)e−sτ )( m 1 (1) ( 2) K m Ti − TCL 2 − TCL 2 − τ m Desired closed loop transfer function = ); (1) ( 2) TCL 2s + 1 TCL 2s + 1 (ηs + 1)e −sτ ; (TCL2 s + 1)2 ) , Ti = ( ) (1) ( 2) (1) ( 2) TCL 2 TCL 2 + Tm TCL 2 + TCL 2 + τ m . Tm − τm m τ τ τ TCL 2 = 0.025 + 1.75 m Tm , m ≤ 0.1 ; TCL 2 = 2 τ m , 0.1 ≤ m ≤ 0.5 ; Tm Tm Tm τ τ TCL 2 = 2 τ m + 5τ m m − 0.5 , 0.5 ≤ m ≤ 0.7 . T T m m b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 442 Rule Kc Minimum ITAE – Chidambaram (2002), p. 156. Model: Method 1 7 Kc T 0.876 m τm 0.898 Sree et al. (2004). Model: Method 1 0.8624 Tm K m τm Manum (2005). Model: Method 1 0.5Tm K m τm Sree and Chidambaram (2006), pp. 19–22. Minimum ITAE – Sree and Chidambaram (2006), pp. 22–23. Model: Method 1 Comment Ti 0.324Tm e 6.46 8 Ti 4τm Tm (2Tm − 3τm ) (Tm − 2τm ) 0.4015Tme Kc 0.898 0.324Tm e 0.2 ≤ τm ≤ 0.6 Tm 0.01 ≤ τm ≤ 0.6 Tm τm < 0.25 Tm 2 2.695 −1.277 Tm e Km T 0.876 m τm Tm 0.9744 τm 9 τm Tm ≤ 0.2 τm τ 5.8 m Tm Model: Method 1 τm Tm ≤ 0.2 6.461τ m Tm 0.2 ≤ τm ≤ 0.6 Tm Direct synthesis: frequency domain criteria De Paor and O’Malley (1989). Model: Method 1 7 8 9 10 Kc = 10 A m = 2; Ti Kc τm <1. Tm τm 1 10.7572 − 35.1 . Km Tm 3 2 τ τ τ Ti = Tm 143.34 m − 73.912 m + 19.039 m − 0.2276 . Tm Tm Tm 1 τm Kc = 10.7572 − 35.1 . Km Tm Kc = 1 cos Km Tm τ τ τ 1 − m sin 1 − m m + T T τm Tm m m Ti = Tm T m 1 − τm tan (0.5φ ) τm Tm τ τ 1 − m m , T T m m , φ = tan −1 Tm τ 1 − m − τ m Tm τ τ 1 − m m . Tm Tm b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Venkatashankar and Chidambaram (1994). 11 Comment Ti 25(Tm − τ m ) Kc 443 Model: Method 1; τ m Tm < 0.67 . Chidambaram τm T 1 < 0.6 1 + 0.26 m (1995b), 25Tm − 27τ m Tm K m τm Chidambaram (1998), p. 91. 2 τ 1 τ Model: Method 1 4.656 − 13.05 m + 13.436 m . Alternatively, K c = Km Tm Tm 12 τm ω p Tm Ti < 0.62 Ho and Xu (1998). Tm AmKm Model: Method 1 Representative results: A m = 2.3 , 0.5775Tm τ m Tm K mτm 0.3212Tm − τ m φ = 250 . m 0.6918Tm Kmτm τ m Tm 0.3359Tm − τ m x 1Tm K m τ m −Tm 1.0472 1.5 30 0 0.5236 3 0.7854 2 450 0.3927 4 Luyben (1998). Model: Method 1 0.31K u 12 Ti = 13 2.2Tu φm 60 0 67.50 TCL = 3.16τ m Maximum closed loop log modulus = 2 dB. Model: Method 1; 13 Ti 0.1 ≤ τ m Tm ≤ 0.6 . Kc Gaikward and Chidambaram (2000). Kc = φ m = 22.50 . Other representative results: coefficient values: Am φm x1 Am x1 11 A m = 1.9 , 2 2 2 1 0.04Tm2 25 τm + x1 Tm x1 (Tm − τm ) 0.98 1 + , 2 2 2 2 Km ( Tm − τm ) τm τm + 625x1 (Tm − τm ) x1 = 1.373, τm Tm < 0.25 ; x1 = 0.953, 0.25 ≤ τ m Tm < 0.67 . 1 2 1.57ωp − ωp τm − Tm −1 . 2 τm τm 1 Tm . Kc = + 8.214 0.2619 + 0.7266 , Ti = Tm 0.193 + 4.19 Tm Tm Km τm b720_Chapter-04 Rule Kc Sree and Chidambaram (2006), pp. 15–16. Paraskevopoulos et al. (2006). Model: Method 1 15 Kc = x1 = FA Handbook of PI and PID Controller Tuning Rules 444 14 04-Apr-2009 14 15 Kc Ti Comment Ti Model: Method 1; 0 ≤ τm Tm < 0.6 . Ti τm < 0.9 Tm 2 1 + x1 Ti x1 1 + Ti 2 x12 2 τm τm 1 0.7266Tm T T 0 . 193 4 . 19 8 . 214 0 . 2619 , + = + + i m T . Tm Km τm m τ τ 2 Ti 1 − m + Ti − m + x 2 T T m m 2Ti 2 (τm Tm ) , 2 τ τ τ τ 2 x 2 = Ti 2 1 − m + Ti − m + 4 m Ti 1 + Ti − m ; Tm Tm Tm Tm φm T φmax τ τ , with φm > 0.2φmax Ti = x 3 1 + m 0.632 m + 0.436 m − 0.0153 m m ; φm τm Tm Tm 1 − max φm 0.0029 − 0.0682 x3 = τm τ + 1.4941 m Tm Tm τ 1.003 − m T m 2 =− , φ max m τm Tm T Tm − 1 + tan −1 m − 1 . τm τm b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Paraskevopoulos et al. (2006) – continued. 16 K min = Ti ωmin 16 K max Am 1 + ωmin 2 Kc Ti Comment or K min A m Ti τm Tm < 0.9 2 1 + Ti ωmin 2 1 + ωmax and K max = Ti ωmax 0.03(τm Tm ) 1.14 − (τm Tm ) ωmin = 1 + 0.05 0.973 + Ti − x1 ( ) − τ 1 T m m 0.006 + 2 Ti − 2 1 + Ti ωmax 2 x1 τm (1 + Ti ) Tm x1 = 445 , with , 0.0029 − 0.0682 τm Tm + 1.4941(τm Tm ) (1.003 − (τm Tm ))2 , 4 2 τ τ x 1.571Tm ωmax = 1 + 0.22 m 1 + 0.1 − 0.3 m 1 Tm Ti τm Tm Ti Ti − 0.9463 − 0.5609 ; (Ti + 1)(τm Tm ) (Ti + 1)(τm Tm ) 1+ (τ m Tm ) max A m − 1 A m − 1 + Ti = x1 1 + 0.65 max 1 − Am − 1 A −1 m [ [ ] ] 2 3 τ τ 3(τm Tm )4 − 2.3(τm Tm )2 τ τ , 0.01 − 0.18 + 5 m − 32 m + 75 m − 51 m + Tm Tm Tm Tm ( 1 − (τm Tm ))3 1.571Tm max max ; A m > 0.2(A m − 1) +1. with A m = 0.4085(τm Tm ) τ m 1 + 1 − 0.2864(τm Tm ) b720_Chapter-04 04-Apr-2009 Handbook of PI and PID Controller Tuning Rules 446 Rule Kc Rotstein and Lewin (1991). Model: Method 1 λ Tm λ + 2 Tm λ2 K m Ti Comment Robust λ λ + 2 Tm λ obtained graphically – sample values below. τ m Tm = 0.2, 0.6Tm ≤ λ ≤ 1.9Tm τ m Tm = 0.2, 0.5Tm ≤ λ ≤ 4.5Tm τ m Tm = 0.4, 1.5Tm ≤ λ ≤ 4.5Tm τ m Tm = 0.6, 3.9Tm ≤ λ ≤ 4.1Tm Tan et al. (1999b). Model: Method 1 Ou et al. (2005a). 17 Kc 18 Kc Kc = λ2 + 2λTm + Tm τm K m (λ + τm ) 2 , Ti = λ2 + 2λTm + Tm τm . Tm − τm K m uncertainty = 50%. K m uncertainty = 30%. Ti τm Tm < 0.5 Ti Model: Method 1 τ Tm 12.7708 m + 3.8019 Tm 1 . 17 Kc = , Ti = τm τm 0.1486 + 0.6564 K m 0.1486 + 0.6564 Tm Tm 18 FA b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 1 + Td s 4.10.2 Ideal PID controller G c (s) = K c 1 + Ti s E(s) R(s) U(s) 1 K c 1 + + Td s Ti s − + K m e − sτ Tm s − 1 m Y(s) Table 92: PID controller tuning rules – unstable FOLPD model G m (s) = Rule Kc Minimum ISE – Visioli (2001). Ti Td K m e − sτ m Tm s − 1 Comment Minimum performance index: regulator tuning 1.18 1.37 τ m τ Model: Method 1 2.42Tm m 0.60τ m K m Tm Tm Minimum ITSE – Visioli (2001). 1.37 τ m K m Tm τ 4.12Tm m Tm 0.90 Minimum ISTSE – Visioli (2001). 1.70 τ m K m Tm τ 4.52Tm m Tm 1.13 0.55τ m Model: Method 1 0.50τ m Model: Method 1 Minimum performance index: servo tuning Minimum ISE – Visioli (2001). τ 4.00Tm m Tm 0.47 0.90 Minimum ITSE – 1.38 τ m Visioli (2001). K m Tm τ 4.12Tm m Tm 0.90 0.95 Minimum ISTSE 1.35 τ m – Visioli (2001). K m Tm τ 4.52Tm m Tm 1 2 3 1.32 τ m K m Tm 0.92 [ ](τ T ) . T = 3.62T [1 − 0.85(T τ ) ](τ T ) . T = 3.70T [1 − 0.86(T τ ) ](τ T ) . Td = 3.78Tm 1 − 0.84(Tm τm ) 0.02 0.95 m m m m m m 0.02 d m m m d m m m 0.93 0.02 0.97 1 Td Model: Method 1 2 Td Model: Method 1 3 Td Model: Method 1 1.13 447 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 448 Rule Kc Ti Comment Td Minimum performance index: servo/regulator tuning Jhunjhunwala and Chidambaram (2001). 4 Ti Kc Direct synthesis: time domain criteria Ti 1.165 Tm 5 T x1 i Km τm Chidambaram (1997). Model: Method 1 6 Valentine and 0.245 Chidambaram 1.165 Tm (1997b). Model: Method 1 K m τ m Kc = τm Tm < 0.6 Ti 4.4Tm + 9.0τm 7 4 0.176Tm + 0.36τm 2.044 τ m Tm 0.6 ≤ τm ≤ 0.8 Tm 0.8 < τm ≤1 Tm Ti 1 τ τ 1.397 Tm 13.0 − 39.712 m , m < 0.2 ; K c = K m Tm Tm K m τm Ti = 0.856Tm e Model: Method 1 Td τ , 0 ≤ m ≤ 1 ; Ti = 0.3444Tme Tm 0.769 2.9595 τ m Tm , 1≤ , 0.2 ≤ τm ≤ 1.4 ; Tm τm ≤ 1.4 ; Tm τ τ Td = Tm 0.5643 m + 0.0075 , 0 ≤ m ≤ 1.4 . Tm Tm 2 5 τ τ τ τ Ti = 0.176 + 0.360 m Tm ; x1 = 0.179 − 0.324 m + 0.161 m , m < 0.6 ; Tm Tm Tm Tm x1 = 0.04 , 0.6 < τm Tm < 0.8 ; x1 = 0.12 − 0.1(τm Tm ) , 0.8 < τm Tm < 1.0 . 6 Ti = 7 Ti = 0.176Tm + 0.36τm 0.179 − 0.324(τm Tm ) + 0.161(τm Tm )2 0.176Tm + 0.36τm . 0.12 − 0.1(τm Tm ) . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Pramod and Chidambaram (2000), Chidambaram (2002), p. 156. Model: Method 1 8 Chidambaram (2002), p. 156. Sree et al. (2004). Model: Method 1 8 9 Ti Ti Kc 9 Ti 10 Ti 449 Comment Td τm Tm < 0.6 0.176Tm +0.36τm 0.6 < τm < 0.8 Tm 0.8 < τm ≤1 Tm 11 Kc Ti Td 12 Kc Ti 0.4917 τ m Model: Method 1 0.01 ≤ τm ≤ 0.9 Tm 2 τm 0.176Tm + 0.36τm 1 τm Kc = 2.121 − 1.906 . + 0.945 2 , Ti = Km Tm T τm m τm − + 0.179 0.324 0.161 Tm Tm Ti = 0.44Tm + 0.90τm (Pramod and Chidambaram, 2000); Ti = 4.4Tm + 9.0τm (Chidambaram, 2002). 10 Ti = 0.176Tm + 0.36τm (Chidambaram, 2002). 0.12 − 0.1(τm Tm ) 11 Kc = 1.397Tm τ 1 τm τm , 0.2 < m ≤ 1.4 ; ≤ 0.2 ; K c = 13.0 − 39.712 , Km Tm Tm K m τm Tm Ti = 0.856Tm e 2.044 τm Tm τm 2.9595 τ τ Tm , 0 < m < 1 ; Ti = 0.344Tm e , 1 < m ≤ 1.4 ; Tm Tm τ τ Td = Tm 0.0075 + 0.5643 m , 0 < m ≤ 1.4 . Tm Tm 12 Kc = 1.4183 Tm K m τm 0.9147 2 τ τ , Ti = Tm 16.327 m + 5.5778 m + 0.8158 , Tm Tm τ 2 τm τ τ 0.01 ≤ ≤ 0.6 ; Ti = Tm 196 m − 247.28 m + 87.72 , 0.6 ≤ m ≤ 0.9 . Tm Tm Tm Tm b720_Chapter-04 FA Handbook of PI and PID Controller Tuning Rules 450 Rule Kc Sree et al. (2004) – continued; Vivek and Chidambaram (2005). Sree and Chidambaram (2006), pp. 117,118. 13 04-Apr-2009 Kc = 1.2824 Tm K m τm Ti = 0.483Tme Td 13 Kc Ti Td 14 Kc Ti Td Comment 0.01 ≤ τm ≤ 1.2 Tm Model: Method 1 Alternative tuning rules: Ti Td Kc TCL 2 τ m Tm 0.02Tm 0.0 101.0000 K m 0.0404Tm 0.0 10.3662 K m 0.3874Tm 0.0435Tm 0.10Tm 0.1 5.3750 K m 0.8600Tm 0.0884Tm 0.20Tm 0.2 3.2655 K m 1.8018Tm 0.1375Tm 0.40Tm 0.3 2.4516 K m 3.0400Tm 0.1868Tm 0.60Tm 0.4 2.0217 K m 4.6500Tm 0.2366Tm 0.80Tm 0.5 1.7525 K m 6.7286Tm 0.2866Tm 1.00Tm 0.6 1.5458 K m 10.337Tm 0.3380Tm 1.30Tm 0.7 1.3723 K m 17.693Tm 0.3910Tm 1.80Tm 0.8 1.2420 K m 33.610Tm 0.4440Tm 2.60Tm 0.9 1.1400 K m 80.620Tm 0.4969Tm 4.20Tm 1.0 1.0895 K m 143.6Tm 0.5975Tm 5.00Tm 1.2 1.0536 K m 277.37Tm 0.6982Tm 6.00Tm 1.4 0.8325 3.3739 Ti τ τ , Ti = Tm 5.5734 m − 0.0063 , 0.01 ≤ m ≤ 0.5 ; Tm Tm τm Tm , (Sree et al., 2004); Ti = 0.483Tme − 3.3739 τm Tm (Vivek and Chidambaram, 2005; Model: Method 5); 0.5 ≤ τm Tm ≤ 1.2 ; Td = 0.507 τm + 0.0028Tm . 14 Desired closed loop transfer function = Kc = − ([Ti − 0.5τm ]s + 1)e−sτ ; ( )( ) 0.5(Ti − 0.5τ m )τ m Ti , Td = , (1) ( 2) Ti K m TCL + T + 1 . 5 τ − T 2 CL 2 m i ( ( ) ) (1) ( 2) (1) ( 2) (1) ( 2) TCL 2TCL 2 + 0.5τ m TCL 2 + TCL 2 + 0.5TCL 2TCL 2 Ti = m (1) ( 2) TCL 2s + 1 TCL 2s + 1 Tm − 0.5τm ( τm (1) ( 2) + Tm TCL 2 + TCL 2 + τ m Tm ) . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Sree and Chidambaram (2006), pp. 117,118. Sree and Chidambaram (2006). Model: Method 1 15 Kc = Kc Ti Td Comment 451 15 Kc Ti 0.5τ m Model: Method 1 16 Kc Ti Td pp. 15–16. Kc Ti Td p. 248; τ 0.01 ≤ m ≤ 1.1 . Tm 17 2 τ τ τ 1 4282 m − 1334.6 m + 101 , 0 < m < 0.2 ; Tm Km Tm Tm Kc = 1.1161 Tm K m τm 0.9427 , 0.2 ≤ τm ≤ 1.0 . Tm , 0.8 ≤ τm ≤ 1.0 . Tm 2 τm τm τ + 0.8288 , 0 ≤ m ≤ 0.8 ; Ti = Tm 36.842 − 10 . 3 T T T m m m τ Ti = 76.241Tm m Tm Desired closed loop transfer function = TCL 2 = 4.5115Tm (τm Tm ) 5.661 ( 6.77 (ηs + 1)e −sτ ; (TCL2 s + 1)2 m , 0.8 ≤ τm Tm ≤ 1.0 , ) TCL 2 = 0.0326 + 0.4958(τm Tm ) + 2.03(τm Tm )2 Tm , 0 ≤ τm Tm ≤ 0.8 . 16 Kc = 2 τm τm 1 0.726Tm + T T 0 . 096 2 . 099 4 . 104 0 . 229 , = + + i m T , τm Tm Km m 2 τ τ τ Td = Tm 0.019 + 0.419 m + 0.822 m , 0 ≤ m < 0.6 . Tm T T m m 17 Kc = 1.214 Tm K m τm 0.8449 1.028 τ , Ti = 3.05Tm m Tm 1.0145 τ , Td = 0.672Tm m Tm . b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 452 Rule Kc Sree and Chidambaram (2006) – continued. 18 Kc Ti Td Comment Ti Td p. 248; τ 0.01 ≤ m ≤ 1.1 . Tm Direct synthesis: frequency domain criteria De Paor and O’Malley (1989). Model: Method 1 Chidambaram (1995b). 19 1.166 Tm Kc = K m τm Kc = Ti = 1 cos Km Kc Ti Td τm <1 Tm 20 Kc 25Tm − 27 τm 0.46τ m τ m Tm < 0.6 Also Chidambaram (1998), p. 93. Model: Method 1 τ 0.1 ≤ m ≤ 0.6 ; 21 Ti Td Kc Tm Model: Method 1 Gaikward and Chidambaram (2000). 18 19 0.6365 1.105 , Ti = 1.0553Tme Tm τm τ , Td = 0.7542Tm m Tm 1 − τm Tm τ τ 1 − m m + sin T T τm Tm m m 0.789 . τ τ 1 − m m , T T m m τm Tm Tm , Td = Tm tan (0.75φ) , 1 − τm Tm 1 − τm Tm tan (0.75φ ) τm Tm φ = tan −1 (Tm τm − 1) − (τm Tm ) − (τm Tm )2 . 20 Kc = 2 τm τm 1 1 T . 1.3 + 0.3 m or K c = 5 . 332 15 . 02 16 . 214 − + T τm Km Tm K m m 21 Kc = 2 τm τ 1 Tm + + 2.099 m + 0.096 , 0 . 299 0 . 726 , T = T 4 . 104 i m τm Km Tm Tm 2 τm τm . + + Td = Tm 0.822 0 . 419 0 . 019 Tm Tm b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti 453 Comment Td Robust Ti Td Kc Rotstein and Lewin (1991), λ determined graphically – sample values provided: Marchetti et al. τm Tm = 0.2 , 0.5Tm ≤ λ ≤ 1.9Tm . K m uncertainty (2001). = 50%. τm Tm = 0.4 , 1.3Tm ≤ λ ≤ 1.9Tm . Model: Method 1 τm Tm = 0.2 , 0.4Tm ≤ λ ≤ 4.3Tm . K m uncertainty τm Tm = 0.4 , 1.1Tm ≤ λ ≤ 4.3Tm . = 30%. τm Tm = 0.6 , 2.2Tm ≤ λ ≤ 4.3Tm . 22 Tan et al. (1999b). 23 Ti Kc τm Tm < 1.0 ; Model: Method 1 Td λ λ Tm λ + 2 + 0.5τm 0.5λ + 2 τ m T Tm m , T = λ λ + 2 + 0.5τ , T = 22 Kc = . d i m T λ2 K m λ m λ + 2 + 0.5τ m Tm 23 Kc = Ti = 1 K m {[− (0.0373 λ ) + 0.1862](τm Tm ) + [0.6508 − 0.1498 log λ ]} , {[(10.5045 λ ) + 0.8032](τm Tm ) + [(3.2312 λ ) − 0.3445]} T , {[− (0.0373 λ ) + 0.1862](τm Tm ) + [0.6508 − 0.1498 log λ]} m τ 1 Td = Tm 0.9065λ0.1632 m − Tm 0.2075λ + 2.0335 − 0.0373 τ + 0.1862 m + (0.6508 − 0.1498 log λ ) , λ not specified. λ T m b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 454 Rule Kc Lee et al. (2000). Model: Method 1 Shamsuzzoha and Lee (2006b). Ti Td Comment 24 Kc Ti Td 0 ≤ τm Tm < 2 25 Kc Ti Td Model: Method 1 26 Ti Td Kc Shamsuzzoha et If τm Tm >> 1 , λ = τm (Shamsuzzoha et al., 2005); λ = x1Tm , with al. (2005), Shamsuzzoha x1 values given below (Shamsuzzoha and Lee, 2007a). and Lee (2007a). Ms τm Tm Ms τm Tm Model: Method 1 3.0 3.5 4.0 3.0 3.5 4.0 0.01 0.01 0.01 0.01 0.40 0.34 0.31 0.3 0.05 0.05 0.04 0.05 0.65 0.55 0.48 0.4 0.10 0.09 0.08 0.1 1.10 0.90 0.75 0.5 0.23 0.20 0.18 0.2 24 Kc = 2 T , x1 = Tm CL + 1 eτ m Tm − 1 , Ti = −Tm + x 1 − x 2 , − K m (2TCL + τm − x1 ) Tm Ti τm (0.167 τm − 0.5x1 ) 2TCL + τm − x1 − x2 . Ti 2 2 x2 = 25 2 TCL + x1τm − 0.5τm , Td = 2TCL + τm − x1 Kc = − λ2 − 0.5τm 2 + x1τm Ti ’ , Ti = −Tm + x1 − x 3 , x 3 = K m (τm − x1 + 2λx 2 ) 2λx 2 + τm − x1 3 − Tm x1 − Td = − Tm x1 − 2 0.167 τm − 0.5x1τm λ2 + 2λx 2Tm + Tm 2 e τ m Tm τm − x1 + 2λx 2 − 1 ; − x 3 , x1 = Tm 2 Ti Tm ( ) x 2 not explicitly specified ( x 2 ≥ 0.707 normally); λ is ‘slightly more’ than τm . 26 Kc = − 3λ2 + 2x1τm − 0.5τm 2 − x12 Ti , , Ti = −Tm + 2 x1 − K m (3λ + τm − 2x1 ) 3λ + τm − 2x1 2 − 2Tm x1 + x1 − Td = λ3 + 0.167 τm 3 − x1τm 2 + x12 τm 3λ2 + 2x1τm − 0.5τm 2 − x12 3λ + τm − 2x1 , Ti 3λ + τm − 2x1 3 λ τ m Tm . e x 1 = Tm 1 + − 1 Tm b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 455 4.10.3 Ideal controller in series with a first order lag U(s) E(s) R(s) + − 1 1 K c 1 + + Td s Ti s Tf s + 1 1 1 G c (s) = K c 1 + + Td s Tf s + 1 Ti s K m e − sτ Tm s − 1 m Y(s) Table 93: PID controller tuning rules – unstable FOLPD model G m (s) = Rule Chandrashekar et al. (2002). 1 Kc = Kc 1 Kc Ti Td K m e − sτ m Tm s − 1 Comment Direct synthesis: time domain criteria Model: Method 1 Ti 0.5τ m 2 τ τ τ 1 4282 m − 1334.6 m + 101 , 0 < m < 0.2 ; Tm Km Tm Tm Kc = 1.1161 Tm K m τm 0.9427 , 0. 2 ≤ τm ≤ 1. 0 . Tm , 0. 8 ≤ τm ≤ 1.0 . Tm 2 τ τ τ Ti = Tm 36.842 m − 10.3 m + 0.8288 , 0 ≤ m ≤ 0.8 ; T T T m m m τ Ti = 76.241Tm m Tm τ τ Tf = 0.1233 m + 0.0033Tm , 0 ≤ m ≤ 1.0 . Tm Tm Desired closed loop transfer function = τ TCL 2 = 4.5115Tm m Tm 5.661 , 0.8 ≤ (ηs + 1)e −sτ ; (TCL2 s + 1)2 m τm ≤ 1.0 , Tm 2 τ m τm τm TCL 2 = 0.0326 + 0.4958 + 2.03 Tm , 0 ≤ T ≤ 0.8 . Tm T m m 6.77 b720_Chapter-04 04-Apr-2009 Handbook of PI and PID Controller Tuning Rules 456 Rule 2 Chandrashekar et al. (2002). Model: Method 1 Kc Ti Td Kc Ti Td Alternative tuning rules: Td Tf τ m Tm 0.02Tm 0.0 0.0435Tm 0.0134Tm 0.10Tm 0.1 0.0884Tm 0.0250Tm 0.20Tm 0.2 3.2655 K m 1.8018Tm 0.1375Tm 0.0435Tm 0.40Tm 0.3 2.4516 K m 3.0400Tm 0.1868Tm 0.0581Tm 0.60Tm 0.4 2.0217 K m 4.6500Tm 0.2366Tm 0.0696Tm 0.80Tm 0.5 1.7525 K m 6.7286Tm 0.2866Tm 0.0784Tm 1.00Tm 0.6 1.5458 K m 10.337Tm 0.3380Tm 0.0885Tm 1.30Tm 0.7 1.3723 K m 17.693Tm 0.3910Tm 0.1005Tm 1.80Tm 0.8 1.2420 K m 33.610Tm 0.4440Tm 0.1124Tm 2.60Tm 0.9 0.1247Tm 4.20Tm 1.0 0.5975Tm 0.1138Tm 5.00Tm 1.2 0.6982Tm 0.0957Tm 6.00Tm 1.4 Ti 101.0000 Km 0.0404Tm 10.3662 Km 0.3874Tm 5.3750 K m 0.8600Tm 0.0 0.0 1.1400 K m 80.620Tm 0.4969Tm Desired closed loop transfer function = Kc = − Model: Method 1; p. 118. ([Ti − 0.5τm ]s + 1)e−sτ ; ( Td = m )( ) (1) ( 2) TCL 2s + 1 TCL 2s + 1 Ti , (1) ( 2) K m TCL + T 2 CL 2 + 1.5τ m − Ti ( ( ) ) (1) ( 2) (1) ( 2) (1) ( 2) TCL 2TCL 2 + 0.5τ m TCL 2 + TCL 2 + 0.5TCL 2TCL 2 Ti = Comment TCL 2 Kc Sree and 1.0895 K m 143.6Tm Chidambaram 1.0536 K m 277.37Tm (2006). 2 FA ( τm (1) ( 2) + Tm TCL 2 + TCL 2 + τ m Tm Tm − 0.5τm (1) ( 2) 0.5TCL 0.5(Ti − 0.5τm )τm 2 TCL 2 τ m , Tf = − . (1) ( 2) Ti Tm Ti − 1.5τ m − TCL 2 − TCL 2 ( ) ) , b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Kc Ti 457 Comment Td Robust Zhang et al. (2000). Model: Method 1 Lee et al. (2006). Model: Method 1 3 4 Kc Ti Td Recommended τm ≤ λ ≤ 2τm . Zhang (2006). Model: Method 1 5 Kc Ti Td Recommended τm ≤ λ ≤ 2τm . ) 2 3 Kc = Kc = 2τm ≤ λ ≤ 5τm , τm Tm < 1 ; λ large, τm Tm > 1 . ( λ2 τm + λ2 + 2λτ m Tm + 2(λ + τm )Tm λ2 τm λ (λ + 2τm ) , Ti = + + 2(λ + τm ) , 2 K mλ[λτ m + (λ + 2τm )Tm ] Tm Tm λ2 τm + λ(λ + 2τm )Tm + (2λ + τm )Tm Td = τm 4 Td λ2 τm + λ(λ + 2τm )Tm + 2(λ + τm )Tm 2 , Tf = 2 λ2 τm Tm . λ2 τm + λ (λ + 2τm )Tm 2 Ti λ2 + x1τm − 0.5τm , Ti = −Tm + x1 − , − K m ( 2λ + τ m − x 1 ) 2λ + τm − x1 τm (0.167 τm − 0.5x1 ) 2 λ2 + x1τm − 0.5τm 2λ + τm − x1 − ; Ti 2λ + τm − x1 2 − Tm x1 − Td = Desired closed loop transfer function = λ3 5 Kc = Tm 2 + 3λ2 + 3λ + τm Tm λ3 3λ2 Km 2 + T Tm m , Ti = λ3 Tm 2 + 2 λ τ m Tm , x = T + 1 e − 1 . (λs + 1)2 1 m Tm e − sτ m 3λ2 + 3λ + τm , Tm λ2 Td = Tm λ3 2 + 3λ +3 Tm 3λ2 + + 3λ + τm 2 Tm Tm λτ m , Tf = λTm . λ + 3Tm b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 458 1 1 + Td s 4.10.4 Classical controller G c (s) = K c 1 + Td s Ti s 1 + N E(s) R(s) + − U(s) 1 1 + Td s K c 1 + Td Ti s s 1 + N K m e − sτ Tm s − 1 m Y(s) Table 94: PID controller tuning rules – unstable FOLPD model G m (s) = Rule Kc Ti Td K m e − sτ m Tm s − 1 Comment Minimum performance index: regulator tuning Minimum IAE – 0.91Tm K m τm 1.70τ m 0.60τ m τ m Tm = 0.1 Shinskey (1988), 1 . 00 T K τ 1 . 90 τ 0 . 60 τ τ m Tm = 0.2 m m m m m p. 381. Method: 0.89Tm K m τm 2.00τ m 0.80τ m τ m Tm = 0.5 Model 1; 2.25τ m 0.90τ m N not specified. 0.86Tm K m τm τ m Tm = 0.67 0.83Tm K m τm Huang and Chen (1997), (1999). 1 1 Kc 2.40τ m 1.00τ m τ m Tm = 0.8 Direct synthesis: time domain criteria Ti Td N =∞; Model: Method 4 0 < τm Tm < 1 ; ‘Good’ servo and regulator response. Kc = 1.02379 T 0.5 , Ti = K c K m Tm 2 x1 + τm , 1 + 1.61519 m T K K 1 T − Km τ c m m m m 2 τ τ with recommended x 1 = Tm − 1.3735.10 −3 + 0.22091 m + 1.6653 m ; Tm Tm 2 τm τm −4 . Td = Tm 1.5006.10 + 0.23549 + 0.16970 Tm Tm b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Comment Td Direct synthesis: frequency domain criteria Paraskevopoulos x1τm Tm N =∞; 2 2 Ti et al. (2006). Ti x 2 1 + x 2 τ > 0.15τm Tm 0.02 < m < 0.9. 3 1 + Ti 2 x 2 2 Model: Method 1 Ti Tm 2 0 ≤ x1 ≤ 0.55 + τm 0.0098 0.31 − ; x1 < 0.15 . Tm 1 − (τm Tm ) Ti 2 (1 − (τm Tm ) + Td ) + Ti − (τm Tm ) + Td + x 3 x2 = 2Ti 2 ((τm Tm ) − Td ) , 2 τ 2 τ τ τ x 3 = Ti 2 1 − m + Td + Ti − m + Td + 4 m − Td Ti 1 + Ti − m + Td ; Tm Tm Tm Tm ( ) 0.632((τ m Tm ) − Td ) + 0.436 (τ m Tm ) − Td − 0.0153 φ m φ max m , Ti = x 4 1 + max (τm Tm ) − Td 1 − φ m φ m x4 = 0.0029 − 0.0682 (τm Tm ) − Td + 1.4941((τm Tm ) − Td ) (1.003 − (τm Tm ) + Td )2 ( ; T τ T max φm > 0.2φmax = − m − Td m − Td − 1 + tan −1 m − Td − 1 . m , φm τm Tm τm 3 x 2 and x 3 given in footnote 2; T Tτ Ti = x 4 1 + m 1 + 0.4 d m − 0.22Td x 5 , x 4 given in footnote 2; Tm τm )1 −(φ(φ φ φ ) ) , = −(τ T ) (T τ ) − 1 + tan ( (T τ ) − 1 ) . ( x 5 = − 0.0153 − 0.436 τm Tm + 0.632(τm Tm ) φmax m m m m m −1 m m m max m max m m ) 459 b720_Chapter-04 FA Handbook of PI and PID Controller Tuning Rules 460 Rule Paraskevopoulos et al. (2006). Model: Method 1 4 04-Apr-2009 0 ≤ x1 ≤ 0.55 + Kc = 4 Kc Ti Td Comment Kc Ti x1τm Tm N=∞ τm 0.0098 0.31 − ; x1 < 0.15 . 0.02 < τm Tm < 0.9 . Tm 1 − (τm Tm ) 2 2 K max 1 + ωmin 1 + ωmax or K min A m , K min = Ti ωmin , K max = Ti ωmax ; 2 2 2 Am 1 + Ti ωmin 1 + Ti ωmax 2 0.03([τm Tm ] − Td ) 1.14 − [τm Tm ] + Td ωmin = 1 + 0.05 0.973 + Ti − x 2 [ ] − τ + 1 T T m m d 0.006 + x2 T i −([τm Tm ] − Td )(1 + Ti ) , 4 2 x τ τ 1.571 ωmax = 1 + 0.22 m − Td 1 + 0.1 − 0.3 m − Td 2 Tm Tm Ti (τm Tm ) − Td Ti Ti − 0.9463 − 0.5609 . (Ti + 1)([τm Tm ] − Td ) (Ti + 1)([τm Tm ] − Td ) 1+ (τ m Tm )− Td max A m − 1 A m − 1 + 0.01 − 0.18 + 5 τm − T Ti = x 2 1 + 0.65 d Tm 1 − A m − 1 A max − 1 m [ [ ] ] [ + 0.01 − 32([τm Tm ] − Td ) + 75([τm Tm ] − Td ) − 51([τm Tm ] − Td ) x2 = 2 ] 3([τm Tm ] − Td )4 − 2.3([τm Tm ] − Td )2 + 0.01 , (1 − [τm Tm ] + Td )3 0.0029 − 0.0682 [τm Tm ] − Td + 1.4941([τm Tm ] − Td ) (1.003 − [τm Tm ] + Td )2 max A m > 0.2(A max = m − 1) +1, A m 3 ; 1.571 τm 0.4085[[τm Tm ] − Td ] T − Td 1 + 1 − 0.2864[[τ T ] − T ] m m d m 1 (7.41 − 3.52[τm Tm ])(TdTm τm )3 1+ 3 1 − (0.47 + 0.49[τm Tm ])(Td Tm τm ) . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Paraskevopoulos et al. (2006) – continued. 5 5 Kc Ti Kc Ti > Td Comment 0.15τm Tm N=∞ 461 K c and x 2 as given in footnote 4, with 0.03([τm Tm ] − Td ) 1.14 − [τm Tm ] + Td ωmin = 1 + 0.05 0.973 + Ti − x 2 [ ] − τ + 1 T T m m d 0.006 + x2 τ T i − m − Td (1 + Ti ) Tm (0.14 + 1.15[τm Tm ])(Td Tm τm )3 x 2 , 1 + 2 T 1 + 2(Td Tm τm ) i 2 4 x 2 τm τm 1.571 ωmax = 1 + 0.22 − Td 1 + 0.1 − 0.3 − Td [ T T T τ m m i m Tm ] − Td Ti Ti − 0.9463 − 0.5609 ; (Ti + 1)([τm Tm ] − Td ) (Ti + 1)([τm Tm ] − Td ) ( 5.53 − 0.41[τm Tm ])(Td Tm τm )4 1+ 3 2 1 − 0.65(Td Tm τm ) 1.27 − 0.4(Ti x 2 ) [ ][ ] 1+ (τ m Tm )− Td max A m − 1 A m − 1 + 0.01 − 0.18 + 5 τm − T Ti = x 3 1 + 0.65 d Tm 1 − A m − 1 A max − 1 m [ [ ] ] [ + 0.01 − 32([τm Tm ] − Td ) + 75([τm Tm ] − Td ) − 51([τm Tm ] − Td ) 2 3 ] 2 3([τm Tm ] − Td )4 − 2.3([τm Tm ] − Td )2 Td Tm + . 1 x4 , + 0.01 3 (1 − [τm Tm ] + Td ) τm T 3 0.367 + 1.78[τ T ] m m x 3 = x 2 1 + Td m , τm 1 − (Td Tm τm )2 3τ m 3 2 (A − 1)3 τm Td Tm Tm τm τ m m 2 − 3.32 x 4 = , 3 T + 1.2 T − 0.27 − max m A m − 1 m τm Tm ( A max as given in footnote 4. 0.02 < τm Tm < 0.9 . m ) b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 462 4.10.5 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N R(s) E(s) + − b + b s + b s 2 U(s) T s 1 d f1 f2 f0 K c 1 + + Ti s 1 + Td s 1 + a f 1s + a f 2s 2 N K m e − sτ Tm s − 1 Y(s) m Table 95: PID controller tuning rules – unstable FOLPD model G m (s) = Rule Kc K m e − sτ m Tms − 1 Td Comment 0.25τm Model: Method 1 Ti Robust 1 Shamsuzzoha et al. (2007). 0.667 τm Kc For τm Tm > 0.6 , λ = τm . Otherwise, λ = x1Tm ; x1 given for various τm Tm and M s values: 1 Kc = − τm Tm M s = 3. 0 M s = 3. 6 τm Tm M s = 3. 0 M s = 3. 6 0.05 0.1 0.2 0.3 0.015 0.04 0.09 0.14 0.015 0.04 0.08 0.13 0.4 0.5 0.6 0.21 0.32 0.41 0.18 0.26 0.36 3 4τm λ τ m Tm e − 1 , bf 2 = 0 , , b f 0 = 1 , b f 1 = Tm 1 + K m (6τm + 18λ − 6b f 1 ) Tm 2 N = ∞ , af1 = 2b f 1τm + τm + 12λτ m + 18λ2 + Tm , a f 2 = 0 . 6τ m + 18λ − 6b f 1 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 463 4.10.6 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s α + β d K c T 1+ d s N E(s) R(s) + − T s 1 d K c 1 + + Ti s Td 1+ s N − U(s) + K m e − sτ Tm s − 1 m Table 96: PID controller tuning rules – unstable FOLPD model G m (s) = Rule Minimum IAE – Huang and Lin (1995). 1 Kc = − Kc 1 Kc Ti Y(s) K m e − sτ m Tm s − 1 Comment Td Minimum performance index: regulator tuning Model: Method 2 Ti Td α = 0 , β = 1 , N = 10; 0.01 ≤ τm Tm ≤ 0.8 . T 1 τ 0.2675 + 0.1226 m + 0.8781 m Km Tm τm 1.004 ( , Ti = Tm 0.0005 + 2.4631(τm Tm ) + 9.5795(τm Tm ) 2.9123 ) , T = 0.0011T + 0.4759τ . d m m b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 464 Rule Kc Minimum ISE – Paraskevopoulos et al. (2004). Minimum IAE – Huang and Lin (1995). Minimum ISE – Paraskevopoulos et al. (2004). 2 Kc 4 (1) Ti Td Comment 3 0 Model: Method 1 Ti Minimum performance index: servo tuning Model: Method 2 Ti Td Kc α = 0 , β = 1 , N = 10; 0.01 ≤ τm Tm ≤ 0.8 . 5 Kc 0 Ti (1) Model: Method 1 Minimum performance index: other tuning Paraskevopoulos et al. (2004). Model: Method 1 2 Kc (1) Kc = Kc (max) = (min) Kc ωmax Ti Km 6 (max) Kc , Kc 0 Ti (1) (min) = 1 + ω max 2 Tm 2 2 1 + ωmax Ti 2 ωmin Ti Km , ωmin = 1 + ω min 2 Tm 2 α =1 , 1 + ωmin 2 Ti 2 τm 1 Ti − , Tm Tm Tm + Ti [4.935 τm][(Ti (Ti + τm )) − (τm Tm )] , T ω 2 >> 1 . (τm Tm ) + π[(Ti (Ti + τm )) − (τm Tm )] i max 1.916 + 4.78(τ m Tm ) 3 Ti = τm , α = 1 , 0.05 ≤ τm Tm ≤ 0.85 . 0.88 − (τm Tm ) 1.0251 4 ), K c = −(1 K m )(− 0.433 + 0.2056(τm Tm ) + 0.3135(Tm τm ) 6.6423 ), Ti = Tm (− 0.0018 + 0.8193(τm Tm ) + 7.7853(τm Tm ) 7.6983(τ T ) ). Td = Tm (− 0.0312 + 1.6333(τm Tm ) + 0.0399e 2.725 + 1.71(τ m Tm ) (1) 5 K c given in footnote 2. Ti = τm , α = 1 , 0.05 ≤ τm Tm ≤ 0.85 . 0.88 − (τm Tm ) ωmax = m 6 Kc (1) m [ ] ∞ ∫ [e (t) + (u( t) − u ) ]dt . given in footnote 2. Minimise performance index = 2 0 Ti = Tm 0.747 + 3.66(τm Tm ) + 18.64(τm Tm ) , 0.01 ≤ τm Tm ≤ 0.3 . 3 4.52 + 0.476(τm Tm )2 τm Ti = τm ≤ 0.85 . , 0.3 ≤ Tm 0.88 − (τm Tm ) ∞ 2 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Paraskevopoulos et al. (2004) – continued. Chidambaram (2000c). Model: Method 1 7 Kc (1) 8 Kc 9 Kc Ti Td Comment Ti 0 α =1 465 Direct synthesis: time domain criteria Ti ‘Small’ τ m Tm . 0 τ 0.1 ≤ m ≤ 0.6 Ti Tm 10 (1) Chidambaram τm Tm < 0.6 Ti 0.245 (2000c), Srinivas 1.165 Tm 4.4Tm + 9 τ m 0.176T τ and m 0.6 ≤ m ≤ 0.8 K m τ m Chidambaram T +0.36τm m (2001), Sree and ( 2) τ Chidambaram Ti 0.8 ≤ m ≤ 0.9 (2006), Tm pp. 182–184. Model: Method 1 ξ = 0.333 ; N = ∞ , β = 1 , α = 0.74 + 0.42(τm Tm ) − 0.23(τm Tm )2 (Chidambaram, 2000c); 2 τ 1 1 Tm τ − m (Srinivas ; β= N =∞, α = m + τm − Ti K c K m Td K c K m 2Ti and Chidambaram, 2001); ξ ≥ 1 ; N = ∞ , β = 1 ; α = 1 (Sree and Chidambaram, 2006). 7 Kc (1) ∞ ∫ [e (t) + (du dt) ] dt. given in footnote 2. Minimise performance index = 2 2 0 2.58 + 3.60(τm Tm )2 τm Ti = τm ≤ 0.85 . , 0.03 ≤ T ( ) 0 . 88 − τ T m m m 8 Kc = 2 Tm Ti 2 2 0.04TS ξ + Ti τm α = 1− , Ti = 0.04TS ξ2 + Tm τm + 0.4TSξ 2 , Tm − τm 0.2TS ξ 2 (Tm − τ m ) 0.04TS 2 ξ 2 + Tm τ m + 0.4TS ξ (Chidambaram, 2000c); α = 1 − 2 0.2TSξ 2 or Ti α = 0.733 + 0.6(τm Tm ) − 0.36(τm Tm ) (Sree and Chidambaram, 2006). 5.8 τ m Tm −1.277 τ m Tm 9 K c = (2.695 K m )e , Ti = 0.4015Tm e , 2 α = 0.733 + 0.6(τ m Tm ) − 0.36(τ m Tm ) ; ξ = 0.333 . 2 10 Ti (1) = 0.176Tm + 0.36τm 0.179 − 0.324(τm Tm ) + 0.161(τm Tm )2 , Ti ( 2) = 0.176Tm + 0.36τm . 0.12 − 0.1(τm Tm ) b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 466 Rule Kc Ti Comment Td τm Tm ≤ 0.2 11 Kc 6.461τ m Jhunjhunwala 0.898 and 0.324Tm e Tm Tm Chidambaram 0.876 τm (2001). 12 Ti Model: Method 1 Kc τm ≤ 0.6 Tm 0 0.2 ≤ Td β =1, N = ∞ . Minimum ISE (Chidambaram, 2002, pp. 157–158). Chandrashekar et al. (2002). Model: Method 1 11 Kc = 13 K c K m (2 − Tm ) + (K c K m − 1)Ti 1 τm (Jhunjhunwala and 10.7572 − 35.1 ; α = Km Tm 2K c K m Chidambaram, 2001); α = 12 Kc = 0 Ti Kc K c K m (2 + τm ) − (K c K m + 1)Ti (Chidambaram, 2002). 2K c K m 1 τ τ 1.397 Tm 13.0 − 39.712 m , m < 0.2 ; K c = K m Tm Tm K m τm 2.044 τ m Ti = 0.856Tm e Tm τ , 0 ≤ m ≤ 1 ; Ti = 0.3444Tme Tm 0.769 2.9595 τ m Tm , 1≤ , 0.2 ≤ τm ≤ 1.4 ; Tm τm ≤ 1.4 ; Tm Td = 0.5643τm + 0.0075Tm , 0 ≤ τm Tm ≤ 1.4 ; α = 0.3137 , τm Tm = 0.1 ; α = 1 − 0.4772e − (1.58τ m Tm ) , 0.2 ≤ τm Tm ≤ 1 . 13 49.535 − 21.644 τ m Tm τm 0.8668 τm e , < 0.1 ; K c = K m Tm Km Tm Ti = 0.1523Tm e7.9425τ m Tm ; −0.8288 Kc = Desired closed loop transfer function = , 0.1 ≤ τm ≤ 0.7 ; Tm (ηs + 1)e −sτ ; (TCL2 s + 1)2 m TCL 2 = 0.025Tm + 1.75τm , τm Tm ≤ 0.1 ; TCL 2 = 2 τ m , 0.1 ≤ τm Tm ≤ 0.5 ; TCL 2 = τm (5(τm Tm ) − 0.5) , 0.5 ≤ τm Tm ≤ 0.7 . α = 0.505 + 3.426(τm Tm ) − 10.57(τm Tm ) , τm Tm ≤ 0.2 ; 2 α = 0.713 + 0.232(τm Tm ) , 0.2 ≤ τm Tm ≤ 0.7 (Chandrashekar et al, 2002); α = 0.67 , τm Tm ≤ 0.1 ; α = 0.6354 + 0.437(τm Tm ) , 0.1 ≤ τ m Tm ≤ 0.7 (Sree and Chidambaram, 2006). b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Chandrashekar et al. (2002). Model: Method 1 14 Kc = Kc 14 Kc Ti Td Ti 0.5τ m 467 Comment 2 τ τ τ 1 4282 m − 1334.6 m + 101 , 0 < m < 0.2 ; T Km T T m m m Kc = [ ] 1.1161 Tm K m τm 0.9427 , 0.2 ≤ τm ≤ 1.0 ; Tm Ti = Tm 36.842(τm Tm ) − 10.3(τm Tm ) + 0.8288 , 0 ≤ τm Tm ≤ 0.8 ; 2 Ti = 76.241Tm (τm Tm ) 6.77 , 0.8 ≤ τm Tm ≤ 1.0 ; α = 0.83 , τm Tm ≤ 0.1 ; α = 0.8053 − 0.1935(τm Tm ) , 0.1 ≤ τm Tm ≤ 0.7 (Chandrashekar et al., 2002); α = 0.505 + 3.426(τm Tm ) − 10.17(τm Tm ) , τm Tm ≤ 0.2 ; 2 α = 0.713 + 0.232(τm Tm ) , 0.2 ≤ τm Tm ≤ 1.0 (Sree and Chidambaram, 2006). (ηs + 1)e −sτ ; (TCL2 s + 1)2 2 TCL 2 = (0.0326 + 0.4958(τm Tm ) + 2.03(τm Tm ) )Tm , 0 ≤ τm Tm ≤ 0.8 ; 5.661 TCL 2 = 4.5115Tm (τ m Tm ) , 0.8 ≤ τm Tm ≤ 1.0 , N = ∞ ; β = 0 m Desired closed loop transfer function = (Chandrashekar et al., 2002; Sree and Chidambaram, 2006); [(1 − α )Tis + 1]e−sτm . Desired closed loop transfer function = TCL 2s 2 + 2ξTCLs + 1 If it is desired to minimise the overshoot (servo response), for ξ ≥ 1 , then α = 1 . Otherwise, the optimum value of α is chosen by minimising the ISE (servo response). Then, α = 1 − 2(TCL Ti ) , ξ = 1 ; α = 1 − 2(TCL ξ Ti ) , ξ < 1 ; 2 3 − ξ + ξ 2 − 1 − ξ − ξ 2 − 1 − − ξ + ξ 2 − 1 + x1 TCL , ξ > 1 with α = 1− 3 Ti − ξ + ξ2 − 1 − ξ − ξ2 − 1 + x 2 3 2 x 1 = − − ξ − ξ 2 − 1 + − ξ + ξ 2 − 1 − ξ − ξ 2 − 1 and 3 2 2 x 2 = − ξ − ξ 2 − 1 − ξ + ξ 2 − 1 − 2 − ξ + ξ 2 − 1 − ξ − ξ 2 − 1 ; N= ∞ ; β = 1 (Sree and Chidambaram, 2005a). b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 468 Rule Chandrashekar et al. (2002). Model: Method 1 Kc Ti Td Kc Ti 0 Comment 6.0000 K m Alternative tuning rules: α Ti 0.6000Tm 0.6667 τ m Tm = 0.1 3.2222 K m 1.4500Tm 0.7246 τ m Tm = 0.2 2.2963 K m 2.6571Tm 0.7742 τ m Tm = 0.3 1.8333 K m 4.4000Tm 0.8182 τ m Tm = 0.4 1.5556 K m 7.0000Tm 0.8571 τ m Tm = 0.5 1.3265 K m 14.6250Tm 0.8974 τ m Tm = 0.6 1.1875 K m 31.0330Tm 0.9323 τ m Tm = 0.7 Kc Alternative tuning rules; TCL 2 = x 5 Tm : x1 K m x 2Tm N = x3 x 4 , x 3Tm β = 0. Sree and Chidambaram (2006). x1 x2 10.3662 5.3750 3.2655 2.4516 2.0217 1.7525 1.5458 1.3723 1.2420 1.1400 1.0895 1.0536 0.3874 0.8600 1.8018 3.0400 4.6500 6.7286 10.337 17.693 33.610 80.620 143.6 277.37 x3 Coefficient values: x4 x5 α 0.0435 0.0134 0.10 0.742 0.0884 0.0250 0.20 0.767 0.1375 0.0435 0.40 0.778 0.1868 0.0581 0.60 0.803 0.2366 0.0696 0.80 0.828 0.2866 0.0784 1.00 0.851 0.3380 0.0885 1.30 0.874 0.3910 0.1005 1.80 0.898 0.4440 0.1124 2.60 0.923 0.4969 0.1247 4.20 0.948 0.5975 0.1138 5.00 0.965 0.6982 0.0957 6.00 0.978 pp. 118, 125. Model: Method 1 τ m Tm 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti > Ti Paraskevopoulos et al. (2004). Model: Method 1; α = 1. 15 Kc (1) > Comment Td (min) 1.35τm 0 (1 − τm Tm ) 2 x1τ m + x2 Coefficient values: x1 x2 ξ ξ 0.7269 2.5362 0.5 3.0696 3.7509 1.7016 3.0585 0.75 6.975 5.7053 (1) x 1 τ m + x 2 Tm + K 1 1.5 c 6 x 3 τ m Tm x2 x3 4.59 6.16 8.21 -0.19 0.028 0.37 8.28 9.90 16.88 Kc 16 (1) x1 x2 Kc (1) τm < 0.9 Tm 12.41 19.91 12.452 8.4358 2 0.2 < τm < 0.7 Tm x2 x3 ξ 1.89 3.44 51.95 95 1.5 2 0 Ti ξ x2 τm Tm < 0.2 x2 1.142 1.177 2.353 2.67 4.07 8.88 4.76 2 10.73 3 τm Tm < 0.2 Ti 1 1.5 0 TCL 2 x2 TCL 2 Coefficient values: x1 x2 TCL 2 x1 x2 TCL 2 1.518 10.54 3.34 16.89 0.5 0.75 5.81 13 23.08 82.87 51.94 176.8 2 3 x1 15 0.1 < x1 17 (1) τm < 0.7 Tm Coefficient values: x1 x2 TCL 2 TCL 2 0.409 0.2817 0.5 0.717 0.6542 0.75 Kc 0.5 0.75 1 x1 0 −5 Coefficient values: ξ x1 x1 0.02 < τm < 0.3 Tm x 2 τ m Tm x1 469 29.17 49.66 1 1.5 given in footnote 2. Ti (min) = Tm [1.3146(τm Tm ) − 2.3805(τm Tm )2 + x1 ], x1 = 57.622(τm Tm ) − 158.99(τm Tm ) + 173.41(τm Tm ) . 3 16 17 x2 . Ti = Tm x1 − (τm Tm ) − τm Tm [ ] Ti = Tm x1 + x 3 (τm Tm ) . 3 4 5 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 470 Rule Kc 18 Prashanti and Chidambaram (2000). Model: Method 1 Ti Comment Td Direct synthesis: frequency domain criteria Ti Td Kc Modified method of De Paor and O’Malley (1989); τ m Tm < 1 ; α = 0.83 ; β = 0.346 ; N = ∞ . Alternatively, 2 α = 0.6873 + 0.3369 2 β = −1.779 + 2.44 3 τ τ τm − 0.0378 m + 0.045 m , Tm T m Tm 3 τ τ τ τm − 5 m + 4.81 m , 0.05 ≤ m ≤ 0.90 . Tm T T T m m m α β τm Tm Representative results: α β τm Tm 0.691 0.719 0.756 0.787 -1.834 -1.605 -1.413 -1.346 0.05 0.10 0.20 0.30 0.818 0.850 0.884 0.920 -1.336 -1.207 -1.093 -0.884 α β 0.40 0.960 -0.508 0.50 0.9909 -0.159 0.60 0.70 τm Tm 0.80 0.90 −6.405 τ m Alternatively, α = 1 − 0.9726e Tm α α τm Tm τm Tm 0.223 0.543 0.773 18 Kc = Ti = 1 cos Km 0.05 0.10 0.20 0.879 0.922 0.954 0.30 0.40 0.50 τ τ 1 − τm Tm 1 − m m + sin T T τm Tm m m , β = 1 . Representative results: α α τm Tm τm Tm 0.974 0.987 0.9943 0.60 0.70 0.80 τ τ 1 − m m , T T m m τm Tm Tm , Td = Tm tan (0.75φ) , 1 − τm Tm 1 − τm Tm tan (0.75φ ) τm Tm φ = tan −1 (Tm τm − 1) − (τm Tm ) − (τm Tm )2 . 0.9978 0.90 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc 19 Prashanti and Chidambaram (2000). Model: Method 1 Kc Ti Td Comment Ti x1Ti N=∞ 2 α = 0.4835 + 0.8482 3 τ τ τm − 0.5133 m + 0.071 m , Tm T m Tm 3 2 τ τ τ β = −2.206 + 3.461 m − 0.984 m − 0.089 m . T T Tm m m α β τm Tm 0.458 0.563 0.664 0.697 -1.916 -1.878 -1.525 -1.307 0.05 0.10 0.20 0.30 Representative results: α α β τm Tm 0.737 0.786 0.836 0.849 -0.948 -0.729 -0.530 -0.297 0.40 0.50 0.60 0.70 β 0.861 -0.093 0.887 0.038 τm Tm 0.80 0.90 2 Alternatively, α = 0.2243 + 1.289 τ τm − 0.6267 m , β = 1 , Tm Tm τm ≤ 0.90 . Representative results: Tm α α α τm Tm τm Tm 0.05 ≤ Hägglund and Åström (2002). Model: Method 1; M s = 1.4; 0.3 < α < 0.5 . 19 Kc = α τm Tm 0.395 0.336 0.450 0.05 0.10 0.20 0.28 K m Tm 0.544 0.30 0.605 0.40 0.678 0.50 8τ m 0.752 0.772 0.791 0.60 0.70 0.80 0.831 0.2 K m Tm 11τ m 10τ m 0.13 K m Tm 7τ m 0.90 τm = 0.02 Tm 7.8τ m 0.28 K m Tm τm Tm τm = 0.05 Tm 0 0.28 K m Tm 19τ m τm = 0.10 Tm 0.28 K m Tm 135τ m τ m Tm = 0.15 2 τ τ 1 T τ 2.121 − 1.906 m + 0.945 m , Ti = m 0.176 + 0.36 m ; Km Tm x1 Tm Tm x1 = 0.179 − 0.324(τm Tm ) + 0.161(τm Tm ) , τm Tm < 0.6 ; 2 x1 = 0.04 , 0.6 < τm Tm ≤ 0.8 ; x 1 = 0.12 − 0.1(τ m Tm ) , 0.8 < τ m Tm ≤ 1.0 . 471 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 472 Rule Kc Ti Td Comment Td N = ∞; β = 0; Model: Method 1 Td N = ∞; β = 0; Model: Method 1 0 τm < Tm Wang and Cai (2002). 20 Kc Ti Xu and Shao (2003b). 21 Kc Ti Kc Ti Robust Zhang and Xu (2002). Model: Method 1 20 Kc = Td = 21 Kc = α= 22 For τm Tm ≤ 0.5 , λ = 2 − 5τ m ; α = 1 . 0.5236Tm − 0.4764 Tm τm 0.5236Tm 0.4764 Tm , − , Ti = K m τm Km τm 0.5236 Tm τm − 1 ( 0.2618τm Tm τm 0.5236Tm − 0.4764 Tm τm , α= ( ) 1.9099 τm Tm 1 + 0.9099 τm Tm ) ( A m = 3 , φ m = 60 0 ). τ (T τ ) + Tm τm 0.5 Tm τm 0.5 Tm Tm , Td = , + , Ti = m m m τm K m τm Tm τm − 1 (Tm τm ) + Tm τm 2 Tm τ m Tm + Tm τ m Kc = 22 ( A m > 2 , φ m > 60 0 ). λ2 + 2λTm + Tm τm K m (λ + τm ) 2 , Ti = λ2 + 2λTm + Tm τm . Tm − τm ( ) b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 473 4.10.7 Two degree of freedom controller 2 T s 1 d 1 + b f 1s (F(s)R (s) − Y(s) ) − K Y(s) , + U(s) = K c 1 + 0 T 1 + a f 1s Ti s 1+ d s N 1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4 F(s) = 1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5s 4 R(s) F(s) + − T s 1 d 1 + b f 1s K c 1 + + Ti s Td 1 + a f 1s 1+ s N + U(s) − K0 Table 97: PID controller tuning rules – unstable FOLPD model G m (s) = Rule Kc Ti Y(s) K m e − sτ m Tms − 1 K m e − sτ m Tm s − 1 Comment Td Direct synthesis: time domain criteria Jacob and Chidambaram (1996). Model: Method 1 Tm − Tm τm K m (Tm + τm ) Tm − Tm τm 0 Tm τm − 1 K0 = 1 Km Tm τm a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , bf 5 = 0 . Arvanitis et al. (2000b). Model: Method 1 1 Kc = 1 Direct synthesis: frequency domain criteria 0 Ti Kc a f 1 = 0 , b f 1 = 0 ; a f 2 = Ti , b f 2 = 0 ; a f 3 = 0 , b f 3 = 0 ; a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . Tm [2φm + π(A m − 1)] ( ) , 2K m τm A m 2 − 1 Ti = 2πTm . 2φm A m + πA m (A m − 1) 2φm A m + πA m (A m − 1) − π π − 3Tm 2 2τm A m 2 − 1 Am2 − 1 ( ) b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 474 Rule Kc Ti Comment Td Robust Tm − Tm τm Tm − τ m λK m Chidambaram (1997). Model: Method 1 Tm τm − 1 + λ > 1.7 ; 0 0.5τm τm <1. Tm a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 , a f 5 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = (Tm τm ) 0.5 2 Lee et al. (2000). Model: Method 1 a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . 3 a f 5 = 0 , bf 4 = 0 , bf 5 = 0 . 4 0.5 2 TCL + a f 2 τm − 0.5τm , 2TCL + τm − a f 2 τm 2 (0.167 τm − 0.5a f 2 ) T 2 + a f 2 τm − 0.5τm 2 2TCL + τm − a f 2 − CL . Ti 2TCL + τm − a f 2 (K 0K m − 1)Ti , T = Tm − K 0K m τm + τm 2 , i K m (λ + τm ) K 0K m − 1 2(λ + τm ) K0 = 4 Kc = Km . 2 T Ti , a f 2 = Tm CL + 1 e τ m Tm − 1 ; − K m ( 2TCL + τm − a f 2 ) Tm − Tm a f 2 − Kc = λ > 1.7 a f 5 = 0 , b f 4 = 0 , b f 5 = 0 , K 0 = (Tm τm ) 2 3 0 Ti Kc a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 , Ti = −Tm + a f 2 − Td = 0 Ti Kc a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 , Sree and Chidambaram (2006), pp. 27–28. Model: Method 1 Kc = 0 ≤ τm Tm < 2 Td a f 1 = 0 , bf 1 = 0 ; N = ∞ ; bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 , Lee and Edgar (2002). Model: Method 1 2 Ti Kc Km . 2Tm − Tm τm − τm 2λK m , Ti = 1 cos Km Tm − Tm τ m Tm τ m − 1 Tm τ τ τ 1 − m sin 1 − m m + T T τm Tm m m + 0.5τm . τ τ 1 − m m . T T m m b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 4.10.8 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) Y(s) Model − F2 (s) Table 98: PID controller tuning rules – unstable FOLPD model G m (s) = Rule Kc 1 Minimum ITSE – Majhi and Atherton (2000). Model: Method 1 1 Kc = ( Kc Ti K m e − sτ m Tms − 1 Comment Td Minimum performance index: servo tuning 0 Ti α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = 2Tm τm K m ; b f 3 = 1 ; b f 4 = 0.5τm Tm ; a f 5 = 0 . 0.8011Tm 1 − 0.9358 τm Tm K m τm ) , T = T 0.1227 + 1.4550 τ − 1.2711 τ . 2 i m m Tm m 2 Tm 475 b720_Chapter-04 04-Apr-2009 Handbook of PI and PID Controller Tuning Rules 476 Rule Minimum ITSE – Majhi and Atherton (2000). Model: Method 3; Ap K= . K mh Kc Ti Td 2 Kc Ti 0 3 Kc Ti 0 Comment 0< bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; bf 3 = 1 ; a f 5 = 0 . m m m Unit impulse noise rejection response. x1 bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; bf 3 = 1 ; a f 5 = 0 . Coefficient values (deduced from graphs): x2 x3 x4 x1 x2 x3 x4 1.8 4.2 6.9 10.0 2.4 5.2 8.4 11.6 0.17 0.22 0.25 0.26 0.07 0.11 0.14 0.16 0.89 0.91 0.88 0.89 1.10 1.14 1.14 1.13 ( Kc = ) 1.08 0.98 0.95 0.94 0.82 0.86 0.93 1.00 13.5 16.8 21.0 24.0 15.0 18.6 23.1 27.2 0.25 0.27 0.26 0.27 0.19 0.20 0.20 0.20 -0.06 -0.05 -0.04 -0.03 -0.09 -0.09 -0.07 -0.06 2 , ln (1 + K ) 0.1227 + 1.4550 ln (1 + K ) − 1.2711[ln (1 + K )]2 ln (1 + K ) Tm , b f 4 = . 4 tanh −1 K 2 tanh −1 K 1 1 0.8011(K + 0.9946 ) 0.7497 K + 0.9946 − , K0 = K K m K + 0.0682 K + 0.0682 Ti = 4 -0.31 -0.16 -0.11 -0.08 -0.22 -0.16 -0.12 -0.10 0.8011 1 − 0.9358 ln (1 + K ) 1 , K0 = K m ln (1 + K ) Km Ti = 3 τm < 0.5 Tm α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; Unit step setpoint response. Kc = 2(K + 0.9946 ) , K + 0.0682 0.0237(K + 34.5338) 0.0145K 3 + 0.5773K 2 + 2.6554K + 0.3488 Tm , b f 4 = . K + 4.1530 K 3 + 5.2158K 2 + 4.4816K + 0.2817 Equations developed from information provided by Atherton and Majhi (1998). 3 (2Tm − τm )2 x 4 Tm − 2(Tm − τm ) 1 (2Tm − τm ) , K 1 x3 . b f 4 = 0. 5 = − 0 2 3 2 K m 2Tm τm 1 − (2Tm − τm ) x 3 2Tm τm ( τm <1 Tm α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; Direct synthesis: time domain criteria (2Tm − τm ) x1 τmTm 4 x2 0 2Tm − τm 2K τ T 2 Atherton and Majhi (1998). Model: Method 1 τm < 0.693 Tm 0.693 < 3 2 FA ( ) ( )) b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Td 477 Comment Atherton and Boz Parameters as specified by Atherton and Majhi (1998), with the (1998), Majhi following coefficient values (deduced from graphs). Model: Method 1. and Atherton τm Tm < 0.8 . (1999). x1 x2 x3 x4 x1 x2 x3 x4 Minimum ISE – servo – Atherton and Boz (1998). 1.2 0.58 -0.81 2.36 8.5 1.02 -0.16 2.18 2.6 0.91 -0.55 2.49 10.8 1.03 -0.14 2.16 4.5 0.89 -0.30 2.18 13.3 1.02 -0.12 2.13 6.4 0.98 -0.22 2.23 16.0 1.00 -0.09 2.10 Minimum ISTE – 1.8 0.17 -0.31 1.08 13.0 0.28 -0.07 0.94 servo – Atherton 4.2 0.22 -0.16 0.96 16.2 0.31 -0.06 0.99 and Boz (1998). 6.9 0.25 -0.11 0.95 20.3 0.29 -0.04 0.93 10.0 0.26 -0.08 0.92 24.0 0.30 -0.04 0.95 Minimum ITSE – 1.8 0.17 -0.29 1.05 13.0 0.28 -0.06 0.94 servo – Majhi 4.2 0.22 -0.16 0.96 16.8 0.27 -0.05 0.90 and Atherton 6.9 0.25 -0.11 0.95 20.3 0.29 -0.04 0.97 (1999). 10.0 0.26 -0.08 0.92 Minimum ISTSE 2.3 0.08 -0.16 0.71 16.5 0.14 -0.04 0.64 – servo – 5.2 0.11 -0.10 0.65 20.4 0.15 -0.03 0.65 Atherton and Boz 8.4 0.14 -0.07 0.68 25.2 0.15 -0.02 0.64 (1998). 12.4 0.13 -0.04 0.63 31.2 0.14 -0.02 0.61 Minimum ISTSE 2.3 0.08 -0.16 0.69 16.5 0.14 -0.03 0.64 – servo – Majhi 5.2 0.11 -0.10 0.65 20.4 0.15 -0.03 0.67 and Atherton 8.7 0.12 -0.06 0.62 25.2 0.15 -0.02 0.65 (1999). 12.4 0.13 -0.04 0.64 Sree and 0 10Tm K m TS 0.4ξ 2TS Chidambaram α = 0 ; χ = 0 ; δ = 0 ; bf 1 = τm , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; (2006), pp. 24–26. b f 2 = τm , a f 3 = 0 , a f 4 = 0 ; K 0 = 1 K m ; b f 3 = 1 ; b f 4 = τm ; Model: Method 1 a =0. f5 Direct synthesis: frequency domain criteria 0 Ti Arvanitis et al. 5 Kc (2000b). Model: Method 1 α = 0 ; χ = 0 ; δ = 0 ; b f 1 = 0.5τm , a f 1 = Ti , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0.5τm , a f 3 = 0 , a f 4 = 0 ; K 0 = 0 . 5 Kc = Tm [2φm + π(A m − 1)] ( ) , 2K m τm A m 2 − 1 Ti = 2πTm . 2φm A m + πA m (A m − 1) 2φm A m + πA m (A m − 1) − π π − 3Tm 2 2τ m A m 2 − 1 Am2 − 1 ( ) b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 478 Rule Padhy and Majhi (2006a). Model: Method 6 6 Kc Ti Td Kc x1 1 + (x1x 2 τm ) 0 Comment α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = (2Tm τm ) 0.5 K m ; bf 3 = 1 ; b f 4 = 0. 5 τ m ; a f 5 = 0 . Robust 7 Park et al. (1998). Model: Method 1 Ti Kc α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 , a f 4 = 0 ; K 0 = (Tm τm ) 0.5 6 Kc = Td K m ; bf 3 = 1 ; bf 4 = 0 ; a f 5 = 0 . ) [T − 0.7071 T τ ] , x = 1.4142 T τ − 1 , 2φm + π(A m − 1) ( 2K m τm A m 2 − 1 m m m 1 Tm − 0.7071 Tm τm m m 2φ + π(A m − 1) 2A m 2φm + π(A m − 1) x 2 = 0.7854A m m 1 − . 2 π 2 A m 2 − 1 2 A m − 1 ( 7 ) ( ) Apply minimum ITAE – regulator tuning rule or minimum ITAE – servo tuning rule defined by Sung et al. (1996) for SOSPD model, ideal PID controller (Table 21). For these formulae, K m , Tm1 and ξ m are replaced by the following parameters from the unstable FOLPD model: Km Tm τm − 1 0.71Tm [ 0.71 Tm 0.25 τm (Unstable FOLPD) → K m (SOSPD) 0.75 (Tm τm )0.5 − 1 0.5 − τm 0.5 ]T 0.25 m (Tm τm )0.5 − 1 (Unstable FOLPD) → Tm1 (SOSPD) τm −0.75 (Unstable FOLPD) → ξ m (SOSPD) b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Td 479 Comment 8 Ti Td Kc Lee and Edgar α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; (2002). Model: Method 1 ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 , a f 4 = 0 ; b f 3 = 1 ; b f 4 = 0 ; af 5 = 0 . 9 10 Kc Ti x1 Kc Ti x1 K 0 = 0.25K u α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b f 1 = 0 , a f 1 = Td , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = Td , a f 4 = 0 ; b f 3 = 1 ; b f 4 = 0 ; af 5 = 0 . 8 9 Kc = 2 2 K 0 K m − 1 Tm − K 0 K m τm τm T − K 0 K m τm τm , + + , Ti = m K m (λ + τm ) K 0 K m − 1 2(λ + τm ) K 0K m − 1 2(λ + τm ) Td = 4 2 3 (Tm − K 0K m τm )τm 2 , K 0 K m − 1 K 0 K m τm τm τm − + + K m (λ + τm ) 2(K 0 K m − 1) 6(λ + τm ) 4(λ + τm )2 2(K 0 K m − 1)(λ + τm ) K0 = 1 cos Km Kc = (K 0K m − 1)Ti , T = Tm − K 0K m τm + τm 2 + x , i 1 K 0K m − 1 2(λ + τm ) K m (λ + τm ) Tm τ τ τ 1 − m sin 1 − m m + T T T τ m m m m τ τ 1 − m m . T T m m K 0K m τm τm 2 (Tm − K 0 K m τm ) x1 = τ m − + + 2 2(K 0 K m − 1)(λ + τm ) 2(K 0 K m − 1) 6(λ + τm ) 4(λ + τm ) K0 = 10 Kc = 1 cos Km Tm τ τ τ 1 − m sin 1 − m m + T T T τ m m m m τ τ 1 − m m . T T m m (1 + 0.25K u K m )Ti , T = Tm − 0.25K u K m τm + K m (λ + τ m ) i 1 + 0.25K u K m 2 τm + x1 , 2(λ + τm ) 2 0.25K u K m (Tm − 0.25K u K m τ m )τ m τm τm 2 − + + x1 = τ m 2 2(1 + 0.25K u K m )(λ + τ m ) 2(1 + 0.25K u K m 6(λ + τ m ) 4(λ + τ m ) 0.5 . 0.5 ; b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 480 4.11 Unstable FOLPD Model with a Zero K (1 + sTm 3 )e −sτ m K (1 − sTm 4 )e −sτ m G m (s) = m or G m (s) = m Tm1s − 1 Tm1s − 1 1 4.11.1 Ideal PI controller G c (s) = K c 1 + Ti s E(s) R(s) + − 1 K c 1 + Ti s U(s) Model Y(s) Table 99: PI controller tuning rules – unstable FOLPD model with a zero K (1 − sTm 4 )e −sτ m G m (s ) = m Tm1s − 1 Rule Sree and Chidambaram (2003c). 1 Kc Ti Comment Direct synthesis: time domain criteria x T 1 1 m1 Model: Method 1 Ti K mTm 4 x1 = the negative of the value of the ‘inverse jump’ of the closed loop system; x1Tm1 [Tm 4 (1 − x 2 ) − 0.5τm (1 + x 2 )] Tm 4 T . x1 ≥ ; Ti = m 4 x1Tm1 Tm1 (1 − x 2 ) + x 2 Tm 4 x2 > x1Tm1 (Sree and Chidambaram, 2003c); x1Tm1 − Tm 4 Tm 4 + 0.5τm x1Tm1 < x2 < (Sree and Chidambaram, 2006, p. 106). Tm 4 − 0.5τm x1Tm1 − Tm 4 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 481 4.11.2 Ideal controller in series with a first order lag 1 1 G c (s) = K c 1 + + Td s Tf s + 1 Ti s E(s) R(s) + U(s) 1 1 K c 1 + + Td s Ti s Tf s + 1 − Y(s) Model Table 100: PID controller tuning rules – unstable FOLPD model with a zero K (1 + sTm3 )e −sτ m K (1 − sTm 4 )e −sτ m G m (s ) = m or G m (s) = m Tm1s − 1 Tm1s − 1 Rule Kc Ti Comment Td Direct synthesis: time domain criteria Sree and Chidambaram (2003b). Sree and Chidambaram (2006). 1 1 2 Kc Ti 0 Model: Method 1 Kc Ti 0 Model: Method 1; pp. 128–130. (1 − sTm 4 )(1 + Tis )e−sτ ; Desired closed loop transfer function = ( m )( (1) ( 2) 1 + TCL 2s 1 + TCL 2s ) Ti Tm 4 τm Ti Kc = − , Tf = , (1) ( 2) (1) ( 2) K m TCL + T + T + τ − T T T + T 2 CL 2 m4 m i m1 CL 2 CL 2 + Tm 4 + τ m − Ti ( ( [ ) ( ) ] ) , τ − τ < 0.62 . (1) ( 2) T T (1) T ( 2 ) − T τ + TCL 2 + TCL 2 + Tm 4 + τ m Tm1 Ti = m1 CL 2 CL 2 2 m 4 m Tm1 + Tm 4 τm − τm + Tm 4 Tm1 2 ( ) m Tm 4 (1 + sTm3 )(1 + Tis )e−sτ . Desired closed loop transfer function = Kc = − m Tm1 ( m )( (1) ( 2) 1 + TCL 2s 1 + TCL 2s ) Ti Tm3τm Ti , Tf = − , (1) ( 2) (1) ( 2) K m TCL Tm1 TCL 2 + TCL 2 − Tm 3 + τ m − Ti 2 + TCL 2 − Tm 3 + τ m − Ti ( ( [ ) ( ] ). (1) ( 2) T T (1) T ( 2) + T τ + TCL 2 + TCL 2 − Tm 3 + τ m Tm1 Ti = m1 CL 2 CL 2 2 m3 m Tm1 − Tm3τm − τm − Tm 3 Tm1 ( ) ) b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 482 Rule Sree and Chidambaram (2006). Sree and Chidambaram (2006). Kc Ti Td 3 Kc Ti Td 4 Kc Ti 0 Kc Ti Kc Ti Comment Model: Method 1; pp. 128–130. Model: Method 1; pp. 137–138, 143. Robust Sree and Chidambaram (2006). 5 6 Tf = Tm3 Td Model: Method 1; λ not specified; pp. 163–165. 3 (1 + sTm3 )(1 + Tis )e−sτ . Desired closed loop transfer function = ( )( Ti Kc = − ; (1) ( 2) K m TCL + T − Tm 3 + 1.5τm − Ti 2 CL 2 ( Ti = m (1) ( 2) 1 + TCL 2s 1 + TCL 2s ) ( (1) ( 2) (1) ( 2) 0.5τmTCL 2TCL 2 + 0.5τ m Tm1 TCL 2 + TCL 2 + Tm 3 (Tm − 0.5τm )(Tm3 + Tm1 ) ) ) + T T T + x + 0.5τ , (1) ( 2) CL 2 CL 2 m1 1 m 2 Td = 4 0.5τm (Ti − 0.5τm ) Ti ( ]) . [ (1) ( 2) 0.5τ m TCL 2 TCL 2 + Tm 3 Ti − 0.5τ m , Tf = − (1) ( 2) Tm1 TCL 2 + TCL 2 − Tm3 + 1.5τ m − Ti ( Desired closed loop transfer function = (1 − sTm 4 )(1 + Tis )e (1 + TCL3s )3 ( − sτ m ) ; 3 Kc = − Ti = Tm 4 τm Ti − TCL 3 Ti , , Tf = K m (3TCL3 + Tm 4 + τm − Ti ) Tm1 (3TCL3 + Tm 4 + τm − Ti ) ( ) TCL33 + Tm1 3TCL 32 − τm Tm 4 + Tm12 (3TCL3 + Tm 4 + τm ) Tm12 + Tm 4 τm − (τm + Tm 4 )Tm1 , τm τ − m < 0.62 . Tm1 Tm 4 5 Kc = − λ2 + 2λTm1 + 0.5τm Tm1 Ti 0.5(Ti − 0.5τm )τm , Td = , Ti = 0.5τm + K m (2λ + τm − Ti ) Tm1 − 0.5τm Ti 6 Kc = − Ti 0.5τm Tm 4 (Ti − 0.5τm ) − λ3 , , Tf = Tm1 (3λ + Tm 4 + τm − Ti ) K m (3λ + Tm 4 + τ m − Ti ) Ti = 0.5τm + ( ) λ3 + (3λ + Tm 4 + 0.5τm )Tm12 + 3λ2 − 0.5Tm 4 τm Tm1 0.5τmTm 4 + Tm12 − Tm1 (Tm 4 + 0.5τm ) ) (1) ( 2) x 1 = Tm1 TCL 2 + TCL 2 − Tm 3 + τ m ; . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 483 4.11.3 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N R(s) E(s) + − U(s) 2 T s 1 d b f 0 + b f 1s + b f 2 s K c 1 + + Ti s T 1 + a f 1s + a f 2 s 2 1+ d s N K m (1 − sTm 4 )e − sτ Tm1s − 1 m Y(s) Table 101: PID controller tuning rules – unstable FOLPD model with a positive zero K (1 − sTm 4 )e −sτ m G m (s) = m Tm1s − 1 Rule Kc Ti Comment Td Direct synthesis: time domain criteria Sree and Chidambaram (2006). 1 1 Ti Kc (1 − sTm 4 )(1 + Tis )e−sτ . (TCL3s + 1)3 0.5τm (Ti − 0.5τm ) , T = , m Desired closed loop transfer function = Kc = − Ti K m (3TCL3 + Tm 4 + 1.5τm − Ti ) 3 Ti = Model: Method 1; pp. 143–145. Td d ( Ti 2 2 0.5τmTCL3 + Tm1 3TCL3 + 1.5τm TCL 3 − 0.5τm Tm 4 (Tm1 − 0.5τm )(Tm 4 + Tm1 ) ) + x + 0.5τ , 1 m x1 = Tm1TCL 3 (TCL3 + 1.5τm ) + Tm1 (3TCL3 + Tm 4 + τm ) ; 2 3 3 af 2 = − 0.5τm TCL3 , bf 0 = 1 , bf 1 = 0 , bf 2 = 0 , N = ∞ , ( Tm1 3TCL 3 + Tm 4 + 1.5τm − Ti ) 3TCL3 + 1.5τm Tm1 + Tm 4 (Ti − 0.5τm ) + 0.5τm (Ti − 0.5τm − Tm 4 ) . 3TCL3 + Tm 4 + 1.5τm − Ti 2 a f 1 = Tm1 + b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 484 4.11.4 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s α + β d K c T 1+ d s N R(s) E(s) + − T s 1 d K c 1 + + Ti s Td 1+ s N − Y(s) U(s) Model + Table 102: PID controller tuning rules – unstable FOLPD model with a zero G m (s) = Rule K m (1 + sTm3 )e −sτ m K (1 − sTm 4 )e −sτ m or m Tm1s − 1 Tm1s − 1 Kc Ti Comment Td Direct synthesis: time domain criteria Sree and Chidambaram (2003b). 1 1 Desired closed loop transfer function = Kc = − Ti = 0 Ti Kc (1 − sTm 4 )(1 + Tis )e−sτ ; τm − τm < 0.62 . ( ( [ (1) ( 2) (1) Tm1 TCL 2TCL 2 − Tm 4 τ m + TCL 2 Tm12 + Tm 4 τm − m )( (1) ( 2) 1 + TCL 2s 1 + TCL 2s ) , α = 1 − 0.5 + T + T + τ ]T ) . ( 2) CL 2 ) Tm1 Tm 4 (1) ( 2) Tm 4 + TCL 2 + TCL 2 Ti (1) ( 2) K m TCL 2 + TCL 2 + Tm 4 + τm − Ti ( Model: Method 1 m4 (τm + Tm 4 )Tm1 m m1 Ti , b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Sree and Chidambaram (2003c). 2 Sree and Chidambaram (2006). Sree and Chidambaram (2006). 2 Kc Ti Td Comment x1Tm1 K mTm 4 Ti 0 Model: Method 1; x1 ≥ Tm 4 Tm1 . 3 Kc Ti 0 4 Kc Ti 0 Model: Method 1; p. 130. Model: Method 1; p. 138. x1 = the negative of the value of the ‘inverse jump’ of the closed loop system; (x1Tm1 Tm 4 )[Tm 4 (1 − x 2 ) − 0.5τm (1 + x 2 )] , x > x1Tm1 ; 2 (x1Tm1 Tm 4 )(1 − x 2 ) + x 2 x1Tm1 − Tm 4 (1 − sTm 4 )(1 + Tis )e−sτ (Sree and Chidambaram, Desired closed loop transfer function = Ti = ( m )( (1) ( 2) 1 + TCL 2s 1 + TCL 2s ) (1) ( 2) Tm 4 + TCL 2 + TCL 2 2003b); α = 1 − 0.5 3 . Ti (1 + sTm3 )(1 + Tis )e−sτ . Desired closed loop transfer function = Kc = − ( Ti , α = 1− (1) ( 2) K m TCL T + 2 CL 2 − Tm 3 + τ m − Ti ( ( [ m )( (1) ( 2) 1 + TCL 2s 1 + TCL 2s ) ) (1) ( 2) TCL 2 TCL 2 + Tm 3 Ti ] ). (1) ( 2) T T (1) T ( 2) + T τ + TCL 2 + TCL 2 − Tm 3 + τ m Tm1 Ti = m1 CL 2 CL 2 2 m3 m Tm1 − Tm3τm − τm − Tm 3 Tm1 4 ( Desired closed loop transfer function = Kc = − Ti = ) (1 − sTm 4 )(1 + Tis )e−sτ . (1 + TCL3s )3 m Ti T , α = 1 − CL3 (pp. 138, 143), K m (3TCL3 + Tm 4 + τm − Ti ) Ti ( ) TCL33 + Tm1 3TCL 32 − τm Tm 4 + Tm12 (3TCL3 + Tm 4 + τm ) Tm12 + Tm 4 τm − (τm + Tm 4 )Tm1 . 485 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 486 4.12 Unstable SOSPD Model (one unstable pole) G m (s ) = K m e − sτm (Tm1s − 1)(1 + sTm 2 ) 1 4.12.1 Ideal PI controller G c (s) = K c 1 + Ti s E(s) R(s) + 1 K c 1 + Ti s − U(s) K m e −sτ m (Tm1s − 1)(1 + Tm 2 s ) Y(s) Table 103: PI controller tuning rules – unstable SOSPD model (one unstable pole) K m e − sτ m G m (s) = (Tm1s − 1)(1 + sTm 2 ) Rule Kc Ti Comment Minimum performance index: regulator tuning Minimum ITAE – 4Tm1 (τ m + Tm 2 ) Coefficients of K c , Ti 1 Poulin and Pomerleau Kc x 1 Tm1 − 4(τ m + Tm 2 ) deduced from graph. (1996). τ m Tm x1 x2 τ m Tm x1 x2 Model: Method 1 0.05 0.9479 2.3546 0.30 1.6163 2.6612 0.10 1.0799 2.4111 0.35 1.7650 2.7368 0.15 1.2013 2.4646 0.40 1.9139 2.8161 Process output step 0.20 1.3485 2.5318 0.45 2.0658 2.9004 load disturbance. 0.25 1.4905 2.5992 0.50 2.2080 2.9826 0.05 1.1075 2.4230 0.25 1.6943 2.7007 Process input step 0.10 1.2013 2.4646 0.35 1.8161 2.7637 load disturbance. 0.15 1.3132 2.5154 0.40 1.9658 2.8445 0.20 1.4384 2.5742 0.45 2.1022 2.9210 0.25 1.5698 2.6381 0.50 2.2379 3.0003 x 2Tm1 1 + 1 Kc = x1Tm 2 2 4(τm + Tm 2 )2 K m (x1Tm1 − 4[τm + Tm 2 ]) . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Poulin et al. (1996). Model: Method 1 Kc 487 Comment Ti Direct synthesis: frequency domain criteria Tm 2 + τ m < 0.16Tm1 ; 3.53 Tm1 Km 250 ≤ φ ≤ 900 . m Ultimate cycle McMillan (1984). Model: Method 1 2 Kc Tuning rules developed from K u , Tu . Ti 2 1.477 Tm1Tm 2 1 2 Kc = , 2 0.65 Km τm (Tm1 + Tm 2 )Tm1Tm 2 1 + (Tm1 − Tm 2 )(Tm1 − τm )τm 0.65 (T + T )T T m1 m 2 m1 m 2 Ti = 3.33τm 1 + . (Tm1 − Tm 2 )(Tm1 − τm )τm b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 488 1 + Td s 4.12.2 Ideal PID controller G c (s) = K c 1 + Ti s E(s) R(s) − + U(s) 1 K c 1 + + Td s Ti s K m e − sτ m (Tm1s − 1)(1 + sTm 2 ) Y(s) Table 104: PID controller tuning rules – unstable SOSPD model (one unstable pole) K m e −sτ m G m (s) = (Tm1s − 1)(1 + sTm 2 ) Rule Kc Wang and Jin (2004). 1 Kc 2 Kc Ti Td Comment Direct synthesis: time domain criteria Model: Method 1 Ti Td Robust Rotstein and Lewin (1991). Model: Method 1; Tm 2 = 0.5τm ; Ti Td λ determined graphically – sample values provided: τm Tm1 = 0.2 , 0.5Tm1 ≤ λ ≤ 1.9Tm1 τm Tm1 = 0.4 , 1.3Tm1 ≤ λ ≤ 1.9Tm1 Tm 2 >> τm (Marchetti et al., 2001). τm Tm1 = 0.2 , 0.4Tm1 ≤ λ ≤ 4.3Tm1 τm Tm1 = 0.4 , 1.1Tm1 ≤ λ ≤ 4.3Tm1 τm Tm1 = 0.6 , 2.2Tm1 ≤ λ ≤ 4.3Tm1 1 Kc = K m Ti (Tm1 − τm )(Tm 2 + τm ) (TCL3 + τm )3 K m uncertainty = 50%. K m uncertainty = 30%. , Ti = TCL3 + 3τm TCL3 + 3(Tm1Tm 2 + τm Tm1 − τm Tm 2 )TCL 3 + τm Tm 2 (Tm1 − τm ) , (Tm1 − τm )(Tm 2 + τm ) Td = (Tm 2 + τm − Tm1 )TCL33 + 3Tm1Tm 2TCL3 (TCL3 + τm ) + Tm1Tm 2τm 2 . Ti (Tm1 − τm )(Tm 2 + τm ) 3 2 λ λ Tm1 λ + 2 + Tm 2 λ + 2 Tm 2 T T λ m 1 m1 , T = λ 2 Kc = . i T + 2 + Tm 2 , Td = λ λ2 K m m1 λ + 2 + Tm 2 Tm1 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Lee et al. (2000). Shamsuzzoha et al. (2005), Shamsuzzoha and Lee (2007a). 3 4 Kc Ti Td Comment Kc Ti Td Model: Method 1 Kc Ti Td Model: Method 1; λ values not defined. 0.25Ti Tuning rules developed from K u , Tu . 489 Ultimate cycle McMillan (1984). Model: Method 1 3 Kc = 5 Ti Kc 2 Ti λ2 + x1τm − 0.5τm , , Ti = −Tm1 + Tm 2 + x1 − − K m (2λ + τm − x1 ) 2λ + τm − x1 τm (0.167 τm − 0.5x1 ) 2λ + τm − x1 2 − Tm1x1 + Tm 2 x1 − Tm1Tm 2 − Td = Ti 2 − λ2 + x1τm − 0.5τm ; 2λ + τm − x1 2 λ λ = desired closed loop dynamic parameter, x1 = Tm1 + 1 e τ m Tm1 − 1 . Tm1 T 4 i Kc = − , Ti = Tm 2 − Tm1 + 2 x1 − x 2 , K m (4λ + τm − 2 x1 ) 2 Td = 0.167 τm 3 − x1τm 2 + x12 τm + 4λ3 4λ + τm − 2 x1 − x2 , Ti x1 − 2x1 (Tm1 − Tm 2 ) − Tm1Tm 2 − 4 τm 2 2 6λ2 − x1 + 2x1τm − 0.5τm λ . x1 = Tm1 + 1 e Tm − 1 , x 2 = T 4 λ + τ m − 2 x1 m1 2 1.111 Tm1Tm 2 1 5 Kc = , 2 0.65 K m τm (T + T )T T m1 m 2 m1 m 2 1 + (Tm1 − Tm 2 )(Tm1 − τm )τm 0.65 (T + T )T T m1 m 2 m1 m 2 Ti = 2τm 1 + . (Tm1 − Tm 2 )(Tm1 − τm )τm b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 490 4.12.3 Ideal controller in series with a first order lag 1 1 G c (s) = K c 1 + + Td s Tf s + 1 Ti s E(s) R(s) − + 1 K c 1 + + Td s Ti s U(s) 1 (Tf s + 1) Y(s) K m e − sτ m (Tm1s − 1)(1 + sTm 2 ) Table 105: PID controller tuning rules – unstable SOSPD model (one unstable pole) K m e − sτ m G m (s) = (Tm1s − 1)(1 + sTm 2 ) Rule Kc Ti Kc Ti Td Comment Td τ m < Tm1 Robust Zhang and Xu (2002). Model: Method 1 1 Kc = 1 For τm Tm1 ≤ 0.5 , λ = 2 − 5(τ m + Tm 2 ) . Tm 2Tm1 (Tm1 − τm ) + λ3 + 3λ2Tm1 + 3λTm12 + τmTm12 ( K m λ3 + 3λ2Tm1 + 3λTm1τm + τm 2Tm1 2 Ti = Tm 2 + Td = Tf = ) 2 λ3 + 3λ2Tm1 + 3λTm1 + τmTm1 , Tm1 (Tm1 − τm ) Tm 2 (λ3 + 3λ2Tm1 + 3λTm12 + τm Tm12 ) Tm1Tm 2 (Tm1 − τ m ) + λ3 + 3λ2Tm1 + 3λTm12 + τmTm12 ( , λ3Tm1 (Tm1 − τm ) Tm1 λ + 3λ2Tm1 + 3λTm1τm + τm 2Tm1 3 ). , b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 491 1 1 + Td s 4.12.4 Classical controller G c (s) = K c 1 + Ti s 1 + Td s N R(s) E(s) − + U(s) Y(s) 1 1 + Td s K m e − sτ m K c 1 + T T s i 1+ d s (Tm1s − 1)(1 + sTm 2 ) N Table 106: PID controller tuning rules – unstable SOSPD model (one unstable pole) K m e − sτ m G m (s) = (Tm1s − 1)(1 + sTm 2 ) Rule Kc 1 Minimum ITAE Kc – Poulin and Pomerleau (1996), (1997). τ m + Tm 2 Tm1 Process output step load disturbance. Process input step load disturbance. x 2Tm1 1 + 1 Kc = Ti Minimum performance index: regulator tuning Model: Method 1 Ti Tm 2 0.05 0.10 0.15 0.20 0.25 0.05 0.10 0.15 0.20 0.25 x1Tm 2 Comment Td Coefficient values (deduced from graph): τ m + Tm 2 x1 x2 x1 Tm1 0.9479 1.0799 1.2013 1.3485 1.4905 1.1075 1.2013 1.3132 1.4384 1.5698 2.3546 2.4111 2.4646 2.5318 2.5992 2.4230 2.4646 2.5154 2.5742 2.6381 0.30 0.35 0.40 0.45 0.50 0.30 0.35 0.40 0.45 0.50 1.6163 1.7650 1.9139 2.0658 2.2080 1.6943 1.8161 1.9658 2.1022 2.2379 x2 2.6612 2.7368 2.8161 2.9004 2.9826 2.7007 2.7637 2.8445 2.9210 3.0003 2 4(τm + Tm 2 )2 K m (x1Tm1 − 4[τm + Tm 2 ]) , Ti = 4Tm1 (τm + Tm 2 ) ; x1Tm1 − 4(τm + Tm 2 ) 0≤ τm (Td N ) ≤ 2 ; 0.1Tm 2 ≤ Td ≤ 0.33Tm 2 . N b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 492 Rule Kc Huang and Chen (1997), (1999). Model: Method 4 2 Ti Comment Td Direct synthesis: time domain criteria Ti Tm 2 Kc ‘Good’ servo and regulator response; N = ∞ . Direct synthesis: frequency domain criteria Chidambaram and Kalyan (1996). Ho and Xu (1998). Model: Method 1; N =∞. 3 ωpTm1 A m K m Kc = 4 Ti Tm 2 Ti Tm 2 N =∞; Model: Method 1 Representative result: −Tm1 Tm 2 0.7854Tm1 K m τm Paraskevopoulos et al. (2006). Ti x1 2 Kc Am = 2 , φ m = 450 . N =∞; Model: Method 1 2 1 + x1 5 1 + Ti 2 x12 T 0.5 1 + 1.11416 m Km τm Tm 2 Ti 1.13076 τ , Ti = K c K m Tm 2 x1 + τm , 0 < m < 0.8 ; Tm1 K c K m − 1 Tm Tm Recommended x 1 = 0.160367Tm e 3 6.06426 τm Tm 1 . K c and Ti are defined by De Paor and O’Malley (1989), Rotstein and Lewin (1991) or Chidambaram (1995b), for the ideal PI control of an unstable FOLPD model [see Table 91]. 1 4 . Ti = 2 1.57ωp − ωp τm − (1 Tm1 ) 5 τm Tm1 < 0.9 ; Tm 2 Tm1 > 0.3 . Ti (1 − (τm Tm1 )) + Ti − (τm Tm1 ) + x 2 2 x1 = 2Ti 2 (τm Tm1 ) , {T (1 − (τ T )) + T − (τ T )} + 4(τ T )T (1 + T − (τ T )) ; (φ φ ) , T = x (1 + (T τ )[0.632(τ T ) + 0.436 τ T − 0.0153]) 1 − (φ φ ) i x3 = 2 2 x2 = i 3 m m1 m m1 i m m m1 m1 2 m m 0.0029 − 0.0682 τm Tm1 + 1.4941(τm Tm1 ) m1 m1 , i i m m1 max m m max m m (1.003 − (τm Tm1 ))2 max φm > 0.2φmax = −(τm Tm1 ) (Tm1 τm ) − 1 + tan −1 ( (Tm1 τm ) − 1 ) . m , φm b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Paraskevopoulos et al. (2006). Model: Method 1 6 Kc Ti Td Comment Kc Ti Tm 2 N=∞ 6 A m > 0.2(A max m − 1) +1; 493 τm T < 0. 9 ; m 2 > 0. 3 . Tm1 Tm1 K c = K max A m or K min A m , K min = Ti ωmin 1 + ωmin 2 1 + Ti 2ωmin 2 K max = Ti ωmax 0.03[τm Tm1 ] 1.14 − [τm Tm1 ] ωmin = 1 + 0.05 0.973 + Ti − x1 1 − [τm Tm1 ] 1 + ωmax 2 1 + Ti 2ωmax 2 0.006 + x1 , Ti −[τm Tm1 ](1 + Ti ) , 4 2 τ τ x 1.571Tm1 ωmax = 1 + 0.22 m 1 + 0.1 − 0.3 m 1 Tm1 Ti τm Tm1 Ti Ti − 0.9463 − 0.5609 , T + 1 τ T T + 1 τ T ( )( ) ( )( ) m m1 m m1 i i x1 = 0.0029 − 0.0682 τm Tm1 + 1.4941(τm Tm1 ) (1.003 − [τm Tm1 ])2 ; 1+ (τ m Tm1 ) max A m − 1 A m − 1 + Ti = x 2 1 + 0.65 A max − 1 1 A 1 − − m m [ [ ] ] 2 3 τ τ 3(τm Tm1 )4 − 2.3(τm Tm1 )2 τ τ , 0.01− 0.18 + 5 m − 32 m + 75 m − 51 m + Tm1 Tm1 (1 − (τm Tm1 ))3 Tm1 Tm1 1 . 571 T m1 = . with A max m 0.4085[τm Tm1 ] τm 1 + 1 − 0.2864[τm Tm1 ] b720_Chapter-04 04-Apr-2009 Handbook of PI and PID Controller Tuning Rules 494 Rule Kc Ti Paraskevopoulos 2 et al. (2006). Ti x 2 1 + x 2 2 2 1 + Ti x 2 Model: Method 1 7 0 ≤ x1 ≤ 0.55 + (τm + Tm 2 ) 0.31 − Tm1 Comment Td x1 ( τm + Tm 2 ) Tm1 Ti N=∞ > Ti 0.15(τm + Tm 2 ) Tm1 0.0098 ; x1 < 0.15 . 8 7 FA 1 − ([τm + Tm 2 ] Tm1 ) Ti (1 − ([τm + Tm 2 ] Tm1 ) + Td ) + Ti − ([τm + Tm 2 ] Tm1 ) + Td + x 3 2 x2 = x3 = 2Ti 2 (([τm + Tm 2 ] Tm1 ) − Td ) , {T (1 − ([τ + T ] T ) + T ) + T − ([τ + T ] T ) + T } + x , 2 2 i m m2 m1 d i m m2 m1 d 4 x 4 = 4(([τm + Tm 2 ] Tm1 ) − Td )Ti (1 + Ti − ([τm + Tm 2 ] Tm1 ) + Td ) ; 2 τ + Tm 2 τ + Tm 2 φm − Td + 0.436 m − Td − 0.0153 0.632 m max Tm1 Tm1 φm , Ti = x 5 1 + [ ( τm + Tm 2 ] Tm1 ) − Td 1 − φm φmax m ( x5 = 0.0029 − 0.0682 ([τm + Tm 2 ] Tm1 ) − Td + 1.4941(([τm + Tm 2 ] Tm1 ) − Td ) (1.003 − ([τm + Tm 2 ] Tm1 ) + Td )2 ) . 0.02 < τm Tm1 < 0.9 ; Tm 2 Tm1 < 0.3 . φm > 0.2φmax m , τm + Tm 2 Tm1 Tm1 − Td − 1 + tan −1 − Td − 1 . φmax − Td m = − τm + Tm 2 τ m + Tm 2 Tm1 8 x 2 , x 3 and x 4 given in footnote 7; Tm1 T (τ + Tm 2 ) 1 + 0.4 d m − 0.22Td x 6 ; x 5 given in footnote 7; Ti = x 5 1 + Tm1 τm + Tm 2 ( ) τ + Tm 2 τ + Tm 2 φm φmax m x 6 = − 0.0153 − 0.436 m + 0.632 m , Tm1 Tm1 1 − φm φmax m ( τ + Tm 2 Tm1 Tm1 φ max = − m − 1 + tan −1 −1 . m Tm1 τ m + Tm 2 τ m + Tm 2 ) b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc 9 K max A m or Paraskevopoulos et al. (2006). Td Comment τm + Tm 2 Tm1 N =∞; Model: Method 1 Ti x3 Ti K min A m 495 A m > 0.2(A max m − 1) +1; 0.02 < τ m Tm1 < 0.9 . 9 K min = Ti ωmin 1 + ωmin 2 2 1 + Ti ωmin , K max = Ti ωmax 2 0.03(x1 − Td ) 1.14 − x1 + Td ωmin = 1 + 0.05 Ti − x 2 0.973 + 1 x1 + Td − 1 + ωmax 2 with 1 + Ti 2ωmax 2 0.006 + x2 Ti −(x1 − Td )(1 + Ti ) , x1 = τm + Tm 2 , Tm1 2 x 1.571 ωmax = 1 + 0.22(x1 − Td )4 1 + 0.1 − 0.3 x1 − Td 2 Ti x1 − Td Ti Ti − 0.9463 − 0.5609 ; (Ti + 1)(x1 − Td ) (Ti + 1)(x1 − Td ) [ ] ( [ ] ) x 1 − Td +1 max A m − 1 A m − 1 + 0.01 5 x − T − 0.18 Ti = x 2 1 + 0.65 1 d max 1 − Am − 1 Am − 1 3(x − T )4 − 2.3(x1 − Td )2 + 0.01 − 32(x1 − Td ) + 75(x1 − Td )2 − 51(x1 − Td )3 + 0.01 1 d , (1 − x1 + Td )3 [ [ ] [ x2 = 0.0029 − 0.0682 x1 − Td + 1.4941(x1 − Td ) (1.003 − x1 + Td )2 ] ] ; T 0.0098 ; x 3 < 0.15 ; m 2 < 0.3 . 0 ≤ x 3 ≤ 0.55 + x1 0.31 − Tm1 1 − x1 b720_Chapter-04 FA Handbook of PI and PID Controller Tuning Rules 496 Rule Kc Paraskevopoulos et al. (2006) – continued. N =∞. 10 04-Apr-2009 K min = Ti ωmin 10 Ti K max A m or > Tm 2 Tm1 < 0.3 ; 0.15(τm + Tm 2 ) τ 0.02 < m < 0.9. Tm T Ti K min A m Comment Td m1 1 + ωmin 2 2 1 + Ti ωmin , K max = Ti ωmax 2 1 + ωmax 2 2 2 1 + Ti ωmax 0.006 + [0.03(x1 − Td ) (1.14 − x1 + Td )] ωmin = 1 + (0.973 + 0.05 [1 − x1 + Td ])Ti − x 2 , x1 = τm + Tm 2 , with Tm1 x2 T i −(x1 − Td )(1 + Ti ) (0.14 + 1.15x1 )(Td x1 ) x 2 0.0029 − 0.0682 x1 − Td + 1.4941(x1 − Td ) , x 2 = ; 1 + 2 1 + 2(Td x1 ) (1.003 − x1 + Td )2 Ti 3 [ ][ ( ] ) ωmax = 1 + 0.22(x1 − Td )4 1 + 0.1 − 0.3 x1 − Td (x 2 T1 )2 [1.571 (x1 − Td )] Ti Ti − 0.9463 − 0.5609 ( )( ) ( )( ) T 1 x T T 1 x T + − + − 1 d 1 d i i ; ( 5.53 − 0.41x1 )(Td x1 )4 1+ 1 − 0.65(Td x1 )3 1.27 − 0.4(Ti x 2 )2 [ ] [ max A m − 1 A m − 1 Ti = x 3 1 + 0.65 max 1 − Am − 1 A − 1 m [ ][ ] x 1 − Td +1 ] + 0.01 5 x − T − 0.18 1 d [ 2 3(x − T )4 − 2.3(x1 − Td )2 Td 1 + x 4 + 0.01 1 d (1 − x1 + Td )3 x1 [ ] ] + 0.01 − 32(x1 − Td ) + 75(x1 − Td )2 − 51(x1 − Td )3 , T 0.367 + 1.78x 1 x 3 = x 2 1 + d , x1 1 − (Td x1 )2 3 (A m − 1)3 , 2x13 − 3.32 x12 + 1.2x1 − 0.27 − 3 A max m −1 1 1.571 = . A max m 3 (x1 − Td )1 + 0.4085[x1 − Td ] 1 + (7.41 − 3.52x1 )(Td x1 ) 3 1 − (0.47 + 0.49x1 )(Td x1 ) 1 − 0.2864[x1 − Td ] T x 4 = d x1 3x1 ( ) b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 497 4.12.5 Two degree of freedom controller 1 βTd s Td s 1 E (s ) − K c α + R (s ) + U (s ) = K c 1 + Td Ti s 1 + Td s 1+ s N N T s β d α + K Td c 1+ s N R(s) E(s) + − T s 1 d K c 1 + + Ti s Td 1+ s N − + U(s) Y(s) K m e − sτ (Tm1 s − 1)(1 + sTm 2 ) m Table 107: PID controller tuning rules – unstable SOSPD model (one unstable pole) K m e − sτ m G m (s ) = (Tm1s − 1)(1 + sTm 2 ) Rule Kc Huang and Lin (1995) – minimum IAE. 1 Ti Td Comment Minimum performance index: regulator tuning α = 0 , β =1; 1 Ti Td N not specified; Kc Model: Method 2 Note: equations continued onto p. 498. 0.05 ≤ τ m Tm1 ≤ 0.4 ; Tm 2 ≤ Tm1 ; Kc = − −1.164 τ τ 1 T − 174.167 − 31.364 m + 0.4642 m − 103.069 m 2 Km Tm1 Tm1 Tm1 2 . 54 1 . 065 1.014 T Tm1 Tm 2 Tm 2 τ m 1 − 83.916 m 2 − − + 66 . 962 59 . 496 T τ T Km Tm1 2 m1 m m1 − Tm 2 τ m Tm 2 τm 2 T 1 − 70.79 m 2 + 23.121e Tm1 + 126.924e Tm1 + 26.944e Tm1 ; Km τm b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 498 Rule Huang and Lin (1995) – continued. 2 Kc Ti Td Comment Kc Ti Td α = 0 ,β =1, N not specified; Model: Method 2 2 2 T τ T τ Ti = Tm1 0.008 + 2.0718 m + 6.431 m + 0.4556 m 2 + 0.7503 m 2 Tm1 Tm1 Tm1 Tm1 3 3 T τ 2 τ T T τ + Tm1 2.4484 m 2 2m − 18.686 m − 2.9978 m 2 − 21.135 m 2 m T 3 Tm1 Tm1 Tm1 m 1 4 τ T 2 T τ 3 τ + Tm1 12.822 m m32 + 39.001 m + 22.848 m 2 4m T T Tm1 m1 m1 4 T 3τ T 2τ 2 T + Tm1 − 4.754 m 2 4 m − 0.527 m 2 4m + 1.64 m 2 ; T T Tm1 m1 m1 2 2 τ T T τ Td = Tm1 1.1766 m − 4.4623 m + 0.5284 m 2 − 1.4281 m 2 Tm1 Tm1 Tm1 Tm1 3 3 T τ τ 2 T τ T + Tm1 4.6 m 2 2m + 11.176 m + 1.0886 m 2 − 5.0229 m 3m 2 T Tm1 Tm1 Tm1 m1 4 τ T 2 T τ 3 τ + Tm1 − 0.0301 − 0.5039 m m32 − 9.8564 m − 7.528 m 2 4m T T Tm1 m1 m1 4 τ T 3 τ 2T 2 T + Tm1 − 2.3542 m m42 + 9.3804 m m4 2 − 0.1457 m 2 . T T Tm1 m1 m1 2 Note: equations continued onto p. 499. 0.05 ≤ τ m Tm1 ≤ 0.25 , Tm1 < Tm 2 ≤ 10Tm1 ; Kc = − 2.1984 τ 1 T τ 1750.08 + 1637.76 m + 1533.91 m − 7.917 m 2 Km Tm1 Tm1 Tm1 − 0.791 3.2927 1.0757 T Tm1 Tm 2 Tm 2 τ m 1 6.187 m 2 − 6 . 451 + 0 . 002452 T τ T Km Tm1 2 m1 m m1 − Tm 2 τ m Tm 2 τm 2 T 1 1.3729 m 2 − 1739.77e Tm1 − 0.000296e Tm1 + 2.311e Tm1 ; Km τm b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 2 2 T τ T τ Ti = Tm1 51.678 − 57.043 m + 1337.29 m + 0.1742 m 2 − 0.1524 m 2 Tm1 Tm1 Tm1 Tm1 3 3 4 τ T τ T τ + Tm1 7.7266 m 2 2m − 6011.57 m + 0.0213 m 2 + 9135.52 m Tm1 Tm1 Tm1 Tm1 2 3 T τ 2 + 0.0645 τ m Tm 2 + 274.851 Tm 2 τ m + Tm1 − 65.283 m 2 m T 3 T 3 T 4 m1 m1 m1 4 T 3τ T 2τ 2 T + Tm1 0.003926 m 2 4 m − 2.0997 m 2 4m − 0.001077 m 2 T T Tm1 m1 m1 Tm 2 τ m τm Tm 2 2 + Tm1 − 49.007e Tm1 + 0.000026e Tm1 + 0.2977e Tm1 ; 2 τm Tm 2 τm + Td = Tm1 − 0.0605 + 4.6998 0 . 0117 − 29.478 Tm1 Tm1 Tm1 2 3 τm Tm 2 Tm 2 τ m + 0.6874 + 140.135 + Tm1 − 0.0129 2 Tm1 Tm1 Tm1 3 2 τ m 2 Tm 2 Tm 2 − 0.1289 τ m Tm 2 + Tm1 0.002455 1 . 4712 + T 3 T 3 Tm1 m1 m1 4 3 Tm 2 τ m 3 τm + 0.007725 τ m Tm 2 0 . 6884 + Tm1 − 238.864 + T 4 T 4 Tm1 m1 m1 4 τ m 2 Tm 2 2 Tm 2 . − 0.000135 + Tm1 − 0.1222 T 4 Tm1 m 1 499 b720_Chapter-04 FA Handbook of PI and PID Controller Tuning Rules 500 Rule Kc Minimum IAE – Huang and Lin (1995). 3 04-Apr-2009 3 Kc Ti Comment Td Minimum performance index: servo tuning α = 0 , β =1; Ti Td N not specified; Model: Method 2 Note: equations continued onto p. 501. 0.05 ≤ τ m Tm1 ≤ 0.4 , Tm 2 ≤ Tm1 ; τ 1 τ Kc = − 10.741 − 13.363 m + 0.099 m Km Tm1 Tm1 T 1 − 708.481 m 2 − T Km m1 −1.344 0.995 + 9.915 + 727.914 Tm 2 τ m Tm1 2 Tm 2 Tm1 T + 84.273 m1 τm 1.031 Tm 2 Tm1 Tm 2 τ m τm Tm 2 2 Tm 2 1 Tm 1 Tm 1 − − 90.959 + 9.034e − 2.386e − 16.304e Tm1 Km τm 2.12 τ T τ Ti = Tm1 − 149.685 − 141.418 m − 88.717 m − 17.29 m 2 Tm1 Tm1 Tm1 0.985 0.286 1.988 T τ Tm 2 T τ + Tm1 20.518 m 2 − 12.82 m 2 2m + 3.611 m Tm1 Tm1 Tm1 Tm1 Tm 2 τ m τm Tm 2 2 Tm 2 Tm1 Tm1 + Tm1 0.000805 + 141.702e − 2.032e + 10.006e Tm1 ; τm 2 τ T τ Td = Tm1 − 0.4144 + 15.805 m − 142.327 m + 0.7287 m 2 Tm1 Tm1 Tm1 2 3 3 T τ T T τ + Tm1 0.1123 m 2 − 18.317 m 2 2m + 486.95 m − 10.542 m 2 Tm1 Tm1 Tm1 Tm1 4 τ 2T τ T 2 τ + Tm1 204.009 m 3m 2 + 47.26 m m32 + 396.349 m T T Tm1 m1 m1 τ 3T τ T 3 τ 2 T 2 + Tm1 − 138.038 m 4m 2 + 52.155 m m42 − 646.848 m m4 2 T T T m1 m1 m1 4 5 τ m 4 Tm 2 τm Tm 2 4731 . 72 425 . 789 + − + Tm1 19.302 T T 5 Tm1 m1 m 1 0.997 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Minimum IAE – Huang and Lin (1995) – continued. 4 Kc Ti Td Comment Kc Ti Td α = 0 , β =1; N not specified; Model: Method 2 τ T 4 τ 3T 2 τ 2 T 3 + Tm1 − 289.476 m m52 − 841.807 m m5 2 + 1313.72 m m5 2 T T T m1 m1 m1 5 6 τ 5 T τ T + Tm1 − 3.7688 m 2 + 6264.79 m − 161.469 m 6m 2 T Tm1 Tm1 m1 τ T 5 τ 4T 2 τ 2 T 4 + Tm1 204.689 m m62 + 25.706 m m6 2 − 791.857 m m6 2 T T T m1 m1 m1 3 6 τ m Tm 2 Tm 2 + Tm1 648.217 − 5.71 . T 2 Tm1 m1 4 Note: equations continued onto p. 502. 0.05 ≤ τ m Tm1 ≤ 0.25 , Tm1 < Tm 2 ≤ 10Tm1 ; Kc = − −0.3055 τ 1 T τ + 10.442 m 2 − 130.2 + 85.914 m + 34.82 m Km Tm1 Tm1 Tm1 − 0.5174 1.0077 0.9879 T Tm1 Tm 2 Tm 2 τ m 1 − 22.547 m 2 − + 14 . 698 52 . 408 T τ T Km Tm1 2 m1 m m1 − Tm 2 τ m Tm 2 τm 2 T 1 − 51.47 m 2 + 53.378e Tm1 − 0.000001e Tm1 + 0.286e Tm1 ; Km τ m 2 τ T τ Ti = Tm1 72.806 − 268.746 m − 4.9221 m 2 + 2468.19 m Tm1 Tm1 Tm1 2 3 3 T τ T T τ + Tm1 0.6724 m 2 + 151.351 m 2 2m − 6914.46 m − 0.0092 m 2 Tm1 Tm1 Tm1 Tm1 4 τ 2T τ T 2 τ + Tm1 − 795.465 m 3m 2 − 14.27 m m32 + 5580.17 m T T Tm1 m1 m1 τ 3T τ T 3 τ 2 T 2 + Tm1 1417.65 m 4m 2 + 0.4057 m m42 + 55.536 m m4 2 T T T m1 m1 m1 501 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 502 Rule Kc Ti Comment Td Direct synthesis: time domain criteria Sree and Chidambaram (2006). 5 Ti Kc Td Model: Method 1; p. 189; β =1; N = ∞ . Tm 2 τ m Tm 2 τm 4 2 T + Tm1 − 0.001119 m 2 − 44.903e Tm1 + 0.000034e Tm1 − 15.694e Tm1 Tm1 19.056 7.3464 τ Tm 2 ; + Tm1 678778 m Tm1 Tm1 1.1798 τm Tm 2 τm − Td = Tm1 175.515 − 86.2 0 . 008207 + 348.727 Tm1 Tm1 Tm1 0.1064 0.0355 0.0827 Tm 2 τm Tm 2 Tm 2 τ m + Tm1 − 55.619 + 0.0418 + 78.959 T Tm1 Tm1 Tm1 2 m1 Tm 2 τ m Tm 2 τm 2 Tm 2 Tm 1 Tm 1 − 187.01e + Tm1 0.005048 + 0.000001e − 0.0149e Tm1 . τm 5 Kc = 1.13076 T 0.5 , Ti = K c K m Tm1 2 x1 + τm ; α = 1 + τm ; 1 + 1.11416 m1 T K K 1 T K c K m Ti Km τ − c m m1 m m1 Recommended x 1 = 0.160367Tm1e 6.06426 τm T m1 ; 0< τm < 0.8 (Huang and Chen, 1999). Tm1 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 4.12.6 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) − Y(s) Model F2 (s) Table 108: PID controller tuning rules – unstable SOSPD model (one unstable pole) K m e − sτ m G m (s) = (Tm1s − 1)(1 + sTm 2 ) Rule Kc Majhi and Atherton (1999), Atherton and Boz (1998). Model: Method 1 Minimum ISE – servo – Atherton and Boz (1998). 1 Ti = x5 = −1 x1 τm + Tm 2 −1 1 Ti x 2x5 K m Comment Td Direct synthesis: time domain criteria 0 Ti α = 0 ; χ = 0 ; δ = 0 ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = [1 − x 5 x 3 ] K m , b f 3 = 1 ; a f 5 = 0 . Coefficient values deduced from graphs: x1 x2 x3 x4 x1 x2 x3 x4 1.2 2.6 4.5 6.4 − Tm1−1 0.58 0.91 0.89 0.98 -0.81 -0.55 -0.30 -0.22 2.36 2.49 2.18 2.23 8.5 10.8 13.3 16.0 1.02 1.03 1.02 1.00 x x − (Tm1 − Tm 2 − τm ) τm − Tm 2 ≤ 0.8 ; , bf 4 = 6 4 , 1 − x 5x 3 Tm1 (Tm1Tm 2 + τmTm1 − τmTm 2 )3 , x = (Tm1Tm 2 + τ m Tm1 − τ m Tm 2 )2 . τm 2Tm12Tm 2 2 -0.16 -0.14 -0.12 -0.09 6 τ m Tm1Tm 2 2.18 2.16 2.13 2.10 503 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 504 Rule Kc Ti Comment Td Parameters as specified by Majhi and Atherton (1999) and Atherton and Boz (1998). x1 x2 x3 x4 x1 x2 x3 x4 Minimum ISTE – servo – Atherton and Boz (1998). 1.8 4.2 6.9 10.0 Minimum ITSE – 1.8 servo – Majhi 4.2 and Atherton 6.9 (1999). 10.0 Minimum ISTSE 2.3 – servo – 5.2 Atherton and Boz 8.4 (1998). 12.4 Minimum ISTSE 2.3 – servo – Majhi 5.2 and Atherton 8.7 (1999). 12.4 0.17 -0.31 1.08 13.0 0.28 -0.07 0.94 0.22 -0.16 0.96 16.2 0.31 -0.06 0.99 0.25 -0.11 0.95 20.3 0.29 -0.04 0.93 0.26 -0.08 0.92 24.0 0.30 -0.04 0.95 0.17 -0.29 1.05 13.0 0.28 -0.06 0.94 0.22 -0.16 0.96 16.8 0.27 -0.05 0.90 0.25 -0.11 0.95 20.3 0.29 -0.04 0.97 0.26 -0.08 0.92 0.08 -0.16 0.71 16.5 0.14 -0.04 0.64 0.11 -0.10 0.65 20.4 0.15 -0.03 0.65 0.14 -0.07 0.68 25.2 0.15 -0.02 0.64 0.13 -0.04 0.63 31.2 0.14 -0.02 0.61 0.08 -0.16 0.69 16.5 0.14 -0.03 0.64 0.11 -0.10 0.65 20.4 0.15 -0.03 0.67 0.12 -0.06 0.62 25.2 0.15 -0.02 0.65 0.13 -0.04 0.64 Direct synthesis: frequency domain criteria Continued on 2 next page. T T K i d Kwak et al. (2000). Model: Method 3 2 c Optimal gain margin: regulator response. 0 ≤ Tm1 τm ≤ 1 − (Tm 2 Tm1 ) . [ + 2.189(T τ ) ξ ] , τ T < 0.9 or K = (1 K )[− 0.365 + 0.260((τ T ) − 1.4 ) + 2.189(T τ ) ξ ] , τ T ≥ 0. 9 ; T = T [2.212(τ T ) − 0.3] , τ T < 0.4 or T = T [− 0.975 + 0.910((τ T ) − 1.845) + x [5.25 − 0.88((τ T ) − 2.8) ] , K c = (1 K m ) − 0.67 + 0.297(Tm1 τm ) 2.001 0.766 m1 m m 2 c m m m m1 m m 0.766 m1 m1 m m1 0.520 i m1 i m1 m m1 m m1 2 m m1 2 1 m m1 τm Tm1 ≥ 0.4 with x1 = 1 − e − (ξ m [0.15 + 0.33(τ m Tm1 )]) ; Td = ( Tm1 1.171 x 2 1.45 + 0.969(Tm1 τm ) )− 1.9 + 1.576(T τ ) 0.530 m1 m , with x 2 = 1 − e − (ξ m [−0.15 + 0.939(Tm1 τ m ) 1.121 ]) . b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kwak et al. (2000) – continued. 3 Kc Ti Td Kc Ti Td 505 Comment α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , a f 1 = 0 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; b f 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; b f 3 = 1 ; a f 5 = 0 .4 Robust 5 Lee et al. (2000). Model: Method 1 Ti Kc 0 ≤ τm Tm1 < 2 Td α = 0 ; β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; bf 1 = 0 , [ ] a f 1 = Tm1 ((λ Tm1 ) + 1) e τ m Tm1 − 1 , a f 2 = 0 ; ε = 0 ; φ = 0 ; ϕ = 0 ; 3 2 bf 2 = 0 , a f 3 = a f 1 , a f 4 = 0 ; K 0 = 0 . Optimal gain margin: servo response. 0 ≤ Tm1 τm ≤ 1 − (Tm 2 Tm1 ) . [ ξ ], ξ ≤ 0.9 or K = (1 K )[− 0.544 + 0.308(τ T ) + 1.408(T τ ) ξ ] , ξ > 0.9 . K c = (1 K m ) − 0.04 + 0.333 + 0.949(Tm1 τm ) 0.983 m m 0.832 c m m m1 m1 Ti = Tm1 [2.055 + 0.072(τm Tm1 )]ξ m , τ m Tm1 ≤ 1 or m m m Ti = Tm1 [1.768 + 0.329(τm Tm1 )]ξ m , τ m Tm1 > 1 ; Td = 4 K0 = [ Tm1 x1 0.55 + 1.683(Tm1 τm ) 1.090 1 G m ( jωu )(1 + jb f 4ωu ) K m , x1 = 1 − e −1.149ξ m (τ m Tm1 ) 1.060 ] . ( ) , b f 4 = x 3 + x 4 (τm Tm1 ) + x 5 (τm Tm1 )2 Tm1 , with 2 3 4 2 3 4 x 3 = −0.0030 + 0.6482x 6 − 2.2841x 6 + 2.6221x 6 − 0.9611x 6 , x 4 = 0.2446 − 1.0410 x 6 − 13.6723x 6 − 16.7622 x 6 + 5.1471x 6 , 2 3 4 x 5 = 0.1685 + 0.8289 x 6 − 9.3630 x 6 + 2.9855x 6 + 7.3803x 6 , x 6 = Tm 2 Tm1 . 5 Kc = Ti , recommended τm ≤ λ ≤ 2τm ; − K m (2λ + τm − a f 1 ) 2 Ti = −Tm1 + Tm 2 + a f 1 − λ2 + a f 1τm − 0.5τm , 2λ + τm − a f 1 τm (0.167 τm − 0.5a f 1 ) 2λ + τm − a f 1 2 − Tm1a f 1 + Tm 2a f 1 − Tm1Tm 2 − Td = Ti Desired closed loop transfer function = e − sτ m (λs + 1)2 2 − λ2 + a f 1τm − 0.5τm ; 2λ + τm − a f 1 (Lee et al., 2006). b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 506 4.13 Unstable SOSPD Model (two unstable poles) G m (s ) = K m e − sτm (Tm1s − 1)(Tm 2s − 1) 1 + Td s 4.13.1 Ideal PID controller G c (s) = K c 1 + Ti s R(s) E(s) − + 1 K c 1 + + Td s Ti s U(s) K m e − sτ (Tm1s − 1)(Tm 2 s − 1) m Y(s) Table 109: PID controller tuning rules – unstable SOSPD model (two unstable poles) K m e − sτ m G m (s) = (Tm1s − 1)(Tm 2 s − 1) Rule Kc Wang and Jin (2004). 1 Kc = 1 Kc Ti K m Ti (Tm1 − τm )(Tm 2 − τm ) (TCL3 + τm )3 Td Comment Direct synthesis: time domain criteria Model: Method 1 Ti Td , Ti = TCL3 + 3τm TCL3 + 3(Tm1Tm 2 + τm Tm1 − τm Tm 2 )TCL 3 + τm Tm 2 (Tm1 − τm ) , (Tm1 − τm )(τm − Tm 2 ) Td = (Tm 2 + τm − Tm1 )TCL33 + 3Tm1Tm 2TCL3 (TCL3 + τm ) + Tm1Tm 2τm 2 . Ti (Tm1 − τm )(Tm 2 − τm ) 3 2 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Kc Ti Kc Ti Td Comment Td Model: Method 1 Robust Lee et al. (2000). 2 Kc = 2 Ti , λ = desired closed loop dynamic parameter; K m (4λ + τm − x1 ) 2 Ti = −Tm1 − Tm 2 + x1 − x 2 , x 2 = 6λ2 − x 3 + x1τm − 0.5τm ; 4 λ + τ m − x1 3 x 3 − (Tm1 + Tm 2 ) x1 + Tm1Tm 2 − Td = 2 4λ3 + x 3τm + 0.167 τm − 0.5x1τm 4λ + τm − x1 − x2 ; Ti x1 , x 3 values determined by solving 1 − (x s + x s + 1)e 3 2 1 (λs + 1) 4 −τms s= 1 1 , Tm1 Tm 2 =0. 507 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 508 4.13.2 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N R(s) + E(s) − U(s) 2 T s 1 d b f 0 + b f 1s + b f 2 s K c 1 + + Model Ti s Td 1 + a f 1 s + a f 2 s 2 s 1+ N Y(s) Table 110: PID controller tuning rules – unstable SOSPD model (two unstable poles) K m e − sτ m G m (s) = (Tm1s − 1)(Tm 2 s − 1) Rule Kc Ti Kc Ti Td Comment Td Model: Method 1 Robust Shamsuzzoha and Lee (2007d). 1 Recommended λ = τm . 4 τm 4 τm λ λ Tm12 + 1 e Tm1 − Tm 2 2 + 1 e Tm 2 + Tm 2 2 − Tm12 Ti Tm1 Tm 2 1 Kc = , Ti = K m (4λ + τm − Ti ) Tm1 − Tm 2 4 τm Tm1 λ + 1 e − 1 − Ti Tm Tm1 , b f 0 = 1 , b f 1 = 0.5τm , b f 2 = 0 , N = ∞ , Ti 2 Tm1 Td = 0.5τm Ti − Ti Td + 2λτ m + 6λ2 a f 2 = 0 , a f 1 = 0.1 + Tm1 + Tm 2 , for large parametric 4 T τ + λ − m i 0.5τm Ti − Ti Td + 2λτ m + 6λ2 + Tm1 + Tm 2 . uncertainties; otherwise, a f 1 = τm + 4λ − Ti b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 509 4.13.3 Two degree of freedom controller 2 T s 1 d 1 + b f 1s (F(s) R (s) − Y(s) ) − K Y(s) , U(s) = K c 1 + + 0 Ti s T 1 + a f 1s 1+ d s N 1 + b f 2 s + b f 3s 2 + b f 4 s 3 + b f 5s 4 F(s) = 1 + a f 2 s + a f 3s 2 + a f 4 s 3 + a f 5 s 4 R(s) F(s) + − T s 1 d 1 + b f 1s K c 1 + + Ti s Td 1 + a f 1 s 1+ s N + U(s) − Kc Ti Kc Ti Y(s) K0 Table 111: PID tuning rules – unstable SOSPD model G m (s) = Rule Model K m e − sτ m (Tm1s − 1)(Tm 2 s − 1) Td Comment Td N=∞ Robust 1 Lee et al. (2000). Model: Method 1 a f 1 = 0 , bf 1 = 0 ; bf 2 = 0 ; bf 3 = 0 ; a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , b f 5 = 0 , K 0 = 0 ; recommended τm ≤ λ ≤ 2τm . 1 Desired closed loop transfer function = e −sτ m (λs + 1)4 (Lee et al., 2006). 2 Kc = 6λ2 − a f 3 + a f 2 τm − 0.5τm Ti , , Ti = −Tm1 − Tm 2 + a f 2 − x1 , x1 = K m (4λ + τm − a f 2 ) 4λ + τ m − a f 2 a f 3 − (Tm1 + Tm 2 )a f 2 + Tm1Tm 2 − Td = a f 2 , a f 3 determined by 1 − 4λ3 + a f 3τm + 0.167 τm 3 − 0.5a f 2 τm 2 4λ + τm − a f 2 − x1 ; Ti (a s + a s + 1)e f3 2 f2 (λs + 1) 4 −τms s= 1 1 , Tm1 Tm 2 =0; 0≤ τm <2. Tm b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 510 Rule 2 Shamsuzzoha et al. (2006b). Model: Method 1 Kc Ti Td Comment Kc Ti Td N=∞ a f 1 = 0 , bf 1 = 0 ; a f 2 = 0 , bf 2 = 0 ; a f 3 = 0 , bf 3 = 0 ; a f 4 = 0 , a f 5 = 0 , bf 4 = 0 , bf 5 = 0 , K 0 = 0 . λ , ξ , a f 1 not specified. 2 Kc = Td = Ti , Ti = x1 − Tm1 − Tm 2 − x 2 , K m (4λξ + τm − x1 ) x 3 − x1 (Tm1 + Tm 2 ) + Tm1Tm 2 − 2 τ 0.167 τm 3 − 0.5x1τm 2 + x 3τm + 4λ3ξ 4λξ + τm − x1 − x2 , Ti 2 τ m m 2 λ2 2λξ Tm1 2λξ Tm1 2 λ 2 + + − + + 1 e + Tm 2 2 − Tm12 1 e T Tm1 m2 2 T 2 T T T m1 m2 m1 m2 , x1 = Tm1 − Tm 2 2 x2 = τ m 2 2λξ Tm1 x 4λ2ξ 2 + 2λ2 − x 3 + x1τm − 0.5τm 2 2 λ , x 3 = Tm1 −1− 1 . 1 e + + T 2 T T 4λξ + τm − x1 m 1 m1 m1 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 511 4.14 Unstable SOSPD Model with a Zero K m (1 + Tm3 s )e −sτ m K m (1 + Tm 3s )e −sτ m G m (s) = or G ( s ) = or m 2 2 − Tm1 s 2 + 2ξ mTm1s + 1 Tm1 s 2 − 2ξ m Tm1s + 1 G m (s) = K m (1 − Tm 4 s )e −sτm 2 Tm1 s 2 − 2ξ m Tm1s + 1 1 4.14.1 Ideal PI controller G c (s) = K c 1 + Ti s R(s) + 1 U(s) K c 1 + Model Ti s E(s) − Y(s) Table 112: PI controller tuning rules – unstable SOSPD model with a zero K m (1 − Tm 4s )e −sτ m G m (s ) = 2 Tm1 s 2 − 2ξ m Tm1s + 1 Rule Kc Sree and Chidambaram (2004). 1 1 Kc Direct synthesis: time domain criteria Model: Method 1 Ti Desired closed loop transfer function = Kc = Ti (1) ( 2) K m TCL + TCL + x1 + Tm 4 + τm − Ti Ti = − x1 = ( ( ) (1 − sTm 4 )(1 + Tis)e−sτ . (1 + sTCL(1) )(1 + sTCL(2) )(1 + sx1 ) m ); (1) ( 2) (1) ( 2 ) Tm12 TCL + TCL + x1 + Tm 4 + τm − TCL TCL x1 ( Comment Ti Tm 4 τm − Tm12 ) ; (1) ( 2) + TCL + Tm 4 + τm (− 2ξ m Tm 4 τm + Tm1 (Tm 4 + τm ) ) − x 2 Tm1 TCL (1) ( 2 ) (1) ( 2) (− 2ξmTm1 + Tm 4 + τm )TCL + TCL + 2ξ m Tm1 ) TCL − Tm12 + (Tm 4 τm − Tm12 )(TCL −1 2 (1) ( 2 ) x 2 = (Tm 4 τm − Tm1 )(TCL TCL − τm Tm 4 ) ; − τm ([2ξ m Tm1 ] + Tm 4 ) < 0.62 . , b720_Chapter-04 04-Apr-2009 Handbook of PI and PID Controller Tuning Rules 512 Rule Kc Ti Sree and Chidambaram (2006). Kc 2 2 K c given in footnote 1; Ti = − x1 = FA ( ) ( Comment Model: Method 1; pp. 147–148. Ti ) (1) ( 2) (1) Tm12 TCL + TCL + x1 + Tm 4 + τm − Tm1TCL Tm 4 Tm 4 τm − Tm12 ( , with )( (1) ( 2) (1) ( 2 ) Tm1 TCL + TCL + Tm 4 + τm (− 2ξ mTm 4 τm + Tm1 (Tm 4 + τm ) ) − Tm 4 τm − Tm12 TCL TCL − τmTm 4 (1) ( 2 ) (1) ( 2) (− 2ξmTm1 + Tm 4 + τm )(TCL TCL − Tm12 ) + (Tm 4 τm − Tm12 )(TCL + TCL − Tm12 ) ) b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models 513 4.14.2 Ideal controller in series with a first order lag 1 1 + Td s G c (s) = K c 1 + Ti s Tf s + 1 R(s) E(s) − + U(s) 1 1 K c 1 + + Td s Ti s Tf s + 1 Y(s) Model Table 113: PID controller tuning rules – unstable SOSPD model with a zero K m (1 − Tm 4s )e −sτ m K m (1 + Tm3s )e−sτ m G m (s ) = or 2 2 2 Tm1 s − 2ξmTm1s + 1 Tm1 s 2 − 2ξ mTm1s + 1 Rule Kc Ti Comment Td Direct synthesis: time domain criteria Sree and Chidambaram (2004). 1 1 Ti Kc Desired closed loop transfer function = Kc = Model: Method 1 Td (1 − sTm 4 )(1 + Tis)e−sτ . (1 + sTCL(1) )(1 + sTCL(2) )(1 + sx1 ) m Ti , K (T + T + x + T + 0.5τ − T ) (1) CL m ( 2) CL 2 m4 m i (1) ( 2 ) 0.5(Ti − 0.5τm )τm 0.5τmTCL TCL x 2 , Td = , Tf = 2 (1) ( 2) Ti Tm1 TCL + TCL + x 2 + Tm 4 + 0.5τm − Ti ( ( (1) ( 2) x − Tm12 TCL + TCL + x 2 + Tm 4 + τm Ti = 1 2 0.5Tm1 Tm 4 τm − Tm14 ) ) with ( ) )(T T + 0.5τ [T + T − T ]T + 2ξ T x ) 2 (1) ( 2 ) (1) ( 2 ) ( 2 ) ( 3) ( 3) (1) (1) ( 2 ) x1 = TCL TCL x 2 + 0.5τmTm1 TCL TCL + TCL TCL + TCL TCL + ξmTm1τmTCL TCL x 2 , x2 = ( 4 2 Tm1 0.5Tm 4 τm − Tm1 (1) ( 2 ) CL CL m (1) CL ( 2) CL m4 ( 2) CL x4 ( 2 (1) ( 2) x 3 = TCL + TCL + Tm 4 + τm , ] )( [ ) + (2ξT − T − 0.5τ ) (T [T T + 0.5τ (T + T )]+ ξ T τ T T − T ) . 2 ( 2 2 m 2 m1 (1) ( 2) (1) ( 2 ) x 4 = −Tm1 0.5Tm 4 τm − Tm1 Tm1 TCL + TCL + 0.5τm + 2ξ m Tm1 − 0.5τmTCL TCL m1 m4 (1) ( 2 ) CL CL ) (1) ( 2 ) Tm1 (− 2ξ m Tm1 + Tm 4 + 0.5τm ) 0.5τm TCL TCL − Tm 4 x 3 , x4 4 + m m4 3 m (1) CL ( 2) CL (1) ( 2 ) m m1 m CL CL 4 m1 b720_Chapter-04 FA Handbook of PI and PID Controller Tuning Rules 514 Rule Sree and Chidambaram (2006). 2 04-Apr-2009 2 Kc Ti Td Comment Kc Ti Td p. 165; Model: Method 1 (1 + sTm3 )(1 + Tis)e−sτ . (1 + sTCL(1) )(1 + sTCL(2) )(1 + sx 2 ) m Desired closed loop transfer function = Kc = Ti , (1) ( 2) K m TCL + TCL + x 2 − Tm3 + 1.5τm − Ti ( ) ( (1) ( 2) x − Tm14 TCL + TCL + x 2 − Tm 3 + τm Ti = 1 2 − 0.5Tm1 Tm3τm − Tm14 ) + 0.5τ , T = 0.5(T − 0.5τ )τ , m (1) ( 2 ) 0.5τm TCL TCL x 2 2 (1) ( 2) Tm1 TCL + TCL + x 2 − Tm3 + 1.5τm − Ti Tf = ( 2 i d m m Ti ); ( ) )(T T + 0.5τ [T + T + T ]T − 2ξ T x ) (1) ( 2 ) (1) ( 2 ) ( 2) (1) (1) ( 2 ) x1 = TCL TCL x 2 + 0.5τ m Tm1 TCL TCL + TCL x 2 + x 2 TCL − ξ m Tm1τ m TCL TCL Tm3 , x2 = ( 4 2 Tm1 − 0.5Tm3τ m − Tm1 (1) ( 2 ) CL CL m (1) CL ( 2) CL m3 ( 2) CL x4 ( (1) ( 2) x 3 = TCL + TCL − Tm 3 + τm , 2 ( 2 )( [ ] 2 (1) ( 2) (1) ( 2 ) x 4 = Tm1 0.5Tm3τm + Tm1 Tm1 TCL + TCL + 0.5τm + 2ξm Tm1 − 0.5τm TCL TCL ( 2 + Tm1 2ξTm1 + Tm3 − 0.5τm ) ) (1) ( 2 ) Tm1 (− 2ξ m Tm1 − Tm3 + 0.5τm ) 0.5τm TCL TCL + Tm3 x 3 , x4 4 + m m3 3 2 ) (T [T T + 0.5τ (T + T )]+ ξ T τ T T − T ) . 2 m1 (1) ( 2 ) CL CL m (1) CL (2) CL (1) ( 2 ) m m1 m CL CL 4 m1 b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Sree and Chidambaram (2006). 3 3 Kc Ti Td Comment Kc Ti Td p. 148; Model: Method 1 Desired closed loop transfer function = Kc = (1 − sTm 4 )(1 + Tis)e−sτ . (1 + sTCL(1) )(1 + sTCL(2) )(1 + sx 2 ) 0.5(Ti − 0.5τm )τm , T = , m Ti (1) ( 2) K m TCL + TCL + x 2 + Tm 4 + 1.5τm − Ti ( (1) ( 2 ) 0.5τm TCL TCL x 2 Tf = 2 (1) ( 2) Tm1 TCL + TCL + x 2 + Tm 4 + 1.5τm − Ti ( Ti = ( (1) ( 2) x1 − Tm14 TCL + TCL + x 2 + Tm 4 + τm 0.5Tm12Tm 4 τm − Tm14 515 ) d Ti ); ) + 0.5τ , with m ( ) )(T T + 0.5τ [T + T − T ]T + 2ξ T x ) 2 (1) ( 2 ) (1) ( 2 ) ( 2) (1) (1) ( 2 ) x1 = TCL TCL x 2 + 0.5τm Tm1 TCL TCL + TCL x 2 + x 2TCL + ξ m Tm1τmTCL TCL Tm 4 , x2 = ( 4 2 Tm1 0.5Tm 4 τm − Tm1 (1) ( 2 ) CL CL m (1) CL ( 2) CL m4 ( 2) CL m m4 3 x4 ( (1) ( 2) x 3 = TCL + TCL + Tm 4 + τm , 2 ( 2 )( [ ] 2 2 (1) ( 2) (1) ( 2 ) x 4 = −Tm1 0.5Tm 4 τm − Tm1 Tm1 TCL + TCL + 0.5τm + 2ξ m Tm1 − 0.5τmTCL TCL ( 2 + Tm1 2ξTm1 − Tm 4 − 0.5τm ) ) (1) ( 2 ) Tm1 (− 2ξ m Tm1 + Tm 4 + 0.5τm ) 0.5τm TCL TCL − Tm 4 x 3 , x4 4 + ) (T [T T + 0.5τ (T + T )]+ ξ T τ T T − T ) . 2 m1 (1) ( 2 ) CL CL m (1) CL ( 2) CL (1) ( 2 ) m m1 m CL CL 4 m1 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 516 4.14.3 Generalised classical controller 2 T s 1 d b f 0 + b f 1s + b f 2 s + G c (s) = K c 1 + 2 T 1 + a f 1s + a f 2 s Ti s 1 + d s N E(s) R(s) + − U(s) 2 T s b + b s + b s 1 d f1 f2 f0 K c 1 + + Model Ti s Td 1 + a f 1 s + a f 2 s 2 s 1 + N Y(s) Table 114: PID controller tuning rules – unstable SOSPD model with a zero K m (1 + Tm 3s )e −sτ m K m (1 − Tm 4s )e −sτ m G m (s ) = or 2 2 2 − Tm1 s + 2ξ m Tm1s + 1 Tm1 s 2 − 2ξ mTm1s + 1 Rule Kc Rao and Chidambaram (2006a), (2006b). 1 1 Ti Kc Comment Td Direct synthesis: time domain criteria x1x 2 − x 3x 4 x 5x 3 − x 6 x 2 Model: Method 1 x x −x x x x −x x 1 5 Desired closed loop transfer function = 6 4 1 2 3 4 (T T s + T s + 1)(1 − sT )e i d 2 i (λs + 1)3 m4 − sτ m , recommended λ = 1.2τm ; b f 0 = 1 , b f 1 = 0.5τ m , b f 2 = 0 , N = ∞ . Kc = 1 Ti 2 , x1 = − τmTm1Tm 4ξ m − Tm1 (Tm 4 + 0.5τm ) , K m 3λ + τm + Tm 4 − Ti ( ) x 2 = 0.5Tm1 6λ2 + 3τm λ − Tm 4 τ m − 2ξm Tm1 (3λ + τm + Tm 4 ) − 0.5τm λ3 , 2 3 x 3 = λ2Tm1 (λ + 1.5τm ) − λ3τm Tm1ξ m − Tm1 (3λ + τm + Tm 4 ) , 2 4 x 4 = Tm1 − 0.5τm Tm 4 , x 5 = −Tm1 (2ξm Tm1 + Tm 4 + 0.5τm ) , 2 2 2 ( 2 ) x 6 = Tm1 0.5τm Tm 4 − Tm1 ; af1 = 0.5τ m Tm1 2 3 λ − Tm 4 Ti Td λ3 0.05τm , Tm 4 = 0 . , Tm 4 ≠ 0 ; a f 1 = 2 3λ + τ m + Tm 4 − Ti Tm1 3λ + τm − Ti b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Td Comment Kc x 5 x 3 − x 6 x1 x 2 x 3 − x 4 x1 x 5 − x 2Ti Ti x1 Model: Method 1; λ not specified. Kc x 3 x 4 − x 6 x1 x 4 x 2 − x1x 5 x 3 − x 2Ti Ti x1 Model: Method 1; λ not specified. Kc Ti 517 Robust Sree and Chidambaram (2006), pp. 168–169. Sree and Chidambaram (2006), pp. 167–168. 2 Kc = 2 3 Ti 2 2 , x1 = Tm1 , x 2 = −Tm1 (0.5τm − 2ξm Tm1 ) , K m (4λ + 0.5τm − Ti ) x 3 = −0.5τm Tm1 , x 4 = −Tm1 , x 5 = λ4 + 6λ2Tm1 + 2ξ m Tm1 (4λ + 0.5τm ) , 2 4 2 3 x 6 = −2ξ m Tm1λ4 + 4Tm1 λ3 − Tm1 (4λ + 0.5τm ) ; b f 0 = 1 , b f 1 = 0.5τm , b f 2 = 0 , N = ∞ , 2 af 2 = − 3 Kc = λ4Tm3 ( 4 ) 4λ + 0.5τm − Ti Tm12 , a f 1 = Tm3 − λ4 (4λ + 0.5τm − Ti )Tm12 . Ti λ5 , bf 2 = 0 , N = ∞ , , af 2 = 2 K m (5λ + Tm 4 + 0.5τm − Ti ) Tm1 (5λ + Tm 4 + 0.5τm − Ti ) x 3 − x 2Ti + Ti (Tm 4 + 0.5τm ) x1 , b f 0 = 1 , b f 1 = 0.5τm , 5λ + Tm 4 + 0.5τm − Ti 10λ2 − 0.5Tm 4 τm − a f 1 = 2ξ mTm1 + ( ) x1 = Tm1 0.5Tm 4 τm − Tm1 , x 2 = Tm1 (Tm 4 + 0.5τm − 2ξ mTm1 ) , 2 2 ( 4 ) x 3 = 2ξm Tm1λ5 − Tm1 10λ2 − 0.5Tm 4 τm + 2ξ mTm1[5λ + Tm 4 + 0.5τm ] + 5Tm1 λ4 , 4 x 4 = Tm1 (Tm 4 + 0.5τm − 2ξ mTm1 ) , 2 2 [ ] x 5 = Tm1 2ξm Tm1 (Tm 4 + 0.5τm ) − 4ξ m Tm1 + Tm1 − 0.5Tm 4 τm , 2 5 x6 = λ 2 ( 3 − 2ξ m Tm1 10λ2 − 0.5Tm 4 τm ) 2 2 2 ( 4 − 4ξm Tm1 5λ + Tm 4 + 0.5τm ) − 10Tm12λ3 + Tm1 (5λ + Tm 4 + 0.5τm ) . 4 b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 518 4.14.4 Two degree of freedom controller 1 U (s) = G c (s)E (s) − F1 (s)R (s) , T s β T s 1 d d E (s) , F1 (s) = K c α + + G c (s) = K c 1 + Td Td Ti s 1+ s s 1+ N N F1 (s) + R(s) E(s) − − G c (s) U(s) Model + Y(s) Table 115: PID controller tuning rules –– unstable SOSPD model with a zero K m (1 − Tm 4 s )e −sτ G m (s) = Rule m 2 2 Tm1 s − 2ξ m Tm1s + 1 Kc Ti Comment Td Direct synthesis: time domain criteria Sree and Chidambaram (2004). 1 1 Desired closed loop transfer function = Kc = Ti (1) ( 2) K m TCL + TCL + x1 + Tm 4 + τm − Ti ( ( ( 2) T 2 T (1) + TCL + x1 + Tm 4 + τm Ti = − m1 CL Tm 4 τm − Tm12 x1 = 0 Ti Kc ( Model: Method 1 (1 − sTm 4 )(1 + Tis)e−sτ . (1 + sTCL(1) )(1 + sTCL(2) )(1 + sx1 ) m ) 2ξ 1 < 0.62 ; , − τm m + Tm1 Tm 4 )− T T x , (1) ( 2 ) CL CL 1 ) (1) ( 2) + TCL + Tm 4 + τm (− 2ξ m Tm 4 τm + Tm1 (Tm 4 + τm ) ) − x 2 Tm1 TCL (1) ( 2 ) (1) ( 2) (− 2ξmTm1 + Tm 4 + τm )TCL + TCL + 2ξ m Tm1 ) TCL − Tm12 + (Tm 4 τm − Tm12 )(TCL ( )( ) (1) ( 2 ) and with x 2 = Tm 4 τm − Tm12 TCL TCL − τm Tm 4 , α = 1 − ( ) (1) ( 2) Tm 4 + 0.5 TCL + TCL . Ti , b720_Chapter-04 04-Apr-2009 FA Chapter 4: Controller Tuning Rules for Non-Self-Regulating Process Models Rule Sree and Chidambaram (2004) – continued. Sree and Chidambaram (2006). 2 Kc Ti Td Comment 2 Kc Ti Td Model: Method 1; β = 0, N = ∞. 3 Kc Ti 0 Model: Method 1; p. 154. ( (1) ( 2) T + 0.5 TCL + TCL Ti , α = 1 − m4 (1) ( 2) Ti K m TCL + TCL + x 2 + Tm 4 + 0.5τ m − Ti 2 (1) ( 2) 0.5 Ti − 0.5τm τm x − Tm1 TCL + TCL + x 2 + Tm 4 + τm , Td = Ti = 1 , 2 4 Ti 0.5Tm1 Tm 4 τm − Tm1 Kc = ( ( ) ) 2 ( ( 519 ), ) ) (1) ( 2 ) (1) ( 2 ) ( 2 ) ( 3) ( 3) (1) (1) ( 2 ) x1 = TCL TCL x 2 + 0.5τmTm1 TCL TCL + TCL TCL + TCL TCL + ξmTm1τmTCL TCL x 2 , x2 = ( )( 4 Tm1 2 x 4 0.5Tm 4 τ m − Tm1 )( ( 4 + Tm1 (1) ( 2) x 3 = TCL + TCL + Tm 4 + τm , 2 ( 2 [ ] ) x )(− 2ξ T + T + 0.5τ )(0.5τ T T − T x ) , (1) ( 2 ) (1) ( 2) ( 2) TCL TCL + 0.5τm TCL + TCL − Tm 4 TCL + 2ξ m Tm 4 x 3 )( [ 4 m m1 m4 ] 2 m (1) ( 2) (1) ( 2 ) x 4 = −Tm1 0.5Tm 4 τm − Tm1 Tm1 TCL + TCL + 0.5τm + 2ξ m Tm1 − 0.5τmTCL TCL ( [ ( 2 (1) ( 2 ) (1) ( 2) − x 5 Tm1 TCL TCL + 0.5τ m TCL + TCL 3 Desired closed loop transfer function = Kc = Ti (1) ( 2) K m TCL + TCL + x1 + Tm 4 + τm − Ti ( Ti = − x1 = ( (1) ( 2) Tm12 TCL + TCL + x1 ( )] 4 (1) ( 2 ) + ξ m Tm1τm TCL TCL − Tm1 ), Tm 4 τm − Tm12 m (1 − sTm 4 )(1 + Tis)e (1 + sTCL(1) )(1 + sTCL(2) )(1 + sx1 ) . ) (1) m1 CL m 4 ) (1) ( 2 ) (1) ( 2) (− 2ξmTm1 + Tm 4 + τm )(TCL TCL − Tm12 ) + (Tm 4 τm − Tm12 )(TCL + TCL − Tm12 ) )( ) (1) ( 2 ) and with x 2 = Tm 4 τm − Tm12 TCL TCL − τm Tm 4 , α = 1 − (1) ( 2 ) TCL TCL − Tm 4 Ti 3 x 5 = −2ξTm1 + Tm 4 + 0.5τm . )+ T + τ − T T T , m4 2 m4 ) − sτ m (1) ( 2) Tm1 TCL + TCL + Tm 4 + τm (− 2ξ m Tm 4 τm + Tm1 (Tm 4 + τm ) ) − x 2 ( (1) ( 2 ) m CL CL . , b720_Chapter-04 04-Apr-2009 FA Handbook of PI and PID Controller Tuning Rules 520 4.14.5 Two degree of freedom controller 3 U (s) = F1 (s)R (s) − F2 (s)Y(s) ( ) 1 − β T s 1 + b f 1s [ ] 1 − χ δ d , F1 (s) = K c [1 − α ] + + + T Ti s 1 + Ti s 1 + a f 1s + a f 2 s 2 d 1 + s N [ ] K (b + b f 4 s ) 1 − ϕ T s 1+ bf 2s [ ] − φ 1 d F2 (s) = K c [1 − ε] + + − 0 f3 2 1 + a f 3s + a f 4 s T Ti s 1+ a f 5s 1+ d s N R(s) + F1 (s) U(s) Y(s) Model − F2 (s) Table 116: PID controller tuning rules –– unstable SOSPD model with a zero Rule Kc Ti Comment Td Direct synthesis: time domain criteria x1x 2 − x 3x 4 x 5x 3 − x 6 x 2 x1x 5 − x 6 x 4 x1x 2 − x 3 x 4 Rao and 1 Kc Chidambaram (2006b). Model: Method 1 β = 0 ; χ = 0 ; δ = 0 ; N = ∞ ; b f 1 = 0.5τm ; a f 2 = 0 ; ε = 0 ; φ = χ ; ϕ = 0 ; bf 2 = 0 , a f 3 = 0 ; a f 4 = 0 ; K 0 = 0 . 1 ( ) Desired closed loop transfer function = TiTds 2 + Tis + 1 (1 − sTm 4 )e−sτ m K c = Ti (K m (3λ + τm + Tm 4 − Ti )) , recommended λ = 1.2τm ; af1 = (λs + 1)3 ; λ3 − Tm 4Ti Td λ3 0.05τm , = , Tm 4 = 0 with a T ≠ 0 ; m 4 f 1 Tm12 3λ + τm + Tm 4 − Ti Tm12 3λ + τm − Ti 0.5τm x1 = − τmTm1Tm 4ξ m − Tm1 (Tm 4 + 0.5τm ) , x 4 = Tm1 − 0.5τm Tm 4 , 2 ( 2 ) x 2 = 0.5Tm1 6λ2 + 3τm λ − Tm 4 τm − 2ξm Tm1 (3λ + τm + Tm 4 ) − 0.5τm λ3 , 2 3 x 3 = λ2Tm1 (λ + 1.5τm ) − λ3τm Tm1ξ m − Tm1 (3λ + τm + Tm 4 ) , 2 4 ( ) x 5 = −Tm1 (2ξm Tm1 + Tm 4 + 0.5τm ) , x 6 = Tm1 0.5τm Tm 4 − Tm1 ; 2 2 2 Formulate φ = (Ti − Tm 4 + b f 1 ) (Ti + K m K cTi − 0.5τm K m K c − Tm 4 K m K c ) ; α = 1 − ((Tm 4 − b f 1 ) Ti ) − (x 7 Ti )[(Ti K m K c ) + Ti − Tm 4 − 0.5τm ] ; if φ > 1 , x 7 = 1 ; if φ ≤ 1 , 0.6 ≤ x 7 ≤ 0.7 . b720_Chapter-05 17-Mar-2009 FA Chapter 5 Performance and Robustness Issues in the Compensation of FOLPD Processes with PI and PID Controllers 5.1 Introduction This chapter will discuss the compensation of FOLPD processes using PI and PID controllers whose parameters are specified using appropriate tuning rules. The gain margin, phase margin and maximum sensitivity of the compensated system as the ratio of time delay to time constant of the process varies, are used as ways of judging the performance and robustness of the system. This work follows the method of Ho et al. (1995b), (1996), in which good approximations for the gain and phase margins of the PI or PID controlled system are analytically calculated. The method will be extended to determine analytically the maximum sensitivity of the compensated system. Insight will be obtained into the range of time delay to time constant ratios over which it is sensible to apply various tuning rules, to compensate a FOLPD process. The chapter is organised as follows. Formulae for calculating analytically the gain margin, phase margin and maximum sensitivity are outlined in Sections 5.2 and 5.3. The performance and robustness of FOLPD processes compensated with a variety of PI and PID tuning rules are evaluated in Section 5.4. In Section 5.5, tuning rules are designed to achieve constant gain and phase margins for all values of delay, for a number of process models and controller structures. Conclusions of the work are drawn in Section 5.6. This work was previously published by O’Dwyer (1998), (2001a). 521 b720_Chapter-05 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 522 5.2 The Analytical Determination of Gain and Phase Margin 5.2.1 PI tuning formulae The controller and process model are respectively given by 1 G c (s) = K c 1 + Ti s G m (s ) = (5.1) K m e −sτ m 1 + sTm (5.2) K m e − jωτ m K c ( jωTi + 1) 1 + jωTm jωTi (5.3) Then G m ( jω)G c ( jω) = From the definition of gain and phase margin, the following sets of equations are obtained: [ ] φ m = arg G c ( jωg )G m ( jωg ) + π (5.4) 1 G c ( jω p )G m ( jω p ) (5.5) Am = where ωg and ωp are given by G c ( j ω g ) G m ( jω g ) = 1 [ ] arg G c ( jωp )G m ( jωp ) = − π (5.6) (5.7) From Equation (5.3), G m ( jω)G c ( jω) = K m K c 1 + ω2 Ti ωTi 1 + ω2 Tm 2 2 ∠ − 0.5π + tan −1 ωTi − tan −1 ωTm − ωτ m (5.8) b720_Chapter-05 17-Mar-2009 FA Chapter 5: Performance and Robustness Issues... 523 Therefore, from Equation (5.4) φ m = π − 0.5π + tan −1 ωg Ti − tan −1 ωg Tm − ωg τ m (5.9) with ωg given by the solution of Equation (5.6), i.e. 2 K m K c 1 + ωg Ti 2 ωg Ti 1 + ωg Tm 2 2 =1 (5.10) Also, from Equations (5.5) and (5.8), Am = 1 G m ( jω p ) G c ( jω p ) = 2 2 2 2 ωp Ti 1 + ωp Tm KmKc 1 + ωp Ti (5.11) with ωp given by the solution of Equation (5.7), i.e. − 0.5π + tan −1 ωp Ti − tan −1 ωp Tm − ωp τ m = − π (5.12) From Equation (5.10), ωg may be determined analytically to be ωg = ( 2 2 ) (K K Ti K c K m − 1 + 2 c 2 m 2Ti Tm ) 2 2 2 2 − 1 Ti + 4 K c K m Tm 2 2 (5.13) An analytical solution of Equation (5.12) (to determine ωp ) is not possible. An approximate analytical solution may be obtained if the following approximation for the arctan function is made: tan −1 x ≈ π π π x , x < 1 and tan −1 x ≈ − , x >1 4 2 4x (5.14) This is quite an accurate approximation, as is shown in Figures 1a and 1b (next page). Considering Equation (5.12), four possibilities present themselves if the approximation in Equation (5.14) is to be used. These possibilities, together with the formula for ωp that may be determined analytically for each of these cases, are (i) 1 1 π ± π 2 − 4πτ m − Ti Tm ωp Ti > 1, ωp Tm > 1 : ω p = 4τ m (5.15) b720_Chapter-05 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 524 π ± π2 − ωp Ti > 1, ωp Tm < 1 : ωp = (ii) π (0.25πTm + τ m ) Ti 2(0.25πTm + τ m ) (5.16) π (iii) ωp Ti < 1, ωp Tm > 1 : ωp = Tm 4τ m − πTi (5.17) (iv) ωp Ti < 1, ωp Tm < 1 : ωp = 2π 4τ m + π(Tm − Ti ) (5.18) The gain and phase margin of the compensated system for each of the tuning rules, as a function of τ m Tm , may be calculated by applying Equations (5.9), (5.11), (5.13) and the relevant approximation for ωp from Equations (5.15) to (5.18). Figure 1a: Arctan x and its approximation (Equation 5.14) arctan x approximation Figure 1b: % error in taking approximation (Equation 5.14) to arctan(x) b720_Chapter-05 17-Mar-2009 FA Chapter 5: Performance and Robustness Issues... 525 5.2.2 PID tuning formulae The classical PID controller structure is considered, whose transfer function is given as 1 1 + sTd G c (s) = K c 1 + Ti s 1 + sαTd (5.19) and the process model is given by Equation (5.2). Substituting Equations (5.2) and (5.19) into Equations (5.4) and (5.5) gives φ m = 0.5π + tan −1 ωg Ti + tan −1 ωg Td − tan −1 ωg Tm − tan −1 ωg αTd − ωg τ m (5.20) and Am = (1 + ω T )(1 + ω α T ) (1 + ω T )(1 + ω T ) 2 ωp Ti p KcKm 2 m 2 2 p i 2 2 p 2 p 2 d 2 (5.21) d From Equation (5.6), ωg is given by the solution of 2 K m K c 1 + ωg Ti 2 ωg Ti 1 + ωg Tm 2 2 2 1 + ωg Td 2 2 1 + α 2 ωg Td 2 =1 (5.22) The equation for ωg may be determined to be 6 4 2 ωg + a 1ω g + a 2 ωg + a 3 = 0 2 with a1 = ( 2 ) 2 2 2 2 2 2 Ti Tm α 2 Td 2 a2 = 2 2 2 Ti Tm + α 2 Td − K c K m Ti Td 2 2 Ti − K c K m (Ti + Td ) 2 2 Ti Tm α 2 Td 2 (5.23) 2 , 2 and a 3 = − Kc Km 2 2 2 Ti Tm α 2 Td 2 . Following the procedure outlined by Ho et al. (1996), the analytical solution of Equation (5.23) is given as ωg = 2 with Q = 3 R + Q3 + R 2 + 3 R − Q3 + R 2 − 3 9a a − 27a 3 − 2a 1 3a 2 − a 1 and R = 1 2 . 9 54 a1 3 (5.24) b720_Chapter-05 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 526 From Equation (5.7), ω p is given by the solution of 0.5π + tan −1 ωp Td + tan −1 ωp Ti − tan −1 ωp Tm − tan −1 ωp αTd − ωp τ m = 0 (5.25) As with Equation (5.12), an analytical solution of this equation is not possible. An approximate analytical solution may be obtained if the approximation detailed in Equation (5.14) is made. Looking at Equation (5.25), twelve possibilities present themselves if the approximation in Equation (5.14) is to be used; these possibilities are (1) ωp Ti > 1, ωp Tm > 1 (2) ωp Ti > 1, ωp Tm < 1 (a) ω p Td ≤ 1, ω p αTd < 1 (a) ω p Td ≤ 1, ωp αTd < 1 (b) ω p Td > 1, ωp αTd < 1 (b) ω p Td > 1, ωp αTd < 1 (c) ω p Td > 1, ω p αTd > 1 (c) ω p Td > 1, ωp αTd > 1 (3) ωp Ti < 1, ωp Tm > 1 (4) ωp Ti < 1, ωp Tm < 1 (a) ω p Td ≤ 1, ωp αTd < 1 (a) ω p Td ≤ 1, ωp αTd < 1 (b) ω p Td > 1, ωp αTd < 1 (b) ω p Td > 1, ωp αTd < 1 (c) ω p Td > 1, ωp αTd > 1 (c) ω p Td > 1, ωp αTd > 1 The formulae for ωp that may be determined for each of these cases are as follows: (1) ωp Ti > 1, ωp Tm > 1 (a) ωp Td ≤ 1, ωp αTd < 1 : 1 1 π ± π 2 − π(4τ m − πTd (1 − α )) − Ti Tm ωp = (5.26) 4τ m − πTd (1 − α ) b720_Chapter-05 17-Mar-2009 FA Chapter 5: Performance and Robustness Issues... 527 (b) ωp Td > 1, ωp αTd < 1 : 1 1 1 + − 2π ± 4π 2 − π(4τ m + πTd α ) Td Ti Tm ωp = (5.27) 4τ m + πTd α (c) ωp Td > 1, ωp αTd > 1 : 1 1 1 1 π ± π 2 − 4πτ m + − − Td Ti αTd Tm ωp = (5.28) 4τ m (2) ωp Ti > 1, ωp Tm < 1 : (a) ωp Td ≤ 1, ωp αTd < 1 : 2π ± 4π 2 + ωp = π (πTd − πTm − παTd − 4τ m ) Ti πTm + 4 τ m + παTd − πTd (5.29) (b) ωp Td > 1, ωp αTd < 1 : 1 1 3π ± 9π 2 − π + (πTm + παTd + 4τ m ) Td Ti ωp = (5.30) πTm + 4τ m + παTd (c) ωp Td > 1, ωp αTd > 1 : 1 1 1 2π ± 4π 2 + π − − (πTm + 4τ m ) αTd Td Ti ωp = (5.31) πTm + 4τ m (3) ωp Ti < 1, ωp Tm > 1 : (a) ωp Td ≤ 1, ωp αTd < 1 : ωp = π Tm (4τ m + π[αTd − Td − Ti ]) (5.32) b720_Chapter-05 528 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules (b) ωp Td > 1, ωp αTd < 1 : 1 1 − π ± π 2 − π − (πTi − παTd − 4 τ m ) Tm Td ωp = (5.33) πTi − 4τ m − παTd (c) ωp Td > 1, ωp αTd > 1 : ωp = π π π − − Td Tm αTd πTi − 4τ m (5.34) (4) ωp Ti < 1, ωp Tm < 1 : (a) ωp Td ≤ 1, ωp αTd < 1 : ωp = 2π π(αTd − Td ) + π(Tm − Ti ) + 4τ m (5.35) (b) ωp Td > 1, ωp αTd < 1 : 2π ± 4π 2 − ωp = π (πTm − πTi + παTd + 4τ m ) Td πTm − πTi + 4τ m − παTd (5.36) (c) ωp Td > 1, ωp αTd > 1 : 1 1 π ± π 2 − π − (πTi − πTm − 4τ m ) αTd Td ωp = (5.37) 4τ m − πTm − πTi The gain and phase margin of the compensated system for each of the tuning rules, as a function of τ m Tm , may now be calculated by applying Equations (5.20), (5.21), (5.24) and the relevant approximation for ωp from Equations (5.26) to (5.37). As Equation (5.19) shows, the procedure applies to tuning rules defined for the classical PID controller structure only; however, Ho et al. (1996) suggest that the method may be applied to tuning rules defined for the ideal PID controller structure, by using equations that approximately convert these tuning rules into their equivalent in the classical controller structure. b720_Chapter-05 17-Mar-2009 Chapter 5: Performance and Robustness Issues... FA 529 5.3 The Analytical Determination of Maximum Sensitivity The maximum sensitivity is the reciprocal of the shortest distance from the Nyquist curve to the (-1,0) point on the Rl-Im axis. It is defined as follows: 1 all ω 1 + G ( jω)G ( jω) m c M max = Max (5.38) For a FOLPD process model controlled by a PI controller, G m ( jω)G c ( jω) = 1 + ω2 Ti 2 KcKm 2 ωTi 1 + ω2 T (5.39) m and arg[G m ( jω)G c ( jω)] = −0.5π − tan −1 ωTm + tan −1 ωTi − ωτ m (5.40) For a FOLPD process model controlled by a PID controller, 2 G m ( jω)G c ( jω) = 2 (1 + ω2 Ti )(1 + ω2 Td ) KcKm 2 2 (1 + ω2 T )(1 + α 2 ω2 T ) ωTi m (5.41) d and arg[G m ( jω)G c ( jω)] = − 0.5π − tan −1 ωTm − tan −1 ωαTd + tan −1 ωTi + tan −1 ωTd − ωτ m (5.42) The maximum sensitivity may be calculated over an appropriate range of frequencies corresponding to phase lags of 100 0 to 260 0 . 5.4 Simulation Results Space considerations dictate that only representative simulation results may be provided; an extensive set of simulation results covering 103 PI controller tuning rules and 125 PID controller tuning rules (for the ideal and classical controller structures) are available (O’Dwyer, 2000). The MATLAB package has been used in the simulations. Figures 2 to 7 show how gain margin, phase margin and maximum sensitivity vary as the ratio of time delay to time constant varies, if some PI tuning rules are used (Figures 2 to 4) and corresponding PID tuning rules for the classical controller structure (with α = 0.1) are used (Figures 5 to 7). In these results, Z-N refers to the process reaction curve method of Ziegler and Nichols (1942); W-W refers to the process reaction curve method of Witt and Waggoner (1990); IAE reg, ISE reg and ITAE reg refer to the tuning rules for regulator applications that minimise the integral of absolute error criterion, the integral of squared error criterion b720_Chapter-05 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 530 and the integral of time multiplied by absolute error criterion, respectively, as defined by Murrill (1967) for PI tuning rules and Kaya and Scheib (1988) for PID tuning rules based on the classical controller structure. Figures 8 to 15 show gain and phase margin comparisons between corresponding PI and PID controller tuning rules. It is clear that the gain margin is generally less when the PID rather than the PI tuning rules are considered, over the ratios of time delay to time constant taken; the difference between the phase margins is less clear cut. This suggests that these PID tuning rules should provide a greater degree of performance than the corresponding PI tuning rules, but may be less robust. Comparing the individual tuning rules, it is striking that the ISE based tuning rules have generally the smallest gain margin and have also a small phase margin, suggesting that this is a less robust tuning strategy. The results in Figures 4 and 7 confirm these comments. Figure 2: Gain margin Figure 3: Phase margin Figure 4: Max. sensitivity 120 4 - = Z-N * = IAE reg + = ITAE reg o = ISE reg 3.5 3 8 7 100 6 80 5 2.5 60 2 40 4 3 20 1.5 1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Ratio of τ m to Tm Ratio of τ m to Tm Ratio of τ m to Tm Figure 5: Gain margin Figure 6: Phase margin Figure 7: Max. sensitivity 2.5 - = W-W * = IAE reg + = ITAE reg o = ISE reg 2 2 100 6 90 5.5 80 5 70 4.5 60 4 50 3.5 40 3 30 2.5 1.5 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Ratio of τ m to Tm 1.8 2 20 2 10 1.5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Ratio of τ m to Tm 1.8 2 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Ratio of τ m to Tm 2 b720_Chapter-05 17-Mar-2009 Chapter 5: Performance and Robustness Issues... FA 531 No general conclusion can be reached as to the best tuning rule (as expected); it is interesting, though, that a full panorama of simulation results (O’Dwyer, 2000) show that many tuning rules may be applied at ratios of time delay to time constant greater than that normally recommended. One example may be seen in Figures 5 to 7, where the gain margin, phase margin and maximum sensitivity (associated with the use of the PID tuning rule for obtaining minimum IAE in the regulator mode) tends to level out when the ratio of time delay to time constant is greater than 1; normally, the tuning rule is used when the ratio is less than 1 (Murrill, 1967). On the other hand, it is clear from Figures 8 and 9 that there is a significant degradation of performance when using the PID tuning rule of Witt and Waggoner (1990) and the PI tuning rule of Ziegler and Nichols (1942) for large ratios of time delay to time constant, which is compatible with application experience. The decision between the use of a PI and PID controller to compensate the process depends on the ratio of time delay to time constant in the FOLPD model, together with the desired trade-off between performance and robustness, as expected. It turns out, however, that the analytical method explored allows the calculation of a far wider range of gain and phase margins for PI controllers; it is also true that stability tends to be assured when a PI controller is used (O’Dwyer, 2000). Thus, a cautious design approach is to use a PI controller, particularly at larger ratios of time delay to time constant. Finally, the volume of tuning rules and data generated means that the use of an expert system to recommend a tuning rule based on user defined requirements is indicated; work is ongoing on such an implementation (Feeney and O’Dwyer, 2002). b720_Chapter-05 17-Mar-2009 Handbook of PI and PID Controller Tuning Rules 532 Figure 8: Gain margin comparison Figure 9: Phase margin comparison 4 120 3.5 100 - = W-W PID o = Z-N PI 3 80 2.5 60 2 40 1.5 20 1 FA 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Ratio of τ m to Tm Ratio of τ m to Tm Figure 10: Gain margin comparison Figure 11: Phase margin comparison 2 100 3 90 2.8 - = IAE reg PID o = IAE reg PI 2.6 80 2.4 70 2.2 60 2 50 1.8 40 1.6 30 1.4 20 1.2 10 1 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Ratio of τ m to Tm 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Ratio of τ m to Tm 1.8 2 b720_Chapter-05 17-Mar-2009 FA Chapter 5: Performance and Robustness Issues... Figure 12: Gain margin comparison 533 Figure 13: Phase margin comparison 2.5 100 - = ISE reg PID o = ISE reg PI 90 80 70 2 60 50 40 1.5 30 20 10 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 2 0.8 1 1.2 1.4 1.6 1.8 Figure 14: Gain margin comparison Figure 15: Phase margin comparison 2.8 90 - = ITAE reg PID o = ITAE reg PI 70 2.2 60 2 50 1.8 40 1.6 30 1.4 20 1.2 10 0.2 0.4 0.6 0.8 1 2 80 2.4 0 0.6 Ratio of τ m to Tm 100 1 0.4 Ratio of τ m to Tm 3 2.6 0.2 1.2 1.4 Ratio of τ m to Tm 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Ratio of τ m to Tm 1.8 2 b720_Chapter-05 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 534 5.5 Design of Tuning Rules to Achieve Constant Gain and Phase Margins, for All Values of Delay Normally, the gain and phase margins of the compensated systems tend to increase as the time delay increases, supporting the common view that PI and PID controllers are less suitable for the control of dominant time delay processes. However, for the PI control of a FOLPD process model, O’Dwyer (2000) shows that the tuning rules proposed by Chien et al. (1952), Haalman (1965) and Pemberton (1972a), among others, facilitate the achievement of a constant gain and phase margin as the time delay of the process model varies. All of these tuning rules have the following structure: K c = aTm K m τ m , Ti = Tm . Following this observation, original approaches to the design of tuning rules for PI, PD and PID controllers are proposed, for a wide variety of process models, which allow constant gain and phase margins for the compensated system. This work is organised as follows. In Section 5.5.1, PI controller tuning rules are specified for processes modelled in FOLPD form and IPD form. In Section 5.5.2, PID controller tuning rules are described for processes modelled in FOLPD form, SOSPD form and SOSPD form with a negative zero. Section 5.5.3 deals with the design of PD controller tuning rules for the control of processes modelled in FOLIPD form. 5.5.1 PI controller design 5.5.1.1 Processes modelled in FOLPD form For such processes and controllers, Equations (5.1) and (5.2) apply, i.e., G m (s) = K m e − sτ m 1 + sTm 1 G c (s) = K c 1 + Ti s with From Section 5.2.1, φ m = π − 0.5π + tan −1 ω g Ti − tan −1 ω g Tm − ω g τ m (Equation 5.9) b720_Chapter-05 17-Mar-2009 Chapter 5: Performance and Robustness Issues... FA 535 with ω g given by the solution of 2 K m K c 1 + ωg Ti 2 ωg Ti 1 + ωg Tm 2 2 =1 (Equation 5.10) and Am = 2 2 2 2 ωpTi 1 + ωp Tm K mK c 1 + ωp Ti (Equation 5.11) with ωp given by the solution of − 0.5π + tan −1 ω p Ti − tan −1 ω p Tm − ω p τ m = −π (Equation 5.12) If K c and Ti are designed as follows: Kc = aTm K mτm (5.43) and Ti = Tm (5.44) Then Equation (5.12) becomes − 0.5π − ωp τm = −π i.e. ω p = π 2τ m (5.45) (5.46) Substituting Equation (5.46) into Equation (5.11) gives A m = πTm 2K m K c τ m (5.47) Substituting Equation (5.44) into Equation (5.10) gives i.e. K m K c ω g Tm = 1 (5.48) ω g = K m K c Tm (5.49) Substituting Equations (5.44) and (5.49) into Equation (5.9) gives φ m = 0 . 5 π − K m K c τ m Tm (5.50) b720_Chapter-05 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 536 Finally, substituting Equation (5.43) into Equation (5.47) gives A m = π 2a (5.51) and substituting Equation (5.43) into Equation (5.50) gives φ m = 0 .5 π − a (5.52) Some typical tuning rules are shown in Table 117. Table 117: Typical PI controller tuning rules – FOLPD process model a Kc Ti Am φm π6 π3 1.047Tm K m τ m Tm 1.5 π 4 0.785Tm K m τm Tm 2.0 π 4 π6 0.524Tm K m τm Tm 3.0 π3 These rules are also provided in Table 9. It may also be demonstrated that, if K c and Ti are designed as follows: Kc = aTm τm Tu K u 2 Tu + 4π 2 Tm 2 (5.53) and Ti = Tm (5.54) where K u and Tu are the ultimate gain and ultimate period, respectively, then the constant gain and phase margins provided in Equations (5.51) and (5.52) are obtained. 5.5.1.2 Processes modelled in IPD form A similar analysis to that of Section 5.5.1.1 may be done for the design of PI controllers for processes modelled in IPD form. The process is modelled as follows: G m (s) = K m e −sτ m s (5.55) Corresponding to Equations (5.4) and (5.9), the phase margin is φ m = π − 0.5π + tan −1 ω g Ti − 0.5π − ω g τ m (5.56) b720_Chapter-05 17-Mar-2009 FA Chapter 5: Performance and Robustness Issues... 537 with ωg given by the solution of 2 K m K c 1 + ω g Ti 2 =1 2 ω g Ti (5.57) If K c and Ti are designed as follows: Kc = a K m τm (5.58) and Ti = bτ m (5.59) then it may be shown that a2 a ωg = ± 2 2 2bτ m 2τ m a b + 4 2 0.5 2 (5.60) and, substituting Equations (5.59) and (5.60) into Equation (5.56), φ m = tan −1 0.5a 2 b 2 + 0.5ab a 2 b 2 + 4 0.5 − 0.5a 2 + 0.5 a a 2b2 + 4 b (5.61) Corresponding to Equations (5.5) and (5.11), the gain margin, 2 Am = ω p Ti 2 K m K c 1 + ω p Ti 2 (5.62) with ω p given by the solution of − 0.5π + tan −1 ω p Ti − 0.5π − ω p τ m = −π (5.63) i.e. ω p is given by the solution of tan −1 ω p Ti = ω p τ m (5.64) b720_Chapter-05 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 538 An analytical solution to this equation is not possible, though if the π (when ω p Ti > 1 ) is used, simple approximation tan −1 ω p Ti ≈ 0.5π − 4ω p Ti calculations show that the following analytical solution for ω p may be obtained: 0.5 π 2 0.5π + 0.25π − τ m b ωp = (5.65) The inequality ω p Ti > 1 may be shown to be equivalent to b > 1.273. Substituting Equations (5.58), (5.59) and (5.65) into Equation (5.62), calculations show that: b π π2 π + − 4a 2 4 b Am = 1+ 2 2 b π π π + − 4 2 4 b 2 (5.66) 2 Some typical tuning rules are shown in Table 118. Table 118: Typical PI controller tuning rules – IPD process model Kc Ti Am φm 0.558 K m τm 1.4τ m 1.5 46.2 0 0.484 K m τm 1.55τm 2.0 45.5 0 0.458 K m τm 3.35τ m 3.0 59.9 0 0.357 K m τm 4.3τ m 4.0 60.0 0 0.305 K m τm 12.15τ m 5.0 75.0 0 It may also be shown that the maximum sensitivity is a constant if K c and Ti are specified according to Equations (5.58) and (5.59). The maximum sensitivity may be calculated to be: a2 M max = 1 − 2a cos(ω r τ m ) − 2 2 ω r τ m −0.5 (5.67) b720_Chapter-05 17-Mar-2009 FA Chapter 5: Performance and Robustness Issues... 539 with ω r τ m being a constant obtained numerically from the solution of the 2 2 equation (a 2 ω r τ m ) + cos ω r τ m = sin(ω r τ m ) / ω r τ m . It may also be shown that, if K c and Ti are designed as follows: K c = aK u (5.68) Ti = bTu (5.69) and then the following constant gain and phase margins are determined: 2b π π2 π + − 4 4b πa 2 Am = 1+ 2 π π2 π + − 4 4b a 2 π 2 2 1 [ and φ m = tan −1 2π 2 a 2 b 2 + 2πab 4π 2 a 2 b 2 + 4 − 0.125π 2 a 2 + 2 (5.70) ] 0 .5 πa 4π 2 a 2 b 2 + 4 16b (5.71) 5.5.2 PID controller design 5.5.2.1 Processes modelled in FOLPD form – classical controller The classical PID controller is given by 1 1 + sTd G c (s) = K c 1 + Ti s 1 + sαTd (Equation 5.19) and the process model is given by G m (s) = K m e − sτ m 1 + sTm (Equation 5.2) From Section 5.2.2, φm = 0.5π + tan −1 ωg Ti + tan −1 ωg Td − tan −1 ωg Tm − tan −1 ωg αTd − ωg τm (Equation 5.20) b720_Chapter-05 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 540 with ωg given by the solution of 2 K m K c 1 + ω g Ti 2 ω g Ti 1 + ω g Tm Also Am = ω p Ti KcK m 2 2 2 1 + ω g Td 2 2 1 + α 2 ω g Td =1 2 (Equation 5.22) (1 + ω T )(1 + ω α T ) (1 + ω T )(1 + ω T ) 2 p 2 2 m 2 p 2 p 2 i 2 p 2 d 2 (Equation 5.21) d with ω p given by 0.5π + tan −1 ω p Td + tan −1 ω p Ti − tan −1 ω p Tm − tan −1 ω p αTd − ω p τ m = 0 (Equation 5.25) If K c , Ti and Td are designed as follows: aTm K mτm Ti = αTm Kc = (5.72) (5.73) and Td = Tm (5.74) then, Equation (5.25) becomes 0.5π − ω p τ m = 0 (5.75) ω p = π 2τ m (5.76) i.e. and Equation (5.21) becomes A m = α πTm 2K m K c τ m (5.77) K m K c ω g Tm = 1 (5.78) ω g = K m K c Tm (5.79) Equation (5.22) becomes i.e. b720_Chapter-05 17-Mar-2009 Chapter 5: Performance and Robustness Issues... FA 541 Finally, substituting Equations (5.73), (5.74) and (5.79) into Equation (5.20) gives φ m = 0 .5 π − K m K c τ m α T m (5.80) Substituting Equations (5.72), (5.73), (5.74) and (5.76) into Equation (5.21) gives A m = πα 2a (5.81) and substituting Equation (5.72) into Equation (5.80) gives φ m = 0.5π − a α (5.82) This design reduces to the PI controller design when α = 1 . 5.5.2.2 Processes modelled in SOSPD form – series controller For such processes and controllers, K m e − sτ m G m (s) = (1 + sTm1 )(1 + sTm 2 ) (5.83) 1 (1 + sTd ) G c (s) = K c 1 + Ti s (5.84) with Therefore, G m (s)G c (s) = K m K c e −sτ m (1 + sTi )(1 + sTd ) Ti s(1 + sTm1 )(1 + sTm 2 ) (5.85) If Ti and Td are designed as follows: Ti = Tm1 (5.86) Td = Tm 2 (5.87) and then, from Equation (5.85) G m (s)G c (s) = K m K c e −sτ m Ti s (5.88) b720_Chapter-05 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 542 which is equal to G m (s)G c (s) in Section 5.5.1.1 when Ti = Tm . Therefore, designing aTm Kc = (5.89) K mτm will allow A m = π 2a and φ m = 0.5π − a , as before. 5.5.2.3 Processes modelled in SOSPD form with a negative zero – classical controller For such processes, K m e −sτ m (1 + sTm 3 ) G m (s) = (1 + sTm1 )(1 + sTm 2 ) (5.90) with G c (s) given by Equation (5.19). Therefore, G m (s)G c (s) = K m K c e −sτ m (1 + sTm 3 )(1 + sTi )(1 + sTd ) Ti s(1 + sTm1 )(1 + sTm 2 )(1 + αsTd ) (5.91) If Ti , Td and α are designed as follows: Ti = Tm1 (5.92) Td = Tm 2 (5.93) α = Tm 3 Tm 2 (5.94) aTm K mτm (5.95) and therefore, designing Kc = will allow A m = π 2a and φ m = 0.5π − a , as in Sections 5.5.1.1 and 5.5.2.2. 5.5.3 PD controller design In this case, the process is modelled in FOLIPD form, i.e. K m e −sτ m s(1 + sTm ) (5.96) G c (s) = K c (1 + Td s ) (5.97) G m (s) = with b720_Chapter-05 17-Mar-2009 Chapter 5: Performance and Robustness Issues... FA 543 Therefore, G m (s)G c (s) = K m K c e −sτ m (1 + sTd ) s(1 + sTm ) (5.98) Therefore, designing Kc = aTm K mτm (5.99) and Td = Tm (5.100) will allow A m = π 2a and φ m = 0.5π − a , as in Sections 5.5.1.1, 5.5.2.2 and 5.5.2.3. 5.6 Conclusions This chapter has considered the performance and robustness of a PI and PID controlled FOLPD process, with the parameters of the controllers determined by a variety of tuning rules. The original contributions of this work are as follows: (a) an expansion of the analytical approach of Ho et al. (1995b), (1996) to determine the approximate gain and phase margin analytically under all operating conditions, (b) the analytical determination of the approximate maximum sensitivity of the compensated system and (c) the application of the algorithms using a wide variety of PI and PID tuning rules. The implementation of autotuning algorithms in commercial controllers means that choice of a suitable tuning rule is an important issue; the techniques discussed allow an analytical evaluation to be performed of candidate tuning rules. Finally, the chapter discusses an original approach to design tuning rules for both PI and PID controllers, for a variety of delayed process models, with the objective of achieving constant gain and phase margins for all values of delay. In one of the cases discussed (PI control of an IPD process model), an analytical approximation is used in the development; this approximation may also be used to determine further tuning rules for other process models with delay, and for other PID controller structures. b720_Appendix-01 17-Mar-2009 FA Appendix 1 Glossary of Symbols and Abbreviations a f 1 , a f 2 , a f 3 , a f 4 , a f 5 , a f 6 = Denominator parameters of a filter associated with some PID controller structures a 1 , a 2 , a 3 , b1 , b 2 , b 3 = Parameters of a third order process model a 1 , a 2 , a 3 , a 4 , a 5 , b1 , b 2 , b 3 , b 4 , b 5 = Parameters of a fifth order process model A 1 , A 2 , A 3 , A 4 , A 5 = Areas calculated from the process open loop step response (see, for example, Vrančić et al., 1996) A m = Gain margin A p = Peak output amplitude of limit cycle determined from relay autotuning b f 1 , b f 2 , b f 3 , b f 4 , b f 5 , b f 6 = Numerator parameters of a filter associated with some PID controller structures D R = Desired closed loop damping ratio d(t) = Disturbance variable (time domain) du dt = Time derivative of the manipulated variable (time domain) e(t) = Desired variable, r(t), minus controlled variable, y(t) (time domain) E(s) = Desired variable, R(s), minus controlled variable, Y(s) FOLPD model = First Order Lag Plus time Delay model FOLIPD model = First Order Lag plus Integral Plus time Delay model F1 (s), F2 (s) = Transfer functions of components of some PID controller structures G c (s) = PID controller transfer function G CL (s) = Closed loop transfer function 544 b720_Appendix-01 17-Mar-2009 FA Appendix 1: Glossary of Symbols and Abbreviations 545 G CL ( jω) = Desired closed loop frequency response G m (s) = Process model transfer function G p (s) = Process transfer function G p ( jω) = Process transfer function at frequency ω G p ( jω) = Magnitude of G p ( jω) G p ( jωφ ) = Magnitude of G p ( jω) , at frequency ω , corresponding to phase lag of φ ∠G p ( jω) = Phase of G p ( jω) ± h = Relay amplitude (relay autotuning) IE = Integral of Error = ∞ ∫ e( t )dt 0 IAE = Integral of Absolute Error = ∞ ∫ e( t ) dt 0 IMC = Internal Model Controller IPD model = Integral Plus time Delay model I 2 PD model = Integral Squared Plus time Delay model ISE = Integral of Squared Error = ∞ ∫ e ( t)dt 2 0 ISTES = Integral of Squared Time multiplied by Error, all to be Squared = ∞ ∫ [t e( t)] dt 2 2 0 ISTSE = Integral of Squared Time multiplied by Squared Error = ∞ ∫ t e ( t)dt 2 2 0 ITSE = Integral of Time multiplied by Squared Error = ITAE = Integral of Time multiplied by Absolute Error = K c = Proportional gain of the controller KH = τm2 ξ m Tm1 τ m 2 − T − (Hwang, 1995) m 1 2 9 2τ m K m 18 9 ∞ ∫ te ( t )dt 2 0 ∞ ∫ t e( t ) dt 0 b720_Appendix-01 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 546 K H1 = τ m 2 τ m Tm 1.884K c K u τ m − + 2 18 9ω u 2K m τ m 18 9 + 2 2 2 2 9 τm 49Tm 7T τ 4.71K u K c (τm + 1.6Tm ) 1.775K c K u − + + m m− 2 2K m τm 324 324 162 81ωu 81ωu (Hwang, 1995) K H2 = 9 2K m τ m τm 1.884K u K c τ m ξ τ T 9 2 − Tm1 − m m m1 + x1 + 2 18 9 9 ω u 2K m τ m 2 2 with 4 2 2 2 2 2 3 3 τ 49τm ξm Tm1 τm Tm1 7Tm1ξm τm 10τm Tm1 ξ m 4 − + + − x2 x1 = Tm1 + m + 324 81 9 81 9 and x2 = 2 2 2 3 0.471K u K c 10τ m 4T τ (8ξ m τ m + 9Tm1 ) 1.775K u K c τ m + m1 m − 2 81 ωu 81ω u 81 (Hwang, 1995) K i = Integral gain of the controller K L = Gain of a load disturbance process model K m = Gain of the process model K p = Gain of the process K u = Ultimate proportional gain ∧ K u = Ultimate proportional gain estimate determined from relay autotuning K φ = Proportional gain when G c ( jω)G p ( jω) has a phase lag of φ K 0 = Weighting parameter used in some two degree of freedom PID controller structures K x % = Proportional gain required to achieve a decay ratio of 0.01x K φ0 = Controller gain when G c ( jω)G p ( jω) has a phase lag of φ 0 min = Minimum max = Maximum b720_Appendix-01 17-Mar-2009 Appendix 1: Glossary of Symbols and Abbreviations FA 547 M s = Closed loop sensitivity M max = Maximum value of closed loop sensitivity, M s M t = Complementary closed loop sensitivity n = Order of a process model with a repeated pole N = Parameter that determines the amount of filtering on the derivative term on some PID controller structures OS = Closed loop response overshoot (often a percentage) PI controller = Proportional Integral controller PID controller = Proportional Integral Derivative controller r = Desired variable (time domain) 2 r1 = 0.7071M max − M max 1 − 0.5M max 2 2 M max − 1 (Chen et al.,1999a) R(s) = Desired variable (Laplace domain) s = Laplace variable SOSPD model = Second Order System Plus time Delay model SOSIPD model = Second Order System plus Integral Plus time Delay model t = Time Tar = Average residence time = time taken for the open loop process step response to reach 63% of its final value Td = Derivative time of the controller TCL = Desired closed loop system time constant TCL 2 = Desired parameter of second order closed loop system response (1) ( 2) ( 3) TCL , TCL , TCL = Desired dynamic parameters of the closed loop system response Tf = Time constant of the lag in some PID controller structures Ti = Integral time of the controller TL = Time constant of a load disturbance FOLPD process model Tm = Time constant of a FOLPD process model Tm1, Tm2, Tm3, Tm4 = Time constants of second, third or fourth order process models, as appropriate b720_Appendix-01 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 548 Tm1i , Tm2i , Tm3i , Tm4i = Time constants of a general process model Tp = Time constant of a FOLPD process Tp1, Tp2, Tp3 = Time constants of second or third order process, as appropriate TR = Closed loop rise time TS = Closed loop settling time Tu = Ultimate period ∧ T u = Ultimate period estimate determined from relay autotuning Tx % = Period of the waveform with a decay ratio of 0.01x , when the closed loop system is under proportional control TOSPD model = Third Order System Plus time Delay model u(t) = Manipulated variable (time domain) u ∞ = Final value of the manipulated variable (time domain) U(s) = Manipulated variable (Laplace domain) V = Closed loop response overshoot (as a fraction of the controlled variable final value) y(t) = Controlled variable (time domain) y ∞ = Final value of the controlled variable (time domain) Y(s) = Controlled variable (Laplace domain) α , β , χ , δ , ε , φ , ϕ = Weighting factors in some PID controller structures 2 ε= 6Tm1 + 4ξ m Tm1 τ m + K H K m τ m ε1 = 6Tm1 + 4Tm1 ξ m τ m + K u K m τ m 2 2τ m Tm1 ω u 2 (Hwang, 1995) 2Tm ωu + K H1 K m τ m ωu − 1.884 K u K c (Hwang, 1995) 0.471K c K u ω H1 τ m 2 ε2 = (Hwang, 1995) 2 2Tm1 τ m ω H 2 ε0 = 2 2 6Tm1 ωu + 4Tm1ξ m τ m ω u + K H 2 K m τ m ωu − 1.884 K u K c τ m 2 2 (0.471K u K c τ m + 2 τ m Tm1 ωu )ω H 2 (Hwang, 1995) b720_Appendix-01 17-Mar-2009 FA Appendix 1: Glossary of Symbols and Abbreviations 549 ± ε 4 = Relay hysteresis width (relay autotuning) φ = Phase lag φc = Phase of the plant at the crossover frequency of the compensated system, with a ‘conservative’ PI controller (Pecharromán and Pagola, 2000) φ m = Phase margin φω = Phase lag at an angular frequency of ω κ = 1 Km Ku λ = Parameter that determines robustness of compensated system. τ = τ m (τ m + Tm ) τ CL = Desired closed loop system time delay τ L = Time delay of a load disturbance process model τ m = Time delay of the process model τ am = τ 'm − τ m , τ 'm is obtained using the tangent and point method of Ziegler and Nichols (1942). Subsequently, τ m is estimated from the process step response (Schaedel, 1997). ω = Angular frequency ω bw = − 3 dB closed loop system bandwidth (Shi and Lee, 2002) ω−6dB = Frequency where closed loop system magnitude is –6dB (Shi and Lee, 2004) ωc = Maximum cut-off frequency ωd = Bandwidth of stochastic disturbance signal (Van der Grinten, 1963) ω CL = 2π TCL ωg = Specified gain crossover frequency (Shi and Lee, 2002) ωH = 1+ KHKm 2T τ ξ K K τ Tm1 + m1 m m + H m m 3 6 2 ω H1 = 2 (Hwang, 1995) 1 + K H1 K m 2 0.942 K c K u τ m τ m Tm K H1 K m τ m + − 3 6 3ωu (Hwang, 1995) b720_Appendix-01 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules 550 1 + K H2 K m ωH 2 = 2 2 Tm1 + 0.942 K c K u τ m 2ξ m τ m Tm1 K H 2 K m τ m + − 3 6 3ωu (Hwang, 1995) ω M max = Frequency where the sensitivity function is maximised (Rotach, 1994) ωn = Undamped natural frequency of the compensated system ωp = A m φ m + 0.5πA m (A m − 1) (A 2 m ) − 1 τm (see, for example, Hang et al. 1993a, 1993b) ω pc = Phase crossover frequency ωr = Resonant frequency (Rotach, 1994) ωu = Ultimate frequency ^ ω u = Ultimate frequency estimate obtained from relay autotuning ω φ = Angular frequency when G c ( jω)G p ( jω) has a phase lag of φ ^ ω φ = Estimate of angular frequency when G c ( jω)G p ( jω) has a phase lag of φ ξ = Damping factor of the compensated system ξ m = Damping factor of an underdamped process model ξ des m = Desired damping factor of an underdamped process model ξ m ni , ξ mdi = Damping factors of a general underdamped process model b720_Appendix-02 17-Mar-2009 FA Appendix 2 Some Further Details on Process Modelling Processes with time delay may be modelled in a variety of ways. The modelling strategy used will influence the value of the model parameters, which will in turn affect the controller values determined from the tuning rules. The modelling strategy used in association with each tuning rule, as described in the original papers, is indicated in the tables in Chapters 3 and 4. Some outline details of these modelling strategies are provided, together with references that describe the modelling method in detail. The references are given in the bibliography. For all models, the label ‘Model: Method 1’ indicates that the model method has not been defined or that the model parameters are assumed known; interestingly, this is the modelling method assigned to a majority of tuning rules. A2.1 FOLPD Model A2.1.1 Parameters estimated from the open loop process step or impulse response Method 2: Parameters estimated using a tangent and point method (Ziegler and Nichols, 1942). Method 3: K m , τ m determined from the tangent and point method of Ziegler and Nichols (1942); Tm determined at 60% of the total process variable change (Fertik and Sharpe, 1979). Method 4: K m , τ m assumed known; Tm estimated using a tangent method (Wolfe, 1951). 551 b720_Appendix-02 552 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules Method 5: Parameters estimated using a second tangent and point method (Murrill, 1967). Method 6: Parameters estimated using a third tangent and point method (Davydov et al., 1995). Method 7: τ m and Tm estimated using the two-point method; K m estimated from the open loop step response (ABB, 2001). Method 8: τ m and Tm estimated from the process step response: Tm = 1.4( t 67% − t 33% ) , τ m = t 67% − 1.1Tm ; K m assumed known (Chen and Yang, 2000). 1 Method 9: τ m and Tm estimated from the process step response: Tm = 1.245( t 70% − t 33% ) , τ m = 1.498t 33% − 0.498t 70% ; K m assumed known (Vítečková et al., 2000b). 2 Method 10: τ m and Tm estimated from the process step response: Tm = 0.910(t 75% − t 25% ) , τ m = 1.262 t 25% − 0.262 t 75% ; K m is estimated from the process step response (Arrieta Orozco and Alfaro Ruiz, 2003; Arrieta Orozco, 2003). 3 Method 11: K m estimated from the process step response; τ m = time for which the process variable does not change; Tm determined at 63% of the total process variable change (Gerry, 1999). Method 12: K m estimated from the process step response; τ m = time for which the process variable has to change by 5% of its total value; Tm determined at 63% of the total process variable change (Kristiansson, 2003). Method 13: K m estimated from the process step response; τ m is the time for which the process variable has to change until it is outside a predefined noise band; Tm = 0.25(t 98% − τ m ) (McMillan, 2005). 4 1 t 67% , t 33% are the times taken by the process variable to change by 67% and 33%, respectively, of its total value. 2 t 70% is the time taken by the process variable to change by 70% of its total value. t 75% , t 25% are the times taken by the process variable to change by 75% and 25%, respectively, of its total value. 4 t 98% is the time taken by the process variable to change by 98% of its total value. 3 b720_Appendix-02 17-Mar-2009 Appendix 2: Some Further Details on Process Modelling FA 553 Method 14: Parameters estimated using a least squares method in the time domain (Cheng and Hung, 1985). Method 15: The model parameters are estimated by minimising a function of the error between the process and model open loop step response (Zhuang, 1992). Method 16: Parameters estimated from linear regression equations in the time domain (Bi et al., 1999). Method 17: Parameters estimated using the method of moments (Åström and Hägglund, 1995). Method 18: Model parameters estimated using the area method (Klán, 2000). Method 19: Parameters estimated from the process step response and its first time derivative (Tsang and Rad, 1995). Method 20: Parameters estimated from the process step response using numerical integration procedures (Nishikawa et al., 1984). Method 21: Parameters estimated from the process impulse response (Peng and Wu, 2000). Method 22: Parameters estimated using a ‘process setter’ block (Saito et al., 1990). Method 23: Parameters estimated from a number of process step response data values (Kraus, 1986). Method 24: Parameters estimated in the sampled data domain using two process step tests (Pinnella et al., 1986). Method 25: A model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters K 'm , Tm' and τ 'm . Subsequently, τ m is estimated from the process step response; then, a parameter labelled τ am = τ 'm − τ m is defined (Schaedel, 1997). Method 26: A model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters K 'm , Tm' and τ 'm . Subsequently, τ m is estimated from the process step response (Henry and Schaedel, 2005). Method 27: Parameters estimated from the open loop step response (Clarke, 2006). b720_Appendix-02 554 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules A2.1.2 Parameters estimated from the closed loop step response Method 28: K m estimated from the open loop step response. T90% and τ m estimated from the closed loop step response under proportional control (Åström and Hägglund, 1988). Method 29: Parameters estimated using a method based on the closed loop transient response to a step input under proportional control (Sain and Özgen, 1992). Method 30: Parameters estimated using a second method based on the closed loop transient response to a step input under proportional control (Hwang, 1993). Method 31: Parameters estimated using a third method based on the closed loop transient response to a step input under proportional control (Chen, 1989; Taiwo, 1993). Method 32: Parameters estimated from a step response autotuning experiment – Honeywell UDC 6000 controller (Åström and Hägglund, 1995). Method 33: Parameters estimated from the closed loop step response when process is in series with a PID controller (Morilla et al., 2000). Method 34: Parameters estimated by modelling the closed loop response as a second order system (Ettaleb and Roche, 2000). A2.1.3 Parameters estimated from frequency domain closed loop information Method 35: Model parameters estimated using a relay autotuning method (Yu, 2006). Method 36: Parameters estimated from two points, determined on process frequency response, using a relay and a relay in series with a delay (Tan et al., 1996). Method 37: Tm and τ m estimated from the ultimate gain and period, determined using a relay in series with the process in closed loop; K m assumed known (Hang and Cao, 1996). Method 38: Tm and τ m estimated from the ultimate gain and period, determined using a relay in series with the process in closed loop; K m estimated from the process step response (Hang et al., 1993b). b720_Appendix-02 17-Mar-2009 Appendix 2: Some Further Details on Process Modelling FA 555 Method 39: Tm and τ m estimated from data obtained when a relay is introduced in series with the process in closed loop; K m assumed known (Padhy and Majhi, 2006a). Method 40: Parameters estimated from data obtained when a relay is introduced in series with the process in closed loop (Padhy and Majhi, 2006b). Method 41: Tm and τ m estimated from the ultimate gain and period, determined using the ultimate cycle method of Ziegler and Nichols (1942); K m estimated from the process step response (Hang et al., 1993b). Method 42: Tm and τ m estimated at ω1800 ; K m estimated at ω 0 0 (Tavakoli et al., 2006). Method 43: Tm and τ m estimated from relay autotuning method (Lee and Sung, 1993); K m estimated from the closed loop process step response under proportional control (Chun et al., 1999). Method 44: Parameters estimated in the frequency domain from the data determined using a relay in series with the closed loop system in a master feedback loop (Hwang, 1995). Method 45: K u and Tu estimated from relay autotuning method; τ m estimated from the open loop process step response, using a tangent and point method (Wojsznis et al., 1999). Method 46: Parameters estimated by including a dynamic compensator outside or inside an (ideal) relay feedback loop (Huang and Jeng, 2003). Method 47: Parameters estimated from measurements performed on the manipulated and controlled variables when a relay with hysteresis is introduced in place of the controller (Zhang et al., 1996b). Method 48: Non-gain parameters estimated using a relay, with hysteresis, in series with the process in closed loop; K m is estimated with the aid of a small step signal added to the reference (Potočnik et al., 2001). Method 49: Parameters estimated using a relay in series with the process in closed loop (Perić et al., 1997). Method 50: Parameters estimated using an iterative method, based on data from a relay experiment (Leva and Colombo, 2004). b720_Appendix-02 556 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules Method 51: K m estimated from relay autotuning method; τ m , Tm estimated using a least squares algorithm based on the data recorded from the relay autotuning method, with the aid of neural networks (Huang et al., 2005a). Method 52: Parameters estimated from data obtained when an asymmetrical relay is introduced in series with the process in closed loop (Majhi, 2005). Method 53: Parameters estimated from data obtained when an asymmetrical relay is introduced in series with the process in closed loop (Prokop and Korbel, 2006). A2.1.4 Other methods Method 54: Parameter estimates back-calculated from a discrete time identification method (Ferretti et al., 1991). Method 55: Parameter estimates determined graphically from a known higher order process (McMillan, 1984). Method 56: Parameter estimates determined from a known higher order process (Skogestad, 2003). Method 57: The parameters of a second order model plus time delay are estimated using a system identification approach in discrete time; the parameters of a FOLPD model are subsequently determined using standard equations (Ou and Chen, 1995). Method 58: Parameter estimates back-calculated from a discrete time identification non-linear regression method (Gallier and Otto, 1968). Method 59: Parameter estimates determined using a recursive least squares method (Bai and Zhang, 2007). b720_Appendix-02 17-Mar-2009 FA Appendix 2: Some Further Details on Process Modelling 557 A2.2 FOLPD Model with a Zero Method 2: Method 3: Non-delay parameters are estimated in the discrete time domain using a least squares approach; time delay assumed known (Chang et al., 1997). Parameters estimated graphically from the open-loop step response (Slätteke, 2006). A2.3 SOSPD Model A2.3.1 Parameters estimated from the open loop process step response Method 2: Method 3: Method 4: Method 5: Method 6: 5 K m , Tm1 and τ m are determined from the tangent and point method of Ziegler and Nichols (1942) (Shinskey, 1988, page 151); Tm 2 assumed known. FOLPD model parameters estimated using a tangent and point method (Ziegler and Nichols, 1942); corresponding SOSPD parameters subsequently deduced (Auslander et al., 1975). Parameters estimated using a tangent and point method (Murata and Sagara, 1977). Tm1 = Tm 2 . τ m , Tm1 estimated from process step response: Tm1 = 0.794( t 70% − t 33% ) , τ m = 1.937 t 33% − 0.937 t 70% . Km 5 assumed known (Vítečková et al., 2000b). Tm1 = Tm 2 . K m , τ m and Tm1 estimated from the process step response; τ m = time for which the process variable does not change; Tm1 determined at 73% of the total process variable change (Pomerleau and Poulin, 2004). t 70% , t 33% are the times taken by the process variable to change by 70% and 33%, respectively, of its total value. b720_Appendix-02 558 Method 7: 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules Parameters estimated from the underdamped or overdamped transient response in open loop to a step input (Jahanmiri and Fallahi, 1997). A2.3.2 Parameters estimated from the closed loop step response Method 8: Method 9: Parameters estimated from a step response autotuning experiment – Honeywell UDC 6000 controller (Åström and Hägglund, 1995). Parameters estimated from the servo or regulator closed loop transient response, under PI control (Rotach, 1995). A2.3.3 Parameters estimated from frequency domain closed loop information Method 10: In this method, Tm1 = Tm 2 = Tm . Tm and τ m estimated from K u , Tu determined using a relay autotuning method; K m estimated from the process step response (Hang et al., 1993a). Method 11: Parameters estimated using a two-stage identification procedure involving (a) placing a relay and (b) placing a proportional controller, in series with the process in closed loop (Sung et al., 1996). Method 12: Parameters estimated using a two-stage identification procedure involving (a) placing a relay in series with the process in closed loop and (b) introducing an excitation to the relay feedback system through an external DC input at the output of the relay (Huang et al., 2003). Method 13: Parameters estimated from data determined by inserting a relay in series with the process in closed loop (Lavanya et al., 2006a). Method 14: Parameters estimated in the frequency domain from the data determined using a relay in series with the closed loop system in a master feedback loop (Hwang, 1995). b720_Appendix-02 17-Mar-2009 Appendix 2: Some Further Details on Process Modelling FA 559 Method 15: Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharromán and Pagola, 1999). Method 16: Parameters estimated from data obtained when the process phase lag is − 90 0 and − 180 0 , respectively (Wang et al., 1999). Method 17: Model parameters estimated by including a dynamic compensator outside or inside an (ideal) relay feedback loop (Huang and Jeng, 2003). Method 18: K m estimated from a relay autotuning method; τ m , Tm1 and ξ m estimated using a least squares algorithm based on the data recorded from the relay autotuning method, with the aid of neural networks (Huang et al., 2005a). Method 19: Model parameters estimated using a two-stage identification procedure involving (a) placing a relay and (b) placing a relay which operate on the integral of the error, in series with the process in closed loop (Naşcu et al., 2006). Method 20: Model parameters estimated using a relay autotuning method (Thyagarajan and Yu, 2003). A2.3.4 Other methods Method 21: Parameter estimates back-calculated from a discrete time identification method (Ferretti et al., 1991). Method 22: Parameter estimates back-calculated from a second discrete time identification method (Wang and Clements, 1995). Method 23: Parameter estimates back-calculated from a third discrete time identification method (Lopez et al., 1969). Method 24: Model parameters estimated assuming higher order process parameters are known (Skogestad, 2003). Method 25: Parameter estimates back-calculated from a discrete time identification non-linear regression method (Gallier and Otto, 1968). b720_Appendix-02 560 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules A2.4 SOSPD Model with a Zero Method 2: Method 3: Parameters estimated from a closed loop step response test using a least squares approach (Wang et al., 2001a). Tm1 = Tm 2 . K m , τ m , Tm1 and Tm 3 (or Tm 4 ) are estimated from the process step response; the latter three parameters are estimated using a tabular approach (Pomerleau and Poulin, 2004). A2.5 TOSPD Model Method 2: Method 3: Parameters estimated using a tangent and point method (Murata and Sagara, 1977). Model parameters estimated using least squares regression in the time domain; the model input is a step signal (Yu, 2006, p. 107). A2.6 General Model Method 2: A FOLPD model is obtained using the tangent and point method of Ziegler and Nichols (1942); label the parameters K 'm , Tm' and τ 'm . ( Method 3: Method 4: ) Then, n = 10 τ 'm Tm' + 1 . Subsequently, τ m is estimated from the open loop step response, and Tar is estimated using ‘known methods’ (Schaedel, 1997). Parameters estimated using a tangent and point method (Murata and Sagara, 1977). Model parameters Taa , τ m and Tar are estimated using the area method (Gorez and Klàn, 2000). A2.7 Non-Model Specific Modelling methods have not been specified in the tables in most cases, as typically the tuning rules are based on K u and Tu . Modelling methods have been specified whenever relevant data have been estimated using a relay method. b720_Appendix-02 17-Mar-2009 FA Appendix 2: Some Further Details on Process Modelling Method 2: Method 3: Method 4: 561 K u and Tu estimated from an experiment using a relay in series with the process in closed loop (Jones et al., 1997). K u and Tu estimated with the assistance of a relay (Lloyd, 1994). ω900 and A p estimated using a relay in series with an integrator (Tang et al., 2002). Method 5: Method 6: ∧ ∧ K u and Tu estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharromán and Pagola, 1999). Parameters estimated from an experiment using a relay in series with the process in closed loop (Chen and Yang, 1993). A2.8 IPD Model A2.8.1 Parameters estimated from the open loop process step response Method 2: Method 3: Method 4: Method 5: τ m assumed known; K m estimated from the slope at start of the process step response (Ziegler and Nichols, 1942). K m , τ m estimated from the process step response (Hay, 1998). K m , τm estimated from the process step response (Clarke, 2006). K m , τ m estimated from the process step response using a tangent and point method (Bunzemeier, 1998). A2.8.2 Parameters estimated from the closed loop step response Method 6: Parameters estimated from the servo or regulator closed loop transient response, under PI control (Rotach, 1995). b720_Appendix-02 562 Method 7: Method 8: Method 9: 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules Parameters estimated from the servo closed loop transient response under proportional control (Srividya and Chidambaram, 1997). Parameters estimated from the closed loop response, under the control of an on-off controller (Zou et al., 1997). Parameters estimated from the closed loop response, under the control of an on-off controller (Zou and Brigham, 1998). A2.8.3 Parameters estimated from frequency domain closed loop information Method 10: Parameters estimated from K u and Tu values, determined from an experiment using a relay in series with the process in closed loop (Tyreus and Luyben, 1992). Method 11: Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharromán and Pagola, 1999). Method 12: Parameters estimated from data determined when a relay is placed in series, in closed loop, with the closed loop compensated process, under PI or PID control (NI Labview, 2001). A2.9 FOLIPD Model Method 2: Method 3: Method 4: Method 5: Parameters estimated from the open loop process step response and its first and second time derivatives (Tsang and Rad, 1995). Parameters estimated using the method of moments (Åström and Hägglund, 1995). Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharromán and Pagola, 1999). Parameters estimated from the open loop response of the process to a pulse signal (Tachibana, 1984). b720_Appendix-02 17-Mar-2009 Appendix 2: Some Further Details on Process Modelling Method 6: Method 7: Method 8: FA 563 Parameters estimated from the waveform obtained by introducing a single symmetrical relay in series with the process in closed loop (Majhi and Mahanta, 2001). K m estimated from the response of the process to a square wave pulse; other process parameters estimated from the waveform obtained by introducing a relay in series with the process in closed loop (Wang et al., 2001a). Parameters estimated using a relay in series with the process in closed loop (Perić et al., 1997). A2.10 SOSIPD model Method 2: Parameters estimated from values determined from an experiment using an amplitude dependent gain in series with the process in closed loop (Pecharromán and Pagola, 1999). A2.11 Unstable FOLPD Model Method 2: Method 3: Method 4: Method 5: Method 6: Parameters estimated by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin, 1995; Deshpande, 1980). Parameters estimated using a relay feedback approach (Majhi and Atherton, 2000). Parameters estimated using a biased relay feedback test (Huang and Chen, 1999). Parameters estimated using a single symmetric relay feedback feedback test (Vivek and Chidambaram, 2005). Tm and τ m estimated from data obtained when a relay is introduced in series with the process in closed loop; K m assumed known (Padhy and Majhi, 2006a). b720_Appendix-02 564 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules A2.12 Unstable SOSPD Model (one unstable pole) Method 2: Method 3: Method 4: Parameters estimated by least squares fitting from the open loop frequency response of the unstable process; this is done by determining the closed loop magnitude and phase values of the (stable) closed loop system and using the Nichols chart to determine the open loop response (Huang and Lin, 1995; Deshpande, 1980). Parameters estimated by minimising the difference in the frequency responses between the high order process and the model, up to the ultimate frequency. The gain and delay are estimated from analytical equations, with the other parameters estimated using a least squares method (Kwak et al., 2000). 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Optimum settings for automatic controllers, Transactions of the ASME, November, pp. 759–768. Zou, H., Hedstrom, K.P., Warrior, J. and Hays, C.L. (1997). Field based process control system with auto-tuning, US Patent Number 5,691,896. Zou, H. and Brigham, S.E. (1998). Process control system with asymptotic auto-tuning, US Patent Number 5,818,714. b720_Bibliography 17-Mar-2009 This page intentionally left blank FA b720_Index 17-Mar-2009 FA Index Åström and Hägglund (2004), 20, 98, 174, 360, 381 Åström and Hägglund (2006), 21, 22, 72, 74, 76, 109, 110–111, 145, 177, 227, 276, 321, 326, 356, 362, 390, 424 Atherton and Boz (1998), 175, 274, 477, 503–504 Atherton and Majhi (1998), 476–477 Atkinson (1968), 341 Atkinson and Davey (1968), 324 Auslander et al. (1975), 206 ABB (1996), 325 ABB (2001), 9, 32, 52, 152, 157, 319 Abbas (1997), 57, 100 Aikman (1950), 321 Alenany et al. (2005), 360, 378 Alfa-Laval, 7, 322 Alfaro Ruiz (2005a), 80, 122, 124, 207, 326, 359 Allen Bradley, 5, 8 Alvarez-Ramirez et al. (1998), 357 Andersson (2000), 74, 357 Anil and Sree (2005), 422, 423 Araki (1985), 156 Arbogast and Cooper (2007), 365, 367, 370 Arbogast et al. (2006), 74 Argelaguet et al. (1997), 156 Argelaguet et al. (2000), 156 Arrieta and Vilanova (2007), 125–127 Arrieta Orozco (2003), 39, 43, 47, 49, 122, 123, 124 Arrieta Orozco and Alfaro Ruiz (2003), 39, 43, 47, 49, 122, 123, 124 Arrieta Orozco (2006), 43, 49, 86, 87–88, 91, 94 Arvanitis et al. (2000a), 221, 232, 261, 275 Arvanitis et al. (2000b), 473, 477 Arvanitis et al. (2003a), 373, 376, 377 Arvanitis et al. (2003b), 400–409, 413 Åström (1982), 322, 325 Åström and Hägglund (1984), 325 Åström and Hägglund (1988), 79, 325 Åström and Hägglund (1995), 21, 56, 108, 164, 248, 319, 322, 325, 331, 347, 351, 359, 389, 413 Åström and Hägglund (2000), 19 Bai and Zhang (2007), 70 Bailey, 5, 6, 7, 8, 10 Bain and Martin (1983), 46, 53 Barberà (2006), 49, 94 Bateson (2002), 322, 327, 349 Bekker et al. (1991), 55 Belanger and Luyben (1997), 370 Benjanarasuth et al. (2005), 353, 379 Bequette (2003), 21, 24, 120 Bialkowski (1996), 74, 283, 287, 357, 436 Bi et al. (1999), 57 Bi et al. (2000), 57, 221 Blickley (1990), 324 Boe and Chang (1988), 77 Bohl and McAvoy (1976b), 210, 230, 241, 250 Borresen and Grindal (1990), 32, 79 Boudreau and McMillan (2006), 223, 357 Brambilla et al. (1990), 74, 112, 203, 227 Branica et al. (2002), 109 Bryant et al. (1973), 20, 52, 197–198 Buckley (1964), 21 599 b720_Index 600 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules Buckley et al. (1985), 354 Bunzemeier (1998), 351, 368 Calcev and Gorez (1995), 319, 325 Callender (1934), 359 Callender et al. (1935/6), 18, 23, 30, 78 Camacho et al. (1997), 100 Carr (1986), 324 Chandrashekar et al. (2002), 441, 455–456, 466–468 Chang et al. (1997), 182 Chao et al. (1989), 189, 190, 194, 239, 240, 241–242, 242–243, 244, 245, 246–247 Chau (2002), 326 Chen and Seborg (2002), 58, 103, 221–222, 357, 361, 389 Chen and Yang (1993), 318, 325, 330 Chen and Yang (2000), 68 Chen et al. (1997), 75 Chen et al. (1999a), 64, 354, 362, 389 Chen et al. (1999b), 64, 226 Chen et al. (2006), 283 Cheng and Hung (1985), 86, 93 Cheng and Yu (2000), 354 Chesmond (1982), 322, 327 Chidambaram (1994), 354 Chidambaram (1995a), 107–108 Chidambaram (1995b), 443, 452 Chidambaram (1997), 440, 448, 474 Chidambaram (1998), 443, 452 Chidambaram (2000a), 161, 176 Chidambaram (2000b), 376 Chidambaram (2000c), 441, 465 Chidambaram (2002), 28, 33, 58, 80, 103, 162, 181, 222, 280, 383, 420, 441–442, 449, 466 Chidambaram and Kalyan (1996), 492 Chidambaram and Sree (2003), 353, 361, 362, 376 Chidambaram et al. (2005), 59 Chien (1988), 74, 130, 146, 285, 287, 357, 366, 369, 394, 396 Chien and Fruehauf (1990), 130, 146, 285, 287, 357, 366, 369, 394, 396 Chien et al. (1952), 31, 78 Chien et al. (1999), 160–161, 376 Chien et al. (2003), 277, 282, 284, 286, 290, 292 Chiu et al. (1973), 52, 220 Chun et al. (1999), 75 Clarke (2006), 72, 356 Cluett and Wang (1997), 66–67, 101, 353, 361 Cohen and Coon (1953), 31, 78 Concept, 6 Connell (1996), 79 Controller structures Classical controller, 6–7, 25, 134–148, 238–250, 286–287, 341–342, 368–370, 395–396, 423, 425, 436, 458–461, 491– 496 Controller with filtered derivative, 6, 122– 133, 236–237, 284–285, 298, 305–307, 316, 336–340, 366–367, 394 Generalised classical controller, 8, 26, 149–151, 251–252, 288, 343–345, 371, 397–398, 462, 483, 508, 516–517 Ideal controller in series with a first order lag, 6, 24, 118–121, 182, 232–235, 282– 283, 297, 315, 332–335, 364–365, 392– 393, 422, 455–457, 481–482, 490, 513– 515 Ideal PI controller, 5, 18–22, 28–29, 30– 77, 180–181, 183–205, 277–278, 293– 295, 310–311, 318–323, 350–358, 383– 384, 385–387, 430, 438, 440–446, 480, 486–487, 511–512 Ideal PID controller, 4, 5, 23, 78–117, 206–231, 279–281, 296, 303–304, 312– 314, 324–331, 359–363, 388–391, 420– 421, 424, 447–454, 488–489, 506–507 Two degree of freedom controller 1, 8–9, 27, 152–167, 253–263, 289–291, 299– 301, 308–309, 317, 346–348, 372–377, 399–415, 426, 431–435, 437, 439, 463– 472, 484–485, 497–502, 518–519 Two degree of freedom controller 2, 9, 168–169, 378–380, 416–417, 427–428, 473–474, 509–510 Two degree of freedom controller 3, 10– 11, 170–179, 264–276, 292, 302, 349, 381–382, 418–419, 429, 475–479, 503– 505, 520 ControlSoft Inc. (2005), 80 Coon (1956), 350, 385 Coon (1964), 350, 385 Cooper (2006a), 165, 179 Cooper (2006b), 120, 179 Corripio (1990), 324, 341 Cox et al. (1994), 323 b720_Index 17-Mar-2009 Index Cox et al. (1997), 67 Cuesta et al. (2006), 60 Davydov et al. (1995), 57, 128, 311, 316 De Oliveira et al. (1995), 74 De Paor (1993), 325 De Paor and O’Malley (1989), 442, 452 Desbiens (2007), 181, 203, 278 Devanathan (1991), 45, 90 Dutton et al. (1997), 330 ECOSSE Team (1996a), 45 ECOSSE Team (1996b), 322, 327 Edgar et al. (1997), 38, 137–138, 153, 319, 341, 346 Eriksson and Johansson (2007), 174, 178 Ettaleb and Roche (2000), 57 Faanes and Skogestad (2004), 33, 58, 74, 136, 147 Farrington (1950), 324 Fanuc, 5 Fertik (1975), 19, 42, 49, 136, 140 Fertik and Sharpe (1979), 32 Fischer and Porter, 6, 7, 10 Fisher–Rosemount, 10 Foley et al. (2005), 59 Ford (1953), 359 Foxboro, 6, 7, 10, 134 Frank and Lenz (1969), 33, 40, 41, 43 Friman and Waller (1997), 64, 109, 319 Fruehauf et al. (1993), 56, 99, 352 Fukura and Tamura (1983), 50, 197 Gaikward and Chidambaram (2000), 443, 452 Gain margin, analytical calculation, 522– 528 Gain margin, figures, 530, 532–533 Gain and phase margins, constant, 534–543 PI controller, FOLPD model, 534–536 PI controller, IPD model, 536–539 PD controller, FOLIPD model, 542–543 PID controller, FOLPD model, classical controller, 539–541 PID controller, SOSPD model, series controller, 541–542 PID controller, SOSPD model with a negative zero, classical controller, 542 FA 601 Gallier and Otto (1968), 45, 90, 191, 214 García and Castelo (2000), 328 Genesis, 9 Geng and Geary (1993), 77, 116 Gerry (1998), 19, 89 Gerry (1999), 74, 113 Gerry (2003), 45 Gerry and Hansen (1987), 19 Gong et al. (1996), 130 Gong et al. (1998), 23, 131 Gong (2000), 113 Gonzalez (1994), 56, 100 Goodwin et al. (2001), 129 Gorecki et al. (1989), 55, 62, 99, 105, 352, 360, 362 Gorez (2003), 58, 104, 109 Gorez and Klàn (2000), 220, 312, 313 Gu et al. (2003), 117 Haalman (1965), 19, 40, 239, 352, 388 Haeri (2002), 163 Haeri (2005), 59 Hägglund and Åström (1991), 322 Hägglund and Åström (2002), 68, 165, 376, 471 Hägglund and Åström (2004), 21, 70–71, 202–203, 205, 321, 355–356 Hang and Åström (1988a), 167 Hang and Åström (1988b), 325 Hang and Cao (1996), 167 Hang et al. (1991), 55, 167 Hang et al. (1993a), 63, 223, 233 Hang et al. (1993b), 63, 134, 318, 341 Hang et al. (1994), 236 Hansen (1998), 275 Hansen (2000), 21, 376, 426 Harriott (1964), 327 Harriott (1988), 39, 47, 184 Harris and Tyreus (1987), 146, 178 Harrold (1999), 135, 342 Hartmann and Braun, 6 Hartree et al. (1937), 25 Hassan (1993), 189, 210–211 Hay (1998), 32–33, 79–80, 322, 327, 351, 352, 359 Hazebroek and Van der Waerden (1950), 30–31, 39–40, 352 Heck (1969), 32 Henry and Schaedel (2005), 65 Hill and Adams (1988), 77 b720_Index 602 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules Hill and Venable (1989), 36–37, 83–85 Hiroi and Terauchi (1986), 152, 170 Ho and Xu (1998), 443, 492 Ho et al. (1994), 223 Ho et al. (1995a), 223–224 Ho et al. (1997), 224 Ho et al. (1998), 87, 92 Ho et al. (1999), 63, 87, 92 Ho et al. (2001), 178 Honeywell, 5, 7, 9, 11, 56, 248 Horn et al. (1996), 120, 151 Hougen (1979), 61, 146, 199, 247, 294, 354, 369, 387, 396 Hougen (1988), 199, 248, 369 Huang and Chao (1982), 136, 138, 139, 140, 141, 142, 143, 238, 240, 242, 243, 244, 245, 246 Huang and Chen (1997), 458, 492 Huang and Chen (1999), 458, 492, 502 Huang and Jeng (2002), 19, 46 Huang and Jeng (2003), 97–98, 218–219 Huang and Jeng (2005), 71, 146, 232, 249 Huang and Lin (1995), 463, 464, 497–502 Huang et al. (1996), 38, 46, 185–187, 191– 193, 264–268, 270–273 Huang et al. (1998), 74, 228 Huang et al. (2000), 165, 176, 261, 276, 290, 292 Huang et al. (2002), 121 Huang et al. (2003), 146, 251 Huang et al. (2005a), 71, 77, 146, 148, 232, 235 Huang et al. (2005b), 379, 416 Huba (2005), 429 Huba and Žáková (2003), 353, 387 Hwang (1995), 44, 48, 89, 96–97, 190, 195– 196, 213, 215–217 Hwang and Chang (1987), 318 Hwang and Fang (1995), 44, 51, 89, 97 Idzerda et al. (1955), 318 Intellution, 5 Isaksson and Graebe (1999), 75 Jacob and Chidambaram (1996), 440, 473 Jahanmiri and Fallahi (1997), 251 Jhunjhunwala and Chidambaram (2001), 448, 466 Johnson (1956), 318 Jones and Tham (2006), 296, 302 Jones et al. (1997), 323, 329 Juang and Wang (1995), 100 Jyothi et al. (2001), 29, 182, 291, 383–384, 420–421 Kamimura et al. (1994), 21, 56 Kang (1989), 208 Karaboga and Kalinli (1996), 326 Kasahara et al. (1997), 160 Kasahara et al. (1999), 131–132 Kasahara et al. (2001), 161 Kaya (2003), 381 Kaya and Atherton (1999), 381 Kaya and Scheib (1988), 136, 138, 139, 140, 142, 171, 172 Keane et al. (2000), 218 Keane et al. (2005), 331 Keviczky and Csáki (1973), 19, 20, 53, 194 Khan and Lehman (1996), 48, 55, 57 Khodabakhshian and Golbon (2004), 278 Kim et al. (2004), 163 King (2006), 73, 112 Kinney (1983), 341 Klán (2000), 58, 312 Klán (2001), 376, 390, 413 Klán and Gorez (2000), 319 Klán and Gorez (2005a), 59 Klán and Gorez (2005b), 59 Klein et al. (1992), 32, 61 Kookos et al. (1999), 354 Kosinsani (1985), 34, 82 Kotaki et al. (2005a), 153–154, 164 Kotaki et al. (2005b), 153–155 Kraus (1986), 134 Kristiansson (2003), 69, 149–150, 204, 228, 234, 251, 320, 332, 344 Kristiansson and Lennartson (1998a), 337– 338 Kristiansson and Lennartson (1998b), 338 Kristiansson and Lennartson (2000), 320, 338–339 Kristiansson and Lennartson (2002a), 320, 340, 366, 394 Kristiansson and Lennartson (2002b), 69, 119, 334 Kristiansson and Lennartson (2003), 320, 321 Kristiansson and Lennartson (2006), 69–70, 119, 120, 321, 335, 392 Kristiansson et al. (2000), 343 b720_Index 17-Mar-2009 FA Index 603 Kuhn (1995), 57, 128 Kukal (2006), 60, 104 Kuwata (1987), 20, 27, 53, 77, 116, 158, 159, 160, 166–167, 198, 204, 231, 259, 262–263, 294, 295, 296, 300, 301, 311, 317, 352, 375, 386, 413, 415, 430, 435, 437, 438, 439 Kwak et al. (2000), 504–505 Loron (1997), 438 Luo et al. (1996), 326 Luyben (1993), 77 Luyben (1996), 370 Luyben (1998), 443 Luyben (2000), 278 Luyben (2001), 74, 120 Luyben and Luyben (1997), 326 Landau and Voda (1992), 231 Larionescu (2002), 314, 316 Latzel (1988), 61, 129–130 Lavanya et al. (2006a), 237 Lavanya et al. (2006b), 73 Lee and Edgar (2002), 121, 168–169, 178– 179, 233, 379, 382, 416, 419, 474, 479 Lee and Shi (2002), 147, 151 Lee and Teng (2003), 313 Lee et al. (1992), 63, 106, 164, 177 Lee et al. (1998), 75, 113, 228 Lee et al. (2000), 454, 474, 489, 505, 507, 509 Lee et al. (2005), 323 Lee et al. (2006), 113, 227, 281, 362, 390, 421, 457, 505 Leeds and Northup, 5, 79 Lennartson and Kristiansson (1997), 337 Leonard (1994), 360 Leva (1993), 322, 336 Leva (2001), 65, 74, 112, 132 Leva (2005), 165 Leva (2007), 74 Leva and Colombo (2000), 132 Leva and Colombo (2004), 165 Leva et al. (1994), 70, 224–225 Leva et al. (2003), 68–69, 132, 202, 226 Leva et al. (2006), 132 Li et al. (1994), 105 Li et al. (2004), 283 Lim et al. (1985), 118, 232 Lipták (1970), 327 Lipták (2001), 33, 80 Litrico and Fromion (2006), 356, 363 Litrico et al. (2006), 356 Litrico et al. (2007), 358 Liu et al. (2003), 417, 427–428 Lloyd (1994), 328 Lopez (1968), 81, 86, 87, 183, 188, 207, 208–209 Lopez et al. (1969), 188, 210 MacLellan (1950), 321, 327 Madhuranthakam et al. (2008), 155, 156, 256, 257, 269, 289–290 Maffezzoni and Rocco (1997), 130 Majhi (1999), 323, 329, 346, 347 Majhi (2005), 50 Majhi and Atherton (1999), 175, 274, 477, 503–504 Majhi and Atherton (2000), 440, 475–476 Majhi and Mahanta (2001), 419 Majhi and Litz (2003), 250 Mann et al. (2001), 54, 102 Mantz and Tacconi (1989), 348 Manum (2005), 21, 222, 442 Marchetti and Scali (2000), 181, 203, 228, 278, 280, 295, 296 Marchetti et al. (2001), 453, 488 Marlin (1995), 38, 46, 81, 91 Mataušek and Kvaščev (2003), 69 Matsuba et al. (1998), 77, 116 Maximum sensitivity, analytical calculation, 529 Maximum sensitivity, figures, 530 Maximum sensitivity, constant, 538 McAnany (1993), 56 McAvoy and Johnson (1967), 187, 240, 324 McMillan (1984), 76, 116, 387, 391, 487, 489 McMillan (1994), 32, 318, 322, 325, 327 McMillan (2005), 22, 59, 322, 325, 335 Mesa et al. (2006), 60 Modicon, 6 Modcomp, 8 Mollenkamp et al. (1973), 52, 220 Moore, 10, 11 Morari and Zafiriou (1989), 130 Morilla et al. (2000), 102 Moros (1999), 31, 80 Murata and Sagara (1977), 34, 46, 183, 191, 293, 310 Murrill (1967), 32, 33, 40, 41, 79, 81, 86, 87 b720_Index 604 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules Naşcu et al. (2006), 235 NI Labview (2001), 323, 329–330, 350, 351, 359 Normey-Rico et al. (2000), 168, 378 Normey-Rico et al. (2001), 378 Nomura et al. (1993), 20, 23, 27, 56, 99– 100, 160 O’Connor and Denn (1972), 90 O’Dwyer (2001a), 63, 146, 249, 287, 355, 390 O’Dwyer (2001b), 135–136, 342 Ogawa (1995), 75, 357 Ogawa and Katayama (2001), 162 Okada et al. (2006), 54 OMEGA Books (2005), 152, 348 Omron, 8 Oppelt (1951), 31 Ou and Chen (1995), 139, 142 Ou et al. (2005a), 364, 446 Ou et al. (2005b), 365, 393 Ou et al. (2005c), 132 Oubrahim and Leonard (1998), 263, 415 Ozawa et al. (2003), 163 Padhy and Majhi (2006a), 177, 478 Padhy and Majhi (2006b), 73 Pagola and Pecharromán (2002), 323 Panda et al. (2004), 204, 228, 232, 234, 249 Panda et al. (2006), 357 Paraskevopoulos et al. (2004), 464–465, 469 Paraskevopoulos et al. (2006), 444–445, 459–461, 492–496 Park et al. (1997), 212, 214 Park et al. (1998), 478 Parr (1989), 32, 79, 318, 322, 323, 324, 327, 328 Paz Ramos et al. (2005), 360 Pecharromán (2000a), 257, 258, 374, 410, 431 Pecharromán (2000b), 258–259, 374–375, 411, 432–433 Pecharromán and Pagola (2000), 257–258, 374, 410, 432 Pemberton (1972a), 34, 220 Pemberton (1972b), 219–220 Peng and Wu (2000), 81 Penner (1988), 358 Perić et al. (1997), 77, 116, 387, 391 Pessen (1953), 341 Pessen (1954), 341 Pessen (1994), 86, 148, 318 Pettit and Carr (1987), 324 Phase margin, analytical calculation, 522– 528 Phase margin, figures, 530, 532–533 Pinnella et al. (1986), 54 PMA (2006), 33, 80 Polonyi (1989), 296 Pomerleau and Poulin (2004), 199, 277, 282 Potočnik et al. (2001), 66 Poulin and Pomerleau (1996), 352, 386, 395–396, 486, 491 Poulin and Pomerleau (1997), 395–396, 491 Poulin and Pomerleau (1999), 353, 387 Poulin et al. (1996), 201, 249, 286, 396, 487 Pramod and Chidambaram (2000), 449 Prashanti and Chidambaram (2000), 470– 471 Process models Delay model, 12, 18–27 Delay model with a zero, 12, 28–29 Fifth order system plus delay model, 13, 303–309 FOLIPD model, 14, 385–419 FOLIPD model with a zero, 14, 420–423 FOLPD model, 13, 30–179 FOLPD model with a zero, 13, 180–182 General model, 14, 310–317 General model with integrator, 15, 438– 439 IPD model, 14, 350–382 IPD model with a zero, 14, 383–384 I 2 PD model, 14, 424–429 Non-model specific, 14, 318–349 SOSIPD model, 14, 430–435 SOSIPD model with a zero, 14, 436 SOSPD model, 13, 183–276 SOSPD model with a zero, 13, 277–292 TOSIPD model, 14, 437 TOSPD model, 13, 293–302 Unstable FOLPD model, 15, 440–479 Unstable FOLPD model with a zero, 15, 480–485 Unstable SOSPD model (one unstable pole), 15, 486–505 Unstable SOSPD model (two unstable poles), 15, 506–510 Unstable SOSPD model with a zero, 15, 511–520 b720_Index 17-Mar-2009 Index Prokop et al. (2000), 57 Prokop et al. (2005), 319 Pulkkinen et al. (1993), 75 Ramasamy and Sundaramoorthy (2007), 330–331, 333, 344–345 Rao and Chidambaram (2006a), 516 Rao and Chidambaram (2006b), 516, 520 Ream (1954), 23, 52, 197 Reswick (1956), 31 Rice (2004), 362, 370 Rice (2006), 377 Rice and Cooper (2002), 364, 369 Rivera et al. (1986), 74, 112 Rivera and Jun (2000), 120, 228, 233, 364, 390, 393 Robbins (2002), 319, 326 Rotach (1994), 327–328 Rotach (1995), 199, 220, 352, 361 Rotstein and Lewin (1991), 446, 453, 488 Rovira et al. (1969), 45, 48, 90, 93 Rutherford (1950), 321, 327 Sadeghi and Tych (2003), 91, 92, 93 Sain and Özgen (1992), 79 Saito et al. (1990), 99 Sakai et al. (1989), 32 SATT Instruments, 11 Schaedel (1997), 65, 118, 128, 144, 201, 248, 297, 298, 311, 315 Schlegel (1998), 21 Schneider (1988), 55 Seki et al. (2000), 325 Setiawan et al. (2000), 158 Shamsuzzoha and Lee (2006a), 151, 252, 288, 363 Shamsuzzoha and Lee (2006b), 454 Shamsuzzoha and Lee (2007a), 115, 166, 229–230, 262, 363, 377, 391, 415, 454, 489 Shamsuzzoha and Lee (2007b), 150 Shamsuzzoha and Lee (2007c), 169 Shamsuzzoha and Lee (2007d), 508 Shamsuzzoha et al. (2004), 151 Shamsuzzoha et al. (2005), 454, 489 Shamsuzzoha et al. (2006a), 398 Shamsuzzoha et al. (2006b), 510 Shamsuzzoha et al. (2006c), 114, 229, 363, 390, 422 Shamsuzzoha et al. (2007), 371, 462 FA 605 Shen (1999), 254 Shen (2000), 255–256 Shen (2002), 158, 347 Shi and Lee (2002), 147 Shi and Lee (2004), 252 Shigemasa et al. (1987), 159 Shin et al. (1997), 329 Shinskey (1988), 19, 35, 39, 137, 138, 153, 156, 208, 238, 351, 368, 372, 458 Shinskey (1994), 19, 39, 138, 157, 184, 239, 254, 351, 368, 372, 385, 395, 399 Shinskey (1996), 19, 38, 39, 137, 152, 184, 187, 239, 253, 254, 368–369, 372 Shinskey (2000), 33, 135 Shinskey (2001), 33 Shinskey (2003), 39, 138 Siemens, 6 Sklaroff (1992), 56, 248 Skoczowski (2004), 59, 249 Skoczowski and Tarasiejski (1996), 314 Skogestad (2001), 20, 352 Skogestad (2003), 20, 21, 22, 58, 248, 352, 353, 396, 425 Skogestad (2004a), 358 Skogestad (2004b), 221, 353, 418, 429 Skogestad (2004c), 222 Skogestad (2006a), 61 Skogestad (2006b), 61, 249 Slätteke (2006), 180, 181 Smith (1998), 67 Smith (2002), 50, 56, 74, 357, 358 Smith (2003), 93, 228, 326 Smith and Corripio (1997), 34, 46, 53, 138, 140, 144 Smith et al. (1975), 53, 220, 247 Somani et al. (1992), 62, 106, 200–201 Square D, 9 Sree and Chidambaram (2003a), 181, 182 Sree and Chidambaram (2003b), 481, 484– 485 Sree and Chidambaram (2003c), 480, 485 Sree and Chidambaram (2004), 511, 513, 518–519 Sree and Chidambaram (2005a), 467 Sree and Chidambaram (2005b), 361 Sree and Chidambaram (2006), 353, 362, 376, 389, 442, 444, 450–452, 456, 465, 466–467, 468, 474, 477, 480, 481–482, 483, 485, 502, 511, 514–515, 517, 519 Sree et al. (2004), 59, 104, 442, 449–450 b720_Index 606 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules Srinivas and Chidambaram (2001), 162, 465 Srividya and Chidambaram (1997), 354 St. Clair (1997), 32, 135, 341 Streeter et al. (2003), 331 Sung et al. (1996), 212, 214 Suyama (1992), 53, 99, 220 Suyama (1993), 159 Syrcos and Kookos (2005), 90 Tachibana (1984), 388 Taguchi and Araki (2000), 157, 260, 299– 300, 375, 411–412, 433 Taguchi and Araki (2002), 157, 412 Taguchi et al. (1987), 156, 257, 410 Taguchi et al. (2002), 375, 412, 433–434 Tan et al. (1996), 70, 108, 201 Tan et al. (1998a), 393 Tan et al. (1998b), 120, 364, 393 Tan et al. (1999a), 135, 326, 342 Tan et al. (1999b), 446, 453 Tan et al. (2001), 326, 342 Tang et al. (2002), 323, 330 Tang et al. (2007), 121, 133, 147–148 Tavakoli and Fleming (2003), 47 Tavakoli and Tavakoli (2003), 123 Tavakoli et al. (2005), 158, 375 Tavakoli et al. (2006), 47 Taylor, 11 Thomasson (1997), 74, 75, 357 Thyagarajan and Yu (2003), 205 Tinham (1989), 324 Toshiba, 7, 9, 159 Trybus (2005), 60, 222–223, 313–314 Tsang and Rad (1995), 144, 397 Tsang et al. (1993), 144, 149, 397 Tuning rules: Direct synthesis, frequency domain criteria, 21, 29, 61–73, 105–112, 118– 120, 129–130, 146, 149–150, 164–165, 177–178, 181, 182, 199–203, 223–227, 232, 236, 249–250, 261–262, 276, 278, 280, 286–287, 290–291, 294, 297, 298, 301, 303–304, 305–307, 308–309, 311, 314, 315, 319–321, 331, 334–335, 336– 340, 347–348, 349, 354–356, 362, 366, 369, 376, 381, 383–384, 387, 389–390, 392, 394, 396, 398, 413–414, 419, 420– 421, 424, 438, 442–445, 452, 459–461, 470–472, 473, 477–478, 487, 492–496, 504–505 Direct synthesis, time domain criteria, 20– 21, 23, 27, 28, 52–61, 98–105, 118, 128–129, 144–145, 149, 158–164, 168, 175–176, 181, 182, 197–199, 219–223, 232, 247–248, 251, 259–260, 274–276, 277, 280, 282–283, 284, 286, 290, 292, 294, 300, 311, 312–314, 316, 317, 330– 331, 333, 344–345, 352–353, 360–362, 375–376, 378–379, 383, 386–387, 388– 389, 396, 397, 413, 416, 418, 420, 423, 425, 426, 429, 430, 435, 438, 439, 440– 442, 448–452, 455–456, 458, 465–469, 473, 476–477, 480, 481–482, 483, 484– 485, 488, 492, 502, 503–504, 506, 511– 512, 513–515, 516, 518–519, 520 Minimum performance index, regulator tuning, 19, 33–45, 81–90, 122–123, 136–139, 152–155, 171, 180, 183–190, 207–213, 238–243, 253–256, 264–269, 289, 293, 310, 341, 346, 351–352, 360, 368–369, 372–373, 385–386, 388, 395– 396, 399–404, 447, 458, 463–464, 486, 491, 497–499 Minimum performance index, servo tuning, 19, 45–51, 90–97, 123–124, 140–143, 156, 172–173, 191–197, 213– 217, 243–247, 257, 269–273, 279, 290, 293, 310, 347, 360, 373, 378, 386, 388, 405–409, 440, 448, 464, 475–476, 500– 501 Minimum performance index, servo/ regulator tuning, 125–127, 257–259, 299–300, 448 Minimum performance index, other tuning, 19–20, 51–52, 97–98, 156–158, 174, 218–219, 296, 332, 343–344, 347, 352, 360, 374–375, 381, 410–412, 431– 434, 464–465 Other tuning rules, 22, 205, 321–323, 327–330, 358 Process reaction curve methods, 18–19, 23, 25, 30–33, 78–80, 134–136, 152, 170, 206–207, 350–351, 359, 368, 385 Robust, 21–22, 23, 24, 74–76, 112–115, 120–121, 130–133, 146–148, 151, 165– 166, 168–169, 178–179, 181, 182, 203– 204, 227–230, 233–235, 236–237, 251– 252, 278, 280–281, 283, 285, 287, 288, 295, 296, 302, 314, 316, 357–358, 362– 363, 364–365, 366–367, 369–370, 371, b720_Index 17-Mar-2009 Index 377, 379–380, 382, 387, 390–391, 393, 394, 396, 415, 416–417, 419, 421, 422, 427–428, 436, 446, 453–454, 457, 462, 472, 474, 478–479, 482, 488–489, 490, 505, 507, 508, 509–510, 517 Ultimate cycle, 22, 26, 76–77, 116–117, 148, 151, 166–167, 169, 204–205, 230– 231, 235, 250, 262–263, 278, 283, 295, 296, 301, 318–319, 324–326, 335, 341– 342, 348, 358, 363, 370, 371, 387, 391, 415, 435, 437, 487, 489 Turnbull, 7 Tyreus and Luyben (1992), 326, 352 Urrea et al. (2006), 22 Valentine and Chidambaram (1997a), 57, 102 Valentine and Chidambaram (1997b), 448 Valentine and Chidambaram (1998), 440 Van der Grinten (1963), 20, 98, 219, 388 VanDoren (1998), 152, 348 Velázquez-Figueroa (1997), 386, 387, 388 Venkatashankar and Chidambaram (1994), 443 Vilanova (2006), 121 Vilanova and Balaguer (2006), 133 Visioli (2001), 360, 447 Visioli (2002), 179 Visioli (2005), 74 Visioli (2006), 61 Vítečková (1999), 53, 220–221, 353, 389 Vítečková (2006), 61, 105, 199, 223 Vítečková and Víteček (2002), 54, 103 Vítečková and Víteček (2003), 54, 103 Vítečková et al. (2000a), 53, 220–221, 353, 389 Vítečková et al. (2000b), 53, 221 Vivek and Chidambaram (2005), 450 Voda and Landau (1995), 53, 64 Vrančić (1996), 294, 301, 305, 319, 336– 337, 348 Vrančić and Lumbar (2004), 307, 308–309 Vrančić et al. (1996), 294, 319 Vrančić et al. (1999), 303–304, 306, 337 Vrančić et al. (2004a), 294, 307, 308–309 Vrančić et al. (2004b), 294 Wade (1994), 322, 327 Wang (2000), 226 FA 607 Wang (2003), 23, 75–76, 113–114, 237 Wang and Cai (2001), 414 Wang and Cai (2002), 414, 472 Wang and Clements (1995), 220 Wang and Cluett (1997), 352–353, 361 Wang and Cluett (2000), 66–67, 101 Wang and Jin (2004), 488, 506 Wang and Shao (1999a), 225 Wang and Shao (1999b), 68 Wang and Shao (2000a), 68 Wang and Shao (2000b), 226 Wang et al. (1995a), 91, 93, 279 Wang et al. (1995b), 319, 334 Wang et al. (1999), 225 Wang et al. (2000a), 67 Wang et al. (2000b), 280 Wang et al. (2001a), 280 Wang et al. (2001b), 419 Wang et al. (2002), 144–145 Wang et al. (2005), 365 Wills (1962a), 214, 219, 230 Wills (1962b), 207, 213 Wilton (1999), 51–52, 218 Witt and Waggoner (1990), 134–135, 137, 139, 141, 143 Wojsznis and Blevins (1995), 116 Wojsznis et al. (1999), 116–117, 326 Wolfe (1951), 19, 31, 350 Xing et al. (2001), 92, 95, 96 Xu and Shao (2003a), 414 Xu and Shao (2003b), 472 Xu and Shao (2003c), 414 Xu and Shao (2004a), 414, 418 Xu and Shao (2004b), 71 Xu et al. (2004), 71 Xu et al. (2005), 60 Yang and Clarke (1996), 63, 223, 249 Yi and De Moor (1994), 336 Yokogawa, 5, 8 Young (1955), 322 Yu (1988), 35, 40–41, 42 Yu (1999), 326 Yu (2006), 21, 22, 24, 26, 130, 151, 295, 357, 366, 371 Zhang (1994), 56, 286 Zhang (2006), 457 Zhang and Xu (2002), 472, 490 b720_Index 608 17-Mar-2009 FA Handbook of PI and PID Controller Tuning Rules Zhang et al. (1996a), 328 Zhang et al. (1996b), 147 Zhang et al. (1999a), 364, 369 Zhang et al. (1999b), 357, 393 Zhang et al. (2000), 457 Zhang et al. (2002a), 379 Zhang et al. (2002b), 121, 132, 147, 233 Zhang et al. (2002c), 169 Zhang et al. (2003), 169 Zhang et al. (2005), 93 Zhang et al. (2006), 380 Zhong and Li (2002), 75, 236, 358 Zhuang (1992), 41, 44, 48, 50, 51, 86, 88, 92, 95, 172–173 Zhuang and Atherton (1993), 41, 44, 48, 50, 51, 86, 88, 92, 95, 107, 172–173 Ziegler and Nichols (1942), 30, 78, 318, 324, 350, 359 Zou and Brigham (1998), 362 Zou et al. (1997), 112, 362