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A multivariable predictive control of greenhouse microclimate
Conference Paper · October 2010
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A multivariable predictive control of greenhouse microclimate
Mouna Boughamsa, Messaoud Ramdani
Department of Electronics, Faculty of Engineering
University Badji-Mokhtar of Annaba
PO. Box. 12, 23000, Annaba, Algeria
E-mail: messaoud.ramdani@univ-annaba.org
Abstract— This paper deals with the problem of modeling
and control of greenhouses inside climate defined by the air
temperature, humidity and CO2 concentration. The control
objective aims to ensure a favourable inside microclimate
for the culture growth to achieve high yield at low expense,
good quality and low production cost. Achieving this goal is
difficult, due to the complexity of the phenomena involved
in the plant growth process and the high sensitivity to the
outside weather. A multivariable predictive control based
on subspace state space description is used in order to
optimise the future behaviour of the greenhouse environment,
concerning the set-point profile and the minimisation of the
control effort energy. The simulation results show that the
proposed controller provides promising performances.
subspace modeling approach. Section 4 outlines the model
based predictive control. Experimental results concerning
the greenhouse climate control are presented in section 5.
Finally, some concluding remarks as well as some possible
improvments are given in section 6.
Outside weather conditions
wind speed
air temperature
air humidity
CO2 concentration
solar radiation
greenhouse
Keyword : Greenhouse climate, predictive control, Subspace
identification.
I. I NTRODUCTION
Greenhouses are designed to provide a protected indoor
microclimate in which crops can be grown under a tightly
controlled climate. The greenhouse climate is one of the key
factors affecting plant production, and it is influenced by
many elements such as the outside weather, the actuators,
and the crop itself. Fig. 1 outlines the general scheme of
a greenhouse process. The automatic climate control, which
is used mainly to maintain a protected environment despite
fluctuations of external climate, has many advantages such
as energy conservation, better productivity of plants, and
reduced human intervention [3].
Many greenhouses use a conventional PID control, but this
control strategy may not be suitable to garantee the desired performance due the interaction between the different
variables and components in a greenhouse. Motivated by
these disadvantages, recent research led to the development
of strategies for further enhancement of greenhouse climate
control. Optimal and predictive control have been proposed
to decide about heating and ventillation and produce control
actions that regulate air temperature and CO2 concentration
[1], [2], [4]. Most techniques are not designed specifically
in order to enable simultaneous control of air temperature
and humidity concentration in the greenhouses. However,
humidity control has a great effect on crop growth and
production.
In this paper, we investigate the use of a constrained multivariable predictive control of the microclimate of a greenhouse.
The rest of this paper is organized as follows : Section 2
describes briefly the analytic greenhouse model to simulate
the greenhouse. Section 3 provides an overview of the
crops
energy
(temperature)
CO2
CO2 concentration
vapor
(air humdity)
ventilation
heating
CO2 enrichment
vapor injection
control
Fig. 1.
Scheme of greenhouse climate model.
II. T HE GREENHOUSE MODEL
The dynamic behaviour of the greenhouse-climate is a
combination of physical processes involving enery trasfer
(radiation and heat) and mass transfer (water vapour fluxes
and CO2 concentration) taking place in the greenhouse and
from the greenhouse to the outside air, Fig. 1.
These processes depend on the outside climate conditions,
structure of the greenhouse, type and state of the crop and
on actuating control signals, such as ventillation and heating
and CO2 injection to influence photosynthesis and cooling
by evaporation for humidity enrichment and decreasing the
air temperature.
The greenhouse climate model describes the dynamic behaviour of the state variables by the use of differential equations
for the air temperature, humidity and CO2 cencentrations.
The model can be written by deriving the appropriate energy
and mass balances, for the inside temperature, Ta (◦ C),
humidity, Ha (g/m3 ) and CO2 concentration (g/m3 ). Based
on the first principle, the evolution of the temperature of the
greenhouse can be described by the following differential
equation :
III. S UBSPACE IDENTIFICATION METHODS
dTa
Vg
ρCp
= Qr + Qh + Qs − Qc − Qv − Qtr
Sg
dt
(1)
where Vg is teh volume of the greenhouse (m3 ), Sg is
the soil surface (m2 ), ρ is the air density (kg/m3 ), Cp
is the specific heat of the air (Jkg −1 K −1 ), Qr is the
energy contribution by solar radiation (W ), Qh is energy
supplied by the heating system (W ), Qs and Qc are the
heat exchanges by convection and conduction between the
soil surface, cover and the greenhouse air (W ), Qv is the
heat lost by ventilation and infiltration (W ), Qtr is the flux
of the latent heat due to crop transpiration. The effect of
ventillation and filtration is given by :
Qv =
ρCp
φvent (Ta − Te )
Sg
(2)
where φvent is the ventillation rate (m3 /s), which depends
on the specific vent type, opening angle and wind speed,
Te is the external air temperature. The heat fluxes due to
crop transpiration can be deduced from the crop canopy
transpiration and from the inside air saturation deficit. The
heat exchanged by conduction and convection between the
cover, soil and internal greenhouse air, depends on the
difference between the air temperature and their surface
temperature.
Qc = ha−e (Ta − Te )
(3)
Qs = hs−a (Ts − Ta )
(4)
where ha−e is the heat transmission coefficient by convection
and conduction, (W m−2 K −1 ) and hs−a is the heat transmission coefficient throught the greenhouse soil, (W m−2 K −1 ).
The energy coming from the solar radiation can be expessed
as Qrad = crad Se , where crad is the heat load coefficient.
The humidity differential equation is given by :
dHa
(5)
= ΦC,AI − ΦAI,AE − Φcond
dt
where Ccap,h (m) is mass capacity of the greenhouse air. The
water vapour exchange throught the ventillation system is
influenced by the ventillation flux and the difference between
inside and outside air humidities :
Ccap,h
ΦAI,AE = φvent (Ha − He )
(6)
The crop transpiration is driven by the difference in water
vapour pressure between the ambient air and the sub-stomatal
cavity, which is assumed to be saturated with respect to water
vapour. The canopy transiration is described by
ΦC,AI = ccanopy,h cm,h (cres,h Hstoma (Ta ) − Ha )
(7)
where cres,h is a parameter, which reflects the stomatal
resistance to the humidity transpiration and cm,h is a mass
transfer coefficient constant. The ccanopy,h represents the
canopy humidity range. When the outside temperature is
much lower than inside air temperature, the condensation
process takes place. This process can be described by
Φcond = ccond,h (Hroof (Te ) − Ha )
(8)
The subspace identification methods refer to a class of
algorithms based on the approximation of subspaces generated by the row spaces of block Hankel matrices of the
input/output data, to calculate a discrete-time state space
model of the following form :
xk+1 = Axk + Buk + wk
yk = Cxk + Duk + vk
Q S
wi
δij ≥ 0
=
E
wjT vjT
vi
ST R
(9)
(10)
where x, u, and y are the process states, inputs and outputs,
respectively, while A is the system (state transition) matrix,
B is the input matrix, C is the output matrix and D is the
direct input to output matrix. w is called the process noise
and v is called the measurement noise. The matrices Q, S
and R are the covariance matrices of the noise sequences
w and v. E denotes the expected value operator and δij the
Kronecker delta. The subscript index k denotes a discrete
time system. Related to equation (9), it is assumed that the
system is asymptotically stable, the pair (A, C) is observable
and the pair (A, B) is controllable.
There are now many different versions of subspace algorithms, and they have reached a certain level of maturity.
All subspace methods consist of three main steps : first
estimating the predictable subspace for multiple future steps,
then extracting state variables from this subspace and finally
fitting the estimated states to a state-subspace model by applying the least squares method. The subspace identification
algorithm considered in this study is the standard Subspace
State-Space System IDentification (N4SID) algorithm.
First, the measured data are arranged to form block Hankel
matrices Yf , Yp , Uf , and Up , where the subscripts "f" and
"p" denote the future and past, respectively. The Hankel matrices can be arranged to form a linear regression equation :


Up
(11)
Y f = R U p R Yp R U f  Y p 
Uf
which can be solved in a least squeres sense. By excluding
the linear combination of the Uf , the matrix of predicted
outputs can be written as :
Up
(12)
Ŷf = RUp RYp
Yp
It can be shown [7] that the input-state-output relations can
be expressed as
Yf = ΓXf + RUf Uf + Ef
(13)
where Γ is the extended observability matrix. Xf is a matrix
of state sequences stored as row vectors, and Ef is a noise
term. By excluding Uf , the matrix of predicted outputs can
be defined.
Ŷf = ΓX̂f
(14)
where X̂f presents the predicted states, which are up to
now known. By performing the singular value decomposition
(SVD) of (12), deleting small singular values, and comparing
to (14) gives
Ŷf = U SV T ≈ U1 S1 V1T = ΓX̂f
1/2
Γ = U1 S1 , X̂f = Γ+ Ŷf ,
(15)
(16)
where+ denotes the Moore-Penrose pseudoinverse.
Once the matrix of states is given by (16), the state space
model matrices can be found by solving a simple set of
overdetermined equations in a least squares sense
A B
X̂k+1
X̂k
=
+ vk
(17)
C D
Yk
Uk
with vk as residual matrix. In addition, the Kalman gain
K can then, if desired, be computed from A, C, and the
covariance matrix of vk .
Fig. 2.
The basic principle of model-based predictive control.
IV. M ODEL PREDICTIVE CONTROL
Model predictive control (MPC) is a powerfull methodology for controlling industrial processes. Since its emergence more than two decades ago, the technique has been
considerably developed. MPC, is perhaps, the most general
way of posing the control problem in the time domain and
applicable to linear and nonlinear systems, especially when
the reference trajectories are known. The MPC consist of four
basic elements ; (i) a model, which describes the process,
(ii) a goal, defined by an objective function, (iii) optional
constraints on the system and control variables, and (iv) an
optimization procedure. The model describes the dynamic
behaviour of the system and may be black-box, gray-box
or white-box model. The objective function depends on the
predicted future system inputs and output. In general, the
difference between system outputs and a reference trajectory
is used in combination with a cost term on the control effort.
A special case is the quadratic form (18), mostly referred to
as generalized predictive control (GPC), which can be solved
analytically for linear systems without constraints. With
constraints, the optimization problem is a convex Quadratic
Programming (QP) problem, which can be efficiently be
solved numerically. The quadratic form is given by :
J=
Hp
X
i=1
2
kr (k + i) − ŷ (k + i)kPi +
Hc
X
2
ku (k + i − 1)kQi
i=1
(18)
where Pi and Qi are positive definite weight matrices.
In linear model based predictive control, a linear model is
used to predict the output ŷ as a function of the control signal
sequence û (k, · · · , k + Hp ), with Hp the prediction horizon.
The objective function given by (18), is minimized for a
given reference trajectory. The signal u may change over
the control horizon Hc (Hc ≤ Hp ) and remains constant
between Hc and Hp . Given a linear model in state-space
description :
x (k + 1) = Ax (k) + Bu (k)
y (k) = Cx (k)
(19)
the constrained linear model based predictive control can be
obtained by solving the quadratic program
1
T
T
∆ũ H∆ũ + c ∆ũ
(20)
min
∆ũ
2
with :
H = 2 RuT P Ru + Q
T T
T
c = 2 Ru P (Rx Ax (k) − r̃)
(21)
and satisfaying the constraints on u, ∆u, and y :
Λ∆ũ ≤ ω
(22)
with

I∆u
 −I∆u 


Λ =  I Hp m  ,
 −I Hp m 

Ru
−Ru



ω=


umax − Iu u (k − 1)
−umin − Iu u (k − 1)
∆umax
−∆umin
y max −Rx Ax(k)
y min −Rx Ax(k)



 (23)


I Hp m is a (Hp m × Hp m) unity matrix. The matrices
Rx , Ru , Iu and I∆u are defined :


C


CA


(24)
Rx = 

..


.
CAHp −1


CB
0
···
0


CAB
CB
···
0


Ru = 

..
..
.
.
..
..


.
.
CAHp −1 B CAHp −2 B · · · CAHp −Hc B
(25)
V. S IMULATION R ESULTS
For a reliable simulation of a greenhouse climate process,
the nonlinear model described in section II is used as a virtual
system to generate the data. However, it is not very easy to
select either the input or the output variables for the process.
In this work, the air tempeature Ta (◦ C), and the humidity
20.957 49.103
C=
11.947 9.1409
0 0 0
D=
0 0 0
4.3965
1.1767
13.664
6.4769



This paper shows that the constrained multivariable predictive control can be applied successfully to control the
greenhouse climate. The control signals are computed in
order to optimize the future behaviour of the greenhouse
environment, concerning the set-point accuracy and the minimization of the energy. The use of subspace identification
methods has proved to be a valuable tool in the estimation
of LTI state-space model for the greenhouse. This model
is asymptotically stable and it can be used for control. Our
future work will be directed toward the use of the local model
network approach in order to improve the performace of the
predictive controller.
R EFERENCES
[1] I. Islovish, P. O. Gutman and I. Seginer, A nonlinear optimal greenhouse control problem with heating and ventillation, Optimal Control
Applications and Methods, vol. 17, pp. 157-169, 1996.
Ta (°C)
Ha (g/m3)
15
time (h)
20
−5
25
5
10
15
time (h)
20
25
15
10
50
1.6
2
Ventilation (m3/m2.h)
20
CO2 injection (g/m .h)
55
45
40
35
30
1.4
1.2
1
25
5
10
15
20
time (h)
25
Fig. 3.
20
5
10
15
20
time (h)
30
30
25
25
15
10
15
20
time (h)
25
20
15
10
0
0
20
40
60
time (h)
80
−5
100
120
10
100
8
80
60
40
20
0
10
5
5
0
25
Data sequences of the process.
20
VI. C ONCLUSIONS
10
25

In Fig. 4 are presented the process reponses to setpoint
changes (tracking case). It can be observed that the system
responds well. The constraints of the controls are defined by
the boundaries of the domain of the control variables, heating
[0 · · · 150 (W/m2 )], air ventilation [2 · · · 100 (m3 /(m2 · h))]
and CO2 enrichment [0 · · · 10 (g/(m2 · h)]. The states of the
greenhouse climate process are constrained to prevent stress
of the plants. The temperature should always be above 16◦ C
and if possible not higher than 35◦ C.
View publication stats
0
Heat (W/m2)
−0.24666
−0.08253
0.6043
0.45966
5
0
10
CO2 injection (g/m2h)
0.01747
0.00547
−0.04574
−0.02855
data
model
10
20
Ta (°C)
−0.06526
 −0.01797
B=
 0.18028
0.096702
data
model
30
Ventilation (m3/m2)

40
Heat (W/m2)
Ha (g/m2 ) inside the greenhouse are selected as ouputs. The
heating Q (W/m2 ), the ventillation LR (W m3 /(m2 ·h)) and
the CO2 (g/(m2 · h)) enrichment are selected as inputs. Instead of using Pseudo-random binary sequences, multi-level
(m-level) sequences allow the user to highlight nonlinear
system behaviour while manipulating the harmonic content
of the signal. So, the control signals correspond to m-level
uniformly distributed random sequences. Their amplitudes
and frequencies were chosen so as to adequately excite the
system, without deviating too much from the operating point
and therefore enabling the identification of a suitable linear
model. All data signals are stored at a sampling time rate
of 3 min. In Fig. 3 are presented the data sequences of the
process. The identified model is :


0.82096 −0.18947 −0.00076 0.16485
 −0.05124 0.94452 −0.00682 0.05162 

A=
 0.30709
0.83751
0.95905 −0.43743 
0.14914
0.37314 −0.00421 0.71948
0
50
100
time (h)
150
200
0
50
100
time (h)
150
200
0
50
100
time (h)
150
200
6
4
2
0
−2
Fig. 4. System response to setpoint changes and control signals for tracking
setpoints.
[2] I. Islovish, I. Seginer, P. O. Gutman and M. Borshchevsky, Sub-optimal
CO2 enrichment of greenhouses, Journal of Agriculture Engineering
Research, pp. 117, 136,1995.
[3] A. Sriraman, and r. V. Moyorga, A fuzzy inference sytem approach
for greenhouse climate control, Environ. Informatics Archioves, vol.
2, pp. 699-710, 2004.
[4] M. Y. El. Ghoumari, H. J. Tantau, and J. Serrano, Nonlinear constrained MPC : real time implementation of greenhouses air temperature,
Comput. Elect. Agric., vol. 49, pp. 345-356, 2005.
[5] F. Fourati, and M. Chtourou, A greenhouse control with feedforward
and recurrent neural networks, Simulation Modeling Pract. and Theory,
vol. 15, pp. 1016-1028, 2007.
[6] F. Lafont, and J. F. Balmat, Optimized fuzzy control of a greenhouse,
Fuzzy Sets and Syst., vol. 128, pp. 47-59, 2002.
[7] Van Overschee, P. and De Moor, B., Subspace identification for linear
systems : Theory, implementation, Applications. Kluwer Academic,
Dordrecht, 1996.