Copyright © 2002 IFAC
15th Triennial World Congress, Barcelona, Spain
ROBUST CONTROL OF GREENHOUSE CLIMATE EXPLOITING MEASURABLE
DISTURBANCES
J.C. Moreno*, M. Berenguel*, F. Rodríguez*, A BaÅ„os!
*
Universidad de Almería. Dpto. Lenguajes y Computación. Ctra. Sacramento s/n, La
CaÅ„ada, E04120, Almería, Spain. E-mail: {jcmoreno,beren,frrodrig}@ual.es
!
Universidad de Murcia. Facultad de Informática. Dpto. de Informática y Sistemas.
Campus Universitario de Espinardo, E30071, Murcia, Spain. E-mail: [e-mail]
Abstract: This paper presents the development and implementation of robust control
techniques based on the quantitative feedback theory (QFT) aimed at achieving adequate
values of inside greenhouse temperature in spite of uncertainties and disturbances acting
on the system. A modification of classical design approaches has been included to
incorporate feedforward action (exploiting the availability of measurements of
disturbances, which in the particular case of greenhouses are the main energy source) and
an antiwindup action to account for frequent saturations in the control signal. Results
obtained with this scheme using a validated nonlinear simulator of greenhouse dynamics
are also included. Copyright © 2002 IFAC
Keywords: Robust control, feedforward compensation, agriculture, control nonlinearities,
bilinear systems.
1. INTRODUCTION effect of the control actuators (typically ventilation
and heating to modify inside temperature and
This paper deals with the development and
humidity conditions, shading and artificial light to
implementation of robust control techniques based
change internal radiation, CO2 injection to influence
on the quantitative feedback theory (QFT) aimed at
photosynthesis and fogging/cooling for humidity
achieving desired values of inside greenhouse
enrichment). The coefficients of the equations vary
temperature in spite of uncertainties and disturbances
with operating conditions in such a way that, from
acting on the system. The main objective of
the system dynamics point of view, the greenhouse
greenhouses crop production is to increment the
can be considered a smooth dynamical system which
economic benefits of the farmer by means of finding
dynamics are operating point dependent. The
a trade-off between the improvement of the quality
classical approach in QFT method is to include the
of the horticultural products and the cost of obtaining
effect of disturbances acting on the system as
adequate climate conditions using new greenhouse
unmodelled dynamics or to formulate the problem as
structures and automatic control strategies. As a
a disturbance rejection one. In the case of greenhouse
basic requirement, climate control helps to avoid
climate, the disturbances have the important role of
extreme conditions (high temperature or humidity
being the main energy source in the system and thus,
levels, etc.) which can cause damage to the crop and
they should be exploited to minimize the energy
to achieve adequate temperature integrals that can
consumption and to help to achieve the desired set
accelerate the crop development and its quality while
points. A modification to the standard formulation
reducing pollution and energy consumption.
has been performed to include a feedforward
controller previously developed by some of the
The greenhouse is a complex dynamical system
authors (Rodríguez et al., 2001a) and antiwindup
which behaviour can be described in terms of a
action in combination with the robust controller to
system of nonlinear differential equations describing
exploit the effect of measurable disturbances.
mass balances (water vapour fluxes and CO2
concentration) and energy transfer (radiation and
The paper is organized as follows. In ż2, a brief
heat) in the plastic cover, soil surface, one soil layer
description of the greenhouse dynamics is performed,
and crop. These processes depend on the outside
including a description of the real greenhouse which
environmental conditions, structure of the
model is used for simulation purposes. ż3 is devoted
greenhouse, type and state of the crop and on the
to explain the robust control approaches developed in
this paper, including feedforward and antiwindup
schemes. In ż4, some simulation results are shown
and finally, ż5 presents some conclusions.
2. GREENHOUSE DYNAMICS AND
EXPERIMENTAL PLANT
The greenhouse climate can be described by a
Fig. 2. A 2DoF feedback system
dynamic model represented by a system of
In QFT, closed loop specifications are given in the
differential equations as a function of state variables
frequency domain, as admissible bounds on closed
(internal air temperature and humidity, cover
loop transfer functions. Then, specifications are
temperature, soil surface temperature, PAR radiation,
combined with the uncertainty of the system (in the
etc.), input variables (natural ventilation, shade
form of templates) to obtain limits or boundaries on
screen and pipe heating systems), system variables,
the frequency shape of the compensator G(s). In
system parameters and disturbances (outside air
addition, nominal specifications are used to shape the
temperature and humidity, outside solar radiation,
pre-compensator F(s).
wind speed and direction, etc.). Disturbance
variables have a dominant role and coherent action
onto the formation of the greenhouse environment.
3.2 Inclusion of a feedforward term in the 2DoF
control system
Under several hypothesis, some authors of this paper
have developed a validated nonlinear model of a
As has been pointed out by (Sigrimis and Rerras,
typical plastic cover Mediterranean greenhouse
1996), solar radiation has a strong immediate effect
including both climate conditions and crop
on the internal conditions and produces frequent
development (Rodríguez et al., 2001b). This model is
oscillations (i.e., under passing clouds) in the
being used as a test-bed for the development and
controlled variables. In practice a time running
simulation of several control schemes, as the one
average filter can be used when the measurements of
presented in this paper. The following physical
this variable are used for control purposes. Outside
processes have been included in the balances: solar
temperature and humidity suffer slow variations and
and thermal radiation absorption, heat convection
their measurements can be directly used for
and conduction, crop transpiration, condensation and
disturbance attenuation. Wind velocity includes a
evaporation. This model has been validated using
steady component, corresponding to the mean wind
one-minute measurements from a real  Araba
speed, and a transient component, corresponding to
greenhouse (Fig. 1) located in El Ejido, Almería
the gusting of the wind about the mean value. Mean
(South-East Spain). It is a plastic-made two
wind velocity affects the air exchanges of the
symmetric curved slope roof with five North-South
greenhouse or else the heat balance and can be also
oriented naves of 7.5 x 40 m (1500 m2 of soil surface
used for control purposes.
and 5.5 m. high), laid on a structure made of
galvanized steel. The control actuators and measured
Although the control objective is to achieve a desired
variables are those indicated in the first paragraph of
temperature integral for crop growing purposes, large
this section.
changes in environmental variables affecting the
greenhouse climate influence the net profit (Tap et
al., 1993) , even leading to dangerous situations (e.g.
condensation) as a consequence of the surpassing of
temperature or humidity limits. Due to this reason, it
is important to exploit the effect of disturbances in
the inside conditions of the greenhouse by using
adequate feedforward controllers. The feedforward
term (Rodríguez et al., 2001a) is based on steady-
state balance using a simplified bilinear model of the
Fig. 1. Detail of the Araba greenhouse
system which coefficients are fixed and calculated
for a certain range of typical operating conditions:
3.DEVELOPMENT OF ROBUST CONTROLLERS dXt,a
cter,a dt =cr Pr,e + ch (Xt,h - Xt,a ) - (Ćv +Ć )(X - Pt,e ) +
c t,a
(1)
3.1 The quantitative feedback theory approach
+ cs (Xt,s - X ) +  Evap
t,a
where Pr,e is the solar radiation, Pt,e is the outside
Quantitative Feedback Theory (QFT) is a robust
temperature, Xt,h is the temperature of the heating
control design method (Horowitz, 1982) that uses a
tubes, Xt,s is the temperature of the soil, Ćv is the heat
two-degrees of freedom (2DoF) feedback scheme
transfer coefficient due to ventilation, Ćc is the heat
(Fig. 2), where it is assumed that the uncertain
transfer coefficient from inside of the greenhouse out
system is represented by a transfer function P(s)
(assumed positive), cr is the solar heating efficiency,
belonging to a set of plants !, while G(s) and F(s)
!
!
!
ch is the a heat transfer coefficient of the heating
are respectively the compensator and pre-
system and , cs is the a heat transfer coefficient from
compensator to be synthesised in order to meet
soil to inside air. A term accounting for latent energy
robust stability and performance specifications.
should be that expected from theoretical results.
fluxes has been included in the balance ( Evap),
Nevertheless, due to model mismatches the real
where  is the vaporisation energy of water and Evap
behaviour presents a different behaviour.
the evapotranspiration. Ćv is calculated by using a
p (disturbances)
p (disturbances)
nonlinear expression (Rodríguez et al., 2001)
feedback
feedback
prefilter
prefilter
including inside and outside temperature, wind
controller
controller
uffsat
uffsat
sp e trff y (temp.)
sp e trff y (temp.)
uff
uff
F(s)
F(s)
G(s)
G(s) PLANT
FF
velocity (Pv,e), volumetric flow rate (Vh,efec, related FF PLANT
feedforward
feedforward
with the vents aperture by a geometrical
transformation) and several constants (length of the
1/FF
1/FF
Simplified model
vents clv, gravity constant cg, etc.) and coefficients Simplified model
(inverse
(inverse
AW
(discharge coefficient cd , and wind effect coefficient AW
feedforward)
feedforward)
cw) that have also been fixed. The value of the fixed
Fig. 3. Control scheme
coefficients in the mentioned equations have been
34
obtained using input/output data obtained at the
Set point
32
greenhouse and by iterative search in the range of
30
values given by different authors using genetic
28
algorithms.
26
By using the simplified representation of the heat
Open-loop simulation-model response
24
Open-loop simple-model (inverse FF) response
balance given in equation (1) and considering a
22
steady state balance, it is possible to derive a
20
correlation for the input variables (ventilation and
18
3000 4000 5000 6000 7000 8000 9000
heating) as function of the environmental conditions
lo c a l t im e (h o u rs )
local time (minutes)
and the inside temperature. The series feedforward
controller is obtained by substituting the air
Fig. 4. Open loop effect of feedforward action
temperature Xt,a by the desired temperature trff. Thus,
each sampling instant the following calculations have
3.3. Inclusion of antiwindup action
to be performed (only calculations for diurnal
operation are included):
Another feature of the system is that it suffers from
frequent saturations of the input signal (vents) due to
1.
cr Pr ,e -Ć (trff - Pt ,e ) + cs ( X -trff ) (2)
c t ,s
Ćv =
disturbances and operating point changes and
(trff - Pt ,e )
deficient sizing of vents (often occurs), strongly
Å„Å‚ 2 / 3 üÅ‚
2.
ôÅ‚îÅ‚
Ćv 3cg (trff - Pt, e) / 2 Å‚Å‚ ôÅ‚ Pt,e
ôÅ‚ïÅ‚
limiting the control bandwidth. Due to this fact, as
Vh, efec = + (cw Pv2e )3 śł - cw Pv2e ôÅ‚
òÅ‚ïÅ‚ żł
cden, a cc - sp, a clv cd Pt, e , śł , ôÅ‚ cg (trff - Pt, e)
ôÅ‚ïÅ‚
śł
ûÅ‚
ôÅ‚ðÅ‚ ôÅ‚
ół þÅ‚ the controller must include integral action to track the
where a low-pass filter has been applied to solar
set point temperature, the use of an antiwindup
radiation and wind speed disturbances to avoid
scheme is of advice. In the classical approach, both
sudden changes in the control signals. Notice that
the vents aperture demanded by the control system
neither the latent heat nor the heating tubes terms
and that provided by the saturation block or actuator
have been included, as data used for the experiments
should feed the antiwindup block. The problem that
shown in this paper were obtained with the crop in
arises in this application is that the control signal
the early stages of its development and without using
provided by the robust controller is the reference
heating systems.
temperature of the feedforward controller (Fig. 3),
which provides the vents aperture depending on the
The inclusion of the feedforward term in series with
measurements of environmental variables. So, the
the plant (Fig. 3) allows to explicitly take into
first input point to the antiwindup block has been
account the measured value of the disturbances in
displaced to the output of the feedback controller.
such a way that the control signal provided by the
Fortunately, when saturation occurs in the vents
feedback controller is the reference temperature to
aperture, the corresponding reference temperature of
the feedforward term. Notice that if the model used
the feedforward controller can be on-line calculated
by the feedforward term was an exact one, the
taking into account the actual value of disturbances,
system constituted by the feedforward term in series
in such a way that the scheme reproduces the
with the plant should have a steady state gain near
classical one. In order to guarantee global stability,
unity. Unfortunately, the simplicity of the models
the results presented in (Bańos and Barreiro, 2000,
(fixed coefficients) in comparison with a large
Moreno et. al., 2002) can be applied with the
complex simulation model of the real system (in
proposed approach.
which several coefficients change depend on
operating conditions) and the uncertainty in the
3.4. Robust control design
system (it is impossible to exactly model the
In order to design the robust controller, the input-
greenhouse dynamics) advices the use of robust
output description of the system composed by the
control techniques to account for the mentioned
feedforward term in series with the plant has been
sources of uncertainties. To demonstrate this, Fig 4.
approximated by an uncertain first order system (step
shows the results obtained when implementing only
response tests shown that this approximation could
the feedforward term in open-loop (without feedback
be adopted), in which typical steady state gain and
controller). As can be seen, if the greenhouse
time constant mainly depend on the step input
dynamics should correspond to model described in
amplitude and can vary between the following
equation (1), the response obtained with the system
bounds:
tem perature (deg)
k Using the algorithm in (Moreno et al., 1997), the
, with k "[0.3,10], Ä " [360,1080] s.
P(s) =
performance and stability boundaries are computed,
Äs + 1
and the nominal open loop transfer function (Fig. 7)
Bode Diagrams
using computer tools (Borguesani et al., 1995).
0
-10
80
-20
-30
0.0001 rps
-40 60
-50
-60
40
-70
0
0.001 rps
20
-50
0.005 rps
-100
0
0.01 rps
-150
-200
-20
-250
-5 -4 -3 -2 -1
10 10 10 10 10
Frequency (rad/sec) -40
-400 -350 -300 -250 -200 -150 -100 -50 0
Phase (degrees)
Fig. 5. Frequency domain specifications
Fig.7. Nominal open-loop and bounds at design
frequencies in W.
Due to the uncertainty in the system, robust control
can be used, and Horowitz s method is chosen, as
The resulting controller G is given by equation:
done in other applications related to greenhouse
0.028 0.021
ëÅ‚10 öÅ‚ëÅ‚ öÅ‚
climate (Linker et al., 1999).
G(s) = +
ìÅ‚ ÷Å‚ìÅ‚ ÷Å‚
s s + 0.021Å‚Å‚
íÅ‚ Å‚Å‚íÅ‚
The first step in this method is to choose
Finally, the precompensator F to achieve the nominal
performance and stability specifications. Fig.5 shows
0.017
ëÅ‚ öÅ‚
specification is:
F(s) =
the performance specifications. ìÅ‚ ÷Å‚
s + 0.0017
íÅ‚ Å‚Å‚
Fig. 8 shows the final result of the design for the
As far as stability specifications is concerned, a gain
considered set of plants !.
margin of 5 dB and phase margin of 45º are desired:
G( jÉ)P( jÉ)
(3) 10
d" 2.3dB, "P "!, "É > 0
1 + G( jÉ)P( jÉ)
0
with != Å„Å‚ k : k "[0.3,10],Ä "[360,1080]üÅ‚ . -10
òÅ‚ żł
ółÄs +1 þÅ‚
50
-20
0
Note that (3) does not guarantee stability for the
-30
-50
closed loop system, due to presence of the actuator
-100
-40
saturation, see for example (Moreno et. al., 2002).
-150
-50 -200
-250
A controller {F,G} must be designed in order to
-60
-300 -5 -4 -3 -2 -1
10 10 10 10 10
w rps
assure that the closed loop transfer function T (from
-70
-5 -4 -3 -2 -1
10 10 10 10 10
reference to output) lies within envelopes in Fig. 5,
w rps
and the stability specification in (3) is achieved, with
Fig. 8. Closed loop specifications (dashdot) and
Å„Å‚ G(s)P(s) üÅ‚
.
T " ! =
òÅ‚F(s) 1+ G(s)P(s) : P "!żł frequency responses (solid) of the controlled
ół þÅ‚
system.
In order to proceed with the design of the controller,
the value sets (Barmish, 1988), which describe the
4. RESULTS
system uncertainty in the Nichols chart, are
computed (Fig. 6).
In this section, some illustrative results of the
20
proposed approach are shown and discussed. Fig. 9
shows the evolution of a test covering 13 complete
10
days in summer time with a fixed set point and a
0
shading screen covering the greenhouse. Although
the control scheme has been developed for operation
-10
during sun-shining conditions, it has not been turned
-20
0.0001 rps off during the night to shown the performance of the
0.001 rps
0.005 rps
-30 antiwindup block even in such strongly adverse
0.01 rps situation (the vents are completely closed during the
-40
-90 -80 -70 -60 -50 -40 -30 -20 -10 0
Phase (degrees) night and so, large feedback errors feed the
controller). The evolution of the outside solar
Fig. 6. System value sets.
radiation corresponds to clear day conditions, except
during the fifth and sixth day in which drops of more
Taking into account the typical time constants
than 100 W/m2 occurs. Outside temperature
involved in this problem and specifications, this is a
conditions are also varying and wind velocity
low frequency problem and so, the selected
experiments quite large variations during all the days,
frequency points (rad/s) for the design are
covering values from 0 to 12 m/s which large
W=[0.0001, 0.001, 0.005, 0.01], leading to values of
influence the system behaviour when vents are
"|T(jÉ)|=[0.0063, 0.6777, 5.5564, 14.7622]
opened. Due to the size of the figures, a zoom of a
respectively.
region has been included.
Phase (deg); Magnitude (dB)
Magnitude (dB)
|FL/(1+L)|dB
phase(FL/(1+L)) degrees
Magnitude (dB)
34
34
As can be seen, the tracking and disturbance
rejection capabilities are adequate in those cases in
32
32
which the vents are not saturated. When saturation
30
30
occurs, no degrees of freedom are available to
28
28
control the temperature. After saturation, the
33.2
33.2
33.15
33.15 performance of the system is quite good, as is
26
26
33.1
33.1
expected due to the use of the antiwindup scheme. As
33.05
33.05
24
24
can be seen in Fig. 9(b), the control signal suffers
33
33
22
22 from large excursions covering the whole control
32.95
32.95
32.9
32.9
range. This figure reflects the main drawback of the
20 4950 500 0 505 0 510 0 5150 5200 5250 5300 535 0 540 0
20 4950 500 0 505 0 510 0 5150 5200 5250 5300 535 0 540 0
approach used in this paper: as the controller tends to
18
18
quickly react to changes in disturbances (mainly due
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
4
4
x 10
x 10
local time (minutes)
to the structure of the feedforward term), the control
system is prone to over-actuate, thus increasing
(a) Set point and inside air temperatures (deg)
electricity costs associated to the motors moving the
40
40
vents (even when filtering the disturbances before
35
35
entering the feedforward term). The design can be
35
35
30
30
improved by finding a trade-off between fast tracking
30
30
25
25
25
25 and associated costs (by including stronger filters
20
20
15
15 within the feedforward term or by including design
20
20
10
10
restrictions in the control effort).
5
5
15
15
0
0
495 0 500 0 505 0 510 0 515 0 520 0 525 0 530 0 535 0 540 0
495 0 500 0 505 0 510 0 515 0 520 0 525 0 530 0 535 0 540 0
10
10
5
5
34
0
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
4
4
x 1 0
x 1 0
local time (minutes)
33.5
(b) Vents aperture (º)
33
1200
1200
32.5
1000
1000
800
800
700 720 740 760 780 800 820 840 860 880 900
local time (minutes)
600
600
90 0
90 0 (a) Set point and inside air temperatures (deg)
80 0
80 0
70 0
70 0
400
400
60 0
60 0
50 0
50 0
3 0
40 0
40 0
30 0
30 0
200
200 2 5
20 0
20 0
10 0
10 0
0
0
5000 5050 5100 5150 5200 5250 5300 5350
5000 5050 5100 5150 5200 5250 5300 5350 2 0
0
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 5
4
4
x 1 0
x 1 0
local time (minutes)
1 0
5
(c) Global solar radiation (W/m2)
5
5
4 .5
4 .5
7 0 0 7 2 0 7 4 0 7 6 0 7 8 0 8 0 0 8 2 0 8 4 0 8 6 0 8 8 0 9 0 0
4
4
3 .5
3 .5 local time (minutes)
3
3
2 .5
2 .5
2
2
1 .5
1 .5
1
1
(b) Vents aperture (º)
0 .5
0 .5
0
0
4 9 0 0 4 9 5 0 5 0 0 0 5 0 5 0 5 1 0 0 5 1 5 0 5 2 0 0 5 2 5 0 5 3 0 0 5 3 5 0 5 4 0 0
4 9 0 0 4 9 5 0 5 0 0 0 5 0 5 0 5 1 0 0 5 1 5 0 5 2 0 0 5 2 5 0 5 3 0 0 5 3 5 0 5 4 0 0
Fig. 10. Response to set point changes
3 7
3 6 . 5
3 6
local time (minutes) 3 5 . 5
(d) Outside wind speed (m/s)
3 5
5 0 6 0 5 0 8 0 5 1 0 0 5 1 2 0 5 1 4 0 5 1 6 0 5 1 8 0 5 2 0 0 5 2 2 0
local time (minutes)
2 8
2 8
2 6
2 6 (a) Set point and inside air temperatures (deg)
2 4
2 4
3 5
3 0
2 2
2 2
2 5
2 4
2 4 2 0
2 0
2 0
2 3. 5
2 3. 5
1 5
2 3
2 3
1 8
1 8
1 0
2 2. 5
2 2. 5
5
1 6 2 2
1 6 2 2
0
4 7 0 0 4 8 0 0 4 9 0 0 5 0 0 0 5 1 0 0 5 2 0 0 5 3 0 0 5 4 0 0 5 5 0 0
4 7 0 0 4 8 0 0 4 9 0 0 5 0 0 0 5 1 0 0 5 2 0 0 5 3 0 0 5 4 0 0 5 5 0 0 5 0 5 0 5 1 0 0 5 1 5 0 5 2 0 0
1 4
1 4
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
local time (minutes)
4
4
x 1 0
x 1 0
local time (minutes)
(b) Vents aperture (º)
(e) Outside temperature (deg)
Fig. 11. Response to set point changes
Fig. 9. Complete 13-days simulation
3 5 . 5
the temperature drop following a cloud (tracking
error less than 0.2ºC).
3 5
5. CONCLUSIONS
3 4 . 5
An approach for robustly controlling the inside
temperature of a greenhouse in the face of
3 4 uncertainties and disturbances acting on the system
6 5 0 0 6 5 2 0 6 5 4 0 6 5 6 0 6 5 8 0 6 6 0 0 6 6 2 0 6 6 4 0 6 6 6 0 6 6 8 0 6 7 0 0
has been presented, including feedforward
local time (minutes)
compensation and antiwindup action. The results
(a) Set point and inside air temperatures (deg)
presented are quite promising. After analyzing the
2 5
results, it has to be pointed out that more
2 0 improvements can be performed by limiting control
efforts which lead to an increase in production costs.
1 5
1 0
ACKNOWLEDGEMENTS
5
Authors would like to acknowledge CICYT for
6 5 0 0 6 5 2 0 6 5 4 0 6 5 6 0 6 5 8 0 6 6 0 0 6 6 2 0 6 6 4 0 6 6 6 0 6 6 8 0 6 7 0 0
partially funding this work under grants QUI99-0663-
local time (minutes)
C02-02, DPI2000-1218-C04-03 and DPI2001-2380-C02-
02.
(b) Vents aperture (º)
9 6 0
REFERENCES
9 4 0
9 2 0
Bańos, A. and A. Barreiro (2000). Stability of nonlinear
9 0 0
QFT designs based on robust absolute stability
8 8 0
criteria. Int. Journal of Control, 73, pp. 74-88.
8 6 0
8 4 0 Barmish, B. R. (1988). New tools for robustness analysis.
8 2 0
In: Proc. of the 27th IEEE Conference on Decision
8 0 0
6 5 0 0 6 5 2 0 6 5 4 0 6 5 6 0 6 5 8 0 6 6 0 0 6 6 2 0 6 6 4 0 6 6 6 0 6 6 8 0 6 7 0 0 and Control, Austin, TX, pp. 1-6.
local time (minutes)
Borguesani, C., Y. Chait and O. Yaniv (1995). The
(c) Outside global solar radiation (W/m2) Quantitative Feedback Theory Toolbox for MATLAB.
The MathWorks, MA.
5 . 5
Horowitz, I. (1982). Quantitative feedback theory. IEEE
5
Proc.,129 (D-6), 215-226.
4 . 5
Linker, R., P.O. Gutman, I. Seginer (1999). Robust
4
controllers for simultaneous control of temperature
3 . 5
and CO2 concentration in greenhouses. Control
3
Engineering Practice, 7, pp. 851-862.
2 . 5
Moreno, J.C., A. Bańos, and F.J. Montoya (1997). An
2
Algorithm for Computing QFT Multiple-valued
6 5 0 0 6 5 2 0 6 5 4 0 6 5 6 0 6 5 8 0 6 6 0 0 6 6 2 0 6 6 4 0 6 6 6 0 6 6 8 0 6 7 0 0
local time (minutes)
Performance Bounds. In: Proc. Symp. on QFT and
other Frequency Domain Methods and Applications,
(d) Wind speed (m/s)
pp. 29-32. Univ. Stratchclyde (Scotland).
Fig. 12. Disburbance response capabilities
Moreno, J.C., A. Bańos and M. Berenguel (2002). Design
of robust compensators for linear systems with actuator
Fig. 10 and 11 show typical responses to set point
saturation using QFT. Submitted to Int. Journal of
changes and Fig. 12 shows the disturbance rejection
Control.
capabilities of the system. Fig. 10 corresponds to
Rodríguez, F., M. Berenguel and M.R. Arahal (2001a).
wind speed conditions of 7 m/s and clear-day solar
Feedforward controllers for greenhouse climate
radiation between 900 and 1000 W/m2. Set point control based on physical models. ECC01, Portugal.
Rodríguez, F., M. Berenguel and A. Baille (2001b). A
changes of Ä…2 ºC have been performed around 33ºC.
Model of Greenhouse Crop Production. Part I: Model
Development. Part II; Model Validation. Internal
Due to the nonlinear nature of the system, different
Report, Univ. Almería, Spain.
closed-loop time constants are obtained, but lying
Sigrimis, N. and N. Rerras (1996)., A Linear Model for
inside the specifications, even in the case in which
Greenhouse Control, Trans. of the ASAE, 39(1) pp.
the model used to develop the feedforward term is
253-261.
not a good approximation of the real system. Fig. 11
Tap, R.F., L.G. van Willigenburg,, G. van Straten and E.J.
shows another test under low wind speed (around 2
van Henten (1993), Optimal control of greenhouse
m/s) conditions (notice that coefficients of the
climate: computation of the influence of fast and slow
feedforward term were calculated for wind speed
dynamics. Proc. 12th IFAC World Cong., 10, pp. 321-
conditions around 6 m/s and thus, modeling errors 324, Sydney.
are larger in this case). As can be seen, the tracking
capabilities are also quite acceptable in this case.
Finally, Fig. 12 shows the response under passing
clouds and varying low wind speed conditions. It
can be seen how the control system quickly reacts to
changes in solar radiation in order to compensate for