Zgryźliwość kojarzy mi się z radością, która źle skończyła.
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 Baños
‡
*
Universidad de Almería. Dpto. Lenguajes y Computación. Ctra. Sacramento s/n, La
Cañada, E04120, Almería, Spain. E-mail:
‡
Universidad de Murcia. Facultad de Informática. Dpto. de Informática y Sistemas.
Campus Universitario de Espinardo, E30071, Murcia, Spain. 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
This paper deals with the development and
implementation of robust control techniques based
on the quantitative feedback theory (QFT) aimed at
achieving desired values of inside greenhouse
temperature in spite of uncertainties and disturbances
acting on the system. The main objective of
greenhouses crop production is to increment the
economic benefits of the farmer by means of finding
a trade-off between the improvement of the quality
of the horticultural products and the cost of obtaining
adequate climate conditions using new greenhouse
structures and automatic control strategies. As a
basic requirement, climate control helps to avoid
extreme conditions (high temperature or humidity
levels, etc.) which can cause damage to the crop and
to achieve adequate temperature integrals that can
accelerate the crop development and its quality while
reducing pollution and energy consumption.
effect of the control actuators (typically ventilation
and heating to modify inside temperature and
humidity conditions, shading and artificial light to
change internal radiation, CO
2
injection to influence
photosynthesis and fogging/cooling for humidity
enrichment). The coefficients of the equations vary
with operating conditions in such a way that, from
the system dynamics point of view, the greenhouse
can be considered a smooth dynamical system which
dynamics are operating point dependent. The
classical approach in QFT method is to include the
effect of disturbances acting on the system as
unmodelled dynamics or to formulate the problem as
a disturbance rejection one. In the case of greenhouse
climate, the disturbances have the important role of
being the main energy source in the system and thus,
they should be exploited to minimize the energy
consumption and to help to achieve the desired set
points. A modification to the standard formulation
has been performed to include a feedforward
controller previously developed by some of the
authors (Rodríguez
et al
., 2001
a
) and antiwindup
action in combination with the robust controller to
exploit the effect of measurable disturbances.
The greenhouse is a complex dynamical system
which behaviour can be described in terms of a
system of nonlinear differential equations describing
mass balances (water vapour fluxes and CO
2
concentration) and energy transfer (radiation and
heat) in the plastic cover, soil surface, one soil layer
and crop. These processes depend on the outside
environmental conditions, structure of the
greenhouse, type and state of the crop and on the
The paper is organized as follows. In §2, a brief
description of the greenhouse dynamics is performed,
including a description of the real greenhouse which
model is used for simulation purposes. §3 is devoted
to explain the robust control approaches developed in
this paper, including feedforward and antiwindup
schemes. In §4, some simulation results are shown
and finally, §5 presents some conclusions.
2. GREENHOUSE DYNAMICS AND
EXPERIMENTAL PLANT
The greenhouse climate can be described by a
dynamic model represented by a system of
differential equations as a function of state variables
(internal air temperature and humidity, cover
temperature, soil surface temperature, PAR radiation,
etc.), input variables (natural ventilation, shade
screen and pipe heating systems), system variables,
system parameters and disturbances (outside air
temperature and humidity, outside solar radiation,
wind speed and direction, etc.). Disturbance
variables have a dominant role and coherent action
onto the formation of the greenhouse environment.
Fig. 2. A 2DoF feedback system
In QFT, closed loop specifications are given in the
frequency domain, as admissible bounds on closed
loop transfer functions. Then, specifications are
combined with the uncertainty of the system (in the
form of
templates
) to obtain limits or
boundaries
on
the frequency shape of the compensator
G
(
s
). In
addition, nominal specifications are used to shape the
pre-compensator
F
(
s
).
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
typical plastic cover Mediterranean greenhouse
including both climate conditions and crop
development (Rodríguez
et al
., 2001
b
). This model is
being used as a test-bed for the development and
simulation of several control schemes, as the one
presented in this paper. The following physical
processes have been included in the balances: solar
and thermal radiation absorption, heat convection
and conduction, crop transpiration, condensation and
evaporation. This model has been validated using
one-minute measurements from a real “Araba”
greenhouse (Fig. 1) located in El Ejido, Almería
(South-East Spain). It is a plastic-made two
symmetric curved slope roof with five North-South
oriented naves of 7.5
x
40 m (1500 m
2
of soil surface
and 5.5 m. high), laid on a structure made of
galvanized steel. The control actuators and measured
variables are those indicated in the first paragraph of
this section.
As has been pointed out by (Sigrimis and Rerras,
1996), solar radiation has a strong immediate effect
on the internal conditions and produces frequent
oscillations (i.e., under passing clouds) in the
controlled variables. In practice a time running
average filter can be used when the measurements of
this variable are used for control purposes. Outside
temperature and humidity suffer slow variations and
their measurements can be directly used for
disturbance attenuation. Wind velocity includes a
steady component, corresponding to the mean wind
speed, and a transient component, corresponding to
the gusting of the wind about the mean value. Mean
wind velocity affects the air exchanges of the
greenhouse or else the heat balance and can be also
used for control purposes.
Although the control objective is to achieve a desired
temperature integral for crop growing purposes, large
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
., 2001
a
) is based on steady-
state balance using a simplified bilinear model of the
system which coefficients are fixed and calculated
for a certain range of typical operating conditions:
Fig. 1. Detail of the Araba greenhouse
3.DEVELOPMENT OF ROBUST CONTROLLERS
c
dX
t
a
c
P
+
c
X
X
φ
+
φ
)
X
−
P
ter
a
dt
=
r
r
e
h
(
t
h
−
t
a
)
−
(
v
c
(
t
a
t
,
e
+
(1)
3.1 The quantitative feedback theory approach
c
X
X
λ
Evap
+
s
(
t
,
s
−
t
a
)
+
Quantitative Feedback Theory (QFT) is a robust
control design method (Horowitz, 1982) that uses a
two-degrees of freedom (2DoF) feedback scheme
(Fig. 2), where it is assumed that the uncertain
system is represented by a transfer function
P
(
s
)
belonging to a set of plants
where
P
r,e
is the solar radiation,
P
t,e
is the outside
temperature,
X
t,h
is the temperature of the heating
tubes,
X
t,s
is the temperature of the soil,
φ
v
is the heat
c
is the heat
transfer coefficient from inside of the greenhouse out
(assumed positive),
c
r
is the solar heating efficiency,
c
h
is the a heat transfer coefficient of the heating
system and ,
c
s
is the a heat transfer coefficient from
soil to inside air. A term accounting for latent energy
φ
, while
G
(
s
) and
F
(
s
)
are respectively the compensator and pre-
compensator to be synthesised in order to meet
robust stability and performance specifications.
,
)
,
,
,
,
,
,
transfer coefficient due to ventilation,
℘
fluxes has been included in the balance (
λ
Evap
),
should be that expected from theoretical results.
Nevertheless, due to model mismatches the real
behaviour presents a different behaviour.
is the vaporisation energy of water and
Evap
the evapotranspiration.
λ
v
is calculated by using a
nonlinear expression (Rodríguez
et al
., 2001)
including inside and outside temperature, wind
velocity (
P
v,e
), volumetric flow rate (
V
h,efec
, related
with the vents aperture by a geometrical
transformation) and several constants (length of the
vents
c
lv
, gravity constant
c
g
, etc.) and coefficients
(discharge coefficient
c
d
, and wind effect coefficient
c
w
) that have also been fixed. The value of the fixed
coefficients in the mentioned equations have been
obtained using input/output data obtained at the
greenhouse and by iterative search in the range of
values given by different authors using genetic
algorithms.
φ
p (disturbances)
prefilter
feedback
controller
sp
e
trf
f
uff
uf
f
sat
y
(
temp.
)
F(s)
G(s)
FF
PLANT
feedforwa
r
d
1/FF
Simplified model
(inverse
feedforward)
AW
Fig. 3. Control scheme
34
32
Set point
30
28
26
By using the simplified representation of the heat
balance given in equation (1) and considering a
steady state balance, it is possible to derive a
correlation for the input variables (ventilation and
heating) as function of the environmental conditions
and the inside temperature. The series feedforward
controller is obtained by substituting the air
temperature
X
t,a
by the desired temperature
trff
. Thus,
each sampling instant the following calculations have
to be performed (only calculations for diurnal
operation are included):
24
Open-loop simulation-model response
Open-loop simple-model (inverse FF) response
22
20
18
3000
4000
5000
6000
7000
8000
9000
lo c a l tim e (ho urs )
local time (minutes)
Fig. 4. Open loop effect of feedforward action
3.3. Inclusion of antiwindup action
Another feature of the system is that it suffers from
frequent saturations of the input signal (vents) due to
disturbances and operating point changes and
deficient sizing of vents (often occurs), strongly
limiting the control bandwidth. Due to this fact, as
the controller must include integral action to track the
set point temperature, the use of an antiwindup
scheme is of advice. In the classical approach, both
the vents aperture demanded by the control system
and that provided by the saturation block or actuator
should feed the antiwindup block. The problem that
arises in this application is that the control signal
provided by the robust controller is the reference
temperature of the feedforward controller (Fig. 3),
which provides the vents aperture depending on the
measurements of environmental variables. So, the
first input point to the antiwindup block has been
displaced to the output of the feedback controller.
Fortunately, when saturation occurs in the vents
aperture, the corresponding reference temperature of
the feedforward controller can be on-line calculated
taking into account the actual value of disturbances,
in such a way that the scheme reproduces the
classical one. In order to guarantee global stability,
the results presented in (Baños and Barreiro, 2000
,
Moreno
et. al.
, 2002) can be applied with the
proposed approach.
1.
φ
c
r
P
r
e
−
φ
c
(
trff
−
P
t
,
e
)
+
c
s
(
X
t
,
s
−
trff
)
(2)
=
v
(
trff
−
P
)
t
,
e
2.
3
c
trff
−
P
)
2
/
3
φ
g
e
2
,
3
2
P
e
v
2
,
V
=
+
(
c
P
)
−
c
P
h
,
efec
w
e
w
e
c
c
c
c
P
c
trff
−
P
)
den
,
a
c
−
sp
,
a
lv
d
,
e
g
e
where a low-pass filter has been applied to solar
radiation and wind speed disturbances to avoid
sudden changes in the control signals. Notice that
neither the latent heat nor the heating tubes terms
have been included, as data used for the experiments
shown in this paper were obtained with the crop in
the early stages of its development and without using
heating systems.
The inclusion of the feedforward term in series with
the plant (Fig. 3) allows to explicitly take into
account the measured value of the disturbances in
such a way that the control signal provided by the
feedback controller is the reference temperature to
the feedforward term. Notice that if the model used
by the feedforward term was an exact one, the
system constituted by the feedforward term in series
with the plant should have a steady state gain near
unity. Unfortunately, the simplicity of the models
(fixed coefficients) in comparison with a large
complex simulation model of the real system (in
which several coefficients change depend on
operating conditions) and the uncertainty in the
system (it is impossible to exactly model the
greenhouse dynamics) advices the use of robust
control techniques to account for the mentioned
sources of uncertainties. To demonstrate this, Fig 4.
shows the results obtained when implementing only
the feedforward term in open-loop (without feedback
controller). As can be seen, if the greenhouse
dynamics should correspond to model described in
equation (1), the response obtained with the system
3.4. Robust control design
In order to design the robust controller, the input-
output description of the system composed by the
feedforward term in series with the plant has been
approximated by an uncertain first order system (step
response tests shown that this approximation could
be adopted), in which typical steady state gain and
time constant mainly depend on the step input
amplitude and can vary between the following
bounds:
where
p (disturbances)
feedback
controller
prefilter
sp
trff
uff
sat
y
(
temp.
)
e
uff
F(s)
G(s)
FF
PLANT
feedforward
1/FF
Simplified model
(inverse
feedforward)
AW
,
(
,
/
,
(
,
P
(
s
)
=
k
, with
k
∈
[0.3,10],
τ
∈
[360,1080] s.
Using the algorithm in (Moreno
et al
., 1997), the
performance and stability boundaries are computed,
and the nominal open loop transfer function (Fig. 7)
using computer tools (Borguesani
et al
., 1995).
τ
s
+
1
Bode Diagrams
0
-10
80
-20
-30
0.0001 rps
-40
60
-50
-60
-70
40
0
0.001 rps
0.005 rps
0.01 rps
20
-50
-100
0
-150
-200
-250
-20
-5
10
-4
-3
-2
-1
10
10
10
10
Frequency (rad/sec)
-40
-400
-350
-300
-250
-200
-150
-100
-50
0
Fig. 5. Frequency domain specifications
Fig.7. Nominal open-loop and bounds at design
frequencies in
W
.
Phase (degrees)
Due to the uncertainty in the system, robust control
can be used, and Horowitz’s method is chosen, as
done in other applications related to greenhouse
climate (Linker
et al
., 1999).
The first step in this method is to choose
performance and stability specifications. Fig.5 shows
the performance specifications.
The resulting controller
G
is given by equation:
G
(
s
)
=
#
10
+
0
.
028
!
#
0
021
!
s
s
+
0
.
021
Finally, the precompensator
F
to achieve the nominal
specification is:
F
(
s
)
=
#
0
017
!
s
+
0
0017
As far as stability specifications is concerned, a gain
margin of 5 dB and phase margin of 45º are desired:
Fig. 8 shows the final result of the design for the
considered set of plants
℘
.
G
(
j
ω
)
P
(
j
ω
)
≤
2
dB
,
P
∈
,
∀
ω
>
0
(3)
10
1
+
G
(
j
ω
)
P
(
j
ω
)
0
with
℘
=
k
:
k
∈
[ ] [
0
3
10
,
τ
∈
360
1080
]
.
-10
τ
s
+
1
-20
50
Note that (3) does not guarantee stability for the
closed loop system, due to presence of the actuator
saturation, see for example (Moreno
et. al.
, 2002).
A controller {
F
,
G
} must be designed in order to
assure that the closed loop transfer function
T
(from
reference to output) lies within envelopes in Fig. 5,
and the stability specification in (3) is achieved, with
0
-30
-5 0
-40
-1 00
-1 50
-50
-2 00
-2 50
-60
-3 00
10
-5
10
-4
10
-3
10
-2
10
-1
w rps
-70
-5
-4
-3
-2
-1
10
10
10
10
10
w rps
T
∈
ℑ
=
F
(
s
)
G
(
s
)
P
(
s
)
:
P
∈
.
Fig. 8. Closed loop specifications (dashdot) and
frequency responses (solid) of the controlled
system.
1
+
G
(
s
)
P
(
s
)
In order to proceed with the design of the controller,
the value sets (Barmish, 1988), which describe the
system uncertainty in the Nichols chart, are
computed (Fig. 6).
4. RESULTS
In this section, some illustrative results of the
proposed approach are shown and discussed. Fig. 9
shows the evolution of a test covering 13 complete
days in summer time with a fixed set point and a
shading screen covering the greenhouse. Although
the control scheme has been developed for operation
during sun-shining conditions, it has not been turned
off during the night to shown the performance of the
antiwindup block even in such strongly adverse
situation (the vents are completely closed during the
night and so, large feedback errors feed the
controller). The evolution of the outside solar
radiation corresponds to clear day conditions, except
during the fifth and sixth day in which drops of more
than 100 W/m
2
occurs. Outside temperature
conditions are also varying and wind velocity
experiments quite large variations during all the days,
covering values from 0 to 12 m/s which large
influence the system behaviour when vents are
opened. Due to the size of the figures, a zoom of a
region has been included.
20
10
0
-10
-20
0.0001 rps
0.001 rps
0.005 rps
-30
0.01 rps
-40
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Phase (degrees)
Fig. 6. System value sets.
Taking into account the typical time constants
involved in this problem and specifications, this is a
low frequency problem and so,
the selected
frequency points (rad/s) for the design are
W
=[0.0001, 0.001, 0.005, 0.01], leading to values of
∆
|
T
(j
ω
.
$
$
$
∀
,
)|=[0.0063, 0.6777, 5.5564, 14.7622]
respectively.
34
As can be seen, the tracking and disturbance
rejection capabilities are adequate in those cases in
which the vents are not saturated. When saturation
occurs, no degrees of freedom are available to
control the temperature. After saturation, the
performance of the system is quite good, as is
expected due to the use of the antiwindup scheme. As
can be seen in Fig. 9(b), the control signal suffers
from large excursions covering the whole control
range. This figure reflects the main drawback of the
approach used in this paper: as the controller tends to
quickly react to changes in disturbances (mainly due
to the structure of the feedforward term), the control
system is prone to over-actuate, thus increasing
electricity costs associated to the motors moving the
vents (even when filtering the disturbances before
entering the feedforward term). The design can be
improved by finding a trade-off between fast tracking
and associated costs (by including stronger filters
within the feedforward term or by including design
restrictions in the control effort).
32
30
28
33.2
33.15
26
33.1
24
33.05
33
22
32.95
32.9
20
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
18
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
local time (minutes)
(a) Set point and inside air temperatures (deg)
40
35
30
35
30
25
25
20
20
15
10
15
5
0
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
10
5
34
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
local time (minutes)
x 1 0
4
33.5
(b) Vents aperture (º)
33
1200
32.5
1000
800
700
720
740
760
local time (minutes)
800
820
840
860
880
900
600
(a) Set point and inside air temperatures (deg)
900
800
400
600
700
400
500
30
300
200
100
200
25
0
20
5000
5050
5100
5150
5200
5250
5300
5350
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
15
4
local time (minutes)
x 1 0
10
(c) Global solar radiation (W/m
2
)
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
700
720
740
760
780
800
820
840
860
880
900
local time (minutes)
(b) Vents aperture (º)
Fig. 10. Response to set point changes
4900
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
37
36.5
36
local time (minutes)
35.5
(d) Outside wind speed (m/s)
35
5060
5080
5100
5120
5140
5160
5180
5200
5220
28
local time (minutes)
26
(a) Set point and inside air temperatures (deg)
24
35
22
30
25
20
24
20
23.5
15
18
23
10
22.5
16
5
22
0
5050
5100
5150
5200
4700
4800
4900
5000
5100
5200
5300
5400
5500
14
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
local time (minutes)
local time (minutes)
x 10
4
(e) Outside temperature (deg)
Fig. 9. Complete 13-days simulation
(b) Vents aperture (º)
Fig. 11. Response to set point changes
34
32
30
28
33.2
33.15
26
33.1
24
33.05
33
22
32.95
32.9
20
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
18
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
40
35
30
35
30
25
25
20
20
15
10
15
5
0
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
10
5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 1 0
4
1200
1000
800
780
600
900
800
600
700
400
400
500
300
200
100
200
0
5000
5050
5100
5150
5200
5250
5300
5350
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
4
x 1 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4900
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
28
26
24
22
20
24
23.5
18
23
22.5
16
22
4700
4800
4900
5000
5100
5200
5300
5400
5500
14
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
zanotowane.pl doc.pisz.pl pdf.pisz.pl hannaeva.xlx.pl
15th Triennial World Congress, Barcelona, Spain
ROBUST CONTROL OF GREENHOUSE CLIMATE EXPLOITING MEASURABLE
DISTURBANCES
J.C. Moreno
*
, M. Berenguel
*
, F. Rodríguez
*
, A Baños
‡
*
Universidad de Almería. Dpto. Lenguajes y Computación. Ctra. Sacramento s/n, La
Cañada, E04120, Almería, Spain. E-mail:
‡
Universidad de Murcia. Facultad de Informática. Dpto. de Informática y Sistemas.
Campus Universitario de Espinardo, E30071, Murcia, Spain. 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
This paper deals with the development and
implementation of robust control techniques based
on the quantitative feedback theory (QFT) aimed at
achieving desired values of inside greenhouse
temperature in spite of uncertainties and disturbances
acting on the system. The main objective of
greenhouses crop production is to increment the
economic benefits of the farmer by means of finding
a trade-off between the improvement of the quality
of the horticultural products and the cost of obtaining
adequate climate conditions using new greenhouse
structures and automatic control strategies. As a
basic requirement, climate control helps to avoid
extreme conditions (high temperature or humidity
levels, etc.) which can cause damage to the crop and
to achieve adequate temperature integrals that can
accelerate the crop development and its quality while
reducing pollution and energy consumption.
effect of the control actuators (typically ventilation
and heating to modify inside temperature and
humidity conditions, shading and artificial light to
change internal radiation, CO
2
injection to influence
photosynthesis and fogging/cooling for humidity
enrichment). The coefficients of the equations vary
with operating conditions in such a way that, from
the system dynamics point of view, the greenhouse
can be considered a smooth dynamical system which
dynamics are operating point dependent. The
classical approach in QFT method is to include the
effect of disturbances acting on the system as
unmodelled dynamics or to formulate the problem as
a disturbance rejection one. In the case of greenhouse
climate, the disturbances have the important role of
being the main energy source in the system and thus,
they should be exploited to minimize the energy
consumption and to help to achieve the desired set
points. A modification to the standard formulation
has been performed to include a feedforward
controller previously developed by some of the
authors (Rodríguez
et al
., 2001
a
) and antiwindup
action in combination with the robust controller to
exploit the effect of measurable disturbances.
The greenhouse is a complex dynamical system
which behaviour can be described in terms of a
system of nonlinear differential equations describing
mass balances (water vapour fluxes and CO
2
concentration) and energy transfer (radiation and
heat) in the plastic cover, soil surface, one soil layer
and crop. These processes depend on the outside
environmental conditions, structure of the
greenhouse, type and state of the crop and on the
The paper is organized as follows. In §2, a brief
description of the greenhouse dynamics is performed,
including a description of the real greenhouse which
model is used for simulation purposes. §3 is devoted
to explain the robust control approaches developed in
this paper, including feedforward and antiwindup
schemes. In §4, some simulation results are shown
and finally, §5 presents some conclusions.
2. GREENHOUSE DYNAMICS AND
EXPERIMENTAL PLANT
The greenhouse climate can be described by a
dynamic model represented by a system of
differential equations as a function of state variables
(internal air temperature and humidity, cover
temperature, soil surface temperature, PAR radiation,
etc.), input variables (natural ventilation, shade
screen and pipe heating systems), system variables,
system parameters and disturbances (outside air
temperature and humidity, outside solar radiation,
wind speed and direction, etc.). Disturbance
variables have a dominant role and coherent action
onto the formation of the greenhouse environment.
Fig. 2. A 2DoF feedback system
In QFT, closed loop specifications are given in the
frequency domain, as admissible bounds on closed
loop transfer functions. Then, specifications are
combined with the uncertainty of the system (in the
form of
templates
) to obtain limits or
boundaries
on
the frequency shape of the compensator
G
(
s
). In
addition, nominal specifications are used to shape the
pre-compensator
F
(
s
).
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
typical plastic cover Mediterranean greenhouse
including both climate conditions and crop
development (Rodríguez
et al
., 2001
b
). This model is
being used as a test-bed for the development and
simulation of several control schemes, as the one
presented in this paper. The following physical
processes have been included in the balances: solar
and thermal radiation absorption, heat convection
and conduction, crop transpiration, condensation and
evaporation. This model has been validated using
one-minute measurements from a real “Araba”
greenhouse (Fig. 1) located in El Ejido, Almería
(South-East Spain). It is a plastic-made two
symmetric curved slope roof with five North-South
oriented naves of 7.5
x
40 m (1500 m
2
of soil surface
and 5.5 m. high), laid on a structure made of
galvanized steel. The control actuators and measured
variables are those indicated in the first paragraph of
this section.
As has been pointed out by (Sigrimis and Rerras,
1996), solar radiation has a strong immediate effect
on the internal conditions and produces frequent
oscillations (i.e., under passing clouds) in the
controlled variables. In practice a time running
average filter can be used when the measurements of
this variable are used for control purposes. Outside
temperature and humidity suffer slow variations and
their measurements can be directly used for
disturbance attenuation. Wind velocity includes a
steady component, corresponding to the mean wind
speed, and a transient component, corresponding to
the gusting of the wind about the mean value. Mean
wind velocity affects the air exchanges of the
greenhouse or else the heat balance and can be also
used for control purposes.
Although the control objective is to achieve a desired
temperature integral for crop growing purposes, large
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
., 2001
a
) is based on steady-
state balance using a simplified bilinear model of the
system which coefficients are fixed and calculated
for a certain range of typical operating conditions:
Fig. 1. Detail of the Araba greenhouse
3.DEVELOPMENT OF ROBUST CONTROLLERS
c
dX
t
a
c
P
+
c
X
X
φ
+
φ
)
X
−
P
ter
a
dt
=
r
r
e
h
(
t
h
−
t
a
)
−
(
v
c
(
t
a
t
,
e
+
(1)
3.1 The quantitative feedback theory approach
c
X
X
λ
Evap
+
s
(
t
,
s
−
t
a
)
+
Quantitative Feedback Theory (QFT) is a robust
control design method (Horowitz, 1982) that uses a
two-degrees of freedom (2DoF) feedback scheme
(Fig. 2), where it is assumed that the uncertain
system is represented by a transfer function
P
(
s
)
belonging to a set of plants
where
P
r,e
is the solar radiation,
P
t,e
is the outside
temperature,
X
t,h
is the temperature of the heating
tubes,
X
t,s
is the temperature of the soil,
φ
v
is the heat
c
is the heat
transfer coefficient from inside of the greenhouse out
(assumed positive),
c
r
is the solar heating efficiency,
c
h
is the a heat transfer coefficient of the heating
system and ,
c
s
is the a heat transfer coefficient from
soil to inside air. A term accounting for latent energy
φ
, while
G
(
s
) and
F
(
s
)
are respectively the compensator and pre-
compensator to be synthesised in order to meet
robust stability and performance specifications.
,
)
,
,
,
,
,
,
transfer coefficient due to ventilation,
℘
fluxes has been included in the balance (
λ
Evap
),
should be that expected from theoretical results.
Nevertheless, due to model mismatches the real
behaviour presents a different behaviour.
is the vaporisation energy of water and
Evap
the evapotranspiration.
λ
v
is calculated by using a
nonlinear expression (Rodríguez
et al
., 2001)
including inside and outside temperature, wind
velocity (
P
v,e
), volumetric flow rate (
V
h,efec
, related
with the vents aperture by a geometrical
transformation) and several constants (length of the
vents
c
lv
, gravity constant
c
g
, etc.) and coefficients
(discharge coefficient
c
d
, and wind effect coefficient
c
w
) that have also been fixed. The value of the fixed
coefficients in the mentioned equations have been
obtained using input/output data obtained at the
greenhouse and by iterative search in the range of
values given by different authors using genetic
algorithms.
φ
p (disturbances)
prefilter
feedback
controller
sp
e
trf
f
uff
uf
f
sat
y
(
temp.
)
F(s)
G(s)
FF
PLANT
feedforwa
r
d
1/FF
Simplified model
(inverse
feedforward)
AW
Fig. 3. Control scheme
34
32
Set point
30
28
26
By using the simplified representation of the heat
balance given in equation (1) and considering a
steady state balance, it is possible to derive a
correlation for the input variables (ventilation and
heating) as function of the environmental conditions
and the inside temperature. The series feedforward
controller is obtained by substituting the air
temperature
X
t,a
by the desired temperature
trff
. Thus,
each sampling instant the following calculations have
to be performed (only calculations for diurnal
operation are included):
24
Open-loop simulation-model response
Open-loop simple-model (inverse FF) response
22
20
18
3000
4000
5000
6000
7000
8000
9000
lo c a l tim e (ho urs )
local time (minutes)
Fig. 4. Open loop effect of feedforward action
3.3. Inclusion of antiwindup action
Another feature of the system is that it suffers from
frequent saturations of the input signal (vents) due to
disturbances and operating point changes and
deficient sizing of vents (often occurs), strongly
limiting the control bandwidth. Due to this fact, as
the controller must include integral action to track the
set point temperature, the use of an antiwindup
scheme is of advice. In the classical approach, both
the vents aperture demanded by the control system
and that provided by the saturation block or actuator
should feed the antiwindup block. The problem that
arises in this application is that the control signal
provided by the robust controller is the reference
temperature of the feedforward controller (Fig. 3),
which provides the vents aperture depending on the
measurements of environmental variables. So, the
first input point to the antiwindup block has been
displaced to the output of the feedback controller.
Fortunately, when saturation occurs in the vents
aperture, the corresponding reference temperature of
the feedforward controller can be on-line calculated
taking into account the actual value of disturbances,
in such a way that the scheme reproduces the
classical one. In order to guarantee global stability,
the results presented in (Baños and Barreiro, 2000
,
Moreno
et. al.
, 2002) can be applied with the
proposed approach.
1.
φ
c
r
P
r
e
−
φ
c
(
trff
−
P
t
,
e
)
+
c
s
(
X
t
,
s
−
trff
)
(2)
=
v
(
trff
−
P
)
t
,
e
2.
3
c
trff
−
P
)
2
/
3
φ
g
e
2
,
3
2
P
e
v
2
,
V
=
+
(
c
P
)
−
c
P
h
,
efec
w
e
w
e
c
c
c
c
P
c
trff
−
P
)
den
,
a
c
−
sp
,
a
lv
d
,
e
g
e
where a low-pass filter has been applied to solar
radiation and wind speed disturbances to avoid
sudden changes in the control signals. Notice that
neither the latent heat nor the heating tubes terms
have been included, as data used for the experiments
shown in this paper were obtained with the crop in
the early stages of its development and without using
heating systems.
The inclusion of the feedforward term in series with
the plant (Fig. 3) allows to explicitly take into
account the measured value of the disturbances in
such a way that the control signal provided by the
feedback controller is the reference temperature to
the feedforward term. Notice that if the model used
by the feedforward term was an exact one, the
system constituted by the feedforward term in series
with the plant should have a steady state gain near
unity. Unfortunately, the simplicity of the models
(fixed coefficients) in comparison with a large
complex simulation model of the real system (in
which several coefficients change depend on
operating conditions) and the uncertainty in the
system (it is impossible to exactly model the
greenhouse dynamics) advices the use of robust
control techniques to account for the mentioned
sources of uncertainties. To demonstrate this, Fig 4.
shows the results obtained when implementing only
the feedforward term in open-loop (without feedback
controller). As can be seen, if the greenhouse
dynamics should correspond to model described in
equation (1), the response obtained with the system
3.4. Robust control design
In order to design the robust controller, the input-
output description of the system composed by the
feedforward term in series with the plant has been
approximated by an uncertain first order system (step
response tests shown that this approximation could
be adopted), in which typical steady state gain and
time constant mainly depend on the step input
amplitude and can vary between the following
bounds:
where
p (disturbances)
feedback
controller
prefilter
sp
trff
uff
sat
y
(
temp.
)
e
uff
F(s)
G(s)
FF
PLANT
feedforward
1/FF
Simplified model
(inverse
feedforward)
AW
,
(
,
/
,
(
,
P
(
s
)
=
k
, with
k
∈
[0.3,10],
τ
∈
[360,1080] s.
Using the algorithm in (Moreno
et al
., 1997), the
performance and stability boundaries are computed,
and the nominal open loop transfer function (Fig. 7)
using computer tools (Borguesani
et al
., 1995).
τ
s
+
1
Bode Diagrams
0
-10
80
-20
-30
0.0001 rps
-40
60
-50
-60
-70
40
0
0.001 rps
0.005 rps
0.01 rps
20
-50
-100
0
-150
-200
-250
-20
-5
10
-4
-3
-2
-1
10
10
10
10
Frequency (rad/sec)
-40
-400
-350
-300
-250
-200
-150
-100
-50
0
Fig. 5. Frequency domain specifications
Fig.7. Nominal open-loop and bounds at design
frequencies in
W
.
Phase (degrees)
Due to the uncertainty in the system, robust control
can be used, and Horowitz’s method is chosen, as
done in other applications related to greenhouse
climate (Linker
et al
., 1999).
The first step in this method is to choose
performance and stability specifications. Fig.5 shows
the performance specifications.
The resulting controller
G
is given by equation:
G
(
s
)
=
#
10
+
0
.
028
!
#
0
021
!
s
s
+
0
.
021
Finally, the precompensator
F
to achieve the nominal
specification is:
F
(
s
)
=
#
0
017
!
s
+
0
0017
As far as stability specifications is concerned, a gain
margin of 5 dB and phase margin of 45º are desired:
Fig. 8 shows the final result of the design for the
considered set of plants
℘
.
G
(
j
ω
)
P
(
j
ω
)
≤
2
dB
,
P
∈
,
∀
ω
>
0
(3)
10
1
+
G
(
j
ω
)
P
(
j
ω
)
0
with
℘
=
k
:
k
∈
[ ] [
0
3
10
,
τ
∈
360
1080
]
.
-10
τ
s
+
1
-20
50
Note that (3) does not guarantee stability for the
closed loop system, due to presence of the actuator
saturation, see for example (Moreno
et. al.
, 2002).
A controller {
F
,
G
} must be designed in order to
assure that the closed loop transfer function
T
(from
reference to output) lies within envelopes in Fig. 5,
and the stability specification in (3) is achieved, with
0
-30
-5 0
-40
-1 00
-1 50
-50
-2 00
-2 50
-60
-3 00
10
-5
10
-4
10
-3
10
-2
10
-1
w rps
-70
-5
-4
-3
-2
-1
10
10
10
10
10
w rps
T
∈
ℑ
=
F
(
s
)
G
(
s
)
P
(
s
)
:
P
∈
.
Fig. 8. Closed loop specifications (dashdot) and
frequency responses (solid) of the controlled
system.
1
+
G
(
s
)
P
(
s
)
In order to proceed with the design of the controller,
the value sets (Barmish, 1988), which describe the
system uncertainty in the Nichols chart, are
computed (Fig. 6).
4. RESULTS
In this section, some illustrative results of the
proposed approach are shown and discussed. Fig. 9
shows the evolution of a test covering 13 complete
days in summer time with a fixed set point and a
shading screen covering the greenhouse. Although
the control scheme has been developed for operation
during sun-shining conditions, it has not been turned
off during the night to shown the performance of the
antiwindup block even in such strongly adverse
situation (the vents are completely closed during the
night and so, large feedback errors feed the
controller). The evolution of the outside solar
radiation corresponds to clear day conditions, except
during the fifth and sixth day in which drops of more
than 100 W/m
2
occurs. Outside temperature
conditions are also varying and wind velocity
experiments quite large variations during all the days,
covering values from 0 to 12 m/s which large
influence the system behaviour when vents are
opened. Due to the size of the figures, a zoom of a
region has been included.
20
10
0
-10
-20
0.0001 rps
0.001 rps
0.005 rps
-30
0.01 rps
-40
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
Phase (degrees)
Fig. 6. System value sets.
Taking into account the typical time constants
involved in this problem and specifications, this is a
low frequency problem and so,
the selected
frequency points (rad/s) for the design are
W
=[0.0001, 0.001, 0.005, 0.01], leading to values of
∆
|
T
(j
ω
.
$
$
$
∀
,
)|=[0.0063, 0.6777, 5.5564, 14.7622]
respectively.
34
As can be seen, the tracking and disturbance
rejection capabilities are adequate in those cases in
which the vents are not saturated. When saturation
occurs, no degrees of freedom are available to
control the temperature. After saturation, the
performance of the system is quite good, as is
expected due to the use of the antiwindup scheme. As
can be seen in Fig. 9(b), the control signal suffers
from large excursions covering the whole control
range. This figure reflects the main drawback of the
approach used in this paper: as the controller tends to
quickly react to changes in disturbances (mainly due
to the structure of the feedforward term), the control
system is prone to over-actuate, thus increasing
electricity costs associated to the motors moving the
vents (even when filtering the disturbances before
entering the feedforward term). The design can be
improved by finding a trade-off between fast tracking
and associated costs (by including stronger filters
within the feedforward term or by including design
restrictions in the control effort).
32
30
28
33.2
33.15
26
33.1
24
33.05
33
22
32.95
32.9
20
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
18
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
local time (minutes)
(a) Set point and inside air temperatures (deg)
40
35
30
35
30
25
25
20
20
15
10
15
5
0
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
10
5
34
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
local time (minutes)
x 1 0
4
33.5
(b) Vents aperture (º)
33
1200
32.5
1000
800
700
720
740
760
local time (minutes)
800
820
840
860
880
900
600
(a) Set point and inside air temperatures (deg)
900
800
400
600
700
400
500
30
300
200
100
200
25
0
20
5000
5050
5100
5150
5200
5250
5300
5350
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
15
4
local time (minutes)
x 1 0
10
(c) Global solar radiation (W/m
2
)
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
700
720
740
760
780
800
820
840
860
880
900
local time (minutes)
(b) Vents aperture (º)
Fig. 10. Response to set point changes
4900
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
37
36.5
36
local time (minutes)
35.5
(d) Outside wind speed (m/s)
35
5060
5080
5100
5120
5140
5160
5180
5200
5220
28
local time (minutes)
26
(a) Set point and inside air temperatures (deg)
24
35
22
30
25
20
24
20
23.5
15
18
23
10
22.5
16
5
22
0
5050
5100
5150
5200
4700
4800
4900
5000
5100
5200
5300
5400
5500
14
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
local time (minutes)
local time (minutes)
x 10
4
(e) Outside temperature (deg)
Fig. 9. Complete 13-days simulation
(b) Vents aperture (º)
Fig. 11. Response to set point changes
34
32
30
28
33.2
33.15
26
33.1
24
33.05
33
22
32.95
32.9
20
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
18
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4
40
35
30
35
30
25
25
20
20
15
10
15
5
0
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
10
5
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 1 0
4
1200
1000
800
780
600
900
800
600
700
400
400
500
300
200
100
200
0
5000
5050
5100
5150
5200
5250
5300
5350
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
4
x 1 0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4900
4950
5000
5050
5100
5150
5200
5250
5300
5350
5400
28
26
24
22
20
24
23.5
18
23
22.5
16
22
4700
4800
4900
5000
5100
5200
5300
5400
5500
14
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 10
4