a nonlinear feedback technique for greenhouse environmental...
TRANSCRIPT
A nonlinear feedback technique for greenhouseenvironmental control
G.D. Pasgianos a,*, K.G. Arvanitis b, P. Polycarpou c,N. Sigrimis b
a Department of Electrical and Computer Engineering, National Technical University of Athens, Zographou,
Athens 15773, Greeceb Department of Agricultural Engineering, Agricultural University of Athens, Iera Odos 75, Athens 11855,
Greecec Agricultural Research Institute, P.O. Box 22016, Nicosia 1516, Cyprus
Abstract
Climate control for protected crops brings the added dimension of a biological system into a
physical system control situation. The plants in a greenhouse impose their own needs,
significantly affect their ambient conditions in a nonlinear way, and add long-time constants
to the system response. Moreover, the thermally dynamic nature of a greenhouse suggests that
disturbance attenuation (load control of external temperature, humidity, and sunlight) is far
more important than is the case for controlling other types of buildings. This paper presents a
feedback�/feedforward approach to system linearization and decoupling for climate control of
greenhouses and more specifically for the operation of ventilation/cooling and moisturizing.
The proposed method consists of three parts: (a) a model-based feedback�/feedforward
compensation of external disturbances (loads) on the basis of input�/output linearization and
decoupling; (b) the transformation of user-defined desired settings for temperature and
humidity into feasible controller setpoints, taking into account the constraints imposed by the
capacities of the actuators and the psychrometric laws; and (c) additional PI outer loops to
compensate for model uncertainties and deviations from expected disturbances (weather).
Moreover, some tuning tests lump together several physical system parameters to be easily
identified, and the method guarantees accuracy in setpoint tracking while simplifying stability
issues. The proposed method is applicable to any air-conditioning system and is expected to
gain wide acceptance in modern climate control systems.
# 2003 Elsevier Science B.V. All rights reserved.
* Corresponding author.
E-mail addresses: [email protected] (G.D. Pasgianos), [email protected] (K.G. Arvanitis),
[email protected] (P. Polycarpou), [email protected] (N. Sigrimis).
Computers and Electronics in Agriculture
40 (2003) 153�/177 www.elsevier.com/locate/compag
0168-1699/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0168-1699(03)00018-8
Keywords: Greenhouses; Environmental control; Psychrometrics; Feedback linearization; Feedforward
decoupling; Nonlinear systems
1. Introduction
It is well recognized that the climate in protected crop cultivation has a great
influence on the plant growth, and hence on fertility, production yield, quality, and
maintenance processes of the plants. Environmental control is a central feature of
modern production systems, whether within plant growth chambers (or rooms),
greenhouses, or totally closed environments such as those envisioned for food
production and waste treatment (bioregeneration) in space.
Environment control for living systems differs greatly from comparable control
for physical systems. Environment requirements for living systems are typically more
complex and nonlinear, and the biological system is likely to have significant and
numerous effects on its physical surroundings. Moreover, greenhouses and other
natural-light growth facilities must be controlled to deal with rapidly changing solar
loads. Plant production systems often lead to problems that are more related to load
control than to traditional setpoint control. The problems may be exacerbated in the
reduced gravity conditions of space where thermal buoyancy effects, plant
morphologies, and cost considerations can be very different.
Several studies and research applications involving environmental control of
greenhouses have been performed by many researchers (Jones et al., 1984; Gates and
Overhults, 1991; Stanghellini and van Meurs, 1992; Young and Lees, 1993; Zhang
and Barber, 1993; Young et al., 1994, 2000; Stanghellini and De Jong, 1995; Chao et
al., 1995, 2000; Chao and Gates, 1996; Lees et al., 1996; Arvanitis et al., 2000; Taylor
et al., 2000; Zolnier et al., 2000). Most of the studies on analysis and control of the
environment inside greenhouses have been based on the concept of energy and mass
balance and physical modeling. These concepts are very effective in order to clarify
the concepts of environmental control, to refine environmental control strategies,
and to gradually lead to economic optimization, the ultimate objective of
environmental control.
Many dynamic models for greenhouse environment exist in the extant literature,
and they are of nonlinear nature. The central state variable is typically air
temperature with relative humidity (or absolute humidity), and carbon dioxide
concentration is also considered. Disturbances to a greenhouse or other plant
thermal environment occur primarily from solar radiation, outside temperature
(conduction heat transfer and ventilation heat transfer) and interactions with
occupants (plants), the controlled heating and ventilating equipment, and the floor.
However, it is useful to note that, for the most part, the system is subjected to
relatively low frequency disturbances. Indeed, most of these disturbances are
considered as ‘‘loads’’ and a quasi-steady-state analysis often suffices for design
purposes. Perhaps the most common transient disturbance is a step change, either
from switching equipment, changing setpoints, or variable cloud cover.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177154
The fact that temperature and humidity are highly coupled through nonlinear
thermodynamic laws, and the actuators (i.e. windows) are usually subject to
changing characteristics (the gain is largely perturbed by cross-product terms with
disturbances such as wind velocity, outside temperature, etc.) has not been treated as
yet explicitly and analytically to provide a robust control scheme. The practical
controllers do meet the control requirements using many expert types of actuator
adjustments and ad hoc compensators.The use of physical laws from psychrometry, as process constraints, defines the
achievable operating temperature�/humidity range, and the use of a cost function
determines the optimum admissible operating point. This technique allows the
process of temperature�/humidity control to be coupled with other decision support
systems that may be using biological models of some sort. For example, some cost
parameters of the cost function may invoke values from other programs, which use
either a complete (i.e. production) or a partial (i.e. nutrient uptake) model. In this
way, the system may become part of an integrated production system in which thetemperature and humidity setpoints may be selected to include certain goals such as
minimum infection risk, maximum salinity tolerance and so on.
In this respect, this paper provides the means to combine biological and physical
models to simultaneously control the coupled temperature and humidity of air, for
plants grown in greenhouses, specialized growth rooms and chambers, and advanced
life support systems such as those under development in space. The case of MIMO
nonlinear systems with actuation constraints is approached in this paper using a
powerful combination of linearizing and non-interacting feedback�/feedforwardcontrollers, outer loop conventional dynamic controllers (e.g. PID controllers or
pseudoderivative feedback controllers) as well as a precompensator and command
generator (PCG) module, which defines the admissible state set. The proposed
technique is superior to other conventional linear multivariable techniques (e.g.
multivariable PID controller), because it maintains accurate performance in the
whole operating range, which is extremely wide and nonlinear in the present
application. The proposed nonlinear decoupling method produces a global
controller solution, with minimum design and tuning effort as compared with themultivariable PID controller, commonly used for a single operating point (local
controller). The proposed method will prove even more important in applications
with strict requirements on temperature and humidity (i.e. HVAC systems, clean
rooms, plant factories, etc.). The presented application of temperature�/humidity
control in greenhouses usually appears as a need in hot summers, as they prevail in
southern European countries, where cooling is very important. Simulation results
obtained after some preliminary identification tests to identify greenhouse thermal
parameters show the effectiveness and good performance of the proposed non-interacting control scheme, which provides smooth setpoint tracking and fast
regulatory control with disturbance rejection capabilities. The described identifica-
tion and tuning tests can be performed by the operator of the facility at startup and
system drift or evolution can be detected and computed on-line using on-line
techniques (Sigrimis and Rerras, 1996; Arvanitis et al., 2000). The proposed method
is applicable to any air-conditioning system and is expected to gain wide acceptance
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 155
in modern virtual-variable-based SCADA systems with extended computational
capabilities. The proposed method is currently implemented in MACQU (Sigrimis et
al., 2000a,b) systems to be placed in field operation.
2. Feedback�/feedforward linearization and decoupling
Consider the analytic nonlinear system
x�a(x; v)�B(x; v)u; yi �hi(x); (1)
where x � /Rn is the state vector, ui; yi �R (i�/1,. . .,p ) is the ith control input and
output, respectively, and v � /Rd is the external disturbance vector. In Eq. (1), a(x,v),
B(x,v), and hi(x) are analytic matrix-valued functions of appropriate dimensions.In the case where system disturbances, v, are unknown (or cannot be measured),
there is no general theoretical framework in order to control a system of form (1).
However, in the case where disturbances can be measured, and system (1) can be
brought to the form
y(ri)i �f i(x; v)�gT
i (x; v)u; i�1; . . . ; p; (2)
where ri is the relative degree of the ith system output (Isidori, 1981), assuming that
matrix D(x,v) of the form
D(x; v)�gT
1 (x; v)ngT
p (x; v)
24
35
is nonsingular, the feedback�/feedforward control law of the form
u�D�1(x; v) �f1(x; v)
nfp(x; v)
24
35� u1
nup
24
35
8<:
9=;; (3)
where ui (i�/1,. . .,p ) is a set of intermediate control inputs, renders the closed-loop
system, I/O linearized, decoupled, and disturbance isolated, having the form (Isidori,
1981)
y(ri)i � ui (4)
provided that the system states are measurable. In this way, each intermediate
control input ui controls directly the rate ri (rith derivative) of the ith output yi . For
example, in the specific case of temperature�/humidity control, u1 is the rate of
temperature change and u2 is the rate of the absolute humidity change, whichultimately will be translated to the desired temperature�/humidity setpoints through
Eq. (5).
Note that, in order to bring system (1) in form (2), it is necessary that, if a
disturbance appears in Eq. (1), a control input must also be present in the same
equation to allow elimination of the disturbance by feedforward action. This
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177156
feedforward action is inherently present due to the terms involved in matrices D(x,v)
and fi(x,v).
Note also that if api�1riBn; then, system (1) contains some additional unobser-
vable states, called the internal dynamics. The zero dynamics of Eq. (1) are the
internal dynamics of the system when the outputs of the system are kept at zero by
the input. For the closed system to be stabilizable, the system zero dynamics must be
stable (Isidori, 1981).Obviously, the closed-loop system (Eq. (4)) can be further controlled by adding an
‘‘outer loop’’ controller, in order to satisfy some control specifications. This outer
control loop may be based on any conventional linear control strategy such as pole
placement, model matching, H�-control, and can be as simple as a PID controller.
For example, in pole placement control, application of the outer control law,
ui��Xri�1
j�0
aijy(j)i �biui; (5)
brings the new closed-loop system to the form
y(ri)i �
Xri�1
j�0
aijy(j)i �biui:
Furthermore, in the case of setpoint tracking, in order to compensate disturbances,
which have not been taken into account in Eq. (1) or parametric uncertainties, and in
order to attain asymptotic convergence of the error to zero, despite these
uncertainty, an additional control loop with integral action (e.g. a PID controller)
must be used in most cases.
3. Greenhouse ventilation model
3.1. Greenhouse dynamic model
The dynamic model of the energy and mass balance of greenhouse air is shown to
be highly nonlinear. A simple greenhouse heating�/cooling ventilating model can be
obtained by considering the differential equations, which govern sensible and latentheat, as well as water balances on the interior volume. These differential equations
are as follows:
dTin(t)
dt�
1
rCpVT
[qheater(t)�Si(t)�lqfog(t)]�VR(t)
VT
[Tin(t)�Tout(t)]�UA
rCpVT
� [Tin(t)�Tout(t)]; (6a)
dwin(t)
dt�
1
VH
qfog(t)�1
VH
E(Si(t);win(t))�VR(t)
VH
[win(t)�wout(t)]; (6b)
where Tin is the indoor air temperature (8C), Tout the outdoor temperature (8C), UA
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 157
the heat transfer coefficient (W K�1), r the air density (1.2 kg m�3), Cp the specific
heat of air (1006 J kg�1 K�1), qheater the heat provided by the greenhouse heater
(W), Si the intercepted solar radiant energy (W), qfog the water capacity of the fog
system (g H2O s�1), r the latent heat of vaporization (2257 J g�1), VR the
ventilation rate (m3 s�1), win and wout the interior and exterior humidity ratios
(water vapor mass ratio, g H2O kg�1 of dry air), respectively, and E (Si,win) the
evapotranspiration rate of the plants (g H2O s�1). It should be noted that the airvolumes VT and VH to be used in the balances are the temperature and humidity
active mixing volumes, respectively (Young and Lees, 1993; Young et al., 2000).
Short circuiting and stagnant zones exist in ventilated spaces and the active mixing
volume is typically significantly less than the calculated total volume. The active
mixing volume of a ventilated space may easily be as small as 60�/70% of the
geometric volume. This, of course, means that indoor air temperature and humidity
are unlikely to be uniform throughout the air space. Moreover, in a model with only
one state for the temperature, the effective heat capacity usually must be taken largerthan that determined by just the air volume, to encompass some of the heat capacity
of construction materials and the plants. Similarly, the effective volume for humidity
may be smaller or larger than the geometric one, depending on the degree of mixing
and other effects such as air and humidity losses. In Section 5.1, we determine the
normalized parameters C0, tv, and V ?, which are related to the effective volumes VT
and VH, by applying some appropriate identification tests (calibrations).
3.2. Greenhouse psychrometric laws and actuator limits
Temperature and relative humidity are commonly measured air properties, highlycoupled through nonlinear thermodynamic laws; for example,
w� f (T ;RH;P)�0:62198Pws(T) RH
P � Pws(T) RH; (7)
where w is the humidity ratio, P the atmospheric pressure (kPa), and Pws thesaturation pressure of water vapor (kPa). This thermodynamic equation, which
constitutes an equality psychrometric law constraint to the problem of calculating
optimized controller setpoints, can be used to convert relative humidity to absolute
water content. This conversion provides a first step towards a state decoupled and
linearized system. The relation between saturation pressure of water vapor (in Pa)
and temperature (in K) can be evaluated by the following polynomial (Albright,
1990), whose coefficients A1�/A7 are shown in Table 1:
ln Pws�A1
T�A2�A3T�A4T2�A5T3�A6T4�A7 ln T : (8)
For a specific environmental condition, i.e. specific temperature T and absolute
humidity w , the enthalpy H0 (in kJ kg�1 of dry air) is given by
H0�1:006T�w(2501�1:805T): (9)
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177158
We define a specific enthalpy change (Hs) as the energy per unit volume (J m�3)
carried by the ventilating air. A thermal balance of Eq. (6a) at steady state,
neglecting enthalpy of incoming air and conductive heat losses from the greenhouse,
yields the following equation:
HsVR�Si[Hs�Si
VR
: (10)
Eq. (10) constitutes an equality thermal balance constraint relative to the problem of
calculating optimized controller setpoints. The next constraint equation (6b) at
steady state yields
qfog�VR(win(t)�wout(t))�E(Si(t);win(t))�qsVR�E(Si(t);win(t)); (11)
where qs is the specific water per unit air volume required to attain win. Eq. (11)
constitutes an equality mass balance constraint.
The actuating capacity qmaxfog is designed to ensure that ventilation air changed (/
Vmax) can be saturated under any load conditions. Moreover, let wssfog be the water
carrying capacity of the saturated air for the fog system operation, and qssfog be the
effective water carrying capacity, from wout to saturation, for the fog system (see Fig.
1). The actuating limit at the selected ventilation rate is
qlimfog�qss
fogVR5qssfogV max
R 5qmaxfog : (12)
Relation (12) constitutes an actuator capacity inequality constraint.
Maximum cooling is achieved when maximum evaporated water is used for agiven ventilation rate; thus, a control’s feasible region is defined based on maximum
ventilation capacity (e.g. 100 air changes per hour). In this condition, the minimum
specific enthalpy is
Hmins �
1
VmaxR
Si: (13)
Eq. (13) defines the feasible regime to the right of line A1A2, drawn as the enthalpy
H0�Hmins ; as shown in Fig. 1. For example, at half capacity, for q� 1
2qmax
fog and
V � 12V max; that is for Hs�2Hmin
s ; starting from outside conditions at point �A0�,
the operating point will be �A3� instead of �A1� at full capacity. Eq. (7) defines the
Table 1
Polynomial coefficients of Eq. (8) for temperature ranges from 0 to 200 8C
A1 �/5.8002206�/103
A2 1.3914993
A3 �/48.640239�/10�3
A4 41.764768�/10�6
A5 �/14.452093�/10�9
A6 0.0
A7 6.5459673
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 159
lower horizontal line of the regime. The upper horizontal line, which transverses
point �A1�, can be defined if we assume saturation in Eq. (7) (i.e. RH�/1) and then
substitute the calculated w (which, in this case, equals wsfog) in Eq. (9). This leads to
an expression of enthalpy at saturation (Hsat) as a function of temperature and
pressure, i.e.
Hsat�1:006T�0:62198Pws(2501 � 1:805T)
P � Pws
: (14)
Relation (14) constitutes an equality constraint due to psychrometric laws. Then, by
setting Eq. (10) equal to Eq. (14), point �A1� is defined (Fig. 1).
3.3. Calculation of realizable controller setpoints
The decision for a desired point of operation, inside the feasible region, which is
defined by the well-defined lines �A1A2� and �A2A6�, and the air vapor saturationline �A1A5� of Fig. 1, can be based on a cost function to include various aspects of
climate targets such as infection risk, nutrition, quality of product, etc. The weights
of such cost parameters may be drawn from other biological models. For the tests of
this paper, the cost function chosen was of the following form:
J?�c1(Tin;sp�Tin;d)2�c2(RHin;sp�RHin;d)2�c3VR�c4qfog; (15a)
where Tin,d and RHin,d are the indoor desired temperature and relative humidity,
Fig. 1. Actuation limits defined by psychrometric properties: point A1 is the operational condition at
maximum capacity of ventilation and misting. Point A3 is achieved if 50% capacity is used. Air properties
at A3 are drier and hotter than at A1.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177160
respectively, as drawn from plants’ physiological requirements, while Tin,sp and
RHin,sp are the temperature and relative humidity setpoints, to be calculated based
on actuator capacities and economical factors.
Depending on the outside air conditions and the load Si, the achievable operating
space, for any cost, may not contain the desirable conditions (Tin,d, RHin,d). A rule
base can be used to assign values for cost parameters c1 and c2 such as to equalize the
risk on the crop for each of the deviations (Tin,sp�/Tin,d) and (RHin,sp�/RHin,d). In anattempt to use complete functionals for cost calculations, without resorting to fuzzy
rules for cost parameter assignments, we used the following extended quadratic cost
function:
J�c1(Tin;sp�Tin;d)2�c1�
½Tin;sp � Tin;max½�c2(RHin;sp�RHin;d)2�
c2�
1 � RHin;sp
�c3VR�c4qfog: (15b)
The added penalty function terms, add steep hilly excursions on the convex
performance surface to ensure that the calculated setpoints for temperature and
humidity are kept away from an absolute maximum temperature (chosen by
intuition and constraints for crop safety) and from the saturation line (risk of
disease).Using Eqs. (7)�/(14), the load Env(Si,Tout,RHout) of Fig. 2 and a gradient descent
method to minimize Eq. (15b), PCG of Fig. 2 calculates the realizable desirable
target conditions Tin,sp and win,sp, the steady-state control values of qfog and VR;which can be used as feedforward values, and other variables useful for the
calculations at the controller level. The optimization problem which need to be
solved here is as follows:
min J (16a)
subject to
Fig. 2. PCG for calculating feasible control targets.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 161
psychrometric equality constraints (7); (9); (14); (16b)
model equality constraints due to thermal and mass balance
equations (10) and (11); (16c)
actuator capacity inequality constraints (12);
VR5V maxR ; H]Hmin
s ; w5wsfog:
(16d)
The PCG has all the required logic to compute realizable setpoints and avoids
pitfalls (i.e. singular values in D(t) calculations of Eq. (20)) by post-processing the
solution of the optimization problem (16a)�/(16d). The pseudocode of the operation
of the PCG block is shown in Table 2.
4. Control of the greenhouse ventilation model
4.1. Control model
In this section, the control method presented in Section 2 is applied to the problem
of greenhouse ventilation/cooling and moisturizing. To this end, a control model is
first derived. For summer operation, qheater in Eq. (6a) is set to zero. It is also worthnoticing that to a first approximation the evapotranspiration rate E (Si(t),win(t )) is in
most part related to the intercepted solar radiant energy, through the following
simplified relation:
E(Si(t);win(t))�aSi(t)
l�bTwin(t); (17)
where a is an overall coefficient to account for shading and leaf area index, and bT
the overall coefficient to account for thermodynamic constants and other factors
affecting evapotranspiration (i.e. stomata, air motion, etc.). In other words, the two
terms account for the single term VPD, used in literature (Stanghellini and van
Meurs, 1992; Stanghellini and De Jong, 1995; Sigrimis et al., 2001). On the basis of
these observations, relations (6a) and (6b) take the forms
Table 2
Pseudocode of the operation of PCG
Steps Operations
1 Read system characteristics (/VmaxR ;//qmax
fog ) and cost parameters (c1�/c4)
2 Read environmental conditions (Si, Tout, RHout)
3 Input desired temperature (Tin,d), RH (RHin,d), and thresholds DT and DRH
4 Solve for VR and qfog from Tout, wout, Si, Tin,sp, RHin,sp by setting Eqs. (6a) and (6b) equal to 0
5 Compute J
6 Call optimization algorithm to minimize J subject to constraints (16a) and (16b)
7 Return optimal Tin,sp and RHin,sp
8 When environmental conditions change by DT or DRH, go to step 2
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177162
dTin(t)
dt�
1
rCpVT
[Si(t)�lqfog(t)]�VR
VT
[Tin(t)�Tout(t)]�UA
rCpVT
� [Tin(t)�Tout(t)]; (18a)
dwin(t)
dt��
bT
VH
win(t)�1
VH
qfog(t)�a
lVH
Si(t)�VR
VH
[win(t)�wout(t)]: (18b)
Eqs. (18a) and (18b) are obviously coupled nonlinear equations, which cannot be put
into the rather familiar form of an affine analytic nonlinear system, due to their
complexity appearing as the cross-product terms between control and disturbance
variables. Other data-based approaches have been successfully applied to reduce the
complexity of the model and design a control system with good disturbance�/
response characteristics (Young et al., 1994). However, in the present case, relations(18a) and (18b) can alternatively be written in the form of (2), where, in the present
case,
x� [x1 x2]T�[Tin win]T; y�x; r1�r2�1; (19a)
u� [u1 u2]T�[VR qfog]T; v� [v1 v2 v3]T�[Si Tout wout]T; (19b)
f1(x; v)��UA
rCpVT
x1(t)�1
rCpVT
v1(t)�UA
rCpVT
v2(t);
f2(x; v)��bT
VH
x2(t)�a
lVH
v1(t);
(19c)
gT1 (x; v)�
1
VT
(v2(t)�x1(t)) �l
rCpVT
" #;
gT2 (x; v)�
1
VH
(v3(t)�x2(t))1
VH
" #:
(19d)
Note that disturbance variables of the greenhouse heating�/cooling ventilating model
can be easily measured by the instrumentation installed in the greenhousemeteorological cage. Furthermore, the complexity of such systems is rather eased
by the fact that the system state changes slowly and some state-dependent
parameters (i.e. bT) can be considered constant (i.e. quasi-static system operation).
Therefore, in the present case, a combined scheme of feedback with simultaneous
feedforward linearization is plausible.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 163
4.2. Application of the proposed control technique
To this end, in the present case, matrix D(x,v) is given by
D(x; v)�
1
VT
(v2(t)�x1(t)) �l
rCpVT
1
VH
(v3(t)�x2(t))1
VH
26664
37775;
whose determinant D(t ) is given by
D(t)�1
VTVH
�v2(t)�x1(t)�
l
rCp
(v3(t)�x2(t))
�; (20)
which must be nonzero, for the system to be I/O linearized, decoupled, and
disturbance isolated. Note that, in the present case, the sum of the relative degreesequals system dimension, and so there is no internal or zero dynamics. Note also
that, in the case where, D(t)�/0, the input u1(t) affects the system states x1(t) and
x2(t), with exactly the same way as u2(t ), and thereby decoupling as well as
feedback�/feedforward linearization are impossible.
By applying the control law of form (3), the closed-loop system takes on the form
y(1)i � ui; i�1; 2: (21)
Moreover, in order to fix the dynamics of the output yi , we apply the outer control
laws of the form
ui��ai0yi�biui��1
ti
(yi� ui); i�1; 2;
where u1�Tin;sp and u2�win;sp: Then, we obtain
y(1)i �
1
ti
yi �1
ti
ui; i�1; 2;
or in transfer function form
Hi(s)�1
tis � 1; i�1; 2;
where ti (i�/1,2) are the time constants of the new closed-loop systems.
The above control algorithm can be summarized in the following two relations:
u1(t)�Q�1(t)
��rCpVT
t1
u1(t)�lVH
t2
u2(t)�(rCpa�1)v1(t)�UA v2(t)
��
UA�rCpVT
t1
�x1�
�bTl�
lVH
t2
�x2
�; (22a)
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177164
u2(t)�Q�1(t)
����
UA�rCpVT
t1
�x1�
rCpVT
t1
u1�v1�UA v2
�[x2(t)�v3(t)]
�rCp[�x1(t)�v2(t)]
��bT�
VH
t2
�x2�
VH
t2
u2�a
lv1
��; (22b)
where
Q(t)�rCp[v2(t)�x1(t)]�l[v3(t)�x2(t)];
and is depicted in Fig. 3.
The greenhouse interior temperature and relative humidity are measured by a
thermometer and a hygrometer, respectively, which usually are located at a certain
distance from the greenhouse ventilators and the fog (or wet-pad system).
Hygrometers also present a lag time themselves. Hence, the changes in the
temperature and absolute humidity are determined after a certain time delay.
Moreover, transport delays as well as unmodeled dynamics contribute to additional
time lags. Therefore, an overall dead time, d1 and d2, must be considered for eachoutput, y1 and y2, respectively. However, one must keep in mind that the nonlinear
feedback�/feedforward control law, which renders the overall system linear and
decoupled, relies on current state and disturbance measurements. Therefore, time
delays may affect the feedback�/feedforward linearization procedure and could
degrade its performance. In order to avoid this problem, one must select t1 and t2,
which are related to the speed of the closed-loop system response, to be large enough,
resulting to a relatively slow closed-loop system. For example, a choice of t1�/4d1
and t2�/4d2 appears to be quite satisfactory compromise between the speed of theclosed-loop system response and the performance of the feedback�/feedforward
linearizing control law.
As it will be shown in the following section, the proposed control algorithm, based
on feedback�/feedforward linearization and outer loop controllers, is quite robust to
system parametric uncertainty as well as load disturbances. In particular, a 10%
uncertainty can easily be tolerated by the proposed controller. However, in the case
of large parameter variations (e.g. plant growth that affects the greenhouse thermal
capacity as well as evapotranspiration), one must apply more sophisticated controlalgorithms (like robust control or adaptive control algorithms) in order to
Fig. 3. Overall control strategy in case of small time delays and/or a slow desired response.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 165
compensate for such variations. Research on these topics (e.g. along the lines
reported in Sigrimis et al., 1999; Arvanitis et al., 2000) is currently in progress.
Moreover, feedback and feedforward decoupling as well as controller tuning are
easily accomplished by simple time-domain open- and closed-loop step responses.
5. Simulation results
In order to illustrate the efficiency and good performance of the proposed non-interacting control scheme, a series of simulation experiments is presented in the
present section. These simulation experiments were conducted by the use of the
Simulink toolbox of MATLAB, and in order to perform them, a complete nonlinear
dynamic model of a greenhouse with fully developed crop was considered (Rerras,
1998). This model includes decomposed process elements (i.e. instead of lumped heat
balance it uses separate equations for floor, side walls, and roof heat exchange sub-
models) and it is more approximate to a real greenhouse. It is worth noticing at this
point that, since the real model of a greenhouse is significantly more complicatedthan the one described by (18a) and (18b), a series of identification experimental tests
were first conducted, in order to identify the parameters involved in (18a) and (18b).
These tests, which are based on simple and easily available measurements, can be
readily applied to a real greenhouse in order to obtain a quite accurate model.
5.1. Preliminary identification tests for the greenhouse model parameters
The term bTwin(t) in Eq. (17) can be neglected, since the conditions of operating
the ventilation/cooling are rather dominated by solar radiation alone (i.e. bT�/0).Furthermore, in order to simplify the identification procedure, the model described
by (18a) and (18b) is re-written in the following simpler form:
dTin(t)
dt�
1
C0
[Si(t)�l?q%fog(t)]�VR;%
tv
[Tin(t)�Tout(t)]�UA
C0
� [Tin(t)�Tout(t)]; (23a)
dwin(t)
dt�
q%fog(t)
V ?�a?Si(t)�
VR;%
tv
[win(t)�wout(t)]: (23b)
In the above equations, parameter C0�/(rCpVT)�1 describes the thermal capacity of
the greenhouse while UA describes the heat losses. In order to normalize the control
variables, we use the convention that the ventilation rate VR is measured as apercentage of the maximum ventilation rate V max
R (i.e. VR�VR;%V maxR ); parameter tv
represents the inverse of the number of air changes per unit time (that is the time
needed for one air change). Similar to VR,%, we define q%fog as a percentage of the
maximum capacity of the fog system qfog,max; then, l ?�/lqfog,max. Parameter V ?�/
VH/qfog,max represents greenhouse volume per unit of the maximal fog water supply.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177166
Finally, parameter a ?�/a (lVH)�1 describes the contribution of evapotranspiration
to the balance of the absolute humidity.
With the above definitions, the experimental tests, which have been performed for
the purpose of identifying the greenhouse model parameters, are as follows.
5.1.1. Identification test for UA, C0, and a ?To identify the parameters UA, C0, and a ?, we set both the fog and ventilation
systems inactive (i.e. qfog�/0, VR�/0). In this case, relations (23a) and (23b) can be
rewritten in the following form:
dTin(t)
dt�
1
C0
Si(t)�UA
C0
[Tin(t)�Tout(t)]; (24a)
dwin(t)
dt�a?Si(t): (24b)
Integrating the above relations for a time interval, say Dt�/t1�/t0, where t0 and t1 are
the time at the beginning and at the end of the identification test, respectively, and
dividing by the time duration Dt of the experiment, yields
Tin(t1) � Tin(t0)
Dt�
1
C0
�g
t1
t0
Si(t) dt
Dt
��
UA
C0
�g
t1
t0
[Tin(t) � Tout(t)] dt
Dt
��
1
C0
Si;1�UA
C0
DT1;in�out; (25a)
win(t1) � win(t0)
Dt�a?Si;1; (25b)
where Si;1 and DT1;in�out are the average intercepted solar radiant energy and the
average of the temperature difference (Tin(t)�/Tout(t)), respectively, during the timeperiod of the test Dt . Then, we proceed as follows:
(i) By performing the above identification test and collecting data for two time
intervals Dt1 and Dt2 (with Dt1"/Dt2), relation (25a) provides the following system
of equations:
Tin;A(tj) � Tin;A(t0)
Dtj
�1
C0
Si;j�UA
C0
DT j;in�out; j�1; 2; (26)
where in general indices A,B,. . . indicate the conducted experiment. The above
system is linear with respect to (1/C0) and (UA/C0). Therefore, one can easilysolve this system of equations to obtain C0 and UA. Note that, in order to
increase the accuracy of the results, the time period Dt2 must significantly differ
from Dt1.
Note that here C0 reflects partially both the thermal capacity of the thermal
mass (plants, metal, soil) and of the air. Here, we assume that we work under the
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 167
influence of one dominant combined time constant and that some effects of
different time scales are considered as slowly varying disturbances.
(ii) Parameter a ? can be identified, using the same identification test, from the
gradient of the absolute humidity, when air is ‘‘dry enough’’. In this case, (25b)
yields
a?�win;A(tj) � win;A(t0)
Si;jDtj
; j�1 or 2: (27)
Note that, for this preliminary identification test, the time of the experimentation
Dtj (j�/1 or 2) can be quite large. The only precaution taken in this experiment is
that relative humidity should not reach saturation.
If the data acquisition system suffers from measurement or other physical noise
sources, it is recommended that a regression fit is applied of acquired data on Eqs.
(24a) and (24b), which will provide a better estimate of C0, UA, and a ?, than
simplified calculation of Eqs. (26) and (27), respectively.
5.1.2. Identification test for tvThis identification test can be performed after the above test A, by turning on the
ventilation system to its maximum rate VmaxR (VR,%�/1), and wait until Tin stabilizes
to its steady-state value.
(iii) In this case, considering (23a) at steady state we can obtain
tv�C0
1
Si;B=(Tin;B � Tout;B) � UA; (28)
where UA and C0 were obtained by the first experiment.
5.1.3. Identification test for l ? and V ?This identification test can be performed after test B, by turning also the fog
system onto its maximum capacity qfog,max (VR,%�/1 and q%,fog�/1), and wait until
win and Tin reach steady-state values.
(iv) Then, the parameter l ? can be calculated, by considering (23a) in its steadystate, as follows:
l?�Si;C��
C0
tv
�UA
�[Tin;C�Tout;C]; (29)
where UA and C0 were obtained by the first identification test.
(v) Finally, parameter V ? can be obtained, considering (23b) in its steady state, asfollows:
V ?�1
(1=tv)[win;C(t) � wout;C(t)] � a?Si;C
; (30)
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177168
where a ? and tv were obtained by the first and second identification tests,
respectively.
It is worth noticing at this point that in the second and third identification tests, it
may not be possible in the real test to have constant values of the external weather
conditions. Nevertheless, these tests can be performed using the average of the
measured values. Furthermore, if qfog,max exceeds saturation by design we should use
another value for qfog less than full capacity (q%,fogB/1) that avoids saturation.
Note also that the identification tests described above are not the only experimentsthat one can perform in order to obtain the parameters of the model. Several other
combinations of experiments can also be used with on-line and off-line techniques.
5.2. Greenhouse parameters and results of the preliminary identification tests
In the present simulation study, we consider a glass greenhouse having an area of
1000 m2 and a height of 4 m. The greenhouse is equipped by a shading screen, which
reduces the transmitted solar radiant energy by 50%. The maximum water capacity
of the fog system is 26 g min�1 m�3. Maximum ventilation rate corresponds to 20changes of the greenhouse air per hour. Furthermore, we consider that unmodeled
system dynamics as well as sensor dynamics contribute an overall dead time of 0.5
min in both temperature and humidity measurements. That is d1�/d2�/0.5 min.
Finally, in order to test the effectiveness of the proposed control technique in the
presence of measurement noise, a white noise signal is added to all measured
quantities. The signal to noise ratio (SNR) was 3%. To filter the noise in all measured
variables, an additional low-pass filter is used with cut-off frequency 0.05 Hz (20 s),
with all additional time response changes modeled-in.The preliminary identification tests described above were implemented using a
more accurate nonlinear dynamic greenhouse model (NDGM), implemented in
MATLAB, of which the identified parameters are presented in Table 3. In this table,
the parameters are expressed per square meter (m2) of greenhouse area.
5.3. Simulation experiments
We will perform three different tests: (a) a setpoint tracking test; (b) a regulatory
control test; and (c) a full-day real weather test. The first two tests are performed at
Table 3
Identified greenhouse model parameters
C0 (min W 8C�1) �/324.67
UA (W 8C�1) 29.81
tv (min) 3.41
l ? (W) 465
a ? (g m�3 min�1 W�1) 0.0033
1/V ? (g m�3 min�1) 13.3
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 169
three different levels of complexity by using: (i) data emulated with the simple model
of the form (18a) and (18b), without measurement noise and without outer feedback
PI controllers; (ii) data emulated with NDGM, with measurement noise and without
outer feedback PI controllers; and (iii) data emulated with NDGM, with measure-
ment noise and with outer feedback PI controllers.
5.3.1. Setpoint tracking test
A first simulation experiment has been conducted in order to demonstrate the
ability of the proposed control scheme, after the precompensator, to provide non-
interacting control and smooth closed-loop response to setpoint step changes. To
this end, the parameters of NDGM were selected such that the identified time
constants of the two closed-loop subsystems be about t1�/t2�/5 min. Then, afterapplying the feedback plus feedforward linearizing and non-interacting control law,
we obtain the decoupled systems of form (21). Moreover, to illustrate the need for an
additional outer PI controller in the closed loop, the same experiment was performed
for three different cases. First, the proposed control law is applied to a greenhouse
model of the form (18a) and (18b) with the identified parameters as given by Table 3
and without measurement noise. The obtained responses are shown in Figs. 4�/7 by
dashed lines. Figs. 4 and 5 illustrate the response for a setpoint step change of
absolute humidity from 18 to 24 g m�3 (which corresponds to a relative humiditychange from 60 to 80%), at t�/100 min, while the temperature setpoint remains
constant at 30 8C until a setpoint step change from 30 to 28 8C, at t�/200 min, with
absolute humidity setpoint remaining constant at 24 g m�3. Figs. 6 and 7 illustrate
the controller outputs q%,fog and VR;%; respectively, for the three experiments. Note
that in performing the simulation, the outside weather conditions were assumed to
be Tout�/35 8C and wout�/4 g m�3 (RH�/10%), while Si�/300 W m�2. The
obtained responses are quite smooth, and non-interacting control is perfectly
Fig. 4. Response of absolute humidity win for step changes in both humidity and temperature.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177170
attained. In the second case, the same experiment, with the same control law, is
performed for the complete NDGM greenhouse model implemented in MATLAB, and
in the presence of measurement noise. The responses obtained in this case are shown
in Figs. 4�/7 with thin solid lines. From these responses, one can easily recognize that
there is a steady-state error in the closed-loop system outputs and that non-
interacting control is not perfect. Finally, in the third case, the proposed linearizing
and non-interacting control law is applied to the NDGM model (with measurement
noise), but in this case, additional outer PI controllers of the form
Fig. 5. Response of temperature Tin for step changes in both humidity and temperature.
Fig. 6. Fog controller outputs for step changes in both humidity and temperature.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 171
Gc(s)�Kc
�1�
1
tis
�
are used to compensate for the system uncertainty. The proportional gains and the
integral times of these PI controller, in both cases, were preliminarily tuned to Kc�/
0.25 and ti �/0.5 min, respectively. The responses obtained in this case are illustratedin Figs. 4�/7 with thick solid lines. In this case, it is easily recognized that both non-
interacting control and setpoint tracking have been asymptotically attained.
5.3.2. Regulatory control test
The purpose of a second simulation experiment is to demonstrate that the closed-
loop system response is not affected by weather conditions, as it is expected, since the
feedforward term of the linearizing/non-interacting controller compensate for system
external disturbances. Here, the desired setpoints are Tin,sp�/30 8C and win,sp�/18 g
m�3. In order to perform the simulation, step changes of Si, from 200 to 300 Wm�2, of Tout from 35 to 32 8C, and of wout, from 4 to 8 g m�3 have been applied, at
time instants t�/100, 150, and 200 min, respectively. The results, obtained by the
implementation of the three experiments described above, are presented in Figs. 8�/
11. In the case, where there is no uncertainty in the model parameters (dashed lines),
there is no effect of weather conditions on Tin and win. In Figs. 10 and 11 one can
easily recognize that the feedforward terms of the proposed controller change very
fast and compensate for the outside whether conditions. In the second experiment
where the complete greenhouse model is used, and no additional outer PI controllersare used, the temperature in the greenhouse is significantly affected by weather
conditions, while the effect of these disturbances in the humidity is negligible,
although a steady-state error occurs. If the additional outer PI controllers are
introduced in the control loops, then fast regulatory control can be achieved, with
zero steady-state error. From the above simulation experiments, it becomes clear
Fig. 7. Ventilation controller outputs for step changes in both humidity and temperature.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177172
that the combined use of the proposed feedback�/feedforward linearizing control law
and the external PI controllers provides non-interacting control, fast setpoint
tracking, and fast regulatory control with disturbance rejection capabilities, even
in the presence of quite large uncertainty (e.g. a 10% uncertainty).
5.3.3. Full-day real weather test
Finally, a simulation study has been accomplished in order to perform
simultaneous temperature and humidity control in a greenhouse, in case of real
Fig. 8. Regulation of absolute humidity win for step changes in external disturbances in case of
uncertainty.
Fig. 9. Regulation of temperature Tin for step changes in external disturbances in case of uncertainty.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 173
weather conditions. To this end, weather data from a full summer day (June 3, 1999)
in Arizona, USA, have been exploited. Setpoints for win and Tin have been obtained
as outputs of the PCG block, and are illustrated in Figs. 12 and 13, together with the
trajectories of win, wout, and Tin, Si, Tout, respectively. The controller outputs are
presented in Fig. 14. Obviously, the tracking performance of the proposed controller
is remarkable. It is worth noticing that, because the weather conditions and the
desired inputs are slowly varying (in comparison to the time constants of the closed-
loop systems, which equal 5 min), the error in both the temperature and humidity is
very small.
Fig. 10. Fog system controller outputs for step changes in external disturbances in case of uncertainty.
Fig. 11. Ventilation system controller outputs for step changes in external disturbances (case of
uncertainty).
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177174
6. Conclusions
The presented method of decoupling a highly nonlinear and coupled system
proved to be very effective in meeting formal requirements for climate control of
greenhouses such as setpoint tracking and disturbance rejection. The PCG block
computes setpoint trade-offs based on psychrometric properties and actuator limits
and costs to provide optimized setpoints that will allow the feedback�/feedforward
controller to operate without hunting or chattering. The feedback�/feedforward
controller achieves global input�/output linearization and decoupling. Finally, the
Fig. 12. Absolute humidity trajectories in case of simultaneous absolute humidity and temperature
tracking.
Fig. 13. Temperature trajectories in case of simultaneous absolute humidity and temperature tracking.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177 175
outer PI feedback controller compensates for model mismatch and deviations from
expected disturbances.
The described identification and tuning tests can be performed by the operator of
the facility at startup, and system drift or evolution can be detected and computed
on-line using on-line techniques. After these tests are performed, the method not
only guarantees extreme accuracy in setpoint tracking but also downgrades stability
issues to the simplistic cases of feedforward and SISO systems.
The use of physical laws from psychrometry, as process constraints, defines the
achievable operating temperature�/humidity range, and the use of a cost function
determines the optimum admissible operating point. Although response speed and
setpoint tracking accuracy in greenhouse climate control are not very important for
the real practice, the method is easy to implement and will be practiced in the real
field by MACQU systems. More importantly, this technique allows the process of
temperature�/humidity control to be coupled with other decision support systems
that may be using biological models of some sort. Therefore, the method can be
easily used for multiobjective optimization of temperature and humidity setpoint
selection, where the weights of the cost parameters may be evaluated against risk or
other cost factors. More practical details using the method in commercial green-
houses, equipped with dynamic ventilators and wet-pad or fog systems, will appear
in a separate paper including real field experiments.
Acknowledgements
This work is supported by the HORTIMED (ICA3-CT1999-00009) project to
enable collaborative management of the root and shoot greenhouse environment.
Fig. 14. Controller outputs in case of simultaneous absolute humidity and temperature tracking.
G.D. Pasgianos et al. / Computers and Electronics in Agriculture 40 (2003) 153�/177176
MACQUD project (EU-DGVI PL98-4310) provides the MACQU technology for
easy field implementation.
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