A nonlinear feedback technique for greenhouse environmental...

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

mailto:[email protected]




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


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


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)


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


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.


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.


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


dTin(t)

dt�

1

rCpVT

[Si(t)�lqfog(t)]�VR

VT


rCpVT


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.


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)


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.


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


C0


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.


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


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)


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


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.


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.


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.


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.


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).


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.


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.


MACQUD project (EU-DGVI PL98-4310) provides the MACQU technology for

easy field implementation.

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