Robust controllers for simultaneous control of temperature and CO concentration in greenhouses

*Corresponding author. Fax:#972-4-822-1529. E-mail address:[email protected] (R. Linker)

Robust controllers for simultaneous control of temperature

and CO

concentration in greenhouses

R. Linker*, P.O. Gutman, I. Seginer

Agricultural Engineering Department, Technion, Haifa, 32000 Israel

Received 13 March 1998; accepted 12 March 1999

Abstract

The present work focuses on the control of greenhouse air temperature and CO

concentration by means of simultaneous

ventilation and enrichment. Such an operation, which may a priori seem contradictory, was shown by several authors to be required to maintain optimal temperature and CO

setpoints. The control process is divided into two distinct control loops, the "rst maintaining the temperature by adjusting the ventilation, and the second maintaining the CO

concentration by adjusting the

enrichment. The CO

concentration controller assumes the ventilation rate to be constant and approximately known over 2-min

intervals. Implementation in an experimental greenhouse shows the ability of the controllers to meet the requirements. 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Greenhouse; Robust control; Neural network; Optimal climate

1. Introduction

It has been known for years that, in addition to tem-perature and solar radiation, the COconcentration of the air has a direct in#uence on crop photosynthesis (e.g.

Idso and Idso, 1994). CO

enrichment, which consists of supplying highly concentrated CO

to the greenhouse, is a daytime procedure widely used in greenhouses to in-crease crop yield (e.g. Kimball, 1986). However, in warm climates, ventilation is required during most of the day to lower the temperature, and a con#ict exists between the

need to ventilate and the desire to enrich. While common practice usually views ventilation and enrichment as mu-tually exclusive, several studies related to optimal green-house control under warm climates (Ioslovich et al., 1995; Linker et al., 1998; Seginer et al., 1986) have shown that in order to maintain, at any time, optimal temperature and CO concentration, simultaneous ventilation and enrichment would be required.

While a wide range of greenhouse climate controllers have been described in the literature, these were not designed speci"cally in order to enable simultaneous use

of ventilation and enrichment. The documented

control-lers range from switching logics (Chao and Gates, 1996) to an improved digital PI controller (Chotai et al., 1991), and LQI controllers (Nishina et al., 1997). Feedforward control has also been studied, especially in connection with greenhouse heating (Davis and Hooper, 1991; Takakura et al., 1994). Because of the di$culty of

obtaing accurate models, adaptive control has also been in-vestigated (Boaventura Cunha et al., 1997; Sigrimis and Rerras, 1996). In the present work, robust feedback con-trollers are used to deal with the modeling errors (so-called_`uncertainties_a). A combination of robust feedback and adaptive feedforward from the disturbances has also been studied (Linker, 1995), but is not described here due to space limitations.

2. Materials and methods

The data required to design the controllers were col-lected in a single-span, pitched-roof greenhouse at the Technion, Haifa, Israel. The greenhouse had a 8 m by 6 m#oor, with a volume of 190 m, the long axis pointing

northeast. No crop was present in the greenhouse at the time of the experiment, but the concrete#oor was

con-tinuously wetted to simulate a greenhouse with a wet soil surface. Therefore, the results presented here are sup-posed to apply to a greenhouse with small seedlings

Nomenclature

c speci"c heat of air at constant pressure,

J/(kg[air]K) D wind direction, deg

F level of speed of the fans, dimensionless H e!ective height of the greenhouse, m h absolute humidity, kg[vapor]/kg[air] O opening of screen, %

Q ventilation#ux, m/(m[ground]s) q shift operator, dimensionless

R CO

#ux, kg[CO

]/(m[ground]s) s Laplace operator, 1/s

S solar radiation#ux, W/m[ground] ¹ temperature, K

t time, s

u control variable de"ned by Eq. (8), W/m

; _{overall heat transfer coe}$cient of cover,

W/(m[ground]K) = wind speed, m/s

X CO

concentration of air, kg[CO]/kg[air]

Greek letters

* delay in the COcontrol loop, s

j latent heat of vaporization of water, J/kg[vapor]

" solar heating e$ciency, dimensionless

o air density, kg[air]/m

h indoor}outdoor temperature di!erence, K

# time average ofh, K

m indoor}outdoor CO

concentration di!erence, kg[CO

]/kg[air]

Subscripts 1 large fan 2 small fan des desired FB feedback

i inside greenhouse nom nominal

o outside greenhouse r reference

s sampling

Superscript opt optimal

Acronyms FB feedback NN neural network

RMS root mean square, de"ned in Eq. (4)

R coe$cient of linear correlation, de"ned in

Eq. (5)

which do not in#uence the greenhouse climate but

re-quire CO

enrichment. The experimental setup has been fully described in a separate paper (Linker et al., 1998), and only the elements relevant to the present study are brie#y described below.

The variables measured in the greenhouse included dry bulb and wet bulb air temperature, and CO

concentra-tion. The outside weather measurements consisted of dry bulb and wet bulb air temperature, solar radiation, and wind speed and direction.

The greenhouse climate control variables consisted of CO

enrichment and forced ventilation. Since this study focused on daytime control under summer conditions, heating was not considered. Two variable-speed fans (extracting the air from the greenhouse) were mounted on the southwest wall. The fans were rated at 3.39 and 0.46 m/s. The discharge ranges of the fans were dis-cretized. The large fan had eight equally spaced operating levels, each increment being slightly smaller than the full capacity of the small fan. The range of the small fan was divided equally into four levels. Altogether, 32 distinct operating levels were possible (labeled 0}31). The control

variablesF

andFrepresent the levels of operation of the large and small fans, respectively. A linear combina-tion of these two variables, namelyF"4F

#F

is also used below. An 80 cm wide opening, spanning the whole length of the northeast wall of the greenhouse, was con-structed. The opening could be obstructed to any degree by a motorized screen, unfolding from the top. In order to achieve"ne control at low ventilation rates, the lower

part of the opening was partly obstructed by solid tri-angles. Ten equally spaced screen positions (every 8 cm), represented by the control variableO, were allowed. Pure CO

from pressurized containers was injected at the four corners of the greenhouse, through an electrically con-trolled valve. The injection rate, 1.78;10\kg[CO

]/s, was calculated from di!erential weighing of the CO

container.

Measurements relative to greenhouse climate and out-side weather were recorded by two separate data loggers. Both data loggers were connected to a Personal Com-puter (PC) which determined the setpoints and control signals. All measurements were performed every second, except for the screen position, which, in order to be accurate, had to be measured more frequently, once every 0.15 s. 10-min averages of the weather measurements and 2-min averages of the greenhouse climate were transmit-ted to the PC. Due to hardware limitations, the control variables were computed only every 2 min, based on averages of the measurements made during the preceding 2 min. Fan operating levels and the screen opening were maintained constant over 2 min intervals, while an ap-proximately constant enrichment rate was achieved by pulse-width modulation: the CO

Fig. 1. Greenhouse optimal control scheme.Qis the ventilation rate,Ris the enrichment rate,¹

Gis the greenhouse air temperature, andXGis the CO concentration.

3. Control-system architecture

Fig. 1 presents a general scheme of greenhouse optimal control, which includes three decision levels (e.g. Tantau, 1991). The global optimization (highest decision level), which considers the whole growing season and produces costate variables, has been treated, for instance, by Seginer et al. (1991), and van Henten (1994). The inter-mediate level, which performs a local optimization, was described, among others, by Gutman et al. (1993), Bailey and Chalabi (1994), and Linker et al. (1998). The local optimization, which can be performed several times an hour, provides either optimal actuator positions or opti-mal setpoints for the controlled variables. In the second case, the setpoints have to be maintained by the lowest control level, on which the present paper focuses.

As mentioned in the previous section, the means of action on the greenhouse climate are the speeds of the

two fans, the opening of the screen, and the rate at which CO

is injected. The latter in#uences only the CO

con-centration, while the"rst two determine the ventilation

rate (Q), which in#uences both the temperature and the

CO

concentration. These observations lead to the fol-lowing schematic form, where;denotes a linear or non-linear dependence (adapted from Ioslovich et al., 1995).

dX G/dt

"

0

0 ;

¹ G

XG#

; 0

; ;

Q

R

#

0 ; 0

¹ M S

M h

G!h M

. (1)

The "rst vector on the right-hand side contains the

state variables (inside air temperature ¹

G, and CO concentration,X

Fig. 2. Cascaded controller for the simultaneous control of temper-ature and COconcentration.Qis the ventilation rate,Ris the enrich-ment rate, ¹

G is the greenhouse air temperature, XG is the CO concentration, and¹

G andXG denote the optimal setpoints.

variables (ventilation rate, Q, and enrichment rate, R), and the third one represents the disturbances, which, in the present case, are measured (solar radiation,S

M,

out-side air temperature,¹

M, and inside and outside

humid-ity, h

G and hM). Note that this simple representation

assumes that heat storage takes place in the greenhouse air only (and not in the structure).

The system described by Eq. (1) can be controlled by two decoupled loops (Fig. 2):

1. A temperature loop which determines the ventilation rate.

2. A CO

loop which determines the COenrichment, for a given ventilation rate.

A direct consequence of this architecture is that venti-lation is not used to lower a CO

concentration that is higher than the setpoint. This sometimes causes the CO

concentration to present overshoots for prolonged peri-ods of time. However, since a higher CO

concentration has always a positive e!ect on photosynthesis, lowering

the CO

concentration by ventilating would be wasteful. The manipulated variable &ventilation rate' Qis

con-trolled through the fan speeds (F and F) and screen opening (O). Since this control variable is not measurable on-line, a model relating the position of the actuators to the resulting ventilation rate is needed. Such a model is presented in the next section.

4. Ventilation rate modeling

A neural network (NN) was trained to predict the ventilation rate from the control variables (F

,F, and

O), and the measured disturbances (wind speed, =_and direction, D). The data required to train such a model were obtained through gas decay tests (e.g. Nederho!

et al., 1984) for which, since no crop was present in the greenhouse, COcould be used as a tracer gas. At the beginning of each test, CO

was introduced into the unventilated greenhouse until a prespeci"ed CO

level was reached. After stopping the enrichment, the ac-tuatorsF

,FandOwere set to the desired level, and the resulting exponential decrease of CO

concentration was used to determine the ventilation rate o!-line, using the

following simple"rst-order model:

oHdXG

dt "!oQ(XG!XM), (2)

whereHis the e!ective height of the greenhouse (3.5 m),

andois the air density (1.2 kg/m). Assuming the outside CO

concentration XM and the ventilation rate Q (i.e.,

indirectly wind speed and direction) to be constant, inte-gration of Eq. (2) leads to

(X_G(t)!X M)"(X

G(0)!X

M)e\/R& (3)

which can be used to determineQgraphically.

A neural network having as inputs wind speed and direction, fan speeds, and screen opening, and as output the ventilation rate, was trained using the Matlab Neural Networks Toolbox (Demuth and Beale, 1992). The neu-ral network had one hidden layer of two nodes with sigmoid activation functions, and a linear output layer. Altogether, 195 tests were available, of which 174 were used as a training set, and 21 as a testing set. Table 1 presents the training and testing results in terms of the root mean square (RMS) of the residual error, and in terms ofR, the coe$cient of linear correlation between

the measured and predicted values. The error RMS is de"ned as

RMS" H

(y

H!y

H ), (4)

wherey_Hdenotes thejth data-point, andy_H denotes the NN prediction for the corresponding data point. The coe$cient of linear correlationRis de"ned as

R"1! H(yH !>

H)

H(yH!y) (5)

wherey denotes the average of they

H's, and >

H denotes

the jth prediction of a linear regression between the measured and NN predicted values.

The results in Table 1 show the ability of the NN to model the ventilation rate. Adding hidden nodes did not improve the performance of the NN. The NN predictions are presented graphically in Fig. 3. The decrease of the predicted ventilation rate each time that the small fan is turned o!and the large fan level is increased, means that

Fig. 3. Ventilation neural-network model predictions.Dand=are the wind direction and speed, respectively.Fis the fan speeds, andQis the predicted ventilation rate. In each graph, each curve corresponds to a di!erent screen opening. The upper curves correspond to a fully opened screen,

the lower ones to closed screen.D"0 corresponds to wind blowing in the_&screen-to-fans_'direction. Table 1

Neural-network ventilation model: performance over the training and testing sets (maximum ventilation rate: 50;10\(m/(ms)). Ventilation NN

Rtraining 0.92

RMS training (m/(ms)) 0.0012

Rtest 0.95

RMS test (m/(ms)) 0.0008

than the discharge due to one additional step of the large fan. This overlap ensures that the ventilation rate can be controlled continuously. A comparison between ventila-tion rates predicted with similar actuator posiventila-tions, but di!erent wind speed and direction, shows wind in#uence.

Wind speed has only a limited in#uence, but wind

direc-tion signi"cantly in#uences the ventilation rate. Using

the situation of a wind blowing in a direction perpendicu-lar to the screen-to-fans direction (D"90₃) as reference,

it may be seen in Fig. 3 that wind blowing in the screen-to-fans direction (left"gures,D"03) increases the

venti-lation rate. The wind e!ect is larger at small ventilation

rates, which leads to the slightly convex pictures. Sim-ilarly, wind blowing in the direction opposite to the forced ventilation (right "gures, D"1803) reduces the

ventilation rate. Here too, the wind in#uence is larger at

small ventilation rates; hence the concave graphs.

5. Temperature controller

5.1. Robust controller design

In order to design the temperature controller, the fol-lowing "rst-order energy balance model of the

green-house is used:

ocHd¹G

dt""SM!(ocQ#;)(¹G!¹M)!ojQ(hG!hM),

(6)

wherecis the speci"c heat of air (1005 J/(kg K)),;is the

overall heat transfer coe$cient of the cover,"is the solar

heating e$ciency, andjis the latent heat of vaporization

of water (2.5;10 J/kg(vapor)). In this model, the left-hand side term describes the sensible heat rate of change, the "rst term on the right-hand side denotes the solar

energy#ux absorbed by the greenhouse, the second term

the heat#ux leaving the greenhouse by ventilation and

through the cover, and the third term describes the latent heat exchange.

The solar radiation heating e$ciency (") was

deter-mined using data with maximum ventilation rate, so that the heat transfer through the cover (characterized by;) could be ignored. Afterwards, the overall heat transfer coe$cient of the cover (;) was determined using data

with very low ventilation rate. This identi"cation led to

Fig. 4. Temperature control loop.

; _{can be explained by the simplicity of the model} (Eq. (6)), which, for instance, does not include the heat storage in the#oor and the greenhouse structure. Such

unmodeled e!ects appear as uncertainty in the existing

parameters. Note that since the parameter "enters the model only through a measured disturbance, its actual value is not important for the design of the controller.

The controller based on this model determines the required ventilation rate, which is translated into fan speeds and a screen opening using the NN model pre-sented in the previous section (Fig. 4). Since there are three actuators (large and small fan speeds, and screen opening), and only one controlled variable,Q, the same ventilation rate can be obtained with di!erent

combina-tions of the actuators. Therefore, it has to be decided a priori which combination will be considered as best. Based on mechanical considerations, it was decided to try to reduce changes in the fan speeds. In order to do so, a "rst search over the screen opening is performed in

order to determine if the desired ventilation rate can be obtained by adjusting the screen opening alone. If this is not the case, a new search is performed over the screen opening, assuming di!erent speeds for the small fan. If

the desired ventilation rate can still not be attained, the whole procedure is repeated with all the possible speeds for the large fan. Since the desired ventilation rate cannot be exactly realized, a di!erence of 10\ m/(ms)

be-tween the desired and predicted values is considered acceptable.

Due to the bi-linearity of the model (6) (term Q¹ G),

linear control methods may not be suitable. However, the structure of the non-linearity makes it suitable for feed-back linearization (Gutman, 1981). Assuming that the inside and outside temperatures and humidity do not change over a 2-min interval, Eq. (6) can be rewritten as

ocHdh

dt""SM!;h!u, (7)

where

u"ocQh#ojQ(h G!h

M) (8)

and

h"¹ G!¹

M. (9)

Considering the solar radiation as a disturbance, Eq. (7) gives the following di!erential equation linkinguandh:

dh

dt"Ah#Bu (10)

withA"!;/(ocH) andB"!1/(ocH). Since the

con-trol variables are changed only every 2 min, the model has to be discretized into the following form:

h((k#1)t

Q)"eRQh(kt Q)#

Q

eTdvBu(kt

Q) (11)

"eRQh(kt Q)#1

A(eRQ!1)Bu(ktQ), (12)

wheret

Q is the sampling time (120 s). The control

vari-ables are determined using 2-min averages and not instantaneous measurements, so that the controlled variable is

H((k#1)t Q)"1

tQ

RQ

h

(kt_Q#v) dv (13)

"1 t

Q

eRQ!1

A h(ktQ)#

B A

eRQ!1!At Q

A u(ktQ)

.

(14)

Eliminating husing Eq. (12), and introducing the shift operatorq,

H(q)" B

At

Q

(eRQ!1!At

Q)q#(1!eRQ#At QeRQ)

q(q!eRQ) u(q).

(15)

This model includes one parameter which is only approx-imately known,;(;3[3; 10]). Furthermore, the actual

ventilation rate is not exactly the desired one and, based on the performance of the NN ventilation model present-ed in Section 4, as well as rapid changes in wind spepresent-ed and direction, an uncertainty of$10% inQ was

con-sidered (Q3 [0.9; 1.1]Q

), leading to

H(q)" B

At_Q

(eRQ!1!At

Q)q#(1!eRQ#At QeRQ)

q(q!eRQ) iu(q)

(16)

"P(q)iu

(q) (17)

withi3[0.9; 1.1].

Due to the modeling uncertainties, robust control could be used, and Horowitz's method (Horowitz and

Sidi, 1972; Gutman et al., 1988) was chosen. In this method, the designer speci"es time-domain speci"cations

which are translated into frequency-domain ones, using a second- or third-order model for the closed-loop sys-tem. Based on these speci"cations and the model

uncer-tainties, the Horowitz bounds are created in the Nichols chart (which allows one to read closed-loop performance from the open-loop gain and phase). These bounds de"ne

Fig. 5. Temperature model value sets.

Fig. 6. Temperature control loop. Nominal open-loop (Eqs. (17) and (18)) and Horowitz bounds at design frequencies.

closed-loop system meets the speci"cations and is stable.

The method ends with the design of a pre"lter.

In the present case, a compromise between the limita-tions imposed by the sampling frequency and the desire to achieve tight control leads to the following reference step response speci"cations: maximum overshoot 20%,

rise time (80%) 240}600 s, settling time ($10%) 960 s.

Following Horowitz's method, these time-domain

speci-"cations are translated into frequency-domain speci"

ca-tions, and, in order to ensure stability of the closed loop, a sensitivity speci"cation of 6 dB is also introduced.

In order to proceed with the design of the controller, the value sets (e.g. Barmish, 1988), which describe the model uncertainty in the Nichols chart, have to be com-puted. Following AsstroKm and Wittenmark (1994), the

mappingq"exp( jut

Q) is used inP(z)i (Eq. (17)) for all

combinations of;_andi_{, leading to the value sets} pre-sented in Fig. 5.

Combining the frequency-domain speci"cations and

the value sets leads to the Horowitz bounds presented in Fig. 6. The same"gure presents the nominal open loop

with the feedback controller:

G(q)"!0.06 120

(q!1)

(q!e\RQ)

(1!e\RQ)

;(q!e\RQ) (1!e\RQ)

(1!e\RQ)

(q!e\RQ). (18)

This controller was designed by trial-and-error loop-shaping such that the nominal open-loop frequency values (denoted byHin Fig. 6) fell on the permissible side of their respective Horowitz bounds.

Note that althoughG(q) has a negative static gain, the open-loopP(q))G(q) has a positive static gain due to the

negative sign ofBin Eq. (17). Note also that, in contrast with other design methods, the loop-shaped controller obtained with the Horowitz method (Eq. (18)) is not explicitly a function of the plant model (Eq. (17)).

Fig. 7 presents the frequency domain speci"cations, as

well as the computed closed loops ¹"F(q)P(q)G(q)/

(1#P(q)G(q)) with the pre"lter

F(q)"(1

!e\RQ)

(q!e\RQ)

(q!e\RQ)

(1!e\RQ) (19)

and it can be veri"ed that the closed-loop gain extent lies

within the speci"cations at all the design frequencies.

5.2. Comparison with a PI controller and the importance of theventilation model

A simulation study was performed in order to quantify the advantage of using the robust controller (Eqs. (18) and (19)) together with the ventilation NN presented in the previous section. This study comprised two stages. In a"rst stage, the greenhouse model (Eq. (6)) and the NN

Fig. 7. Temperature control loop. Closed-loop speci"cations (bold),

nominal closed loop (solid line), and gain extent (circles), with the controller Eq. (18) and pre"lter Eq. (19).

Fig. 8. Setpoint, measured (bold) and predicted (gray) temperatures (top frame), and actual (bold) and simulated desired ventilation rates (bottom frame), for June 14, 1997. The simulated greenhouse was described by Eq. (6) with;"3, and the ventilation NN.

compare the performance of the robust controller with that of a PI controller. The sensitivity of the closed loop to the accuracy of the ventilation model was also tested. Fig. 8 presents the setpoints, measured temperature, and desired ventilation rate recorded in the greenhouse on June 14, 1997. The temperature and ventilation rate predicted by the closed-loop simulation, using the same controller, are shown in the same "gure. It can be seen

that both the predicted and actual temperature and ven-tilation rates are very similar, which validates the con"

g-uration&model (6) and ventilation NN'as a model of the

greenhouse.

Using this simulator, the performance of the robust controller was compared to that of the PI controller

resulting from an internal model control (IMC) design (Rivera et al., 1986), with a bandwidth of 1/200 rad/s:

G

.'(s)" 1

200B !s/A

#1

!s/A

(20)

and, after discretization using the zero-order-hold method, and computingA

with ;"5:

G .'(q)"

!24.12q#21.12

q!1 . (21)

Fig. 9 presents the results of the simulations with the robust controller (top left frame), and the PI controller (top right frame), assuming the following constant out-side climate: S

M"500 W/m,¹

M"203C,h M"h

G. It can

be seen that the method used to design the controller, as well as its structure, has only a limited e!ect on the

performance of the loop. The bottom frame of Fig. 9 pres-ents the results obtained with the robust controller if it is assumed that the NN ventilation model is not available to translate the desired ventilation rate into actuator positions. For this simulation, the ventilation rate was translated into actuator positions using the following relations, which, as shown in Fig. 10, are an approxima-tion of the mapping created by the neural network.

ifQ

(0.005,F"2, andO"(Q

/0.005)*50, (22) if 0.005(Q

(0.030, F"(Q

!0.005)/(0.03!0.005)_*31, andO"50, (23)

if 0.030(Q

,

F"31, andO"(Q

!0.03)/(0.05!0.03)_*50#50.

(24)

A comparison between frames (a) and (c) (robust con-troller with and without NN ventilation model) shows that imperfect modeling of the ventilation rate strongly a!ects the loop performance. A similar deterioration was

observed on the performance of the control loop with the PI controller (not shown).

From these results, it can be concluded that in order to achieve tight temperature control, an accurate relation between the ventilation rate and the actuator positions is required. As long as this relation is available, similar performances can be obtained with di!erent types of

controllers, corresponding to di!erent designs.

6. CO2controller

In order to design the CO

controller, the following

"rst-order model, of which Eq. (2) is a particular case, is

used:

oHdm(t)

Fig. 9. Setpoint and simulated temperatures with the robust controller (Eqs. (18) and (19)) with NN ventilation (frame (a)); PI controller (Eq. (21)) with NN ventilation model (frame (b)); and robust controller (Eqs. (18) and (19)) without NN ventilation model (frame (c)). In each case, the bold line corresponds to;"10 W/mK and the thin one to;"3 W/mK. A constant outside climate is assumed (S

M"500 W/m,¹

M"20₃C,

q M"q

G).

Fig. 10. Comparison between the desired ventilation rate and the ventilation rate resulting from relations 22}24.

wheremdenotes X G!X

M, * is the delay present in the

control loop, and it has been assumed that the outside CO

concentration is constant.

This model has two uncertain parameters (Qand *), and here again the Horowitz method was chosen to design the controller. The loop delay*, which includes the time to transfer the gas to the sensor and the sensor response time, was determined experimentally to lie be-tween 50 and 110 s. The uncertainty inQwas estimated as $10% of the desired ventilation rate (Q3[0.9;

1.1]Q

). Such a description of the uncertainty allows for the inclusion in the controller of an inverse nominal model (s#Q

/H), where Q3[0.5; 20];10\. At

this point the original system to be controlled can be replaced by

%(s)" 1

oH (s#Q

/H)

Fig. 11. COmodel value sets: comparison between the value sets of the original model (Eq. (25)) (upper value set of each frame) and those of the model with cancellation of the nominal plant (Eq. (27)).

Fig. 12. CO

control loop. Nominal open-loop (Eqs. (27) and (28)) and Horowitz bounds at design frequencies.

Fig. 13. CO

control loop. Closed-loop speci"cations (bold), nominal

closed loop (solid line), and gain extent (circles), with the controller described by Eq. (28).

which has the advantage of having its value sets

&centered' at e\ Q/oH, regardless of Q

. Note that the controlled system now has three uncertain parameters: Q,Q, and*.

This nominal system cancellation can be performed only in continuous time. Therefore, in contrast to the temperature controller, which was designed in the dis-crete-time domain, the CO

controller has to be designed in the continuous-time domain. In a second stage, the controller (containing the inverse nominal model) is translated into the discrete form using the matched pole-zero transformation (Franklin and Powell, 1980). Note that an additional delay of 60 s (t

Q/2) has to be introduced

to model the averaging operation over the two-minute sampling intervals, so that the controller is designed

using the model

%(s)" 1

oH (s#Q

/H)

(s#Q/H) e\ >Q. (27)

The controller is designed to meet the following refer-ence-step speci"cations: maximum overshoot 20%, rise

time (80%) 240}600 s, settling time ($10%) 960 s.

Fig. 11 presents typical value sets. In order to show the advantage of introducing the inverse nominal model, the same"gure presents the value sets of the original model

(Eq. (25)). The value sets are clearly reduced by the nominal system cancellation, which allows for the meet-ing of tighter speci"cations, or alternatively, for using

a weaker controller. Fig. 12 presents the Horowitz bounds (the sensitivity bounds were computed with a 6 dB speci"cation), and the nominal open loop with the

controller

G

!-(s)"0.013

1 s

((3s/0.015#1))

(s/(0.015(3)#1)

(1!e\QRQ)

120s . (28)

Here again, this controller was designed by trial-and-error loop shaping such that the nominal open-loop frequency values (denoted by H in Fig. 12) fell on the permissible side of their respective Horowitz bounds.

Note that the last factor in Eq. (28) is the transfer function for the sample-and-hold operation. The dis-crete-domain equivalent of the controller, including the inverse nominal model, is given by

C

!-(q)"0.013

120 (q!1)

(q!e\(RQ) (q!e\RQ()

(q!e\/&RQ) (q!1) .

Fig. 14. Outside air temperature (top frame), and outside solar radi-ation (bottom frame) recorded on June 4, 1997.

Fig. 15. Implementation results. The top frame presents setpoint (bold) and measured temperature (thin line). The second frame presents the ventilation rate prescribed by the controller. The third frame presents the setpoint (bold) and measured CO

concentration (thin line), and the bottom frame presents the enrichment rate. All"gures present 10-min

averages.

Fig. 13, which presents the frequency-domain speci"

ca-tions together with the nominal closed loop and the gain extent, shows that no pre"lter is required.

7. Implementation results

The controllers described in the previous sections were implemented in the experimental greenhouse. The tem-perature and CO concentration setpoints were com-puted according to the optimization method described by Linker et al. (1998). Fig. 14 presents the outside solar radiation and temperature recorded on June 4, 1997 (the humidity, not shown here, was almost constant at 0.015 kg(vapor)/kg(air)). Fig. 15 presents the computed setpoints, and the control and measured variables for the same day. After a few hours during which no ventilation is applied, and the only ventilation is due to leakage, the air temperature reaches its maximum allowed value (323C in this case). Ventilation is then applied, and it can

be seen that by carefully balancing ventilation and CO enrichment, both the temperature and CO

setpoints can be maintained. Because of the heat storage of the green-house, the optimal operation is not symmetrical around noon. For a given level of solar radiation, maintaining the same temperature in the afternoon requires more ventilation than in the morning. This asymmetry in the ventilation rate is re#ected in the enrichment rate, with

a higher enrichment rate required to maintain the same setpoint. The CO

overshoot at the end of the day is due to the sudden reduction of the ventilation rate, which is responding to the low solar radiation. The enrichment rate is also reduced, but since, as stressed in Section 3, ventilation is not used to lower the CO

concentration, the overshoot remains.

8. Conclusions

The possibility of simultaneously controlling the tem-perature and CO

concentration in a greenhouse has been demonstrated. In particular, it has been shown that, if carefully balanced, ventilation and CO enrichment can be used together, in order to maintain the optimal temperature and a high CO

concentration. Decoupling the temperature and CO

control loops, and using the ventilation rate predicted by the temperature controller as the nominal one when computing the enrichment rate, reduced the CO

model uncertainty, and allowed for tighter control to be achieved.

predicted ventilation rate is used in the CO

control loop, inaccurate modeling of the ventilation would also de-grade the performance of this loop. It has also been shown that if such a relationship is available, the method used to design the controller, as well as the type of the resulting controller, has only a limited in#uence on the

loop's performance.

The CO

concentration proved to be more di$cult to

control accurately than the temperature, mainly because of the delay existing in the feedback loop. This delay resulted from the response time of the sensor, which could not be changed, and the time required to transfer the gas to the sensor. The latter could be reduced by placing the sensor closer to the sampling points (its present location is about 20 m from the sampling points). However, in the absence of special cooling arrangements this would expose the sensor to the excessively high greenhouse temperature. The results presented here sug-gest that such costly arrangements are not required. Due to the decoupling between the temperature and CO

control loop, the ventilation is not used to lower the CO

concentration and, as a result, the CO

concentration sometimes presents large overshoots. However, since a high CO

concentration always has a positive e!ect on

crop photosynthesis, using the ventilation to lower the CO

concentration would indeed be wasteful.

Acknowledgements

This study was supported by BARD, Project IS-1995-91RC, by the Israel Ministry of Agriculture, Project 838-0375-92, and by the Technion Fund for the Promotion of Research.

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