Control Engineering Practice 19 (2011) 354–366

Contents lists available at ScienceDirect

Control Engineering Practice

0967-06

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/conengprac

Nonlinear MPC based on a Volterra series model for greenhouse temperaturecontrol using natural ventilation

J.K. Gruber a,�, J.L. Guzman b, F. Rodrıguez b, C. Bordons a, M. Berenguel b, J.A. Sanchez b

a Dep. de Ingenierıa de Sistemas y Automatica, Universidad de Sevilla, Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092 Sevilla, Spainb Dep. de Lenguajes y Computacion, Universidad de Almerıa, Ctra. Sacramento s/n, 04120 Almerıa, Spain

a r t i c l e i n f o

Article history:

Received 15 April 2010

Accepted 3 December 2010Available online 24 December 2010

Keywords:

Nonlinear model predictive control

Greenhouse climate control

Volterra series model

Identification

61/$ - see front matter & 2010 Elsevier Ltd. A

016/j.conengprac.2010.12.004

esponding author. Tel.: +34 954 488 161; fax

ail addresses: jgruber@cartuja.us.es, jorngrube

a b s t r a c t

Suitable environmental conditions are a fundamental issue in greenhouse crop growth and can be

achieved by advanced climate control strategies. In different climatic zones, natural ventilation is used to

regulate both the greenhouse temperature and humidity. In mild climates, the greatest problem faced by

far in greenhouse climate control is cooling, which, for dynamical reasons, leads to natural ventilation as a

standard tool. This work addresses the design of a nonlinear model predictive control (NMPC) strategy for

greenhouse temperature control using natural ventilation. The NMPC strategy is based on a second-order

Volterra series model identified from experimental input/output data of a greenhouse. These models,

representing the simple and logical extension of convolution models, can be used to approximate the

nonlinear dynamic effect of the ventilation and other environmental conditions on the greenhouse

temperature. The developed NMPC is applied to a greenhouse and the control performance of the

proposed strategy will be illustrated by means of experimental results.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Greenhouse crop growth needs suitable environmental condi-tions, especially appropriate temperatures and an adequate humid-ity, achievable by advanced control techniques. In different climates,a standard tool to obtain the necessary environmental conditions isthe regulation of the natural ventilation of the greenhouse. Duringrecent years, several advanced control techniques for greenhousetemperature control have been developed, such as adaptive control(Sigrimis, Paraskevopoulos, Arvanitis, & Rerras, 1999), feedforwardcontrol (Rodrıguez, Berenguel, & Arahal, 2001), optimal control (vanStraten, van Willengenburg, & Tap, 2002) and feedback linearizingcontrol based on physical (Pasgianos, Arvanitis, Polycarpou, &Sigrimis, 2003) and empirical (Berenguel, Rodrıguez, Guzman,Lacasa, & Perez Parra, 2006) models. There are other many importantpapers on the temperature control (Fuchs, Dayan, Shmuel, & Zipori,1997; Martin-Clouaire, Schotman, & Tchamitchian, 1996; Seginer& Sher, 1993; Tantau, 1993) which are not mentioned here due tospace constraints.

From a modeling point of view, a proposed model of thegreenhouse climate links the output variable (temperature) tothe control variable (vent position) and to the disturbances (mainlyoutside weather and crop). There exists a correlation betweenvent position and output temperature that can be ascribed to the

ll rights reserved.

: +34 954 487 340.

r@gmail.com (J.K. Gruber).

ventilation rate. As a contributing factor, other researchers haveidentified this relationship by constructing semi-physical models,such as Boulard and Baille (1995) and Kittas, Boulard, andPapadakis (1997), and input/output empirical models (Gruberet al., 2008; Perez Parra, Baeza, Montero, & Bailey, 2004; PerezParra, Berenguel, Rodrıguez, & Ramırez-Arias, 2006; Rodrıguez,Arahal, & Berenguel, 1999), and even by using an online estimatorof the ventilation rate based on an unknown input observer with anoutput linearizing feedback, as done by Bontsema, van Henten,Hemming, Budding, and Rieswijk (2006). The physical processesincluded in the energy and mass balances are solar and thermalradiation absorption, heat convection and conduction, crop tran-spiration, condensation, and evaporation (Rodrıguez, 2002).

In all the previous models, it can be seen how the models of thenatural ventilation rate are quite difficult to obtain. Therefore, thiswork presents an approach based on Volterra series models in orderto find a trade-off between the difficulty to obtain the model andthe potential use for control purposes. Generally, Volterra seriesmodels represent the logical extension of finite impulse response(FIR) models with the nonlinearity considered in an additional term(Chen, 1995; Doyle, Pearson, & Ogunnaike, 2001; Volterra, 1959).It is important to mention that the used Volterra series modelconsiders as sole input variable the aperture of the roof and lateralwindows. The possible influence of a heater on the temperaturedynamics has not been taken into account in the identified model.Nevertheless, the used Volterra series model can be extended easilyin order to include the effect of a heater. The resulting model isused for diurnal temperature control and applied to a typical parral

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366 355

greenhouse structure of the Southeast of Spain. Volterra seriesmodels represent an extension of convolution models which havebeen applied successfully in linear model-based predictive control(MPC) (Camacho & Bordons, 2004; Cutler & Ramaker, 1980;Richalet, Rault, Testud, & Papon, 1978). These models exhibitgenerically a good behavior and their structure can be exploitedin the design of controllers, especially in the case of second-orderVolterra series models. Notice that Volterra series models havebeen frequently used to model bilinear systems in such a way that,as the greenhouse system presents this kind of dynamics(Berenguel et al., 2006; Rodrıguez, 2002), this formulation canbe used for modeling greenhouse temperature dynamics, includingthe disturbances in the nominal formulation of second-orderVolterra series models.

The use of a nonlinear model changes the predictive controlproblem from a convex quadratic program to a non-convex non-linear problem, which is much more difficult to solve. Furthermore,in this situation there is no guarantee that the global optimum canbe found, especially in real time control, when the optimum has tobe obtained in a prescribed time. The solution of this problemrequires the consideration (and at least a partial solution) of a non-convex, nonlinear problem (NLP), which gives rise to computa-tional difficulties related to the expense and reliability of solvingthe NLP online. Nevertheless, when the process is described by aVolterra series model, efficient solutions for the model predictivecontrol problem can be found (Doyle et al., 2001; Ling & Rivera,1998; Maner, Doyle, Ogunnaike, & Pearson, 1996). This solutionmakes use of the particular structure of the model, giving an onlinefeasible solution. The main advantage about the use of Volterraseries models relies on the fact that being a natural extension oflinear convolution models, they are quite straightforward to obtainfrom input/output data without any prior consideration about theprocess model structure. Being linear in the parameters, Volterraseries models can be identified using the least squares method.Hence, in this paper the ability to capture nonlinear dynamics of theprocess by a Volterra series model and its use in nonlinear modelpredictive control (NMPC) of the greenhouse temperature areshown. Preliminary versions of this work presented a Volterraseries model of the greenhouse climate only for autumn seasons(Gruber et al., 2008), which was used to design an NMPC strategybeing validated by simulations for this same period (Gruber,Guzman, Rodriguez, Bordons, & Berenguel, 2009). Therefore, thiswork presents a new Volterra series model capturing the system

Fig. 1. Greenhouse facilities used for the

dynamics for the entire year with exception of the months July andAugust. The two hottest months have not been considered in themodel as in this period no cultivation takes place in the greenhouselocated in Almerıa. Furthermore, the development of an NMPCstrategy to control the diurnal greenhouse temperature during themonths September to June is shown and experimental results forthe validation of the modeling and the control issues are presented.It is important to mention that Volterra series models have beenused to approximate the dynamics of a wide range of differentprocesses. However, to the best of the authors’ knowledge, therehave not yet been reported any applications of NMPC based onVolterra series models to real greenhouses.

The paper is organized as follows: Section 2 gives a generaloverview over the greenhouse and its main characteristics. Section 3presents the identification of a suitable Volterra series model andthe nonlinear model predictive control strategy. In Section 4, thecontrol strategy is validated with a detailed first principlessimulation model of the greenhouse. Afterwards, the controlstrategy is applied to a real system and the control performanceis illustrated by experimental results. Finally, in Section 5 the majorconclusions are summarized.

2. System description

Experiments on modeling, control, and optimization were carriedout in a parral greenhouse located at The Cajamar Foundation(El Ejido, Almerıa, South-East Spain). The covering material is a200 mm thick PE film, laid on a galvanized steel structure (see Fig. 1).The control actuators of the greenhouse are: a flap roof window with amaximum aperture angle of 451, a rolling lateral continuous windowwith length of 37 m and aperture of 1.2 m. The installed 95 kW heaterwas neither used in the experiments to obtain suitable input–outputdata for the identification nor during the application of the proposedcontrol strategy, i.e. the heater is not considered as an input to thegreenhouse system. Soil temperature measurements were madeusing semiconductor sensors at different depths, on both sides ofthe mulch, immediately under the soil surface layer, and at a depth of50 mm. The air temperature thermoresistance sensor and the airrelative humidity capacitive sensor were placed at the top of the crop.Eight-semiconductor contact sensors were installed on both sides ofthe cover to obtain measurements of cover temperature, that wereaveraged. A meteorological station was installed outside at a height of

experiences performed in this work.

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366356

6 m for measurements of temperature, relative humidity, globalradiation and photosynthetically active radiation (PAR), rain, andwind speed and direction. The system also registered the state of theactuators (vent positions, shade positions, and heater operation).Some additional crop data were logged, such as substrate watercontent, water supply, amount of drainage, EC of the substrate, planttranspiration measured with an electronic weight, leaf temperature,and CO2 concentration. A uniform sampling time of 1 min wasestablished. A detailed description of the installations and theresearch project in which this work has been performed can befound on the homepage of the Automatic Control, Electronics andRobotics Research Group (2004–2010).

The experimental greenhouse is equipped with a data acquisi-tion system to collect measurements from the sensors installed inthe interior and exterior of the greenhouse. The data acquisitionsystem is based on an industrial Ethernet using Compact Field-Points (National Instruments), which satisfy the standards forautomatization in production environments. During the develop-ment of the control system, particular attention has been paid tothe devices due to the specific climate conditions in greenhouses,which include relative humidities close to the saturation and hightemperatures. The industrial specifications of Compact FieldPointsguarantee a correct operation for temperatures over 70 3C. Themodular structure of systems based on Compact FieldPoints allowsan easy integration of the devices in the software package LabView(National Instruments) and facilitates the measurement and con-trol of industrial processes.

3. Control design for greenhouse temperature control

The main objective of this work is the design of a controlstrategy for the greenhouse temperature in order to achievesuitable climate conditions for crop growth. The idea is theregulation of the greenhouse temperature by natural ventilation,which itself can be manipulated through the aperture of the lateraland roof windows of the greenhouse. Hence, the aperture of thegreenhouse windows will be used as input signal and has to becalculated by the control strategy to be designed. Note that theeffect of a heater on the temperature dynamics is not considered asthe used greenhouse in the Southeast of Spain is usually operatedwithout an additional heating. However, the control design pro-posed in the following sections can be easily extended to includethe heating as a second control variable of the greenhouse.

A model predictive control strategy will be used due to theoptimality of the computed control signal and the possibility toconsider constraints in the control action based on the physicallimitations of the system. Although detailed physical greenhousemodels can be found in the literature (Boulard & Baille, 1995; Kittaset al., 1997; Rodrıguez, 2002), these models cannot be used ascontrol-oriented models in predictive control strategies due to itshigh complexity. As a consequence, a Volterra series model relatingthe aperture of the windows and the main disturbances with thegreenhouse temperature has been developed.

3.1. Model structure and experimental data

Generally, greenhouse temperature modeling is a challengingtask due to its nonlinear dynamic behavior with respect to changesin the aperture of the windows. Furthermore, the greenhousetemperature is strongly influenced by the surrounding environ-mental conditions (e.g. solar radiation and outside temperature),which can be considered as measurable disturbances. Due to thepossibility to consider nonlinear dynamics and to include distur-bances, a Volterra series model has been chosen for the generalmodel structure. This model represents the simple and logic

extension of linear convolution models with an additive nonlinearterm. Another advantage of Volterra series models is the linearity inthe parameters, which allows parameter identification from experi-mental data by well-known linear identification techniques. It hasto be mentioned that the approximation by a Volterra series modelcan be used only for stable fading memory systems (Boyd & Chua,1985). However, with the greenhouse being a stable system, thislimitation does not prevent the use of Volterra series models toapproximate the temperature dynamics. Higher-order Volterraseries models generally allow a better approximation of the systemdynamics, but at the cost of an increasing number of parameters tobe identified. Therefore, a second-order Volterra series model,which represents a trade-off between the nonlinearity of the modeland the parameter number, has been chosen.

A single-input single-output second-order Volterra series model,with the truncation of terms (truncation orders N1 and N2), can bedefined as

yðkÞ ¼ h0þXN1

i ¼ 1

aðiÞuðk�iÞþXN2

i ¼ 1

XN2

j ¼ i

bði,jÞuðk�iÞuðk�jÞ ð1Þ

which corresponds to a linear convolution model with a nonlinear-ity as additional and additive term. In that model, y(k) and u(k)represent the last measured output and input to the system,respectively (k is the current sampling instant). The offset is denotedwith h0 and the linear and nonlinear term parameters are given bya(i) and b(i, j), respectively.

Analogously, the chosen second-order Volterra series modelwith 1 input and n measurable disturbances can be given by

yðkÞ ¼ h0þXN1,u

i ¼ 1

auðiÞuðk�iÞþXN2,u

i ¼ 1

XN2,u

j ¼ i

buði,jÞuðk�iÞuðk�jÞ

þXn

m ¼ 1

XN1,wm

i ¼ 1

awm ðiÞwmðk�iÞ

þXn

m ¼ 1

XN2,wm

i ¼ 1

XN2,wm

j ¼ i

bwm ði,jÞwmðk�iÞwmðk�jÞ ð2Þ

where wmð�Þ with m¼ 1, . . . ,n represent the measurable distur-bances, and N1,wm

and N2,wmdenote the corresponding truncation

orders. The linear and nonlinear truncation orders with respect tothe input u are denoted with N1,u and N2,u, respectively. The linearand nonlinear input term parameters are denoted with au(i) andbu(i, j), respectively. In the same way, the parameters awm ðiÞ andbwm ði,jÞ are used in the linear and nonlinear terms depending on thedisturbances wm.

From previous studies (Gruber et al., 2008, 2009), it is knownthat the greenhouse temperature is mainly influenced by theaperture of the roof and lateral windows, the outside temperature,the outside wind speed, the soil surface temperature, and theoutside global radiation. As the used greenhouse is usuallyoperated with the same values for the aperture of the roof windowsand the aperture of the lateral windows, the opening of the roof andlateral windows has been regarded as a combined variable. Thewind direction has not been considered as measurable disturbancebecause the greenhouse used in this work only has roof ventsoriented to the West with high-density anti-insect screens, whatdecreases the wind effect on the ventilation process and thus thewind direction influence is despicable. Furthermore, this fact hasbeen also observed from the real open-loop tests performed at thisgreenhouse, where it was concluded that the main contribution tothe ventilation process comes from the lateral vents. On the otherhand, the greenhouse is located very close to other greenhousesand therefore the wind direction is slightly affected. Evidently, inother greenhouses with other location and vent distribution, the

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366 357

wind direction should be included as an input to the Volterraseries model. The leaf area index (LAI) is another variable affectingthe greenhouse dynamics, by the crop transpiration. However,the dynamics of this variable is quite slower in comparison to theclimatic variable ones, and thus it has not been included in thegreenhouse model (Gruber et al., 2008). As a consequence, the mainvariables considered for modeling purposes are:

�

me

output: greenhouse temperature Xt,a;

� input: aperture of the roof and lateral windows alat,rf ; �

Table 1Chosen truncation orders for the first- and second-order terms, i.e. the linear and

nonlinear part of the model, for the input and the considered disturbances.

Variable 1st order 2nd order

alat,rf 20 10

Pt,e 1 1

disturbances: outside temperature Pt,e, outside wind speed Pws,e,soil surface temperature Pt,ss,

1 outside global solar radiationPsol,e;

and the model (2) of the greenhouse temperature is defined with1 input and n¼4 disturbances. Note that the dynamic behavior of theroof and lateral windows has been considered in one sole inputvariable. Although the greenhouse possesses independent actuatorsand can be regulated independently, usually the same value is used forboth actuators in the studied greenhouse. Therefore, the combinedeffect of roof and lateral windows has been identified and, as aconsequence, only one input signal is calculated. It has to be mentionedthat during the identification process additional inputs were tested, e.g.the product alat,rf Pws,e, but no large improvements in the parameteridentification stage were detected while the number of parametersincreased. Moreover, the use of variables being cross products betweenthe input and disturbances augments the complexity of the optimiza-tion problem associated to model predictive control and, as aconsequence, complicates the computation of the input signal.

In order to identify the parameters of the Volterra series model ofthe greenhouse temperature, experimental data of autumn (fromAugust to February) and spring (from March to June) seasons from2007 and 2008 has been used. It is important to mention that the dataof the autumn and spring seasons was combined to one set andallowed the identification of one model for the entire year (withexception of July and August). A second set of experimental data of along season from September 2008 to June 2009 has been used forvalidation purposes. Note that both in the model identification andvalidation the months July and August have not been considered.These months have been neglected as the high temperatures inAlmerıa prevent the cultivation of crop in greenhouses provided onlywith natural ventilation (which are the most common ones). Theremaining months (September to June) used in the identification andvalidation represent the period of crop cultivation. The used sets foridentification and validation have rich input signals, necessary for theidentification of a second-order Volterra series model. The behavior ofthe greenhouse climate dynamics is quite different in spring andwinter conditions, and thus using these data sets all the boundaryconditions are considered. Furthermore, the sample time for both setswas 1 min and the data of the wind speed and the global solarradiation were filtered through a first-order filter with a time constantof 5 min before using them for calibration purposes. The data used foridentification and validation were normalized to the interval [0,1].To identify the parameters of the Volterra series model, both leastsquares methods and constrained nonlinear optimization usingsequential quadratic programming have been used.

3.2. Parameter identification

After the choice of the input and the different disturbances whichmainly influence the greenhouse temperature (see Section 3.1), aVolterra series model in the form given in (2) was identified. The

1 The soil surface temperature Pt,ss considered in the greenhouse model is

asured in a depth of 5 cm.

parameter estimation and the model validation were carried out withthe input/output data sets described above. One of the initialproblems was to select the truncation orders for the input and thedisturbances influencing the greenhouse inside temperature. Due tothis reason, in a first step, least squares identification was performedwith long truncation orders for all considered variables. As thegreenhouse is a fading memory system, the parameters tend to zeroalong the horizon and the truncation orders were reduced in the caseof parameters being zero or nearly zero. The choice of a suitablesampling time for the model is a trade-off between the number ofparameters to be identified and the dynamics of the greenhouse. It hasto be mentioned that the ambient conditions, especially the windspeed and the solar radiation (due to passing clouds), can change in afew seconds. The finally chosen sampling time for the model ists¼1 min and the determined truncation orders for the first- andsecond-order terms, i.e. the linear and nonlinear part of the model, forthe input and the different disturbances are the ones given in Table 1.The truncation orders show that the linear and nonlinear part of themodel associated to the aperture of the roof and lateral windows has alagged effect on the greenhouse temperature. The same applies to thelinear influence of the soil surface temperature on the insidetemperature. It is noteworthy with respect to the given truncationorders that the soil surface temperature and the outside global solarradiation have no nonlinear influence on the greenhouse tempera-ture. Furthermore, the outside temperature and the outside windspeed have an almost immediate effect on the temperature.

In a second step, as many parameters identified with the leastsquares method had noisy profiles, constraints in the parametershave been included using sequential quadratic programming. Fromprevious identifications, based on data of a detailed simulationmodel (Gruber et al., 2008) or measured data from the samegreenhouse (Gruber et al., 2009), the dynamics of the input anddisturbances were partially known. The imposed constraints werechosen on the basis of the results from the identification with theleast squares method and the knowledge about the systemdynamics obtained in Gruber et al. (2008, 2009). For the apertureof the roof and lateral ventilations the constraints

aalat,rfðiþ1Þoaalat,rf

ðiÞ for i¼ 1, . . . ,3

aalat,rfðiþ1ÞZaalat,rf

ðiÞ for i¼ 4, . . . ,19

aalat,rfðiÞr0 for i¼ 1, . . . ,20 ð3Þ

and

balat,rfðiþ1,iþ1Þrbalat,rf

ði,iÞ for i¼ 1, . . . ,9

balat,rfði,jþ1Þrbalat,rf

ði,jÞ for i¼ 1, . . . ,9, j¼ i, . . . ,9

balat,rfði,jÞZ0 for i¼ 1, . . . ,10, j¼ 1, . . . ,10 ð4Þ

were imposed in the linear term and nonlinear term parameters. Forthe solar radiation, the following linear term parameters were used:

aPsol,eðiþ1ÞraPsol,e

ðiÞ for i¼ 1, . . . ,4

aPsol,eðiÞZ0 for i¼ 1, . . . ,5 ð5Þ

Pws,e 2 2

Psol,e 5 0

Pt,ss 10 0

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366358

In an analogous way, the constraints

aPt,ssðiþ1ÞraPt,ss

ðiÞ for i¼ 1, . . . ,9

aPt,ssðiÞZ0 for i¼ 1, . . . ,10 ð6Þ

were considered in the linear term parameters of the soil surfacetemperature.

With the constraints in the parameters, a second-order Volterraseries model of the greenhouse temperature was identified. In themodel validation, carried out with the second data set containingthe period from September 2008 to June 2009, a mean square errorin the temperature of 0.93 was obtained. As a representative result,a comparison of the greenhouse temperature and the output ofthe identified model is given in Figs. 2 and 3, both for the autumn(10–19 January 2009) and spring conditions (15–24 May 2009),respectively. As can be seen in the results, the model output shows apromising fit with the measured greenhouse temperature. Inautumn conditions, the model presents a mean value of the absoluteerror of 0:6 3C, a standard deviation of 0:5 3C, a mean relative errorless than 4%, and a maximum error of 2:1 3C in a range of 11.1 to26:4 3C. In spring conditions, similar results were obtained, result-ing in a mean value of the absolute error of 0:68 3C, a standarddeviation of 0:63 3C, a mean relative error less than 4%, and amaximum error of 2:5 3C in a range of 15:4231:4 3C. Thus, theseresults represent acceptable values when they are compared withthe nonlinear model used as reference in this work (Rodrıguez,2002). The validation results from the model (Rodrıguez, 2002) for

Fig. 2. Data set used for the model validation with the greenhouse temperature Xt,a (so

2009), the input alat,rf (aperture of the roof and lateral windows) and the disturbance

radiation), and Pt,ss (soil surface temperature).

the same collection data give a mean value of the absolute error of0:51 3C, a standard deviation of 0:5 3C, and a mean relative error lessthan 3.3% for the autumn season; and a mean value of the absoluteerror of 0:39 3C, a standard deviation of 0:4 3C, and a mean relativeerror less than 2.4% for the spring season. Notice also that, theaccuracy of the temperature sensors is 70:3 3C.

The identified linear term parameters of the aperture of thelateral and roof ventilations alat,rf , outside global solar radiationPsol,e, and soil surface temperature Pt,ss are shown in Fig. 4, while thesecond-order term parameters balat,rf

of the aperture of the lateraland roof ventilations are shown in Fig. 5. It can be seen that thelinear and nonlinear term parameters identified under considera-tion of the constraints (3)–(6) show a coherent and reasonableshape. It is noteworthy that the mean square errors obtained in theunconstrained identification and validation were only insignif-icantly lower than the corresponding mean square errors of theconstrained identification and validation.

3.3. Nonlinear control law

The nonlinear second-order Volterra series model (2) is used topredict the future behavior of the greenhouse temperature con-sidering disturbances. This type of model can be considered as alogic extension of the models widely used in predictive controllersin industry, as Dynamic Matrix Control (DMC) (Cutler & Ramaker,1980; Qin & Badgwell, 2003). The future output of the identifiednonlinear model (2) with the prediction horizon N, the control

lid line) and the model output (dashed line) for autumn conditions (10–19 January

s Pt,e (outside temperature), Pws,e (outside wind speed), Psol,e (outside global solar

Fig. 3. Data set used for the model validation with the greenhouse temperature Xt,a (solid line) and the model output (dashed line) for spring conditions (15–24 May 2009), the

input alat,rf (aperture of the roof and lateral windows) and the disturbances Pt,e (outside temperature), Pws,e (outside wind speed), Psol,e (outside global solar radiation), and Pt,ss

(soil surface temperature).

Fig. 4. Identified linear term parameters for the aperture of the roof and lateral ventilations alat,rf (top), solar radiation Psol,e (bottom left), and soil surface temperature Pt,ss

(bottom right).

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366 359

Fig. 5. Identified second-order term parameters balat,rffor the aperture of the roof

and lateral ventilations.

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366360

horizon Nu, and the truncation order Nt can be calculated in a matrixform as (Doyle et al., 2001)2

y¼ Guuþfuþc ð7Þ

with the predicted output vector yARN and the future inputsequence uARNu defined as

y¼ ½yðkþ1jkÞ,yðkþ2jkÞ, . . . ,yðkþNjkÞ�T

u¼ ½uðkjkÞ,uðkþ1jkÞ, . . . ,yðkþNu�1jkÞ�T ð8Þ

The term Guu with GuARN�Nu represents the linear part dependingon the future input. Analogously, the vector fuARN containsthe future–future cross terms depending on the input sequenceu. The term cARN contains only terms which do not depend on thecurrent or future inputs and is defined as

c¼ cuþXn

m ¼ 1

cwmþh0þd

cu ¼Huupþgu

cwm ¼Hwm wm,pþgwmþGwm wmþfwm for m¼ 1, . . . ,n ð9Þ

where cuARN represents the constant term depending on thevector of past input values and cwm ARN is the vector containing theterms depending exclusively on the disturbances. The vectorsh0ARN and dARN include the offset of the identified model (2)and the estimation error in the prediction, and are defined withh0 ¼ ½h0 h0 . . .h0�

T and d¼ ½dðkÞ dðkÞ . . . dðkÞ�T , respectively. Theterms Huup with HuARN�Nt and guARN are the linear and non-linear parts depending on the past input values, respectively.Analogously, Hwm wm,p with Hwm ARN�Nt and gwm

ARN representthe linear and nonlinear terms depending on the past valuesof the disturbances. Furthermore, Gwm wm with Gwm ARN�Nu andfwm ARN denote the linear and nonlinear terms depending onthe future disturbance vectors. The past input vector upARNt and

2 For the sake of simplicity of the notation and without loss of generality,

a unique truncation order Nt has been used. In the case of a truncation order N1,koNt

or N2,koNt with k¼ fu,w1 ,w2 ,w3 ,w4g, the missing parameters are added with

akðiÞ ¼ 0 8i¼N1,kþ1, . . . ,Nt and bkði,jÞ ¼ 0 8i¼N2,kþ1, . . . ,Nt3j¼N2,kþ1, . . . ,Nt .

the disturbance vectors wm,pARNt for m¼ 1, . . . ,n are defined as

up ¼ ½uðk�1Þ,uðk�2Þ, . . . ,uðk�NtÞ�T

wm,p ¼ ½wmðk�1Þ,wmðk�2Þ, . . . ,wmðk�NtÞ�T for m¼ 1, . . . ,n ð10Þ

For the future values of the disturbances constant values areassumed (the assumption to consider constant the future valuesof the disturbances is performed because short prediction hori-zons are used in the proposed control law, as will be shown inthe following section; for long prediction horizons, forecast dis-turbance models should be used), i.e. the vectors wmARNu form¼ 1, . . . ,n are defined by

wm ¼ ½wmðkjkÞ,wmðkjkÞ, . . . ,wmðkjkÞ�T for m¼ 1, . . . ,n ð11Þ

As a consequence, the linear and nonlinear terms Gwm wm and fwm

depending on the future values of the disturbances are calculatedwith the current values of the disturbances. A detailed descriptionof the matrices Gu, Hu, Gwm and Hwm , as well as how to calculate thevectors fu, fwm , gu and gwm

, can be found in Appendix Appendix Aand in Doyle et al. (2001).

For the design of a model predictive control strategy based on theidentified second-order Volterra series model (2), the followinggeneral model predictive control optimization problem is considered:

u� ¼ argminu

J s:t: Aurb ð12Þ

where the variable to be optimized is the future input sequence u. Thelinear constraints of the optimization problem are defined with thematrix AARnc�Nu and the vector bARnc where nc is the number ofconstraints. The quadratic cost function J is defined generally as

J¼XN

i ¼ 1

ðyðkþ ijkÞ�rðkþ ijkÞÞ2þlXNu�1

i ¼ 0

Duðkþ ijkÞ2 ð13Þ

where rðkþ ijkÞ denotes the desired reference for k+i and theparameter l represents the weighting factor for the control incre-ments and allows the tuning of the controller towards a moreaggressive or a smoother behavior. Furthermore, Duðkþ ijkÞ repre-sents the control increment for k+i calculated at k and is defined as

Duðkþ ijkÞ ¼uðkjkÞ�uðk�1Þ for i¼ 0

uðkþ ijkÞ�uðkþ i�1jkÞ for i¼ 1, . . . ,Nu�1

(ð14Þ

The sequence DuARNu of future control increments is definedgenerally with

Du¼ ½DuðkjkÞ,Duðkþ1jkÞ, . . . ,DuðkþNu�1jkÞ�T ð15Þ

and can be expressed as a function of the input sequence in thefollowing form:

Du¼

1 0 . . . 0

�1 1 & 0

^ & & 0

0 0 �1 1

26664

37775uþ

�1 0 . . . 0 0

0 0 . . . 0 0

^ ^ & 0 0

0 0 0 0 0

26664

37775up

¼ LuuþLpup ð16Þ

with LuARNu�Nu and LpARNu�Nt . Using the definition (16) in (13), thecost function can be expressed directly in matrix form with

J¼ ðy�rÞT ðy�rÞþlDuTDu

¼ ðy�rÞT ðy�rÞþlðLuuþLpupÞTðLuuþLpupÞ ð17Þ

with rARN being the reference trajectory.With the prediction model (7) based on a second-order Volterra

series model and the cost function (17), an iterative approach tocalculate the control action has been chosen. This approach,originally presented in Doyle et al. (2001) and Maner et al.(1996), represents an unconstrained nonlinear model predictive

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366 361

control. This approach has been modified to consider constraints inthe input sequence to be optimized as well as a weighting for thecontrol increments as described in (17). The scheme to calculate thecontrol action is the following:

�

Figalat,

(soi

Step 1: Set i¼1 and uð0Þ ¼ u0, calculate c (9).

� Step 2: Calculate fu using uði�1Þ. � Step 3: Solve the constrained least squares control problem (12)

minimizing the cost function

J¼ 12uðiÞT HqpuðiÞ þfT

qpuðiÞ ð18Þ

with quadratic programming (QP) and

Hqp ¼ GTuGuþlLT

uLu ð19Þ

fTqp ¼ ðr�c�fuÞ

T GuþuTpLT

pLu ð20Þ

�

Step 4: Verify if the first element of the calculated inputsequence uðiÞ satisfies the desired tolerance d in the condition:

juðiÞðkjkÞ�uði�1ÞðkjkÞjod ð21Þ

�

Step 5: If the previous condition is satisfied, set u� ¼ uðiÞ andapply the new control action u�ðkjkÞ to the system. If theprevious condition is not satisfied, set i¼ i+1 and return tostep 2.

Note that the nonlinear term fu is calculated in step 2 andconsidered as a constant in the constrained optimization in step 3.In case of failure to fulfill the convergence condition in step 4, a newnonlinear term fu is calculated in the i-th step with the approximatesolution of the previous step i�1. This procedure is repeated until

Xt,a

[°C

]a l

at,r

f [%

]P

t,e [

°C]

Pw

s,e

[m/s

]P

sol,e

[W

/m2 ]

Pt,s

s [°C

]

0

00

00

0

00

0

1

1

1

1

1

1

2

2

2

2

2

2

3

3

3

3

3

3

5

10

20

20

20

30

50

100

300

600

900

15

25

25

16

24

. 6. Simulations results of the proposed controller for a data set from May 2007. From

rf (aperture of the roof and lateral windows) and disturbances Pt,e (outside tempe

l surface temperature).

the difference between the control actions of two successiveiterations satisfies the convergence condition.

The initial candidate input sequence u0ARNu used in the firststep of the optimization routine can be chosen by shifting the inputsequence u� calculated in the previous sampling period by oneelement. Therefore, the optimized input sequence u� is stored atthe end of the optimization routine to be available for its use as aninitial candidate input sequence in the next sampling period.

4. Control results

The nonlinear control law (see Section 3.3) based on theidentified second-order Volterra series model (see Section 3.2)was implemented in the Matlab/Simulink environment. The pro-posed nonlinear model predictive controller was applied to adetailed simulation model for validation purposes. After successfulvalidation, the proposed controller was applied to the greenhouseand tested under normal weather conditions.

4.1. Simulation results

A nonlinear model based on an energy and mass balance(Rodrıguez, 2002; Rodrıguez, Yebra, Berenguel, & Dormido,2002), validated for the proposed greenhouse, has been used insimulation to check the performance of the proposed nonlinearmodel predictive control. The simulation model takes into accountsolar radiation, heat transfer by convection with the soil surfaceand the heating system, the heat transfer by conduction andconvection between the inside and outside air, the heat exchange

4

4

4

4

4

4

5

5

5

5

5

5

6

6

6

6

6

6

7

7

7

7

7

7

8

8

8

8

8

8

t [d]

top to bottom: output Xt,a (greenhouse temperature) and desired reference, input

rature), Pws,e (outside wind speed), Psol,e (outside global solar radiation), and Pt,ss

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366362

with the outside air due to natural ventilation and infiltration, andthe latent heat produced by the crop transpiration.

The proposed controller was implemented in Matlab/Simulinkand used with a sampling time of ts¼1 min, corresponding to thesampling time used during the identification (see Section 3.2).The prediction and control horizons were chosen in such a way thatthe length of the considered dynamic behavior corresponds to thetime constant of the identified model (2). Finally, based on the usedtruncation orders (see Table 1) of the Volterra series model, valuesof N¼9 for the prediction horizon and Nu¼9 for the control horizonwere chosen, respectively. The weighting factor used in the controlstrategy was set to a value of l¼ 0:01 and constraints both in thecontrol signal and its increments were considered:

0%ruðkþ ijkÞr100%, i¼ 0, . . . ,Nu�1

�25%rDuðkþ ijkÞr25%, i¼ 0, . . . ,Nu�1 ð22Þ

The constraints in the control signal and its increments have beenchosen according to the physical limitations of the actuators. Thelower and upper bounds of the control signal represent completelyclosed or opened windows. The constraints in the control incre-ments were necessary as the windows need several minutes tocompletely open or close. Furthermore, the following hysteresis:

uðkjkÞ ¼u�ðkjkÞ if ju�ðkjkÞ�uðk�1ÞjZeuðk�1Þ if ju�ðkjkÞ�uðk�1Þjoe

(ð23Þ

Fig. 7. Detailed simulations results of the proposed controller for one day. From top to

(aperture of the roof and lateral windows) and disturbances Pt,e (outside temperature), P

temperature).

for the control signal has been used reducing the number of controlchanges in order to preserve the electric motors of the roof andlateral windows. With the given hysteresis, the control signalcalculated by the controller is applied only if the difference to thelast applied control signal is greater than or equal to e. Otherwise,the control signal applied in the previous sampling period will beapplied again in the current sampling period. Based on the range ofthe input signal (from 0% to 100%, see in (22) the imposedconstraints) and the experience with the real system, a value ofe¼ 3% has been used.

In order to validate the proposed nonlinear controller, data fromseveral agricultural seasons (spring and autumn) have been usedcovering different weather conditions. Spring season is the periodwhere the ventilation system is required to work more adequatelydue to the warm external conditions. Therefore, some representa-tive days from May 2007 have been used to illustrate the appro-priate performance of the proposed controller. Fig. 6 shows theevolution of the main variables for a period of 8 days. Duringthe simulations, a value of 23 3C has been used for the setpoint ofthe greenhouse temperature Xt,a. It can be seen that both underclear day and under strong disturbances mainly in solar radiationand wind speed, the system is able to regulate the greenhousetemperature Xt,a around the given setpoint while rejecting dis-turbances. To give a more detailed view of the controller perfor-mance under strong disturbances in outside weather, a zoom of theresults of the seventh day is shown in Fig. 7. The proposed nonlinear

bottom: output Xt,a (greenhouse temperature) and desired reference, input alat,rf

ws,e (outside wind speed), Psol,e (outside global solar radiation), and Pt,ss (soil surface

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366 363

model predictive controller is able to obtain a good performanceeven after long periods where the control signal is saturated. Moresimulations with different data sets have been carried out, yieldingsimilar results.

Notice that as mentioned above, the nonlinear dynamics of thegreenhouse changes during the year. Therefore, the proposed non-linear model predictive controller based on the second-order Volterraseries model identified with data from different year seasons has beensuccessfully used to control the greenhouse temperature using datafrom spring 2007. Hence, the proposed controller covers adequatelywith this greenhouse changing dynamics.

4.2. Experimental results

After validation the proposed control algorithm in simulation,several tests were performed on the real system described inSection 2. The controller parameters were set with the same valuesas in simulations, as well as the different constraints commentedabove, (22) and (23). Several tests were carried out in autumn andwinter yielding promising results. Fig. 8 shows the control resultsand the main system disturbances for the test performed in theperiod of 19–24 January 2010. It is important to mention that thepresented results have been obtained with a tomato cultivationinside the greenhouse. During the shown experiment, the setpoint

Fig. 8. Experimental results of the proposed controller for the period of 19–24 January 20

input alat,rf (aperture of the roof and lateral windows) and disturbances Pt,e (outside t

Pt,ss (soil surface temperature).

was set to 24 3C for most of the days and it was changed to 22 3Cbetween days 2 and 3 in order to evaluate the reaction of the controlalgorithm. As it can be observed from the figure, the controlalgorithm properly follows the setpoint despite of system dis-turbances. Notice that most of the experiments are under cloudydays and with important variations in the wind speed. In the firstday, the control system is not able to reach the setpoint. The reasonis because it is a very cloudy day and thus there is not enoughenergy source to increase the inside temperature. Notice how thecontrol algorithm closes the windows in order to try increasingthe controlled variable. On the other hand, during the third day, theinside temperature is slightly over the setpoint for all the day.The control algorithm fully opens the windows in order to decreasethe inside temperature and the control signal is saturated all thetime. However, there is nothing to do since it is a warm dayand the air exchange through the windows is not enough todecrease the inside temperature below the desired setpoint.

Fig. 9 shows a zoom of the results for the fifth day (23 January2010). This is the clearest day of the experiments as observed fromthe global radiation curve. It can be seen how the control systemaccurately follows the setpoint until 2pm. Afterwards, the dayclouds over and the controllers closes the windows in order tocompensate the fall on the global radiation. However, the dayremains cloudy for the rest of the afternoon and it is not possible toincrease the inside temperature again.

10. From top to bottom: output Xt,a (greenhouse temperature) and desired reference,

emperature), Pws,e (outside wind speed), Psol,e (outside global solar radiation), and

Fig. 9. Detailed experimental results of the proposed controller for one day (23 January 2010, day 5 in Fig. 8). From top to bottom: output Xt,a (greenhouse temperature) and

desired reference, input alat,rf (aperture of the roof and lateral windows) and disturbances Pt,e (outside temperature), Pws,e (outside wind speed), Psol,e (outside global solar

radiation), and Pt,ss (soil surface temperature).

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366364

5. Conclusions

Linear controller design techniques are widely employed inindustry, although a great deal of processes is nonlinear. In manysituations the process is operating in the vicinity of a nominaloperating point and therefore a linear model can provide goodperformance. The simplicity and the existence of tested identifica-tion techniques for linear models allow an easy and successfulimplementation of linear controllers in many situations. However,there exist many situations in which nonlinear effects justify theneed of nonlinear models, such as in the case of strongly nonlinearprocesses subject to disturbances or setpoint tracking problemswhere the operating point is continually changing, showing thenonlinear process dynamics (Qin & Badgwell, 2003). In the case ofthe greenhouse treated in this work, the relationship between thegreenhouse inside temperature and the ventilation rate is ofbilinear nature and highly nonlinear when relating that tempera-ture with the control signal. Moreover, the system under study issubject to strong disturbances, mainly in solar radiation (the mainenergy source for photosynthesis purposes), wind speed, outsidetemperature and soil surface temperature.

In this work, a nonlinear model predictive control strategybased on a second-order Volterra series model has been used tocontrol the system. This model represents the simple and logicalextension of convolution models and can be used to approximatethe nonlinear relationship between vents aperture and insidetemperature, while also accounting for measurable disturbances,being able to include disturbance compensation features. With theVolterra series model being linear in the parameters, linearidentification techniques can be used. The proposed NMPC strategyconsiders the nonlinear dynamics of the controlled system andallows a constrained optimization of the control sequence.

From experimental data, a second-order Volterra series modelrelating the greenhouse inside temperature with the aperture ofthe roof and lateral windows, the outside temperature, the outsidewind speed, the soil surface temperature, and the outside globalsolar radiation was identified. The model was validated with asecond data set and the nonlinear model predictive controlstrategy. The controller, applied to a proven nonlinear model ofthe greenhouse based on first order principles, showed promisingresults in simulation and underlined the feasibility of the proposedapproach. Finally, the proposed control strategy was tested on thegreenhouse and resulted in an adequate control performanceregulating the diurnal temperature in the greenhouse around thedesired reference. The procedure to develop the controller based ona second-order Volterra series model is simple and straightforwardand allows the inclusion of constraints in the control signal. Thecomputation of the control action has been performed using aniterative procedure that converges in a few iterations. The proce-dure has a low computational complexity as in every iteration onlya quadratic programming (QP) problem has to be solved.

Currently, the authors are working on obtaining adequate forecastmodels for measurable disturbances. Future works will be focused oncombining these forecast disturbance models with the proposedpredictive control strategy. On the other hand, new variables, such asCO2 concentration and humidity, and new actuators, such as pipeheating and CO2 enrichment systems, will be considered.

Acknowledgments

Financial support by the Spanish Ministry of Education andScience under grants DPI2007-66718-C04-01, DPI2007-66718-C04-04 and DPI2010-21589-C05-04 is gratefully appreciated.

J.K. Gruber et al. / Control Engineering Practice 19 (2011) 354–366 365

Appendix A. Matrix and vector definitions

This section presents the detailed definitions of the matricesand vectors used in Section 3.3 to calculate the future outputof the prediction model (7) based on a second-order Volterra seriesmodel (Doyle et al., 2001; Maner et al., 1996). The parametersN, Nu, and Nt denote the prediction horizon, the control horizon,and the truncation order of the identified model (2), respectively.For the definition of the vectors and matrices it has been assumedthat the truncation order and the horizons satisfy NurNrNt .

The matrix GkARN�Nu with k¼ fu,w1,w2,w3,w4g represents thelinear influence of the future control actions and disturbanceson the future output. For the future control actions uðkþ jjkÞ ¼ uðkþ

Nu�1jkÞ 8j¼Nu, . . . ,N�1 is considered, i.e. a constant control actionafter reaching the end of the control horizon. Furthermore, due tothe lack of knowledge about the future behavior of the disturbances,the disturbances are considered constant with wmðkþ jjkÞ ¼wm

ðkjkÞ 8j¼ 1, . . . ,N�1 with m¼ 1, . . . ,4. Then, the matrix Gk can bewritten as (Doyle et al., 2001)

Gk ¼

akð1Þ 0 . . . 0

akð2Þ akð1Þ & 0

^ ^ & akð1Þ

^ ^ & akð1Þþakð2Þ

^ ^ & ^

akðNÞ akðN�1Þ . . .XN�Nuþ1

i ¼ 1

akðiÞ

26666666666664

37777777777775

ðA:1Þ

Analogously, the matrix HkARN�Nt with k¼ fu,w1,w2,w3,w4g

represents the linear effect of the past input and disturbance valuesover the model and is defined generally as (Doyle et al., 2001)

Hk ¼

akð2Þ akð3Þ . . . akðNt�1Þ akðNtÞ 0

akð3Þ akð4Þ . . . akðNtÞ 0 0

^ ^ ^ ^ ^ ^

akðNt�1Þ akðNtÞ . . . 0 0 0

akðNtÞ 0 . . . 0 0 0

0 0 . . . 0 0 0

2666666664

3777777775

if N¼Nt

Hk ¼

akð2Þ akð3Þ akð4Þ akð5Þ . . . akðNtÞ 0

akð3Þ akð4Þ akð5Þ . . . akðNtÞ 0 0

^ ^ ^ ^ ^ ^ ^

akðNþ1Þ . . . akðNtÞ 0 0 0 0

266664

377775 if NoNt

ðA:2Þ

The vector fkARN with k¼ fu,w1,w2,w3,w4g contains the future–future and future–past cross terms and is calculated in the followingway (Doyle et al., 2001):

fkðkþ1Þ ¼ ½kðkÞ 0 0 . . . 0� � Bk � ½kðkÞ kðk�1Þ . . . kðk�Ntþ1Þ�T

fkðkþ2Þ ¼ ½kðkþ1Þ kðkÞ 0 . . . 0� � Bk

�½kðkþ1Þ kðkÞ kðk�1Þ . . . kðk�Ntþ2Þ�T

fkðkþ3Þ ¼ ½kðkþ2Þ kðkþ1Þ kðkÞ 0 . . . 0� � Bk

�½kðkþ2Þ kðkþ1Þ kðkÞ kðk�1Þ . . . kðk�Ntþ3Þ�T

fkðkþ4Þ ¼ � � � ðA:3Þ

and the vector gkARN with k¼ fu,w1,w2,w3,w4g containing thepast–past cross terms is defined as (Doyle et al., 2001)

gkðkþ1Þ ¼ ½0 kðk�1Þ kðk�2Þ . . . kðk�Ntþ1Þ� � Bw

�½0 kðk�1Þ kðk�2Þ . . . kðk�Ntþ1Þ�T

gkðkþ2Þ ¼ ½0 0 kðk�1Þ kðk�2Þ . . . kðk�Ntþ2Þ� � Bw

�½0 0 kðk�1Þ kðk�2Þ . . . kðk�Ntþ2Þ�T

gkðkþ3Þ ¼ ½0 0 0 kðk�1Þ kðk�2Þ . . . kðk�Ntþ3Þ� � Bw

�½0 0 0 kðk�1Þ kðk�2Þ . . . kðk�Ntþ3Þ�T

gkðkþ4Þ ¼ � � � ðA:4Þ

where the matrix BkARNt�Nt with k¼ fu,w1,w2,w3,w4g has theform (Doyle et al., 2001):

Bw ¼

bkð1,1Þ bkð1,2Þ bkð1,3Þ . . . bkð1,NtÞ

0 bkð2,2Þ bkð2,3Þ . . . bkð2,NtÞ

0 0 bkð3,3Þ . . . bkð3,NtÞ

^ ^ ^ & ^

0 0 0 0 bkðNt ,NtÞ

26666664

37777775

ðA:5Þ

With the matrix and vector definitions, the future output of theprediction model (7) can be calculated using the matrices Gk (A.1)and Hk (A.2) of the linear part, the vectors fk (A.3) and gk (A.4) of thenonlinear part as well as the vectors d and h0 defined in Section 3.3.

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