Implementation of feedback linearization GPC control for a solar furnace

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ARTICLE IN PRESS Model

JPC-1552; No. of Pages 10

Journal of Process Control xxx (2013) xxx– xxx

Contents lists available at ScienceDirect

Journal of Process Control

jo u r nal homep age: www.elsev ier .com/ locate / jprocont

mplementation of feedback linearization GPC control for a solar furnace

anuel Beschia,∗, Manuel Berenguelb, Antonio Visioli c, José Luis Guzmánb, Luis José Yebrad

Dipartimento di Ingegneria dell’Informazione, University of Brescia, ItalyDepartamento de Lenguajes y Computación, University of Almería, SpainDipartimento di Ingegneria Meccanica e Industriale, University of Brescia, ItalyCIEMAT-PSA, Tabernas, Almería, Spain

r t i c l e i n f o

rticle history:eceived 4 October 2012eceived in revised form1 December 2012

a b s t r a c t

In this paper the temperature control of a solar furnace is addressed. In particular, we propose the use ofa feedback linearization generalized predictive control strategy where both the reference tracking taskand the rejection of disturbances (represented by the variation of the input energy provided by the Sun,mainly because of the solar daily cycle and passing clouds) are considered. This allows the physical and

ccepted 14 February 2013vailable online xxx

eywords:olar furnaceredictive control

security constraints to be explicitly taken into account in the design. Simulation and experimental resultsshow the effectiveness of the methodology and that this kind of plants can be considered as a cheap oralternative option for the material treatment and testing in the industrial context.

© 2013 Elsevier Ltd. All rights reserved.

eedback linearization

. Introduction

In the last forty years there has been an increasing interest fromhe industry in the use of the solar energy because it represents areen and cheap source of energy (see, for example, [1–3]). Amonghe most promising plants in this context are solar furnaces [4–7],hich consist of a collector system with tracking (usually made ofat-faceted heliostat) and a static parabolic system which concen-rates a high percentage of the solar energy in its focal spot (seeig. 1). Possible applications are generally in the field of materialreatment (the case in this paper), for example by sintering, melt-ng and casting different samples (steel, cast-iron, etc.), in order tomprove their mechanical properties such as hardness and wearesistance [8]. These operations are performed by heating the sam-les following many different temperature patterns, where thealues of the required temperatures are so high that they cannote obtained by conventional heating processes. Another interest-

ng application is the development of concentration photovoltaicPV) technology [9], where solar furnaces can be used to obtain theppropriate cell spectral response directly, and the flash tests can

Please cite this article in press as: M. Beschi, et al., Implementation of feed(2013), http://dx.doi.org/10.1016/j.jprocont.2013.02.002

e combined with prolonged PV-cell irradiation on large surfaces,o the thermal response of the PV cell can be evaluated simulta-eously.

∗ Corresponding author. Tel.: +39 0303715510.E-mail addresses: [email protected] (M. Beschi), [email protected]

M. Berenguel), [email protected] (A. Visioli), [email protected]. Guzmán), [email protected] (L.J. Yebra).

959-1524/$ – see front matter © 2013 Elsevier Ltd. All rights reserved.ttp://dx.doi.org/10.1016/j.jprocont.2013.02.002

Due to the complexity and diversity of sample materials andtemperature patterns, these plants are usually controlled manu-ally by trained operators and the obtained performance obviouslydepends on the operators’ skill. It is evident that the design of aneffective control system would allow a more widespread use ofthese plants as the desired performance is obtained under differentoperating conditions.

For this reason, different control strategies have been publishedin literature. Among them, it is worth mentioning PI control plusa feed-forward compensator [10], PI control plus a time-optimalset-point generation [11], adaptive control [12], fuzzy control [13],nonlinear control [14] and model predictive control [14]. This paperdeals with the combination of the feedback linearization strategywith a similar (GPC) algorithm (see [15–18]) to control the sampletemperature. The solar furnace control system must maintain thedesired temperature, despite disturbances in irradiance and systemuncertainties. It is also important to avoid thermal shocks, whichcan damage the equipment. The developed control strategy explic-itly treats the process nonlinearities and the safety constraints. Inparticular, a new safe condition to avoid the thermal shocks andto reduce unfeasibility situations is introduced. An adaptive lawfor the process gain is implemented in order to increase the sys-tem performances when there are model mismatches. In this paper,which is an extension of [19] by especially including experimentalresults.

back linearization GPC control for a solar furnace, J. Process Control

The specific application considered in this paper is the coppersintering, where a sample is exposed to long high temperatureset-points (near the melting point), so that small particles bondtogether and the aggregate shrinks resulting in a decrease of surface

dx.doi.org/10.1016/j.jprocont.2013.02.002


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ARTICLE ING Model

JJPC-1552; No. of Pages 10

2 M. Beschi et al. / Journal of Process

asTot

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2

htdsmsso

T

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•

••••••

••

Fig. 1. PSA solar furnace layout.

rea and energy. The considered plant is the the CIEMAT-PSAolar furnace located in the Plataforma Solar de Almeria (PSA) inabernas, Almería, South-East Spain (an excellent description andverview of different solar furnaces can be found in [20]). However,he main concepts devised can be applied to any solar furnace.

The paper is organized as follow: in Section 2 the solar furnacend its model are presented; in Section 3 the control algorithm isescribed; simulation results are shown in Section 5 while exper-

mental results are discussed in Section 6; finally, in Section 7 theonclusions are drawn.

. Description of the system

The solar furnace addressed in this paper mainly consists of aeliostat which tracks the Sun using an azimuth and pitch posi-ioning mechanism and that reflects sunlight onto a concentratorisk. The amount of incoming energy is modulated using a louvredhutter (control actuator). The concentrator convex mirror gathersost of incoming sunlight from the outdoor heliostat onto a focal

pot of 22 [cm] diameter, located inside a vacuum chamber whereamples are placed for thermal tests. Fig. 1 shows a representationf the plant layout.

The sample’s temperature model is described in details in [10].he model consists of the following first-order nonlinear equation:

h�cSc

Sf (90%)Ss˛aI(t)

[1 − sin[˛0(1 − u(t)/100)]

sin ˛0

]

− ˛e�Ss(T4(t) − T4e ) − ˛cSs(T(t) − Te) = d(mCeT(t))

dt(1)

here:

�h and �c are the reflectivity coefficients of the heliostat and theconcentrator [–];Sc, Ss and Sf(90%) are the surfaces of the concentrator, of the sample,and the focus area where the 90% of the solar input energy isconcentrated [m2];I(t) is the input solar irradiance [W m−2], which is measured byusing a pirheliometer, (its signal has been filtered with a low-passfilter with a time constant equal to 5 [s]);˛a is the absorption capacity of the sample [–];˛0 is the maximum angular aperture of the shutter [rad];u(t) is the shutter angular percentage aperture [%].˛e is the capacity of the sample to exchange heat with the air [–];� is the Stephan–Boltzmann constant (5.67 × 10−8 [W m−2 K−4]);


T(t) is the sample temperature [K], which is assumed to be uni-form into the body;Te is the environmental temperature (300 [K]);˛c is the emissivity of the sample [W m−2 K−1];

PRESS Control xxx (2013) xxx– xxx

• m is the sample mass [kg];• Ce is the specific heat [J kg−1 K−1];

In this work, all the physical properties are supposed to be con-stant. Indeed, it has been observed in control tests that taking intoaccount their dependence on the temperature does not consider-ably improve the obtained results and on-line adaptation of onlyone parameter is also enough [11]. The environmental tempera-ture is supposed constant because it varies slowly with respect tothe system dynamics. In this way the parameters can be groupedtogether and Eq. (1) becomes:

T(t) = GUI(t)

[1 − sin[˛0(1 − u(t)/100)]

sin ˛0

]+ GT4 (T4(t) − T4

e ) + GT (T(t) − Te) (2)

where:

GT4 = −˛e�Ss

mCe, GT = −˛cSs

mCe, GU = �c�h

ScSs˛a

Sf (90%)mCe.

In practical cases, �h and �c assume typical values are 75% and95%, while GU, GT4 and GT can be estimated with a standardleast squares method. It is important to note that the systemparameters depend on the sample ageing and the cleanness ofthe mirrors. In fact, the presence of dust on the mirror surfacesreduces their reflectivity. Historical data show that the parame-ter GU changes significantly (because the cleanness of the mirrorsstrongly depends on the weather) in a range bounded between3 × 10−3 [m2 K J−1] and 5 × 10−3 [m2 K J−1] (where not otherwisespecific, its value is considered equal to 3.109 [m2 K J−1]), while GT

and G4T have a reduced variability and their typical values are: GT4 =

−8.336 × 10−13 [s−1 K−3], and GT = −4.772 × 10−4 [s−1]. Notice thatGT4 assumes a small value because of it is multiplied by the forthpower of the temperature.

The sample temperature is controlled by regulating the apertureof the shutter, namely by regulating the internal solar irradiance Ic,which is the fraction of incoming solar irradiance which reachesthe concentrator and it is calculated as:

Ic(t) = I(t)

[1 − sin(˛0(1 − u(t)/100))

sin ˛0

].

The control task has to compensate for model mismatches and dis-turbances, in particular variations on the solar irradiance mainlycaused by clouds. Another important requirement is to limit thetemperature overshoot, especially when the set-point is near to themelting point of the material, as usual in many material treatmentsas the sintering processes [21]. The control task has also to take inaccount the presence of constraints in the shutter motor, in fact theshutter aperture has a range [0–100%] and, in addition, the maxi-mum velocity of the shutter aperture is limited to [5%/s] (slew rateconstraints). In any case, the presence of a vacuum chamber (MINI-VAC) introduces security constraints which reduce the operativerange of the shutter (see Section 2.1).

2.1. Constraints

The plant presents physical and security constraints which haveto be considered in control design. It is important to stress that insome cases the constraints are too stringent and therefore the setof the admissible solutions collapses into an empty one. How thissituation influences the control strategy is discussed in Section 4.


As mentioned before, the shutter motor presents physical con-straints on the amplitude, namely u(t) ∈ [u, u], and on the velocity,namely u(t) ∈ [u, u]. Due to the presence of the MINIVAC, it is nec-essary to limit the rate of the internal solar irradiance in such a way


ARTICLE ING Model


M. Beschi et al. / Journal of Process

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Constraints

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Fig. 2. Feedback linearization GPC scheme.

o avoid thermal shocks on the upper glass, namely ˙Ic(t) ∈ [ ˙Ic, ˙Ic].bviously, the internal solar irradiance reaching the secondaryoncentrator must be smaller than external one, therefore Ic(t) ∈Ic, Ic] = [0, I(t)]. Another security constraint consists to limit the

emperature derivative, namely T(t) ∈ [T , T]. However, the last con-traint in some cases cannot be satisfied, for instance, when theolar irradiance decreases from the clean sky value Ick(t) (this valuehanges during the year and can be estimated from clear-day mod-ls using astronomical formulae, see [22]) to very cloudy sky valueless than 200 [W m−2]), the control system cannot keep the tem-erature derivative in the desired bound. Nevertheless, it is possible

o guarantee a security temperature derivative constraint when theolar irradiance presents strong increments. This is achieved byimiting, each sampling time, the maximum aperture of the shuttero a suitable value usec(t). This value is determined in such a wayhat if the external solar irradiance increases instantaneously to thelean sky value Ick(t) (after passing clouds), then the temperature

erivative is in any case kept below the safe threshold value T sec .ormally, we have:

˙ (t) = GUIck(t)

[1 − sin[˛0(1 − usec(t)/100)]

sin ˛0

]+ GT4 (T4(t) − T4

e ) + GT (T(t) − Te) ≤ T sec (3)

r, equally:

sec(t) ≤ 100

(1 − 1

˛0arcsin

(sin ˛0

·(

1 + GT4 (T4(t) − T4e ) + GT (T(t) − Te) − T sec

GUIck(t)

))). (4)

emark 1. It is important to note that the numerical values of theonstraints depend on the equipment and the sample used duringhe test.

. Feedback linearization

v(t) = [4GT4 T(t)3 + GT ]v(t) + GUI(t)

[1 − sin(˛0(1

s︸


The feedback linearization (FL) method is an approach to non-inear control design which transforms a nonlinear system into ainear one, so that a linear control method can be used. In other

ords, the control system is constituted by a linear controller, with

PRESS Control xxx (2013) xxx– xxx 3

acts on a fictitious control variable, and a nonlinear term obtainedfrom the transformation, which transforms the fictitious controlvariable into the real one, as shown in Fig. 2.

In this work, input–output feedback linearization is used to con-trol the solar furnace described by the model (2). In particular, thesystem is transformed into an integrator by introducing a fictitiousvariable v(t), equal to the temperature derivative (note that there isthe advantage that the fictitious variable has a clear physical mean-ing and this makes the handling of the constraints easier, as it isexplained below):

v(t) : = T(t) = GUI(t)

[1 − sin[˛0(1 − u(t)/100)]

sin ˛0

]+ GT4 (T4(t) − T4

e ) + GT (T(t) − Te) (5)

Thus, the relation between the real and fictitious control variablesis:

u(t) = 100

(1 − 1

˛0arcsin

(sin ˛0

·(

1 + GT4 (T4(t) − T4e ) + GT (T(t) − Te) − v(t)

GUI(t)

))). (6)

Considering that the solar irradiance is always positive, no numer-ical problems occur in the transformation between v(t) and u(t).The constraints on u(t), Ic(t) and Ic(t) have to be rewritten into thefictitious variable by using both the following relation

t)/100))]

+ GUu(t)I(t)˛0

100 sin ˛0cos(˛0(1 − u(t)/100))︷︷︸

GU Ic(t)

(7)

and (5), that can be written as:

v(t) = GUIc(t) + GT4 (T(t)4 − T4e ) + GT (T(t) − Te) (8)

where the environmental temperature has been considered con-stant as it changes smoothly and its value is small when comparedwith the sample temperature T(T) (mostly when it is considered ina time window defining a prediction horizon, as will be treated inSection 4).

Note that the constraints on the temperature derivative aredirectly applied to the fictitious variable. It is important to notethat relations (5)–(8) are nonlinear and they depend to the output.This aspect is discussed better in Section 4.

4. Generalized predictive control strategy

A generalized predictive control strategy (GPC) [17] is used inthis work. The GPC algorithm computes the future values of ficti-tious control variable v(t) minimizing (taking account directly theconstraints), a cost function made by the sum of two terms, thefirst depends of the future errors, and the second depends on thefuture control effort. The future errors are calculated by using, aspredictor, a Controller Auto-Regressive Integrated Moving Average(CARIMA) model, described by equation [17]:

�A(z−1)T(t) = z−dB(z−1)�v(t − 1) + C(z−1)e(t) (9)

where d is the dead time of the system, T(t) is the output of themodel, v(t) is the control action, e(t) is a zero mean white noise, A,B, C are adequate polynomials in the backward shift operator z−1

and �(z−1) = 1 − z−1. Considering that the linearized system is anintegrator model, it is easy to calculate the polynomials A(z−1) and


B(z−1) by implementing a zero-order-hold discretization (with asample time h), and they can be written as:

A(z−1) = 1 − z−1, B(z−1) = h.


ING Model

J

4 rocess

Ts

T

wsrcttti

J

wttahtNr

J

wiawtmaopsot

aa

g

f

AaotvpcntpAIltwli

where T(kh) is the actual system temperature and T(kh|(k − 1)h) isthe last predicted temperature. The parameter � is used to selectthe adaptation velocity, which has to be greater than the process

ARTICLEJPC-1552; No. of Pages 10

M. Beschi et al. / Journal of P

he output prediction for the linearized case is composed by theum of the forced and free responses (see [23]),

= Gv + f

here T is the vector containing future ahead predictions of theystem output on data up to discrete time kh (with k ∈ N the cur-ent step), matrix G is built by the process open-loop step responseoefficients gi, f represents the plant free response and v is the vec-or of the future control increments obtained from the solution ofhe GPC optimization problem described below. The GPC cost func-ion J to be minimized in this case with respect to the future controlncrements is

=N2∑

j=N1

[T((k + j)h|kh) − r((k + j)h)

]2 + �

Nv∑j=1

[�v((k + j − 1)h|kh)]2

(10)

hich represents the sum of two contributions: the first one to trieso minimize the distance between the actual and the theoreticalemperature trajectory in a temporal horizon between times N1nd N2; the second one measures the control effort in a controlorizon Nv. The second term is weighted with the parameter �. Ashe dead time of the process is negligible, suitable values for N1,2 and Nv are N1 = 1, N2 = N and Nv ≤ N. The cost function J can be

ewritten as:

= vT (GT G + �I)v + 2(f − w)T Gv + (f − r)T (f − r).

here r is the vector of the future reference signals. In this way, tak-ng into account the constraints, the algorithm can be considereds a linear quadratic optimization problem, which can be solved byell developed techniques. Notice that the control action used in

he cost function is the fictitious variable, therefore the algorithminimizes the control effort on v(t), which is actually the temper-

ture derivative and not the control action. Thus, it is possible tobtain smooth profiles of fictitious control variable but aggressiverofiles of the real control action. In this application, obtaining amooth profile of the temperature derivative is considered by theperators a good option, because it reduces the thermal stress ofhe equipment.

The matrices G and f, which are respectively a N × Nv matrix and N × 1 matrix, are calculated by using the Diophantine equation,nd they result in:

ij ={

(1 + i − j)h if i≥j; i = 1. . .N

0 otherwise

i = (i + 1)T(kh) − iT((k − 1)h).

s mentioned above, predictive control provides a clear advantage,s it is possible to include systematic constraint handling in theptimization algorithm. Nevertheless, as said before, the predic-ive algorithm uses the fictitious control signal as the control input,(t), which is mapped using transformations (5) and (6) to com-ensate for process nonlinearities. This method implies that theonstraints presented in Section 2.1 must be mapped as variableonlinear constraints on the fictitious variable and they depend onhe solution of the algorithm. For this mapping, it is necessary toredict the value of the solar irradiance along the horizon (see [18]).s it is difficult to obtain an accurate model of the solar irradiance

(t), it is considered constant along the horizon and equal to theast measured value. Actually, it is possible to take into account all


hese nonlinear constraints in the optimization problem, but in thisay the complexity of the algorithm increases significantly, thus

oosing the most significant advantage of the FL-GPC strategy, thats, its computational efficiency. A simple solution to preserve the


computational efficiency would be to place a conservative limit onthe range of the virtual control signal so the system never reachessaturation for all the admissible solutions. However, if the ficti-tious signal limits are not properly chosen, the controlled systemmay be unstable, or the real control action may be too conserva-tive or too aggressive. In this work, the solution proposed by [24],and used with success in other solar plants [18,25,26] has beenused. In this approach, the constraints on v(t) are calculated everysampling time along the whole control horizon, using the futurecontrol predictions found with the predictive controller at the pre-vious sampling instant. This results in two column vectors vmin andvmax of variable constraints on v(t). This approximation allows thesolver to obtain quickly a suboptimal solution, which is as close tothe optimal solution as the plant conditions do not change signif-icantly between two sample periods. As described in Section 2.1,there are situations where the set of available solutions collapseto an empty set. This situation appears when the solar radiationchanges too fast and therefore the actuator is unable to compensateit or when it assumes small values and therefore the temperaturedecreases because of the insufficient input power. The constraint(4) allows the controller to avoid the unfeasibility when the radi-ation increases from a small to an high value, which is the mostcritical situation because it could generate an high positive thermalgradient.

In the other cases, when the available solutions collapse toan empty set (namely some elements of vmax are less thanthe corresponding elements of vmin), the algorithm imposes thatvmax = max(vmax, vmin), element by element. In this way, there is atleast a solution of the problem, and this solution corresponds to areduction of the temperature, which is the safest operation in theseparticular cases, which are generally caused by a insufficient solarirradiance.

4.1. Gain adaptation algorithm

In all model predictive control strategies, the obtained perfor-mance strongly depends on the accuracy of the model. If thereare model mismatches, a way to improve the performance is toimplement an on-line adaptation of the parameters (see [23]). Thisstrategy must be applied with care because it is possible that thefrequency content of the data is not sufficient for a good estima-tion.

In a solar furnace, as in other concentrated solar radiation plants(see [27]), the parameter GU has the greatest influence on the sys-tem performance (as already mentioned, its value is proportional tothe reflectivity coefficients of the heliostat and of the mirrors whichstrongly depend on their cleanness) and therefore it is importantto have an accurate estimation of its value. It has been observedthat the estimation error of the other parameters does not affectsignificantly the obtained performance. For this reason, a gain adap-tation algorithm is introduced. The parameter GU is updated asfollows:

GU((k + 1)h) =(

1 + T(kh) − T(kh|(k − 1)h)�

)GU(kh)


time constant.The system gain GU can not increase more than the double of the

nominal value and it must be greater than the half of the nominalvalue. This limitation is done for safety reasons.


ARTICLE IN PRESSG Model


M. Beschi et al. / Journal of Process Control xxx (2013) xxx– xxx 5

00:00:00 00:30:00 01:00:00 01:30:00 02:00:00400

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800

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1200

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pera

ture

[K

]

01:00:00 01:05:00 01:10:0011701172117411761178

00:00:00 00:30:00 01:00:00 01:30:00 02:00:000

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ter

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rtur

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]

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tual

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trol

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ion

[K/m

in]

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0

5

00:00:00 00:30:00 01:00:00 01:30:00 02:00:00899.5

900

900.5

901

901.5

Time [hour]

Sola

r R

adia

tion

[W

m-2]

01:00:00 01:05:00 01:10:00899

900

901

Fig. 3. Simulation results for constant irradiance. First plot (from the top): processvariable (solid line) and set-point signal (dashed line). Second plot: control variable.Third plot: virtual control variable. Fourth plot: solar radiation.

5

t

Tzssouav

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rR

adia

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[Wm

−2]

01:00:00 01:05:00 01:10:00840

850

860

870

GT4,m = GT4 (1 − 0.10 + 0.10n(t))

. Simulation results

The proposed control algorithm has been tested by consideringhe model described by (2) with the following parameters [14]:

GU = 3.109 × 10−3 [m2 K J−1], GT = −4.772 × 10−4 [s−1 K−3],

GT4 = −8.336 × 10−13 [s−1], Te = 300 [K].

he sample period, the prediction horizon N, the control hori-on, and the parameter � are set, respectively, equal to 1 [s], 120amples, 20 samples and 105. The high value of � was neces-ary in order to reduce the great control effort caused by the usef the fictitious variable v(t) instead of the real control variable(t) in the cost function. Note also that the temperature values


re three orders of magnitude greater than the fictitious variablealues.

Fig. 4. Simulation results for clear sky condition irradiance. First plot (from the top):process variable (solid line) and set-point signal (dashed line). Second plot: controlvariable. Third plot: virtual control variable. Fourth plot: solar radiation.

The constraints are set equal to:

u = 100%, u = 0%,

u = 4%, u = −4%,

˙Ic = 2 [W m−2], Ic = −2 [W m−2],

T = 20 [K/min], T = −20 [K/min].

T sec = 40 [K/min].

Model mismatches are simulated introducing constant and vari-able errors in the values of the model parameters. In particular thesimulation parameters (GU,m, GT4,m, GT,m) are defined as:

GU,m = GU (1 + 0.10 + 0.10n(t))

GT,m = GT (1 − 0.20 + 0.10n(t))


where n(t) a normal distributed noise signal with standard devia-tion of 1%.




6 M. Beschi et al. / Journal of Process Control xxx (2013) xxx– xxx

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r R

adia

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[Wm

−2]

Fpv

[

•

•

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e [%

]

00:00:00 00:30:00 01:00:00 01:30:00 02:00:00−80

−60

−40

−20

0

20

40

Vir

tual

Con

trol

Act

ion

[K/m

in]

00:00:00 00:30:00 01:00:00 01:30:00 02:00:000

200

400

600

800

1000

Time [hour]

Time [hour]

Time [hour]

Sola

r R

adia

tion

[W

m−

2]

Fig. 7. Simulation results for square wave irradiance profile. First plot (from the
ig. 5. Simulation results for bad sky condition irradiance. First plot (from the top):rocess variable (solid line) and set-point signal (dashed line). Second plot: controlariable. Third plot: virtual control variable. Fourth plot: solar radiation.
The reference signal is a step between 475.13 [K] and 1173.15K]. Four different solar irradiance profiles are used:


solar irradiance constant and equal to 900 [W m−2] (ideal case),shown in Fig. 3;solar irradiance of a day with clear sky conditions, shown in Fig. 4;

00:00:00 00:30:00 01:00:00 01:30:00 02:00:00−100

−50

0

50

Time [hour]

Tem

pera

ture

der

ivat

ive

[K/m

in]

Fig. 6. Temperature derivative trend for square wave irradiance profile.

top): process variable (solid line) and set-point signal (dashed line). Second plot:control variable. Third plot: virtual control variable. Fourth plot: solar radiation.

• solar irradiance of a day with bad sky conditions, shown in Fig. 5;• solar irradiance profile described by a square wave between

100 [W m−2] and 900 [W m−2] with a period of 10 [min] and aduty cycle of 75%, shown in Fig. 7.

Note that this last case is quite unrealistic but it is useful to test theproposed method in a possible worst case.

By looking at Figs. 3 and 4, it is possible to note that the perfor-mance obtained with ideal conditions (constant solar irradiance)and good sky condition are quite similar, therefore the control sys-tem is able to compensate well small irradiance changes and modelmismatches, that are responsible of the variations observed in theshutter aperture in steady or quasi-steady state, as can be seen inthe zooms of Figs. 3 and 4. In addition, the step responses presenta small overshoot (less than 0.6%).

The virtual control signal trends are shown in Figs. 3 and 4. It is


possible to notice that the virtual control action is saturated duringthe transient, while it is affected by the noise and the changes insolar radiation when the system is in steady-state conditions.





00:00:00 00:30:00 01:00:00 01:30:00 02:00:003

3.2

3.4

3.6

3.8

4x 10

−3

Time [hour]

GU

,ad

Fig. 8. Adaptation of the gain for clear sky irradiance profile. On-line adapted gain(solid line), a posteriori estimated gain (dashed line).

1 3 5 7 90

1

2

3

4

σT

[K]

1 3 5 7 90

1

2

3

4

Log10 λ

Log10 λ

σU

[%]

Fp

ssipbtot

dtpstrsstiwtGfis

Fig. 10. Sample used in the tests made of packed wires (courtesy of PSA).

12:01:19 12:43:05 13:24:51 14:06:37 14:48:22400

600

800

1000

1200

Local Time [hour]

Tem

pera

ture

[K]

12:01:19 12:43:05 13:24:51 14:06:37 14:48:220

20

40

60

80

Local Time [hour]

Shut

ter

Ape

rtur

e[%

]

12:01:19 12:43:05 13:24:51 14:06:37 14:48:22−10

−5

0

5

10

Local Time [hour]

Virtu

alC

ontr

olA

ctio

n[K

/min

]

12:01:19 12:43:05 13:24:51 14:06:37 14:48:22900

950

1000

1050

1100

Local Time [hour]

Sola

rR

adia

tion

[Wm

−2]

Fig. 11. Experimental results when the reference is a series of step changes. Firstplot (from the top): process variable (solid line) and set-point signal (dashed line).Second plot: control variable. Third plot: virtual control variable. Fourth plot: solarradiation (solid line) and theoretical clear sky solar radiation (dashed line).

ig. 9. Influence of parameter � on the standard deviations of the temperature (toplot) and of the shutter command (bottom plot).

By looking at Figs. 5 and 7, it is possible to note that when theolar irradiance is not large enough to allow the system to reach theet-point value, the algorithm increases the shutter amplitude, butt does not open the shutter at its maximum level because of theresence of constraint (4) and in this way thermal stresses causedy irradiance increments are avoided. In these cases, the referenceracking performance does not decrease significantly. Indeed, thebtained overshoot with a bad sky condition irradiance is again lesshan 1%.

Fig. 6 shows the temperature derivative profile when the irra-iance signal is the square wave. It is possible to note that, despitehe presence of model mismatches, the maximum temperatureositive derivative is almost everywhere less than the maximumecurity constraint. Conversely, the temperature negative deriva-ive cannot be limited when there is an insufficient level of solaradiation. In these cases, the optimization problem becomes unfea-ible, and therefore it is necessary to relax the constraints. Theatisfactory obtained results are possible thank to the gain adap-ation algorithm, which adapts the parameter GU and thereforemproves the model accuracy. The adapted gain has been compared

ith its theoretical value, which can be found by imposing thathe on-line model (described by the triple (GU,adth, GT , GT4 ), where

is the theoretical adaptive gain) and the a posteriori identi-

Please cite this article in press as: M. Beschi, et al., Implementation of feedback linearization GPC control for a solar furnace, J. Process Control(2013), http://dx.doi.org/10.1016/j.jprocont.2013.02.002

U,adthed model (described by the triple (GU,pos, GT,pos, GT4,pos)) gives theame temperature derivative for each sample, or equally by solving




8 M. Beschi et al. / Journal of Process Control xxx (2013) xxx– xxx

12:46:07 12:49:22 12:52:36 12:55:50 12:59:04620

630

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Local Time [hour]

Tem

pera

ture

[K]

12:54:06 12:56:35 12:59:04

672

674

12:46:07 12:49:22 12:52:36 12:55:50 12:59:0410

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20

25

Local Time [hour]

Shut

ter

Ape

rtur

e[%

]

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25

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2

4

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8

10

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Virtu

alC

ontr

olA

ctio

n[K

/min

]

12:54:06 12:56:35 12:59:040

5

10

12:46:07 12:49:22 12:52:36 12:55:50 12:59:04920

940

960

980

1000

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1040

Local Time [hour]

Sola

rR

adia

tion

[Wm

−2]

12:54:06 12:56:35 12:59:04940

945

950

Fig. 12. Zoom of experimental results when the step change from 623.15 [K] and673.15 [K] is applied. First plot (from the top): process variable (solid line) andstr

t

T

G

Gtc

t

13:41:12 13:47:44 13:54:15 14:00:47 14:07:18

950

1000

1050

Local Time [hour]

Tem

pera

ture

[K]

13:56:08 14:01:43 14:07:18

1070

1075

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Shut

ter

Ape

rtur

e[%

]

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55

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65

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15

Local Time [hour]

Virtu

alC

ontr

olA

ctio

n[K

/min

]

13:56:08 14:01:43 14:07:18−10

0

10

13:41:12 13:47:44 13:54:15 14:00:47 14:07:18

940

960

980

1000

1020

1040

1060

1080

Local Time [hour]

Sola

rR

adia

tion

[Wm

−2]

13:56:08 14:01:43 14:07:18

952

954

956

Fig. 13. Zoom of experimental results when the step change from 923.15 [K] and1073.15 [K] is applied. First plot (from the top): process variable (solid line) and

et-point signal (dashed line). Second plot: control variable. Third plot: virtual con-rol variable. Fourth plot: solar radiation (solid line) and theoretical clear sky solaradiation (dashed line).

he following equation for each sample time:

GU,adth(kh)I(kh)

[1 − sin[˛0(1 − u(kh)/100)]

sin ˛0

]+ GT4 (T4(kh) − T4

e ) + GT (T(kh) − Te)

= GU,posI(kh)

[1 − sin[˛0(1 − u(kh)/100)]

sin ˛0

]+ GT4,pos(T

4(kh) − T4e ) + GT,pos(T(kh) − Te)

hus, the theoretical value of GU,adth results:

U,adth(kh) = GU,pos +(GT4,pos − GT4 )(T4(kh) − T4

e ) + (GT,pos − GT )(T(kh) − Te)

I(kh)[

1 − ((sin[˛0(1 − u(kh)/100)])/(sin ˛0))] (11)

Fig. 8 shows the comparison between the adapted gain andU,adth. It can be noted that the trends are similar and therefore


he value of the parameter is conveniently updated in order toompensate for the estimation error.

In order to understand the influence of the parameter � onhe system performance, simulations with different values of

set-point signal (dashed line). Second plot: control variable. Third plot: virtual con-trol variable. Fourth plot: solar radiation (solid line) and theoretical clear sky solarradiation (dashed line).

� have been performed. The simulation model parameters are(GU,m, GT4,m, GT,m) as in the previous example, the solar radiation isthat of a day with clear sky condition (see Fig. 4), the set-point sig-nal is constant and equal to 773.15 [K] and the initial temperatureis equal to the set-point. Twenty simulations have been performedwith � logarithmically spaced into a interval [10, 109], and for eachsimulation the standard deviations of the temperature and of thereal control action are calculated. The results are shown in Fig. 9,where it is possible to note that the value � = 105 corresponds toa minimum in the standard deviation of the temperature and thestandard deviation of the control action is comparable with thevalues obtained with greater values of �.

6. Experimental results

The presented algorithm was applied to the CIEMAT-PSA solarfurnace. The test sample was made of copper wires with a diameter


of 2 [mm], which are layed in package with a 80 [mm]×80 [mm]exposed surface and an height of 30 [mm] (see Fig. 10).

The controller parameters were set as follows: � = 105, N = 120,Nv = 20 and h = 1 [s]. The output derivative constraint was set





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800

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Local Time [hour]

Tem

pera

ture

[K]

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824

826

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ter

Ape

rtur

e[%

]

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0

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Virtu

alC

ontr

olA

ctio

n[K

/min

]

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0

10

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960

980

1000

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1080

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Sola

rR

adia

tion

[Wm

−2]

14:38:41 14:43:31 14:48:22942944946948950952

Fig. 14. Zoom of experimental results when the step change from 923.15 [K] and823.15 [K] is applied. First plot (from the top): process variable (solid line) andstr

efGG

scrTtot

ss21itsd

12:02:13 12:48:18 13:34:22 14:20:26 15:06:302

2.5

3

3.5x 10

−3

Local Time [hour]

GU

,ad

et-point signal (dashed line). Second plot: control variable. Third plot: virtual con-rol variable. Fourth plot: solar radiation (solid line) and theoretical clear sky solaradiation (dashed line).

qual to ±10 [K/min] by the operator. The model parameters usedor the feedback linearization were set as in Section 2, namely:U = 3.109 × 10−3 [m2 K J−1], GT4 = −8.336 × 10−13 [s−1 K−3], andT = −4.772 × 10−4[s−1].

Fig. 11 shows the experimental results obtained using a series oftep changes as temperature set-point. In addition to the process,ontrol, and virtual control variables, and to the measured solaradiation, the theoretical clear sky solar radiation is also shown.his is calculated using the sky model presented in [22]. Note that,he real solar radiation is lower than the theoretical value becausef the absorption and scattering caused by different components ofhe atmosphere.

It is possible to note that the temperature profile did not presentignificant overshoot, as can be seen in Figs. 12–14 which showome zooms of the experimental results. As already mentioned in, the small overshoot achievable with the controller (less than.35%) is an important characteristic, especially when the work-


ng temperature is close to the melting point. This allows the usero avoid to use specific additional techniques to avoid large over-hoots [11]. Another interesting aspect is the different influence ofisturbance and model mismatches at the different temperatures

Fig. 15. Adaptation of the gain in the test. On-line adapted gain (solid line), a pos-teriori estimated gain (dashed line).

(see, for instance, the zooms of Figs. 12 and 13). It is possible toexplain this aspect considering the expression of the time constant� and the gain K of the linearized gain (see [10]):

� = − 14GT4 T3(t) + GT

,

K = �GUI(t)˛0

100 sin ˛0cos(˛0(1 − u(t)/100)).

The time constant decreases when the temperature increases whilethe gain decreases. For these reasons, the control effort required forthe disturbance rejection task increases with respect to the tem-perature and therefore the error band achievable by the controllerincreases from less than 0.5 [K] to 2 [K]. In any cases, the obtainederror band is less than the other control techniques (which take intoaccount the presence of MINIVAC) implemented in the CIEMAT-PSA Solar Furnace (see [14,11,8]). In fact, a PID controlled systemhas an error band of 5 [K], while the implementation of a linearizedmodel predictive control method allows the system to have an errorband of 2 [K], which is equal to the worst case obtained with thefeedback-linearization control strategy. However the control effortis more relevant using the developed algorithm, because of the useof the virtual variable.

Remark 2. It is important to note that the controllers mentionedabove are tuned for different objectives (set-point tracking, dis-turbance rejection, safety constraints compliance) and the testsare performance with different sample agings. Further, a compari-son with other control strategies which do not take in account theMINIVAC constraints is not completely fair.

The good performance of the proposed strategy depends also onthe gain adaptation algorithm. In order to highlight the effective-ness of the algorithm, the model of the plant has been a posterioriestimated by using the same least square method of Section 2,resulting in the set of parameters: GU,pos = 2.627 [m2K J−1], GT4,pos =−6.418 × 10−13 [s−1 K−3], and GT,pos = −6.326 × 10−4 [s−1].

Fig. 15 shows the comparison between the on-line adapted gainand its theoretical value GU,adth calculated with Eq. (11).

It is possible to note that the gain mismatch was significantreduced during the test, increasing the model accuracy.

7. Conclusions

In this paper, a feedback linearization GPC algorithm for a solarfurnace has been proposed. In particular, a linear GPC approachis applied to a system linearized by introducing a fictitious vari-able. A suboptimal technique to handle the constraints in the model


predictive control strategy has also been highlighted.Simulation and experimental results have shown that the

methodology provides a good performance even in case of modelmismatches and variations of the solar irradiance, by satisfying at


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ARTICLEJPC-1552; No. of Pages 10

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he same time the plant constraints due to physical and safety rea-ons. The achievable temperature error band is smaller with respecto other control strategies implemented in copper sintering pro-ess in the CIEMAT-PSA Solar Furnace but the algorithm has theisadvantage represented by a greater control effort.

cknowledgements

This work has been developed under projects DPI2010-21589-05/04 and DPI2011-27818-C02-01, funded by Spanish Ministryf Economy and Competitiveness and EU-ERFD. The authors thanktaff of the Plataforma Solar de Almería, especially Lidia Roca anduan Luis Rivas.

eferences

[1] R.N. Silva, L.M. Rato, J.M. Lemos, F. Coito, Cascade control of a distributed col-lector solar field, Journal of Process Control 7 (2) (1997) 111–117.

[2] J.D. Álvarez, L.J. Yebra, M. Berenguel, Repetitive control of tubular heat exchang-ers, Journal of Process Control 17 (9) (2007) 689–701.

[3] M. Bar ao, J.M. Lemos, R.N. Silva, Reduced complexity adaptive nonlinear controlof a distributed collector solar field, Journal of Process Control 12 (1) (2002)131–141.

[4] F. Trombe, Solar furnaces and their applications, Solar Energy 1 (2–3) (1957)9–15.

[5] P.E. Glaser, A solar furnace for use in applied research, Solar Energy 1 (2-3)(1957) 63–67.

[6] G. Flamant, A. Ferriere, D. Laplaze, C. Monty, Solar processing of materials:opportunities and new frontiers, Solar Energy 66 (2) (1999) 117–132.

[7] R. Pitz-Paal, High temperature solar concentrators, in: J.B. Gálvez, S.M.Rodríguez (Eds.), Solar Energy Conversion and Photoenergy Systems, EOLSSPublishers, Oxford, UK, 1999.

[8] E.F. Camacho, M. Berenguel, F.R. Rubio, D. Martínez, Control of Solar EnergySystems, Springer, London, UK, 2012.

[9] L. Fernández-Reche, I. Canadas, M. Sánchez, J. Ballestrín, L. Yebra, R. Monterreal,


J. Rodríguez, G. García, M. Alonso, F. Chenlo, PSA solar furnace: a facility fortesting PV cells under concentrated solar radiation, Solar Energy Materials &Solar Cells 90 (2006) 2480–2488.

10] M. Berenguel, E.F. Camacho, F. García-Martín, F.R. Rubio, Temperature controlof a solar furnace, IEEE Control System Magazine 19 (1) (1999) 8–24.

[


11] M. Beschi, A. Visioli, M. Berenguel, L.J. Yebra, Constrained temperature controlof a solar furnace, IEEE Transactions on Control Systems Technology 20 (5)(2012) 1263–1274.

12] B. Andrade da Costa, J.M. Lemos, An adaptive temperature control law for asolar furnace, Control Engineering Practice 17 (2009) 1157–1173.

13] D. Lacasa, M. Berenguel, L. Yebra, D. Martínez, Copper sintering in a solarfurnace through fuzzy control, in: IEEE International Conference on ControlApplications, Munich, Germany, 2006.

14] M. Beschi, M. Berenguel, A. Visioli, L. Yebra, Control strategies for disturbancerejection in a solar furnace, in: Preprints IFAC World Congress, Milan, Italy,2011.

15] Q.M. Zhu, K. Warwick, J.L. Douce, Adaptive general predictive controller fornonlinear systems, in: IEE Proceedings – Control Theory and Applications, vol.138, 1991, pp. 33–40.

16] M. Botto Ayala, T.J.J. Van Den Boom, A. Krijgsman, J. Sa Da Costa, Predictivecontrol based on neural network models with I/O feedback linearization, Inter-national Journal of Control 72 (17) (1999) 1538–1554.

17] E.F. Camacho, C. Bordóns, Model Predictive Control, Springer, London, UK, 2004.18] L. Roca, J.L. Guzman, J.E. Normey-Rico, M. Berenguel, L. Yebra, Robust

constrained predictive feedback linearization controller in a solar desali-nation plant collector field, Control Engineering Practice 17 (2009)1076–1088.

19] M. Beschi, M. Berenguel, A. Visioli, L. Yebra, J.L. Guzmán, A feedback lineariza-tion GPC control strategy for a solar furnace, in: Preprints of American ControlConference (ACC12), Montreal (CND), 2012.

20] D. Martínez, Solar Furnace Technologies, CIEMAT, Madrid, Spain, 1996.21] B. Alexander, R. Balluffi, The mechanism of sintering of copper, Acta Metallur-

gica 5 (1957) 667–677.22] W.B. Stine, M. Geyer, Power from the Sun, 2001 http://www.

powerfromthesun.net/book.html23] E.F. Camacho, M. Berenguel, Application of generalized predictive control to a

solar power plant, in: IEEE International Conference on Control Applications,Glasgow, UK, 1994.

24] M.J. Kurtz, M.A. Henson, Input–output linearizing control of constrained non-linear processes, Journal of Process Control 7 (1996) 3–17.

25] C.M. Cirre, M. Berenguel, M.L. Valenzuela, E.F. Camacho, Feedback linearizationcontrol for a distributed solar collector field, Control Engineering Practice 15(2007) 1533–1544.

26] L. Roca, M. Berenguel, L. Yebra, D. Alarcón-Padilla, Solar field control for desali-


nation plants, Solar Energy 82 (2008) 772–786.27] J.M. Igreja, J.M. Lemos, R.N. Silva, Adaptive receding horizon control of

a distributed collector solar field, in: 44th IEEE Conference on Decisionand Control and 2005 European Control Conference, CDC-ECC05, 2005, pp.1282–1287.


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Implementation of feedback linearization GPC control for a solar furnace

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