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Solar Energy 85 (2011) 1056–1067

Filtered Smith predictor with feedback linearization andconstraints handling applied to a solar collector field

Lidia Roca a,⇑, Jose Luis Guzman b, Julio E. Normey-Rico c, Manolo Berenguel b,d,Luis Yebra a,d

a CIEMAT, Plataforma Solar de Almerıa, Tabernas, Almerıa, Spainb Departamento de Lenguajes y Computacion, Universidad de Almerıa, Spain

c Departamento de Automac�ao e Sistemas (DAS), Universidade Federal de Santa Catarina, Florianopolis-SC, CEP 88040-970, Brazild CIESOL, Centro Mixto Universidad de Almerıa-CIEMAT, 04120 Almerıa, Spain

Received 17 June 2010; received in revised form 10 December 2010; accepted 26 February 2011Available online 21 March 2011

Communicated by: Associate Editor P. Gandhidasan

Abstract

Feedback linearization (FL) has recently been used to control the temperature in a solar field using the water flow rate as the manip-ulated variable. Nevertheless, the system shows undesirable control signal oscillation caused by a delay uncertainty. In this paper, thedynamics resulting from applying the FL transformation to the nonlinear process model is approximated by a first-order model plusdead-time, and a filtered Smith predictor (FSP) is proposed as the control strategy. This controller deals with dead-time errors but alsoincludes control signal saturation caused by flow rate constraints. Simulations demonstrate the robustness of the FSP structure in com-parison to a simple PID and a Smith predictor (SP) approach when a faster closed-loop is required in a system with delay uncertainties.Real experimental results included show the improved behavior of the proposed control scheme.� 2011 Elsevier Ltd. All rights reserved.

Keywords: Dead-time compensator; Delay system; Feedback linearization; Robustness; Solar energy

1. Introduction

Several papers have been published dealing with solarfield control algorithms where the main goal is to reach adesired outlet-temperature using the water flow rate asthe manipulated variable (Camacho et al., 2007a,b). Thesesystems are difficult to control because the daily solar cycle,cloud cover, and weather conditions must be taken intoaccount. In addition, solar fields are nonlinear systemswhich may also present dead-time when there is a passivepipe between the solar collector output and the tempera-ture sensor. This delay varies because the transport time

0038-092X/$ - see front matter � 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.solener.2011.02.026

⇑ Corresponding author.E-mail addresses: lidia.roca@psa.es (L. Roca), joguzman@ual.es (J.L.

Guzman), julio@das.ufsc.br (J.E. Normey-Rico), beren@ual.es (M.Berenguel), luis.yebra@psa.es (L. Yebra).

the heat transfer fluid takes to reach the sensor dependson the fluid flow rate. If small changes around an operatingpoint are considered, solar field dynamics may be approx-imated by a first-order system with a delay which dependson the flow rate (Camacho et al., 1997). Dead-time in solarfields can also be caused by hydraulic upsets Roca et al.(2009) as it will be commented later.

Dead-time compensators (DTC) can be used to controlsolar fields. In Normey-Rico et al. (1998), a filtered predic-tive PI controller (FPPI) was successfully tested in a para-bolic-through collector solar field in which the delay variedwith water flow in the pipe. The algorithm is composed of aPI controller combined with a Smith predictor (SP) anda dead-time estimator. Moreover, a low-order filter isincluded in the predictor to improve robustness. This filteris easy to tune since its bandwidth is related to the nominaldelay.

L. Roca et al. / Solar Energy 85 (2011) 1056–1067 1057

Although the FPPI provided good results in the partic-ular case of the parabolic-through collector solar fieldstudied in Normey-Rico et al. (1998), its performance dete-riorates when the process exhibits high nonlinearitiesbecause the primary controller is a simple PI. Oneapproach that can be used in this case is to combine a feed-back linearization control (FLC) with a dead-time compen-sator structure in a cascade configuration. The FLC acts inthe inner loop giving an equivalent linear model for thetuning of the main loop, therefore simplifying the maincontroller tuning. Although FLC has been applied in solarfields with good results (Barao et al., 2002; Silva et al.,2002, 2003; Igreja et al., 2003, 2005; Cirre et al., 2007; Rocaet al., 2008), in these works problems related to delayuncertainties and constraints are not addressed. Recently,in Roca et al. (2009) FLC approach was used together witha dead-time compensator generalized predictive controller(DTC-GPC) to deal with delay uncertainties and copesystematically with system constraints.

The purpose of our work is to improve the previousresults in Roca et al. (2008) by using a simpler controlstrategy than the DTC-GPC. In Roca et al. (2008), sincethe control signal showed an oscillatory response causedby delay mismatches, the PID controller had to be detunedto maintain stability, slowing down behavior. In that case,the system dynamics after the FLC transformation wasapproximated by an integrative system. In this case weuse an FLC, but with a different algebraic transformationto get a first-order system instead of an integrative one.Then a filtered Smith predictor (FSP) is used on the linearmodel, characterized by a new virtual control signal, todeal with delay errors. An anti-wind-up strategy is alsoused to take into account the constraints in the manipu-lated variable. Disturbances are taken into account totransform the virtual control signal to the water flow rateembedding feedforward action in the control scheme.

The paper is organized as follows. Section 2 describesthe solar collector field and process delays. The FL tech-nique and the model approximation to a first-order systemare presented in the next section. The FSP control strategyis described in Section 4. Comparative simulation resultsand experimental tests are analyzed in Section 5. Finally,some conclusions are summarized.

2. System description

The solar collector field dealt with in this work (seeFig. 1a) is made up of 252 Compound Parabolic Concen-trators (CPC) (AoSol 1.12X) arranged in four rows, orloops, of 63 collectors with a total surface area of approx-imately 500 m2. Seven groups of nine collectors each areconnected in parallel in each loop (see Fig. 1b). In eachgroup, subgroups of three collectors each are connectedin parallel to each other and in series to another two sub-groups. CPCs are non-tracking collectors with a reflectivesurface around the cylindrical absorber tubes that concen-trates irradiance. These CPCs each have seven absorber

tubes each one. The collectors are oriented east-west tomaximize energy collection. The most important distur-bance in the system is the continuous variation in irradia-tion due to the daily cycle and unpredictable cloudtransients. Irradiance is measured using a pyranometertilted at a 36� angle, which is the inclination of the solarcollectors. This solar field feeds two different seawaterdesalination systems: a multi-effect plant (Alarcon-Padillaet al., 2005) and a membrane distillation system (Guillenet al., 2009). Thus, control may be either step-referenceor ramp depending on the experiment.

As explained in Roca et al. (2009), two different delayscontribute to the behavior of the collector water outlet-temperature, (a) a transport delay and (b) an outlet-temperature inlet-flow delay as described in the following:

(a) As the inlet temperature sensor is located at thebeginning of the loop of the selected group, a trans-port delay, dTiF, for the inlet water could be estimatedat each sampling time as a flow-dependent delay withthe approximation described in Normey-Rico et al.(1998).It is important to mention that this delay isnot included directly in the control algorithm. Distur-bance, TiF, is estimated every sampling time usingpast measures of the inlet temperature, TsiF (seeFig. 1b), and by evaluating the delay dTiF using pastflow rates Roca et al. (2009). This temperature esti-mation is then an approximation of the solar collec-tor inlet water temperature.

(b) Experimental tests have demonstrated the existenceof a outlet-temperature inlet-flow delay with a nomi-nal value of dn = 40 s which can vary between 28 and50 s. This observed delay comes from different causes:(i) a water transport delay in the group collector con-nections, (ii) an output-flow delay caused by thedynamics of the heat transfer in the differential flowelements inside the absorber tubes, and (iii) in somecases, water flow rate changes produce hydraulicupsets in the different collector groups and absorbertubes. Since the process is modeled as a collectionof similar absorber tubes with similar characteristicsand behavior, these real hydraulic imbalances pro-duce different dynamics in the absorber tubes, andan apparent delay appears in the system. From a gen-eral point of view, these delays are included in thementioned output temperature-flow-related delay d.Fig. 2 illustrates this variable delay in two differentstep changes of the water flow at different operatingpoints.

3. Feedback linearization control

The main idea behind feedback linearization technique(Isidori, 1995) is treatment of nonlinear systems as if theywere linear by means of algebraic transformations andfeedback.

Fig. 1. Solar field.

3740 3750 3760 3770 3780 3790 3800 3810 3820 3830 3840 3850

48.6

48.8

49

49.2

49.4

49.6

49.8

T oF [º

C]

3740 3750 3760 3770 3780 3790 3800 3810 3820 3830 3840 38502.4

2.6

2.8

3

3.2

3.4

Step 1: water flow rate from 2.5 to 3.4 l/s

Wat

er fl

ow ra

te [l

/s]

2200 2210 2220 2230 2240 2250 2260 2270 2280 2290 230055.2

55.4

55.6

55.8

56

2200 2210 2220 2230 2240 2250 2260 2270 2280 2290 23001.5

2

2.5

3

3.5

4

4.5

5Step 2: water flow rate from 1.6 to 3.5 l/s

Relative time [s]

d = 28 s

d = 43 s

Fig. 2. Inlet flow-outlet water temperature related delay.

1058 L. Roca et al. / Solar Energy 85 (2011) 1056–1067

Coordinate transformation (or mapping) is representedby a function U which depends on a new virtual controlsignal, v, and the system state, x. Moreover, if the non-

linear system includes linearly dependent disturbances(R), this algebraic transformation, U, also depends on thesedisturbances. Following this interpretation, feedback

L. Roca et al. / Solar Energy 85 (2011) 1056–1067 1059

linearization may be applied to track a desired referencesolar field outlet-temperature using the water flow as thecontrol signal.

The outlet-temperature model (Roca et al., 2009) is:

q � Cp � Aa �@T oF ðtÞ@t

¼ bI � IðtÞ �HLeq� ðT ðtÞ � T aðtÞÞ

� Cp � _meqðt � dÞ � T oF ðtÞ � T iF ðtÞLeq

;

ð1Þ

where the parameters (see Table 1) were found experimen-tally by identification techniques under different operatingconditions and validated with real data Torrico et al.(2010).

This model is transformed into canonical form (2) usingfunctions g and b which depend on system disturbances,R = (TiF,Ta,I):

_x ¼ gðx; .Þ þ bðx; .Þ � v ð2Þy ¼ hðxÞ;

where

Table 1Nomenclature.

Name Definition (value-units)

Aa CPC collector absorber cross-section area (1.7453e-004m2)Cp Specific thermal capacity (4190 J/(kg �C))d Inlet water flow-outlet water temperature related delay (30–50 s)ds Delay d expressed in sampling timesdTiF Inlet water temperature outlet water temperature related

transport delay(s)f Water flow rate (m3/s)H Global thermal losses coefficient (4 J/(s K))I Irradiance (W/m2)k Sampling timeKp Proportional gainLeq Equivalent absorber tube length (5.67 m)_meq Equivalent mass flow (kg/s)_mF Water mass flow (kg/s)na Number of absorber tubes in a CPC collector (7)nc Number of collector groups in a loop (7)ncp Number of parallel-connections in a CPC collectors group (3)nl Solar field operative loops (1–4)t Continuous time (s)T Equivalent absorber tube mean temperature (�C)Ta Ambient temperature (�C)Td Derivative time (s)Tf Robustness filter parameter (s)Ti Integral time (s)TiF Solar field one-loop inlet temperature (�C)ToF Solar field one-loop outlet-temperature (�C)Ts Sampling time (s)TsiF Solar field inlet temperature measurement (�C)Tt Anti-reset wind-up time (s)v Virtual control signal (�C/s)bI Irradiance model parameter (0.1024 m)q Water density (975 kg/m3)

x ¼ T oF ;

gðx; .Þ ¼ � H � x2 � Leq � q � Aa � Cp

;

bðx; .Þ ¼ 1

q � Aa � Cp;

vðt � dÞ ¼ bI � IðtÞ �H � T iF ðtÞ

2 � Leqþ H � T aðtÞ

Leq

� Cp � ðT oF ðtÞ � T iF ðtÞÞ � _meqðt � dÞLeq

;

hðxÞ ¼ x:

The following input–output relationship is found:

A � _wðtÞ þ B � w ¼ vðt � dÞ; ð3ÞgðtÞ ¼ wðtÞ;

where w = x = ToF, A = q�Cp�Aa, and B ¼ H2�Leq

.As shown, v(t) does not only depend on future distur-

bances R(t + d), but also on future output, ToF(t + d).For disturbances it is assumed that R(t) � R(t + d), so thesevariables remain constant in a prediction horizon of lengthd (from t to t + d). ToF(t + d) may be predicted by using (3)with a discrete Euler approximation as:

T oF ðk þ dsÞ ¼ T oF ðkÞ � 1� BA

T s

� �ds

þ T s

A

Xds�1

i¼0

vðk � ds þ iÞ � 1� BA

T s

� �ðds�1�iÞ

ð4Þ

where ds is the delay expressed in sampling times (d = dsTs).The equivalent mass flow rate can be computed as:

_meqðtÞ ¼bI � IðtÞ � Leq � H � T iF ðtÞ=2þ H � T aðtÞ � vðtÞ � Leq

Cp � ðT oF ðt þ dÞ � T iF ðtÞÞ;

ð5Þwith no thermal inversion in the solar field, ToF(t + d) –TiF(t).

It is well-known that the use of an input–output linear-ization method in a controllable system of order n requiresa relative degree rd such that rd 6 n (Slotine and Li, 1991),where rd characterizes the effective delay between the inputu and output y. Nevertheless, this requirement may be soft-ened by effective evaluation of yðrd Þ, such as the one pro-posed in (4), which results in quite good responses as willbe shown below.

FL makes it possible to use a linear control (forinstance, a PID controller such as the one used by Rocaet al. (2008)) as shown in the Fig. 3. A filter F(s) has beenincluded to improve the set-point response. To account foractuator saturation, a standard anti-reset wind-up mecha-nism (Astrom and Hagglund, 1995) with integrator wind-up integral time Tt has been implemented. Since anti-wind-up is applied to the virtual signal, flow rate limitsare transformed into virtual signal limits using the inversetransformation of (5). Thus, in this case, the limits in thevirtual control signal at each sampling time are given by:

Fig. 3. FL-PID scheme.

Fig. 4. SP and FSP schemes Normey-Rico and Camacho (2007).

1060 L. Roca et al. / Solar Energy 85 (2011) 1056–1067

vminðkÞ ¼ vb �Cp � _meq;max � ðT oF ðk þ dsÞ � T iF ðkÞÞ

Leq;

vmaxðkÞ ¼ vb �Cp � _meq;min � ðT oF ðk þ dsÞ � T iF ðkÞÞ

Leq;

where

vb ¼ bI � IðkÞ þH � T aðkÞ

Leq� H � T iF ðkÞ

2 � Leq;

_meq;min ¼_mF ;max

ðnl � nc � ncp � naÞ6 _meq 6

_mF ;min

ðnl � nc � ncp � naÞ¼ _meq;max;

_mF ¼ q � f ;

(see Table 1).Notice that a first-order system is obtained with the pro-

posed FL algebraic transformation (see Eq. 3), instead ofthe integrator system used in Roca et al. (2008). This makesit easier to meet control specifications and the usual tuningrules may be applied.

4. Filtered Smith predictor

The well-known SP (see Fig. 4a) is a DTC widely used inthe industry Normey-Rico and Camacho (2007) in which adead-time nominal process model is used P nðsÞ ¼GnðsÞe�Lns.1 Nevertheless, the main drawback of this algo-rithm is that dead-time errors can destabilize the system.A robust solution is the FSP (see Fig. 4b), in which a filterFr(s) is included to attenuate the oscillation caused by delaymismatches (Normey-Rico and Camacho, 2007).

In the nominal case (P(s) = Pn(s)) the closed-loop trans-fer function for set-point changes is the same for the SPand FSP:

HrðsÞ ¼Y ðsÞRðsÞ ¼

CðsÞP nðsÞ1þ CðsÞGnðsÞ

: ð6Þ

Note that the delay is eliminated from the characteristicequation and Fr(s) does not affect Hr(s). Assuming thatthe real plant differs from the nominal case (Pn(s) – P(s))and considering a family of plants P(s) such that

1 The nominal delay dn in Normey-Rico and Camacho (2007) is calledLn.

P(s) = Pn(s)[1 + dP(s)] = Pn(s) + DP(s), the characteristicequation is given by 1 + C(s)[Gn(s) + P(s) � Pn(s)] =1 + C(s)[Gn(s) + DP(s)]. The condition for closed-loop SProbustness is that for all frequencies and all plants in thefamily, the distance between C(s)Gn(s) and the -1 point inthe Nyquist diagram (j1 + C(s)Gn(s)j) is greater than jC(s)DP(s)j. Thus, for the SP:

jdP ðsÞj < dP SP ðsÞ ¼j1þ CðsÞGnðsÞjjCðsÞGnðsÞj

; s ¼ jx; 8x > 0;

ð7Þ

where j is the imaginary unit and x the frequency.Therefore, when the closed-loop transfer function 6 is

defined, dPSP(s) is also fixed and, if C(s) is chosen for ahigh performance then robustness will be poor. Thus, ifC(s) is not appropriately chosen, small uncertainties maydestabilize the system.

Using the FSP structure, the condition is different:

jdP ðsÞj < dP FSPðsÞ ¼j1þ CðsÞGnðsÞjjCðsÞGnðsÞF rðsÞj

; s ¼ jx; 8x > 0:

ð8Þ

Therefore,

dP FSPðsÞ ¼dP SP ðsÞjF rðsÞj

; ð9Þ

and if Fr(s) is a low-pass filter, it can be used to improve therobustness of the system at the desired region of frequency.Although Fr does not affect Hr (see Eq. 6) it modifies thedisturbance rejection response defined by:

HqðsÞ ¼Y ðsÞQðsÞ ¼ P nðsÞ 1� CðsÞP nðsÞF rðsÞ

1þ CðsÞGnðsÞ

� �: ð10Þ

L. Roca et al. / Solar Energy 85 (2011) 1056–1067 1061

Thus Fr(s) must be tuned for a compromise betweenrobustness and disturbance rejection performance.

Fig. 5 shows the controller diagram described in thispaper. The FSP includes a nominal model Gn(s), nominaldead-time, dn, and the filter Fr(s) to attenuate oscillationscaused by delay mismatches. This FSP is combined witha PI, which includes anti-reset wind-up, to control the lin-ear system obtained with FL. The virtual control signal, v,obtained with the FSP-PI, is mapped to the real control sig-nal, _mF , in the block called “FL constraint mapping” tak-ing into account the disturbances (I,Ta, TiF) whereascontrol signal saturation is included in the FSP plant modelinput. Filter F(s) is a simple first-order reference filter,F(s) = (1 + bTrs)/(1 + Trs), used to adjust the response

Fig. 5. FL-PID-

0 500 1000 1500

0 500 1000 150045

50

55

60

65

0 500 1000 150023

24

25

26

Relative

T a [ºC

]

400

500

600

700

I [W

/m2 ]

T iF [º

C]

Fig. 6. Disturbances u

when there is a set-point change Normey-Rico and Cama-cho (2007). This is equivalent to use a PI controller with aset-point weighting b.

5. Results

5.1. Simulation results

FL-PID and FL-PID-FSP are compared in simulationusing the nonlinear model (1) as the real plant and a setof real plant inputs. The goal is to evaluate FL-PID pre-sented in Roca et al. (2008), in which the linear systemobtained from FL was an integrator, and the FL-PID-FSP proposed in section 3, in which the linear system

FSP scheme.

2000 2500 3000 3500

2000 2500 3000 3500

2000 2500 3000 3500

time [s]

sed in simulation.

1062 L. Roca et al. / Solar Energy 85 (2011) 1056–1067

obtained from FL is a first-order model. For model simu-lation and temperature control, MathWorks MATLAB�

and Simulink� were used.Fig. 6 shows real plant inputs from October 5 2009 used

to evaluate the control schemes. Solar irradiance followsthe solar cycle, the inlet temperature varies slowly from45 �C to 60�C, and the ambient temperature is in the rangeof 23–27 �C. Four operative solar field loops are assumedfor this simulation study, nl = 4, and the maximum andminimum inlet-flow rates, f, are set to 4.5 l/s and 1.2 l/s,respectively.

5.1.1. FL-PID with anti-wind-up

PID controllers may be used to control dead-time pro-cesses because the derivative action can be viewed as a lin-ear prediction. Nevertheless, when the system presents avariable dead-time, tuning rules are limited to allow a goodcompromise between performance and robustness (Nor-mey-Rico and Camacho, 2007).

Therefore, following the ideas proposed in Roca et al.(2008), where the system obtained from FL was approxi-mated by an integrative linear system:

P nðsÞ ¼1

se�40; ð11Þ

a PID controller can be designed to the tuning rules pro-posed in Normey-Rico and Camacho (2007) to allow agood compromise between performance and robustness:Kp = 0.0225 �C/s, Ti = 135 s, Td = 17 s, and Tt = 30 s.Fig. 7 shows the response of the solar field control whentwo different dead-times are considered. We can see this de-lay in the Fig. 7 zoom, at 1800 s water flow rate goes downto the minimum and after 40 or 50 s the output starts to in-crease. As observed with this simulation, although a delay-error is considered, the controller response is robust due tothe conservative tuning.

However, when fast responses are required and the con-troller parameters are set to reach these new fast specifica-

0 500 1000 1500

56

58

60

62

64

66

Sola

r fie

ld o

utle

tte

mpe

ratu

re [º

C]

0 500 1000 15001

2

3

4

5

Relativ

d=40sd=50s

setpointd=40sd=50s

setpointd=40sd=50s

Con

trol s

igna

l[l/

s]

Fig. 7. FL-PID results wit

tions using the nominal plant dynamics, the system couldbecome unstable if the delay is different from the nominalone. For instance, the results for Kp = 0.0225 �C/s,Ti = 60 s, Td = 17 s, and Tt = 30 s are represented inFig. 8. The response is the desired when the plant delayis the nominal one, however, there is a more oscillatoryresponse with the 25% plant delay-estimation error.

5.1.2. FL-PID-FSP with anti-wind-up

Following the control structure presented in Fig. 5 andthe model presented in Eq. 3, Pn(s) can be described by atransfer function with static gain, K, time constant, s,and nominal dead-time, dn:

P nðsÞ ¼K

ssþ 1e�dns ¼ 1=B

ðA=BÞsþ 1e�dns

¼ 2:835

2021:4sþ 1e�40s; ð12Þ

where A and B are related to the linear system 3.Controller parameters are defined to obtain a set-point

nominal closed-loop response with almost no-overshootand rise time similar to the one obtained with the fasterPID (Fig. 8). Using the root locus tool: Kp = 106W/m �C, Ti = 25 s, Tt = 15 s and

F ðsÞ ¼ 10sþ 1

25sþ 1; ð13Þ

are obtained. In the second step, the robustness filter istuned using the rule proposed in Normey-Rico and Cama-cho (2007):

F rðsÞ ¼1

1þ sT f: T f ¼ dn=2 ¼ 20s ð14Þ

To show the effect of the filter using this tuning and in thecase of modeling errors, Fig. 9 shows the closed-loop re-sponses of the SP (no filter), FSP for a 25% plant delay-estimation error and FL-PI from Fig. 8. If the filter is tunedwith Tf = 20 s, the oscillatory response disappears, as

2000 2500 3000 3500

2000 2500 3000 3500e time [s]

1800 1820 1840 186059

60

61

62

h conservative tuning.

0 500 1000 1500 2000 2500 3000 3500

56

58

60

62

64

66

0 500 1000 1500 2000 2500 3000 35001

2

3

4

5d=40sd=50s

setpointd=40sd=50s

Relative time [s]

Sola

r fie

ld o

utle

tte

mpe

ratu

re [º

C]

Con

trol s

igna

l[l/

s]

Fig. 8. FL-PID results with faster tuning.

0 500 1000 1500 2000 2500 3000 3500

56

58

60

62

64

66

64.9

64.95

65

65.05

2000 2050 2100

64

65

66

67FL−PI−SP FL−PI−FSP FL−PID

Sola

r fie

ld o

utle

tte

mpe

ratu

re [º

C]

0 500 1000 1500 2000 2500 3000 35001

2

3

4

5

Relative time [s]

FL−PI−SPFL−PI−FSPFL−PID

Con

trol s

igna

l[l/

s]

Fig. 9. System response considering the 25% plant delay-estimation error.

L. Roca et al. / Solar Energy 85 (2011) 1056–1067 1063

shown in Fig. 9. Outlet-temperature decreases slowly withFL-PI-FSP, although it seems to cause a steady-state errorof 0.05 �C. A longer time axis should be used to observehow the temperature finally reaches the set-point.

Notice that by using the FL-PI-FSP, the rise timeobtained in the second step (when the delay-error is 10 s)is 193 s, lower than with the conservative FL-PID(Fig. 7), which was 215 s. The rise time is also 193 s whenthe faster PID is used (Fig. 8), but there is an oscillatoryresponse and a longer overshoot. In that case, the virtualcontrol signal from the PID (Fig. 10) oscillates and theseoscillations are transferred to the water flow rate afterthe FL-mapping. Another improvement with FL-PI-FSP

is better operating points robustness. Observe that withthe FL-PI-FSP scheme, closed-loop dynamics are similarat all operating points, while oscillation is wider at the sec-ond operating point with the FL-PID scheme.

Also notice how the limits of the virtual control signalvary in time depending on the operating point. These limitsare included in the saturation block, so an anti-wind-upmechanism can be applied to the virtual signal. The differ-ence between controller output, vo, and saturation output,v, feeds the input of the PID integrator gain through gain1/Tt (see Fig. 5). It is important to mention that althougha water mass flow rate saturation block is included in thescheme, it is unnecessary because FL constraint mapping

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

0 500 1000 1500 2000 2500 3000 3500

FL−PID

FL−PI−FSP

FL−PI−SP

v vmaxvmin d=50s

−0.1

−0.05

0

0.05

0.1

−50

0

50

100

−50

0

50

100

Virtu

al c

ontro

lsi

gnal

[W/m

]

Virtu

al c

ontro

lsi

gnal

[ºC

/s]

Virtu

al c

ontro

lsi

gnal

[W/m

]

Relative time [s]

Fig. 10. Virtual control signal considering the 25% plant delay-estimation error.

0

101

102

103

itude

|δP|dP FL−PIDdP FL−PI−SPdP FL−PI−FSP

1064 L. Roca et al. / Solar Energy 85 (2011) 1056–1067

calculates virtual limits properly as a function of flow ratelimits, so the output of the FL-mapping block (water massflow rate) is always inside the limits. For instance, as shownin Fig. 9, 1.2 6 f 6 4.5 l/s.

Therefore, as shown in these simulations, it is possible toobtain a faster and more robust behavior using the FL-PI-FSP scheme.

10−3 10−2 10−1 10010−3

10−2

10−1

10

frequency

mag

n

Fig. 11. Modeling error jdP(s)j and robustness index dP.

5.1.3. Robustness analysis

A robustness analysis can be done to justify the obtainedsimulation results by using the Small Gain Theorem (Sko-gestad and Postlethwaite, 2001) and assuming that: (i) themodel Pn[1 + dP(s)] describes the behavior of the real pro-cess, where dP(s)is the multiplicative modeling error, and(ii) that small movements of the control action avoidsaturations.

In this situation, to illustrate the effect of the filter andthe dead-time compensation, the robustness conditionsfor the three different cases (FL-PI-FSP, FL-PI-SP andFL-PID) are compared:

dP PIDðxÞ ¼j1þ CPIDðsÞP nðsÞjjCPIDðsÞP nðsÞj

> jdP ðsÞj;

dP SPðxÞ ¼j1þ CPIðsÞGnðsÞjjCPIðsÞGnðsÞj

> jdP ðsÞj; ð15Þ

dP FSPðxÞ ¼j1þ CPIðsÞGnðsÞjjCPIðsÞGnðsÞF rðsÞj

¼ dP SP ðsÞjF rðsÞj

> jdP ðsÞj;

where s = jx, "x P 0 and CPI(s) is the primary controllerin the SP and FSP schemes Normey-Rico and Camacho(2007).

Fig. 11 shows jdP(s)j assuming the 25% delay estimationerror, and dP curves for the three controllers. This method

shows that the FSP structure causes better results, confirm-ing previous simulations.

5.2. Experimental tests

The first experiment is used to show that FL-PI-FSP canbe tuned for faster responses than the ones obtained usingthe FL-PI in the experiments shown in Roca et al. (2008).With the disturbances in Fig. 12 and parameters Kp =106 W/m �C, Ti = 28 s, Tt = 15 s, Tf = 35 s and set-pointfilter 13, the response is as shown in Fig. 13. There arefour operative loops and the water flow is limited to1.4 6 f 6 4.2 l/s. Outlet-temperature follows the referencedespite irradiance variations and set-point changes.

12 12.5 13 13.5 14 14.5 15

12 12.5 13 13.5 14 14.5 15

12 12.5 13 13.5 14 14.5 15

GMT

700800900

100011001200

I [W

/m2 ]

T iF [º

C]

T a [ºC

]

50

60

70

80

18

19

20

21

22

Fig. 12. Experimental disturbances. February, 23rd 2010.

12 12.5 13 13.5 14 14.5 15

12 12.5 13 13.5 14 14.5 151

2

3

4

12 12.5 13 13.5 14 14.5 150

50

100 vmax v vmin

ToF setpoint

70

80

90

Wat

er fl

ow[l/

s]

Virtu

al c

ontro

lsi

gnal

[W/m

]Te

mpe

ratu

re[ºC

]

GMT

Fig. 13. Experimental results with FL-PI-FSP. February, 23rd 2010.

L. Roca et al. / Solar Energy 85 (2011) 1056–1067 1065

Nevertheless, between 13.2 and 14 GMT the system is notable to follow the reference because water flow rate its atthe maximum. Note that a conservative value of Tf hasbeen tuned because in the real case the modeling errorsare greater than in the simulated one. In this case, the risetime is 112 s, which is less than in Roca et al. (2008), whichwas around 150 s. Nevertheless, control signal oscillatesbecause of delay-errors. Therefore, although oscillationmay be eliminated with FSP, a compromise betweenrobustness and velocity may be found. Notice also thatthe performance does not vary with the operating pointdespite the variable delay due to the use of a bilinear model

for control purposes and the use of FSP. The next experi-ment shows that less aggressive behavior is possible underreference changes using a slower PI (Kp = 50 W/m �C,Ti = 70 s) and tuning the filter Fr for better robustness(Tf = 55 s). This simple tuning parameter is very importantin practice as Tf can be used intuitively to reduce oscilla-tions by increasing its value.

On this day (Fig. 14), the rise time is around 150 s(Fig. 15), which is similar to Roca et al. (2008), but witha smoother control signal. This experiment also showsthat when higher variations in irradiance appear, the con-troller maintains the temperature close to the reference.

11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.4

11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.440

50

60

11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.420

22

24

I [W

/m2 ]

T iF [º

C]

T a [ºC

]

400600800

10001200

Local time (Almería, Spain)

Fig. 14. Experimental disturbances. April, 23rd 2010.

11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.45560657075

11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.41

2

3

4

11.8 12 12.2 12.4 12.6 12.8 13 13.2 13.4

vmin v vmax

ToF setpoint

−50

0

50

100

Tem

pera

ture

[ºC]

Wat

er fl

ow[l/

s]

Local time (Almería, Spain)

Virtu

al c

ontro

lsi

gnal

[W/m

]

Fig. 15. Experimental results with FL-PI-FSP. April, 23rd 2010.

1066 L. Roca et al. / Solar Energy 85 (2011) 1056–1067

The maximum error (3 �C) occurs when irradiance fallsfrom 1200 W/m2 to 400 W/m2 (between t = 13 h andt = 13.3 h), and the controller is unable to follow the refer-ence because the water flow rate is saturated at its mini-mum value. Notice that variations in irradiance changethe water flow rate because this disturbance is included inthe feedforward action. This situation causes the limits ofthe virtual signal to vary and although v may be modifiedto remain inside saturation, variations in disturbances aremainly rejected by the feedback linearization block.

In regard to the ramp reference, some considerationsmay be mentioned. In the case Roca et al. (2008), the inte-grator system with a PI form an open-loop nominal trans-fer function with two integrators. Therefore, the system istype 2 and the theoretical steady-state error is 0 with rampinputs. Nevertheless, with the FL transformation proposedin this paper, the nominal system is type 1 (a first-order sys-tem with a PI controller) and the steady-state error is

ess = Ti/(KpK). As shown in Fig. 15, ramp reference inthe experiment is 0.0019 �C/s. Therefore, ess = 9 � 10�4 �C,which may be considered negligible compared to type 2 sys-tem. Moreover, the steady-state error measured in Fig. 15is under 0.2 �C which is similar to Roca et al. (2008).

Notice that, as expected, the experimental results arequite similar than those obtained in simulation. Small dif-ferences between simulations and real experiments arecaused by model mismatches not included in the simula-tions. For this reason a slower PI and a higher Tf was usedin the second experiment (Fig. 15). This issue confirms thevalidity and robustness of the proposed control system.

6. Conclusion

The controller proposed in this paper combines FL withFSP to control the water outlet-temperature in a solar fieldwith variable delay. This controller shows the following

L. Roca et al. / Solar Energy 85 (2011) 1056–1067 1067

advantages: (i) it is simple to understand and tune; (ii) itsimplementation is straightforward; (iii) feedforward actionis embedded in the FL scheme when the virtual control sig-nal is mapped into the water flow, and (iv) it deals withdelay-errors allowing for a robust behavior. Simulationresults have shown that it is possible to obtain a bettercompromise between robustness and performance thanusing a PID controller and a SP scheme without loosingsimplicity. Real experiments on a solar field have beenincluded to show the obtained promising results.

Acknowledgments

This work has been partially funded by the followingprojects: PHB2009-0008 financed by the Spanish Ministryof Education; CNPq-BRASIL; CAPES-DGU 220/2010;Spanish CICYT and EU-ERDF funds under contractDPI2007-66718-C04-04, DPI2010-21589-C05-02; AQUA-SOL Project (Contract No. EVK1-CT2001-00102) financedby CIEMAT and the European Commission.

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