Robust control of the distributed solar collector field ACUREX using MPC for tracking

Robustcontrolofthedistributedsolar collectorfieldACUREXusingMPC for tracking.⋆ D.Limon∗I.Alvarado∗T.Alamo∗M.Ruiz∗ E.F. Camacho∗ ∗Dpto.deIngenier´ıa deSistemasyAutom´atica,Universidad de Sevilla,Avda delosDescubrimeintos s/n41092,Sevilla(limon, alvarado,alamo,arahal,[email protected]). Abstract: Thispaper presentsthe applicationofarobustmodel predictivecontrolfortracking ofpiece-wiseconstantreferences (RMPCT) to a distributed collectorfield, ACUREX,at the solarpower plantofPSA (SolarPlantofAlmer´ıa).The maincharacteristicofasolarpower plant isthat the primaryenergysource, solar radiation, cannotbemanipulated. Solar radiationvaries throughout the day,causing changes inplantdynamics and strong disturbances inthe process. The realplantisassumed tobemodeled asalinearsystem withadditivebounded uncertainties onthe states.Under mildassumptions,the proposed RMPCT cansteer the uncertainsystem inanadmissibleevolutionto anyadmissiblesteady state, thatis,under anychangeofthe set point.Thisallowsus toreject constantdisturbances compensating the effect ofthen changing the setpoint. 1.INTRODUCTION ACUREXisasolarplantlocated atTabernass desert (Spain) and itisone ofthe availableexperimentalplants ofthe PSA complex (SolarplantofAlmer´ıa).Itconsists ofaseries ofparabolicmirrorsthatreflect solar radiation onto a pipewhereoil getsheated whilecirculating.The objectiveofthe controlsystem inadistributed collector fieldistomaintainthe outlet oil temperatureatadesired level despiteofdisturbances suchaschanges inthe solar irradiance level (caused byclouds),mirror reflectivityor inlet oil temperature. Since solar radiationcannotbe adjusted, thiscanonlybeachieved byadjusting the oil flow. Inthispaper,the controlofthe ACUREXplanthasbeen carried out bymeans ofaRMPCT recentlyproposed in(Limonet al. [2007]),the feasibility,stabilityand asymptoticalconvergence ofthe proposed controller for anyadmissiblesetpointisachieved byadding anartificial steady stateasdecisionvariable, penalizing the deviation between thisartificialsteady stateand the realone in the costfunctionand using asaterminalconstraintan invariantset fortracking.Robustconstraintsatisfaction isguaranteed bytube-based approach; anominalcontrol problem isdefined whosesolution(atrajectory)defines the center ofatube; and wherethe ’cross-section’ ofthe tube isaninvariantset.The constraintsforthe nominalsystem would bethosethat the tubesatisfies the realconstraints. The statetrajectoryofthe controlled system will beforced, bythe controltolieinthistube. Ateachtime, anew tube isdetermined bysolving acontroloptimizationinwhich, the decisionvariables arethe initialstateofthe nominal ⋆TheauthorsacknowledgeMCYT-Spainforfundingthiswork DPI2007-66718-C04-01. system,the controlsequence over afinitehorizonand the artificialsteady state. The paper isorganized asfollows.First, insection2 a brief descriptionofthe distributed solarcollectorfieldisgiven. The section3isdedicated topresenting the RMPCT. After that,section4presentsthe identificationprocedure. Section5showsthe resultsobtained inthe plantusing he RMPCT.Section6showsresultsobtained inthe plant using the RMPCT plus the cancellationoffset loop, and the conclusions arepresented inSection7. Notation:AdefinitepositivematrixTisdenotedasT>0 and T>PdenotesthatT−P>0.Fora givensymmetricmatrix P>0,kxkPdenotestheweightedEuclidean normofx, i.e. kxkP=√xTPx.(a,b)∆ =[a′,b′]′.Considera∈IRna,b∈IRnb, and setΓ⊂IRna+nb,then projectionoperationisdefinedas Proja(Γ)={a∈IRna:∃b∈IRnb,(a,b)∈Γ}.GiventwosetsUand V,suchthatU⊂IRnand V⊂IRn,theMinkowskisumisdefined byU⊕V∆ ={u+v|u∈U,v∈V},thePontryaginsetdifference is: U⊖V∆ ={x|x⊕V⊆U}.fora givenλ,λX={λx:x∈X}.Lettbe a genericvectordefinedast∆ ={t(0),t(1),...}. 2.DESCRIPTION OF THESOLARPOWERPLANT ACUREX The distributed collectorfieldinvolves the collectionof solarenergyand itstransfer to a fluidpiped throughthe system.The energycollected istransferred to a storage tank,whichcan betapped when conditions demand, on toeither asteamgeneratorforelectricalpower generation orthe heatexchanger ofadesalinationplant.Itmainly consistsofapipeline throughwhichoil isflowing and ontowhichthe sun’sraysareconcentrated bymeans of parabolicmirrors,whichfollowthe sun byrotating onone axis, inorder toheat the oil. Itiscomposed by480 modules Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-3-902661-00-5/08/$20.00 © 2008 IFAC 958 10.3182/20080706-5-KR-1001.2589 arranged in20 lines whichform10 parallel loops.Figure 2showsadiagramofthe solarcollectorfield. The fieldis alsoprovided withasun-tracking mechanismwhichcauses the mirrorstorevolvearound anaxisparallel tothatofthe pipeline. Eachofthe loops mentioned aboveisformed by four 12-modulecollectors,suitablyconnected inseries.The loopis172 mlong,the activepartofthe loop(exposed to concentrated radiation) measuring 142 mand the passive part30 m.Afundamentalfeatureofasolarpower plantis Fig.1.Acurex scheme. that the primaryenergysource, whileitisvariable, cannot bemanipulated. The intensityofsolar radiationfromthe sun, inadditiontoits seasonaland dailycyclicvariations, isalsodependentonatmosphericconditions suchascloud cover,humidityand airtransparency.Inorder to obtain asuitableexperimentaldataforthe identification, the experimentsmustbeexecuted inaperiodwherethe environmentalvariables wereconstant, i. e. the changes inthe output areonlyproduced bythe control input. The dynamicalcharacterizationofthe fieldhasbeen done in boththe timeand frequency domains.Open-loopstep responses havebeen obtained atseveraloperating points. One oftheseresponses is showninFigure2(achangein oil flowfrom8to 7 l swasperformed).Ascan beseen, the step responseshowsthat the behaviorofthe fieldcan be modeled byalow-order model as showninprevious works done inthe plant (F.et al. [1994a]),althoughthe plant dynamics aremorecomplex (presence ofantiresonance modes)ascan beseen inthe frequency responseplotshown inthe figure2. The useofhigh-order modelswithstandardparameter estimationalgorithmsoften produces wrong resultsbecause ofthe divergence ofthe estimated parameters(F.et al. [1994b]).Using low-order modelswithModel Predictive controlalgorithms,acceptablebehaviorcan beobtained. 2.1Seriesfeedforwardcompensation The modelsrelate changes inoutlet temperatureto changes inoil flow.The outlet temperatureofthe plant, Fig.2.Dynamicalcharacterizationofthe field. however, isalsoinfluenced bychanges insystem variables suchas solar radiationand fluidinlet temperature. Inorder to accountforthesedisturbances,aseries feedforwardcontroller hasbeen introduced. The feedforward iscalculated fromsteady staterelationships,whichmake anadjustmentinthe fluidflowinput,aimed toeliminate the changeinoutlet temperaturecaused bythe variations insolar radiationand inlet temperature. The calculation employed is: flow=a1Irr −a2(FFref −a3)−a4 FFref −Tin wherea1,a2,a3,a4areparameters,flowisthe oil flow, FFref isthe temperaturesetpoint,Tinisthe inlet oil temperatureand Irr isthe effectivesolar radiation. The feedforwardisplaced inseries withthe robustcontroller whichprovides the signalF Fref (see figure5). Althoughexact eliminationobviouslycannotbeachieved, thistermhelps topreservethe validityofthe assumed system modelsand provides controlbenefitswhen disturbances insolar radiationand fluidinlet temperatureoccur. The feedforwardisconsidered asapartofthe modelsand thus the input signalforthe plantmodel isthe setpoint temperatureforthe feedforwardtermand not the oil flow(thisiscalculated bythe feedforwardcontroller).The feedforwardtermisexplained extensivelyin(Berenguel et al. [1993], F.et al. [1994a,b]). 3.RMPCT 3.1Problemdescription Consider the following uncertaindiscrete-timelineartime- invariantsystem: x+=Ax+Bu+wy=Cx+Du(1) Wherex∈X⊆IRn,u∈U⊆IRmand y∈Y⊆IRpisthe currentstate, the currentinput and the currentoutput of the system;x+isthe successorstate;w∈W⊆IRnis anunknownand bounded statedisturbance. x(k),u(k), w(k)denotethe state, the input and the disturbance at sampling timekofthe system (1).X,U,Y,Wareconvex polyhedracontaining the origininitsinterior. Assumption1.System (1)shouldverify: •(A,B,C,D)areknownand (A,B)controllable. •The stateisaccessible. The proposed controller will bebased onthe responseof the nominalsystem obtained from(1)byneglecting the disturbances w. ¯x+=A¯x+B¯u¯y=C¯x+D¯u(2) ¯x(i)=¯ φ(i;¯x,¯ u)represent the evolutionofthe nominal stateat timeiifitsinitialnominalstateis¯xand the controlsequence isu Tocounteract the disturbances itisdesirabletoforce the trajectorytolieclosetothe nominaltrajectory;thiscan bedone bychoosing the controlutosatisfy: u=¯u+Ke e ∆ =(x−¯x) (3) 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 959 whereedenotes the controlerrorbetween the stateand the stateofthe nominalsystem. The erroresatisfies the difference equation: e+=AKe+w;AK=(A+KC) (4) Assumption2.The feedbackcontrolmatrixKis suchthat AKis stable(Hurwitz). BecauseAKisHurwitz, thereexistsarobustpositively invariantset Z(Kolmanovsky and Gilbert[1998], Rakovic et al. [2005])forthe system (4) thatsatisfies AKZ⊕W⊆Z Thereareseveralmethods fordetermining the robustpositivelyinvariantset asthe ones introduced in(Blanchini [1999], Rakovicet al. [2005], Rakovic[2007]). Since e(0)∈Zimplies e(i)∈Z∀i∈IN,anobvious consequence is: Proposition1.Ifthe initialrealand nominalsystem states,satisfye(0)=x(0)−¯x(0)∈Z,then x(i)∈¯x(i)⊕ Z∀i∈IN,all admissibledisturbance sequences w. Theorem1.(Mayne et al. [2005]).Supposethe initialsystem and nominalsystem states all lieinXand satisfy e(0)=x(0)−¯x(0)∈Z.Inaddition, ifthe initialstate ¯x(0)and controlsequence ¯ uofthe nominalsystem satisfy the tighter constraints¯x(i)=¯ φ(i;x(0),¯ u)∈X⊖Zand ¯u(i)∈U⊖KZforall i∈IN.The statex(i)and the controlu(i)=¯u(i)+K(x(i)−¯x(i)) ofthe realsystem (1) satisfythe originalconstraintsx(i)∈Xand u(i)∈Ufor all i∈INand all admissibledisturbance sequences w. 3.2Controllerformulation Inorder toensures the feasibilityofthe problem when the set pointinchanged, anartificialreachablesteady state isintroduced (¯xs,¯us)asadecisionvariableinour optimizationproblem.Then ifthe new setpointpointisnot reachablethe system will beleaded to a differentreachable setpoint (the artificialsteady state).Itisclearthatdoing thisthe feasibilityisensured but the convergence can be lost.To avoidthis,atermk¯xs−xsk2 T(the offset cost) that penalizes the deviation between the desired steady state and the artificialone isadded inthe costfunction: VN(x,xs;¯ u,¯x,θ)= N−1 X k=0 k¯x(i)−¯xsk2 Q+k¯u(i)−¯usk2 R +k¯x(N)−¯xsk2 P+k¯xs−xsk2 T(5) Thiscostpenalizes the deviation between the predicted trajectoryand the artificialsteady statealong the horizon Nand the deviation between the artificialand the desired steady statexs.θisavariablethatdetermine the artificial steady state(¯xs,¯us)=Mθ(Limonet al. [2005b]). The formulationofthe optimalproblem PN(x,xs)is: min ¯ u,¯x,θ VN(x,xs;¯ u,¯x,θ) (6) s.t.¯x∈x⊕(−Z) ¯x(i)∈¯ X=X⊖Z ¯u(i)∈¯ U=U⊖KZ (¯x(N),θ)∈Ωw ¯ K whereΩw ¯ Kisaninvariantset fortracking associated tothe linearcontrol lawdescribed in(Limonet al. [2005b])with a gain¯ Kthatcan bedifferent toK. Due tothe constraints set does notdepend onxs,the optimizationproblem PN(x,xs)hasafeasiblesolution ∀x∈XN⊂IRn.The size ofthatset depends,among other things,onthe size ofΩ¯ K=Projx(Ωw ¯ K).Asbigger isthis set,bigger isXN(Limonet al. [2005a]).Becauseof that∀x∈XN,The optimizationproblem hasasolution denoted by*,so,V∗ N(x,xs)would bethe optimalcost, ¯ u∗(x,xs),¯x∗(x,xs)and θ∗(x,xs)would bethe optimal value ofthe decisionvariables,¯ x∗(x,xs)would bethe nominaloptimaltrajectoryand (¯x∗ s(x,xs),¯u∗ s(x,xs)) would be the optimalartificialreference. The controlactioniscalculated fromthe optimalsolution asfollows κN(x,xs)=K(x−¯x∗(x,xs)) +¯u∗(0;x,xs) (7) where¯u∗(0;x,xs)isthe firstcomponentof¯ u∗(x,xs). Remark2.(controller properties).Thiscontroller ensures the stability,the convergence and the constraints satisfactionforanychangeofthe setpointatanysampletime. Moreover, ifthe provided setpointisnotadmissiblethe system will beleaded tothe closestequilibrium stateinthe sensethat the offset costisminimized (Alvarado [2007]) Itisimportant that, inthe casethat the disturbances set, the constraints set,the robustinvariantset Zand the invariantset Ωw ¯ Kwerepolyhedra,the proposed problem can beformulated asaQP (QuadraticProgramming Problem),thatcan beeasilysolved using efficientalgorithms (Camacho and Bordons [2004], Luenberger [1989]). Moreinformationabout the controller and howthe set can becalculated can befound in(Limonet al. [2005a,2007], Alvarado [2007]) 4.IDENTIFYINGTHE PLANTPLUSTHE FEEDFORWARD Due tothe chosen controller isaRMPCT,aspace state representationwithadditivedisturbances ofthe plant (1) isneeded . InthiscaseuisFFref ,wisthe model errorand yisthe output temperatureTout. Toidentifythe plant,asitwasaforementioned, aperiod wherethe environmentalvariables wereconstantischosen, thus,the changes inthe output aregoing tobeonly produced bythe changes inthe input.(see anexample infigure3).Afew loworder modelswithdifferentdelays havebeen identified bythe leastsquaremethodand finally the model withthe smaller identificationerrorwasafirst order model without delay: A=0.8656 B=0.1251 C=1D=0 The set ofpossiblevalues ofwhasbeen calculated computing the errorbetween the realdata and the modeled one fromsomeexperiments(see figure4).The obtained set is|w|≤5The constraints set forthatsystem are: 100 ≤u≤350 0≤y≤300 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 960 12.8 13 13.2 13.4 13.6 13.8 14 14.2 14.4 14.6 14.8 150 160 170 180 190 200 210 local time (h) temperatures (oC) FFref tsal tmodel 12.8 13 13.2 13.4 13.6 13.8 14 14.2 14.4 14.6 14.8 20 40 60 80 100 120 A2=8.685678e−001, B2=1.280501e−001, C2=1, D2=0 local time (h) Enviroment Irr/10 Tamb Tin Fig.3.Plantidentification. 14 14.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16 150 160 170 180 190 200 local time (h) temperatures (oC) FFref tsal tmodel 14 14.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16 −3 −2 −1 0 1 2 3 Maximal error = 2.681207e+000 local time (h) Error Model Error 14 14.2 14.4 14.6 14.8 15 15.2 15.4 15.6 15.8 16 0 50 100 150 Environment variables local time (h) Enviroment Irr/10 Tamb Tin Fig.4.Maximalerrorwmax. The supervisionmoduleofthe loopadds acoupleof constraints: •The maximaldifference between the temperatures of eachcollectoris800, Tout−Tin≤80 •The constraintset forthe flowis 2≤flow≤10 l s Theseconstraintslimitsthe reachableset ofoutput temperatures ofthe plant. 5.APPLYINGTHEROBUST MPCFOR TRACKING The proposed RMPCT isused tomanipulateFFref ensuring robustadmissibility,stability,disturbance rejection and setpoint tracking.Figure5showsthe generalscheme ofthe controlled plant. The RMPCT hasbeen adjusted, viasimulation, tocon- Fig.5.RMPCT scheme. trolthe plantinorder tobeasfastasposiblewithout overshooting.The provided nonlinearmodel isbased on partialdifferentialequations (Berenguel et al. [1993]). The controller parametersare: MPC#1 Qz=1,Rz=1,K = −0.36726 Z=(−28.0174,28.0174) Q = 1000,R=1,¯ K = −6.40848 ¯ Yeq ∈(107.34,271.71) ¯ U=U⊖KZ =(110.29,339.71) Xn=(0.00,300.00) •The localcontroller gaintodetermined Zisthe one corresponding tothe LQR withQ=Qzand R=Rz. Itisnot the one who makes Zminimal inorder not toproduce overshooting. •Qand Rarethe matrices weightsofthe costfunction, they arealsoused tocalculatethe controller gain, using the LQR,tocalculatethe invariantset for tracking used asthe terminalconstraint. •¯ Yeq isthe set ofthe admissiblesetpoints. •¯ Uisthe set ofthe admissiblecontrolactions forthe nominalsystem. •Xnisthe domainofattraction. Inthe following figures, inthe upper graphicis shown: the reference (solidline),the artificialreference (dasheddotline) and the output (dashed line) and inthe lower one, the realdisturbances (the radiation, the corrected radiation(bothindashed line),the temperatureofthe input Tin(dashed-dotline))and the anestimationofthe disturbance vest(solidline). Figure6showsthe evolutionofthe plantcontrolled bythe MPC#1under goodconditions. 14.8 14.9 15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 230 232 234 236 238 240 242 244 Local time MPC#1 tsal tref trefart 14.8 14.9 15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 0 20 40 60 80 100 Local time Disturbances rad/10 radcor/10 west*10 tin/10 Fig.6.MPC#1without disturbances. Inthe following subsectionthe behaviorofthe controlled plantunder differentsituations areshow. 5.1Radiationdisturbance Figure7showsthe evolutionofthe plantinpresence of clouds. The parametersofthe controller arethe sameasinthe previous case The maximalpossibleerrorbetween the realstateand the reference in permanentregimeisthe 280(the size ofZ) 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 961 14.2 14.4 14.6 14.8 15 15.2 15.4 250 255 260 265 270 275 280 Local time MPC#1 tsal tref trefart 14.2 14.4 14.6 14.8 15 15.2 15.4 −50 0 50 100 Local time Disturbances rad/10 radcor/10 vest*10 tin/10 Fig.7.MPC#1withradiationdisturbance. thatcan beanonadmissibleerror,to avoidthis,thereare twopossiblesolutions. •Trytodo the localcontroller Kmoreaggressive, with the riskofovershooting. •Lookforamethodthat,changing the setpoint,cancel the error.(See section6) 5.2Pyrometersensorerror Inthiscasethe controlled used ismoreaggressive, the parametersare: MPC#2 Qz=10,Rz=1,K= −1.81804 Z=(−13.7275,13.7275) Q = 1000,R=1,¯ K = −6.40848 ¯ Yeq ∈(121.62,285.89) ¯ U=(124.96,325.04) Xn∈(0.00,300.00) Figure8showsthe evolutionofthe plantcontrolled by the MPC#2.Inorder tointroduce moredisturbances the measureofthe pyrometer wasaffected. Itcan beseen how the evolutionremains inside the predicted limitswiththe exceptionwherethe estimated errorbecomes bigger than 5,thisisbecausethe junctionofthe end ofthe periodwere the measureofthe pyrometer wasdisturbed and the effect ofacloud. Anywaythe errorcan be13.7275 (the size ofZ) that 11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 215 220 225 230 235 240 245 250 255 Local time MPC#2 tsal tref trefart 11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 −60 −40 −20 0 20 40 60 80 100 120 Local time Disturbances rad/10 radcor/10 vest*10 tin/10 Sensor error Fig.8.MPC#2withanerrorinthe pyrometer. can beadmissibleornot, inthe following sectionaprocedureto avoidthiserrorifthe disturbance isconstantis introduced. 6.CANCELLATION OF THETRACKINGERROR Due tothiscontroller allowstochangethe setpoint,this one can bemodified, taking into account the value of the disturbance (provided byanestimator),toremove the effect ofthe disturbance onthe output (Limonet al. [2007]).The modified setpointhasthe expression: ˆs(k)=s−[C+DK](In−(A+BK))−1ˆw(k) ˆs(k)=s−Hsˆw(k) whereˆsisthe setpoint thatshould beprovided tomake the realsystem tends todesired one s,ˆwisanestimation ofthe disturbance w. Inorder toprovide ameasureofthe erroranestimatoris needed. Inthispaper the disturbance estimatorproposed in(Normey-Rico and Camacho [2007])isgoing tobeused (DTC discretedisturbance observer): w+=b·w−A·η+(1−b)·(y+−y+ model) C η=(1−b)·(y−ymodel) C x+ model =Axmodel +Buymodel =Cxmodel +Du Wherebisthe constantofafirstorder filter withthe appropriatevalue. Figure9showsthe generalscheme. Fig.9.Tracking errorcancellationloop. The controller thatisgoing tobeused isthe MPC#1, the MPC#2was so aggressiveand working withthe DTC generates overshooting. 6.1Tindisturbance Figure10 showsthe plantcontrolled bythe MPC#1 using the methodproposed insection(6).Itcan beseen howisfollowing the reference inadaywithaverygood conditions.Inorder tosee the disturbance rejectionthe temperatureinthe input wasdisturbed. Itcan beseen howthe feedforwardoverreactsunder thisdisturbance and howthe RMPCT isattenuating this effect. 6.2Pyrometersensordisturbance Figure11 showsthe responseofthe controlled plantwhen anerrorinthe asensorhappens.Using the proposed method, the effect ofthe disturbance isrejected, without using it,the effect isattenuated but notremoved (see figure8). 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 962 13 13.5 14 14.5 15 15.5 230 235 240 245 250 255 260 265 270 MPC#1 with constant disturbance rejection tsal tref trefart 13 13.5 14 14.5 15 15.5 −5 0 5 10 15 20 Disturbances rad/100 radcor/100 vest tin/10 7º Input disturbance 17º Input disturbance 7º 4º Fig.10.MPC#1withconstantdisturbance rejection. 13.2 13.4 13.6 13.8 14 14.2 240 245 250 255 Local time MPC#1 with constant disturbance rejection tsal tref trefart 13.2 13.4 13.6 13.8 14 14.2 0 20 40 60 80 100 Local time Disturbances rad/10 radcor/10 vest*10 tin/10 Sensor error Fig.11.MPC#1withconstantdisturbance rejection. 6.3Radiationdisturbance Figure12 showsthe responseofthe controlled plantwhen thereisastrong disturbance inthe radiation. Beforethe 12.4 12.6 12.8 13 13.2 13.4 220 230 240 250 260 270 MPC#1 with constant disturbance rejection tsal tref trefart 12.4 12.6 12.8 13 13.2 13.4 −50 0 50 100 Disturbances rad/10 radcor/10 vest*10 tin/10 Fig.12.MPC#1withconstantdisturbance rejection. second reference change, itlookslikethe trajectorygoes out ofthe tube, but asitwascommented before, inthe transient thiscanhappen due tothistubeisnotcentered inthe nominaltrajectory, iscentered inthe reference trajectory. 7.CONCLUSIONS Arobustcontroller forprocesses withunknownbut bounded errorshasbeen presented. The proposed control schemeuses arobustidentificationmethodand afinite horizonrobustmodel predictivecontroller fortracking. The robustcontrolalgorithm hasbeen tested under simulationand inadistributed solarcollectorfield, whichisa processcharacterized bylargeperturbations and changes indynamics caused byclouds and operating conditions. 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