On the design of Robust tube-based MPC for tracking

OnthedesignofRobust tube-basedMPC fortracking⋆ D.Limon∗I.Alvarado∗T.Alamo∗E.F. Camacho∗ ∗Dpto.deIngenier´ıa deSistemasyAutom´atica,Universidad de Sevilla,Avda delosDescubrimeintos s/n41092,Sevilla(limon, alvarado,alamo,[email protected]). Abstract: Thispaper dealswiththe designprocedureofthe recentlypresented robustMPC fortracking ofconstrained linearsystemswithadditivedisturbances.Thiscontroller isbased onnominalpredictions and itiscapabletosteer the nominalpredicted trajectoryto anytarget admissiblesteady state, thatisretaining feasibilityunder anyset pointchange. Bymeans of the notionoftubeoftrajectories,robuststabilityand convergence isachieved. The controller formulationhas someparameterswhichprovides extradegrees of freedomtothe designprocedureofthe predictivecontroller.Theseallowtodealwithcontrolobjectives such asdisturbance rejection, output offset prioritizationorenlargementofthe domainofattraction. Inthispaper,output prioritizationmethod, LMIbased designprocedures and algorithmsfor the calculationofinvariantsetsarepresented. The proposed enhanced designofthe MPCis demonstrated byanillustrativeexample. 1.INTRODUCTION Model predictivecontrol(MPC)isacontroltechnique capabletodealwithhardconstraintsofthe system and the optimizationofaperformance index.Thisisachieved byposing the controlproblem asamathematicalprogramming problem and applying the optimalsolutionina receding horizonmanner.Thiscontroller typicallyrequires aterminalstatepenalizationand constraint, inorder to ensureclosed loopstability.The stabilizing ingredients and the finitepredictionhorizonmakethat the predictive controller mayloosefeasibilityunder set pointchanges. Recently,anovel RobustMPCfortracking hasbeen presented (Alvarado et al. [2007]).The maincharacteristics ofthiscontroller are: (i)adds anartificialsteady state asadecisionvariable(artificialsteady state) (ii)uses an invariantset fortracking asaterminalconstraint (iii) considersacostfunction penalizing the errorw.r.t the artificialsteady stateand anadditionaltermpenalizing the deviation between the artificialsteady stateand the target steady state(iv)itisbased onthe tube-based MPC controller (Mayne et al. [2005]).Thiscontroller,under mildassumptions,cansteer the uncertainsystem inan admissibleevolutionto a neighborhoodofanyadmissible steady state. The robustMPCfortracking has someparametersto betuned, whichprovides extradegrees of freedomtobe exploited according tothe controlobjectives.Thisallows us todealwithdisturbance rejection, enlargementofthe domainofattractionand output tracking prioritization when the set pointisnotadmissible. Inthispaper,the effect ofeachparameter ofthe MPCis analysed and procedures fortheirselectionarepresented. ⋆TheauthorsacknowledgeMCYT-Spainforfundingthiswork (contractsDPI2007-66718-C04-01 and DPI2005-04568) First, itis shownhowtodesignthe proposed MPCto ensurethe evolutionto anoptimized steady statewhen the target isnotadmissible. Then the robustlocalcontroller isdesigned tominimize the effect ofthe disturbances on the stateevolutionand hence toenlargede domainof attractionofthe MPC.Thiscan beposed asaset oflinear matrixinequalities whichsolutionprovides the control gain. Finally,existing resultsonthe computationofan approximationtothe minimalrobustpositivelyinvariant set arespecialized fordisturbances thatcan berepresented asazonotope(anaffine mapping ofaunitarybox). The paper is structured asfollows:In§2the problem to solveisdescribed and in§3the robustMPCfortracking ispresented, in§4the designprocedureisintroduced, in §5the toolthatminimize the invariantset.The paper finishes withanillustrativeexampleinsection5 and some conclusions. Notation:AdefinitepositivematrixTisdenoted as T>0 and T>Pdenotes thatT−P>0.For a given symmetricmatrixP>0,kxkPdenotes the weighted Euclideannormofx, i.e. kxkP=√x⊤Px. (a,b)∆ =[a⊤,b⊤]⊤.Consider a∈IRna,b∈IRnb,and set Γ⊂IRna+nb,then projectionoperationisdefined as Proja(Γ)={a∈IRna:∃b∈IRnb,(a,b)∈Γ}.Given twosetsUand V,suchthatU⊂IRnand V⊂IRn,the Minkowskisum isdefined byU⊕V∆ ={u+v|u∈U,v∈V}, the Pontryaginset difference is:U⊖V∆ ={u|u⊕V⊆U}. Fora given matrixM∈IRn×mand aset V⊂IRm,the set MV⊂IRndenotes the set {y=Mv,v∈V}.Fora given λ,λX={λx:x∈X}.Let tbea genericvectordefined as t∆ ={t(0),t(1),...}.Aset isaCset ifiscompact,convex notempty.Amatrix0n,m∈IRn×mdenotes amatrixof zeros.The set ofvertexes ofa given Cset Γisdenoted 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 15333 10.3182/20080706-5-KR-1001.3054 asvert(Γ).The set BN⊂IRNdenotes the unitaryball BN={b∈IRN:kbk∞≤1}. 2.PROBLEMDESCRIPTION Consider the following uncertaindiscrete-timeLTIsystem: x+=Ax+Bu+w,y=Cx+Du(1) subject tofollowing constraints: x∈X⊂IRnu∈U⊂IRm(2) and disturbances set: w∈W⊂IRn(3) where: xisthe currentstate, uisthe currentcontrol action, x+isthe successorstate, wisanunknownstate disturbance, y∈IRpisthe currentmeasured output,and (A,B,C,D)∈IRn×n×IRn×m×IRp×n×IRp×m.W, isa compact,convex notemptyset and X,Uarepolyhedral and polytopicsetsrespectively. Let φ(i;x,u,w)denotethe solutionof(1)at timeiifthe initialstateisxand the controland disturbance sequences are, respectively,uand w. The overall objectiveistostabilize the constrained system and steer the stateto a neighborhoodofthe setpoint fulfilling the constraintsforanypossibledisturbance. The following standing assumptionismade: Assumption1.The couple(A,B)iscontrollable. 3.ROBUST MPCFOR TRACKING Forthe sakeofaclearand self-contained expositionofthe contributions ofthe paper,the robustMPCfortracking is succinctlypresented inthis section. Amoredetailed explanationofthe controller can befound in(Alvarado et al. [2007], Alvarado. [2007]) 3.1Tube oftrajectories The proposed controller isbased onthe responseofthe nominalsystem,obtained from(1)byneglecting the disturbances w.The nominalsystem isdescribed by: ¯x+=A¯x+B¯u,¯y=C¯x+D¯u(4) Choosing aninitialstate¯xand acontrolsequence ¯ uyields astatesequence ¯ xobtained bysolving (4) (¯xi=¯ φ(i;¯x,¯ u)). Tocounteract the disturbances itisdesirabletoforce the trajectorytolieclosetothe nominaltrajectory;thiscan bedone bychoosing the controlutosatisfy: u=¯u+Ke e ∆ =(x−¯x) (5) whereedenotes the errorbetween the stateand the state ofthe nominalsystem.The erroredynamics isgiven by e+=AKe+w;AK=(A+BK) (6) IfmatrixAKisHurwitzthen thereexistsarobustpositivelyinvariantset Z(Kolmanovsky and Gilbert[1998], Rakovicet al. [2005])forthe system (6) thatsatisfies AKZ⊕W⊆Z The invariance propertyallowstostatethe notionoftube oftrajectories given inthe following proposition: Proposition1.(Mayne et al. [2005]).Ifthe initialrealand nominalsystem states,satisfye(0)=x(0)−¯x(0)∈Z,then x(i)∈¯x(i)⊕Z∀i∈IN,forall disturbance sequences w suchthatw(i)∈W∀i=1,2,.... Fromthispropositionitisinferred thatifthe nominal controlactions arecalculated toensurethat the nominal predicted states and inputs satisfythe following tighter constraints ¯ X=X⊖Z¯ U=U⊖KZ(7) then the system controlled by(5)maintains all the possible trajectories admissible, thatis satisfying the constraintson states and inputsforall possibleuncertainty(Mayne et al. [2005]). 3.2Setpointcharacterizationandinvariantsetfortracking Inthis section, asuitableparametrizationofthe steady states and inputsforthe nominalsystem ispresented. Consider a given output target yt,then anysteady state ofthe nominalsystem zs=(xs,us)associated tothis setpointmustsatisfythe following equation A−InB0n,1 CD−Ip"xs us yt#=0n,1 0p,1(8) Becauseofthe pair(A,B)is stabilizable, the solutionto thisproblem can beparameterized as zs=Mθθyt=Nθθ(9) whereθ∈IRnθisaparameter vectorwhichcharacterizes anysolution, and Mθand Nθaresuitablematrices (Limon et al. [2008]). The existence ofconstraints(7)limitsthe set ofreachable steady states and inputs.The set ofadmissiblesteady states isdenoted asXsand itisapolyhedrongiven by Xs={xs∈¯ X:∃us∈¯ U|(A−In)xs+Bus=0n,1}. Now,the notionofinvariantset fortracking ispresented Definition1.Let xebethe extended state(x,θ)∈ IRn+nθ, let KΩbeacontrolgainsuchthatA+BKΩ isHurwitzand let Kθbegiven byKθ=[−KΩIm]Mθ. Then, aset Ωe t⊂IRn+nθisaninvariantset fortracking admissiblefor(7), if forall (x,θ)∈Ωe t,then x∈¯ X, KΩx+Kθθ∈¯ Uand ((A+BKΩ)x+BKθθ,θ)∈Ωe t. The roleofthe presented parametrizationistosimplifythe proposed predictivecontroller aswellasthe calculationof invariantsetsfortracking. 3.3Predictive controller The mainobjectiveofthiscontroller istorobustlysteer the output system to a (neighborhoodof)atarget yt. Itisassumed that thistarget can be characterized bya parameter vectorθbymeans of(9) (i.e. itisapossible steady output forthe nominalsystem). Thiscontroller,asthe tube-based controller (Mayne et al. [2005]),considersasdecisionvariables the initialnominal state¯xand the sequence of futurenominalcontrolactions ¯ u.Inorder todealwiththe set pointchangeproblem an 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 15334 artificialsteady stateand input (¯xs,¯us)=Mθ¯ θisconsidered asdecisionvariable. Thesevariables arecalculated tominimize the following nominalpredictionbased cost function: VN(x,θ;¯ u,¯x,¯ θ)= N−1 X i=0k¯x(i)−¯xsk2 Q+k¯u(i)−¯usk2 R +k¯x(N)−¯xsk2 P+k¯ θ−θk2 T(10) where¯x(i)=¯ φ(i;¯x,¯ u)and (¯xs,¯us)=Mθ¯ θ.The current statexand the target operation pointgiven byθare parametersofthe costfunction. Thiscostpenalizes the deviation between the predicted trajectoryand the artificialsteady statealong the horizon N.The deviation between the artificialsteady conditions (given by¯ θ)and the target one (given byθ)ispenalized bymeans ofthe so-called offset costk¯ θ−θk2 T. The costfunctionmustbeminimized considering the constraintsderived fromstabilityand admissibilityconditions, leading tothe following quadraticproblem PN(x,θ): min ¯x, ¯ u,θ VN(x,θ;¯ u,¯x,¯ θ) (11) s.t.¯x∈x⊕(−Z) (12) ¯x(i)∈¯ X,i=0,···,N(13) ¯u(i)∈¯ Ui=0,···,N−1(14) (¯x(N),¯ θ)∈Ωe t(15) where: ¯ X=X⊖Zand ¯ U=U⊖KZ. PN(x,θ)is solved online toyieldanoptimal initialstate ¯x∗(x,θ),anoptimalnominalcontrolsequence ¯ u∗(x,θ)= {¯u∗(0,x,θ),¯u∗(1,x,θ),...,¯u∗(N−1,x,θ)}and the optimal artificialsteady conditions parameter θ∗(x,θ).Fromthis, the controlapplied tothe plantisgiven by: kN(x,θ)=¯u∗(0,x,θ)+K(x−¯x∗(x,θ)) (16) Notice that the set ofconstraintsthatdefines the feasibilityregionofPN(x,θ)does notdepend onθ(thatis, the target operating point),but onlyonthe currentstate xand the decisionvariables.Therefore, thisfeasibility regiononlydepends onthe currentstatex.Thus,defining ¯ XN⊂IRnasthe set ofthe admissiblenominal initial states ¯xsuchthat (13),(14)and (15)hold, the set of feasiblestates isgiven byXN=¯ XN⊕Z. 3.4Stabilizing conditionsofthe controller Inorder toensurerobuststabilityofthe closed loop system,the following sufficientconditions onthe defining ingredientsareassumed: Assumption2.The matrices Q,R,T,P,K,KΩ,and sets Ωe tand Zfulfil: •Q>0 and R>0 •Thereexistsaconstantσ>0suchthatσT≥MT xMx, whereMx=[In,0n]Mθ. •The gainmatrixKis suchthatA+BKisHurwitz. •The pairofmatrices KΩand ParesuchthatA+BKΩ isHurwitzand P>0where P−(A+BKΩ)⊤P(A+BKΩ)=Q+K⊤ ΩRKΩ •Set Z⊂Xisanadmissiblerobustpositivelyinvariant set suchthat (A+BK)Z⊕W⊆Zand KZ⊂U. •Ωe tisaninvariantset fortracking (aslargeaspossible) forthe nominalsystem (4)subject tothe constraints ¯ Xand ¯ Uand using ascontrolgainmatrixKΩ. Nowitispossibletoestablishthe following result: Theorem1.(Alvarado et al. [2007]).Consider system (1) subject tothe constraints(2)and suchthatverifies Assumptions 1 and 2.Let kN(x,θ)bethe control lawresulting fromthe solutionofthe optimalproblem PN(x,θ).Let Ytbethe set ofreachabletarget outputsgiven by Yt ∆ ={y=Nθθ:Mθθ∈Projx(Ωe t)ׯ U} Then ∀x(0)∈XNand atarget yt∈Ytthe closed loop system fulfilsthe constraints(2)along itsevolutionand converges asymptoticallytoxs⊕Z. The proofofthistheorem can befound in(Alvarado. [2007]) 3.5Cancellationoftheoutputoffset The proposed controller isablesteer the system stateto aneighborhoodofthe target steady state, thatisxs⊕Z. Ifthe disturbances aredecaying,the system converges asymptoticallytothe target,but ifthe disturbance tends to a steady value, the the closed loopsystem presentsoffset onthe outputs, i.e. the errory(k)−yttends to a constant value. Thiscan becancellated bythe following methods.One possibletechnique isaugmenting the model ofthe system withintegrating disturbances toremoveoffset (Pannocchia [2004]);alinearcontrol lawensures robustoffset-free controlatexpenseofenlarging three times the order ofthe plantwhichmakes the controller designand calculation moredifficult. Asecond methodproposed bythe authorsin(Alvarado et al. [2007]),achieves the offset free without augmenting the plant,but adding anouter loopconsisting ofadisturbance estimatorand aset pointcorrectiongiven by ˆyt(k)=yt−(C+DK)(In+(A+BK))−1ˆw(k) whereˆw(k)isthe estimated disturbance. The estimator dynamics should beadjusted according the dynamics of the disturbance signaland the closed loopsystem. 4.SYNTHESISOF THE PROPOSEDCONTROLLER The presented predictivecontroller isdesigned bypicking the defining parameters(horizon, matrices and sets)fulfilling Assumption2.Since thisdoes notfix the parameters, thesecan be chosen according tocontrolobjectives,such asclosed loop performance, disturbance rejection, domain ofattraction, etc. Matrices Qand Rdefine the stagecostwhichisclosed loop performance index.ControlhorizonNistypicallychosen aslargeaspossible, since alargeNprovides alarger domainofattractionand anenhancementofthe closed loop performance. However,the dimensionofthe problem growswithN,and hence requires alarger computational time. Therefore, atradeoffmustbeachieved. 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 15335 Inthis section, the chosen ofthe extraparameters specific ofthe proposed predictivecontroller:the matrices T,K, KΩ,Pand the setsZand Ωe t. 4.1TheparameterT Thismatrixdefines the weightofthe tracking errorcost k¯ θ−θk2 Tinthe costfunction. The effectsofthisparameter onthe closed loopsystem arethe following : •Transientdynamicsandsetpointfiltering: ifmatrix Tischosen topenalize moreheavilythe tracking error cost,then the convergence of¯ θtoθismade faster, and hence the transientofthe closed loopsystem to the target. •Localoptimality: itiswellknownthatMPCcan be designed tobelocallyinfinitehorizonoptimal(forthe nominalsystem),however the additionofthe artificial steady stateand input makes that thispropertydoes notholdinthe presented controller.The optimality loss can bearbitrarilyreduce bypicking alarge enoughmatrixT. •Offsetminimization:thisisaremarkableproperty ofthe controller (Alvarado. [2007]).Consider that the target ytisnotreachablethatis,the desired output ytdoes notfulfil anyofthe constraints, i.e. Mθθ6∈ Ωtׯ U,∀θ:yt=Nθ.Under theseconditions, the target cannotbereached and then, the output will presentoffset. Inthiscasethe proposed controller drives the system tothe neighborhood˜xs⊕Z,wherethe steady state˜xs (given bythe parameter ˜ θ)isthe one whichminimizes the offset cost.Thatis ˜ θ=argmin ¯ θk¯ θ−θk2 T Mθ¯ θ∈Ωtׯ U Therefore, matrixTallowsus toprioritize some outputs(byweighting moreheavilyitscorresponding terminmatrixT) to achieveaminimum offset on thisoutputs.See that thisprioritizationdoes not varywiththe scaling ofmatrixT,and hence the effectspreviouslypresented can besimultaneously considered inthe designofT. Then the sensiblewaytodesignTisfirstlypickthe structureaccording tothe offset minimization, and then, scalethe matrixto achieveaquicktransientwithasmall enoughoptimalityloss.Finallynotice thatsince the value ofTisindependentofthe restofparameters,thiscan be tuned online maintaining the stabilizing properties. 4.2TheparametersKΩ,PandΩe t The terminalcostand terminalset stronglydepend on the controlgainKΩ.Thiscan bedesigned according the following aspects:fromaperformance pointofview, itis desirablethat the terminalcostistaken asthe cost-to-go, whichcan beensured ifKΩischosen asthe LQregulator gain. Fromadomainofattractionviewpoint, itisdesirable totakeKΩto achievealargeterminalset Ωe t. Inthiscase, the largesize ofthe invariantset fortracking provides alargedomainofattractionXN,even forsmall values ofthe controlhorizon. Then, the mostsensibleway topickKΩisthe LinearQuadraticRegulatorand the matrixPasthe solutionofthe Riccatiequation. The invariantset fortracking Ωe tcan betaken asapolyhedralset whichapproximates arbitrarilytothe maximal invariantset fortracking thanksto a simpleprocedure (Limonet al. [2008]. Thisprovides aset ofreachable targetsYtpracticallyequaltothe maximalone (i.e. the difference between bothsetscan bemade arbitrarilysmall). 4.3TheparameterK Thisparameter hasanimportantroleinthe proposed controller since thiscontrolgainisused tocompensatethe deviationfromthe nominalpredictions inthe controller (16),and therefore, itcharacterizes the dynamics ofthe closed loopsystem inthe presence ofdisturbances.This makes thatKischosen according to a robustnessor disturbance rejectioncriterium. Inthispaper,weconsider asrobustnesscriterium the minimizationofthe minimum admissiblerobustpositively invariantset.Thus,Kischosen to:(i)ensurethe existence ofanadmissiblerobustpositivelyinvariantset Zsuch that the setsX⊖Zand U⊖KZarenotemptysets;(ii) minimize the size ofZ.Thiscriterium provides alarger domainofattractionand alarger set ofreachabletargets aswellasminimizes the effect ofthe disturbance onthe trajectoryofthe system. Consider w.l.o.g.thatset X={x:|h⊤ ix|≤1,i= 1,···,nrx}and set U={u:|ℓ⊤ ju|≤1,j=1,···,nru}. Then, the synthesisproblem tosolveistocalculatethe control lawu=Kxsuchthat the size ofthe ellipsoid E(P,1)={x∈IRn:x⊤Px≤1}isminimized fulfilling that: (i)E(P,1)isarobustinvariantset forsystem (6).This conditioncan bereformulated asfollows: (x+)⊤Px+≤1,∀x∈E(P,1),∀w∈W(17) Applying the S-procedure, and considering the convexitywithrelationtow,equation(17)is satisfied if existsλ≥0suchthat: ((AK)x+w)⊤P((AK)x+w)+λ(1−x⊤Px)<1 AK=A+BK∀x∈IRn,∀w∈vert(W) (18) wherevert(W)denotes the set ofvertexes ofW.This can berewritten as: λP−(A+BK)⊤P(A+BK)−(A+BK)⊤Pw −w⊤P(A+BK)1−λ−w⊤Pw>0 ∀w∈vert(W) (ii)Forall x∈E(P,1),the control law|ℓ⊤ jKx|≤ρjfor all j=1,···,nruand ρj∈(0,1]. The roleofthe parameter ρjistorestrict the set ofadmissiblecontrol inputsto guarantee a given controlrangeofthe the MPCcontroller i.e. the set ¯ U=U⊖KZhasnot emptyinterior. Thisconditioncan beposed as: ℓ⊤ jKP−1K⊤ℓj≤ρ2 j,j=1, . . . , nru Applying the Schur complement,thisyields to ρ2 jℓ⊤ jK K⊤ℓjP>0,j=1, . . . , nru 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 15336 (iii)Forall x∈E(P,1),|h⊤ ix|≤1.Considering similar argumentstothe previous fact,thisconditionis equivalent to 1h⊤ i hiP>0,i=1,...,nrx Inorder tominimize the size ofthe ellipsoidE(P,1), asuitablemeasureofthis set mustbe chosen. Inthis paper,weproposeasmeasureaparameter γ>0such thatE(P,1)⊆√γX.Therefore, minimize the size of E(P,1)isposed asminimizing the parameter γ.Obviously, admissibilityofthe solutionrequires thatγ≤1. Applying standardoperations ofLMIs(Boydet al. [1994]), the proposed synthesisprocedurecan beformulated asthe solutionofthe following convex optimizationproblem: min Y,W,γγ s.a. "λW∗ ∗ 0 1 −λ∗ AW+BYwW#>0,∀w∈vert(W) ρ2 i∗ Y⊤ℓiW>0,i=1,...,nru γ∗ WhiW>0,i=1,...,nrx fora given λ≥0.If feasible, the ellipsoidisgiven by P=W−1and the controlgainisK=YW−1.Itisworth remarking thatanyrobustcriterium thatcan beposed as LMIscan beadded tothis synthesisproblem. 4.4CalculationoftherobustinvariantsetZ Once the controlgainKisdesigned, anadmissiblerobust positivelyinvariant (RPI)set (as small aspossible) must becalculated. See that the proposed synthesisofKensures the existence ofthis set.Itwould bedesirabletocompute the minimum robustpositivelyinvariantset (mPRI)F∞ (Kolmanovsky and Gilbert[1998], given by Fs= s M k=0 (A+BK)kW when stends toinfinity.Unfortunately,F∞canonlybe calculated forsomespecialcases,suchas,when adeadbeatcontrol lawisused. Inthe recentpaper (Rakovicet al. [2005], aprocedure forthe determinationofaninvariantapproachofF∞is presented. Thisallowsone tocomputeaRPIZsuchthat F∞⊆Z⊆F∞⊕ǫBnfora given bound ofthe absolute errorǫ.Tothisaim,the following functions arecalculated α(s)=minα:(A+BK)sW⊆αW β(s)=minβ:Fs⊆βBn Thesevalues arecalculated bysolving anumber ofLinear Programming problems.Ifalargeenoughvalue ofsis taken suchthat (1−α(s))−1α(s)β(s)≤ǫthen the set Z=(1−α(s))−1FsisanapproximationofF∞withan errorbound lessthanǫ. Fromapracticalpointofview, itmightbedesirableto calculatethe approximationforarelativeerrorbound, whichdoes notrequireanaprioriestimationofthe size of the invariantset tochoosethe errorbound. Thisis stated inthe following lemma. Lemma 1.Let sbeapositiverealnumber suchthat α(s)≤λ 1+λ fora given relativeerrorbound λ∈(0,1).Then the set Z=(1−α(s))−1FsisaRPIsuchthat F∞⊆Z⊆(1+λ)F∞ On the other hand, wespecialize thisresultforthe case that the uncertaintyset isazonotopeofthe form W=HBn⊕w0 whereH∈IRn×nisnonsingular.Thisismotivated becausethisclass ofset isfrequentlyused tobound additive uncertainties inpractice. Forthisclass ofuncertainties we canstatethe following lemma. Lemma 2.Consider aset W=HBn⊕w0whereHisa nonsingularmatrix.Denote(A+BK)asAKand define the matrixHz(s)=[As−1 KH,As−2 KH,···,H]. Then kH−1As KHk∞=minα:As KHBn⊆αHBn kHz(s)k∞=minβ: s−1 M k=0 Ak KHBn⊆βBn Based onthislemma,anapproximationtothe mRPI witharelativeerrorboud λforthe caseofzonotopic uncertainties ispresented inthe following lemma. Lemma 3.Let sbesuchthat kH−1(A+BK)sHk∞≤λ 1+λ fora given relativeerrorbound λ∈(0,1)and denote ˆα(s)=kH−1(A+BK)sHk∞.Then the zonotopeZgiven by Z=(1−ˆα(s))−1Hz(s)Bsn⊕(In−(A+BK))−1w0 isaRPIsuchthatF∞⊆φK⊆(1+λ)F∞. The proofsoftheselemmascan befound in([Alvarado., 2007,Chapter 6]. The zonotopeexpressionofthe invariantset makes easier the calculationofthe linearmapping and Pontryagindifference, and consequentlythe calculationofthe polytopes X⊖Zand U⊖KZ.However,constraint (12)requires the calculationofthe set ofinequalities (hyperplanes) that define Z.Azonotopeisacompact expressionofapolytope withanumber of facetsthatgrowsexponentiallywith the dimensionofthe unitarybox(sn).Althoughthere exists specialized algorithmsto obtainthe facets,theseare onlytractableforareduced dimensionofthe zonotope. In([Alvarado., 2007,Chapter 6])can befound several practicalprocedures forthiscalculation. 5.ILLUSTRATIVE EXAMPLE Consider aconstrained sampled doubleintegrator 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 15337 x+=1 1 0 1 x+0 0.5 1 0.5u+w y=[0 1 ]x. wherethe disturbances arebounded inw∈W=0.1B2. The system mustfulfil the following constraints:|x1|≤5, |x2|≤5,|u1|≤0.3,|u2|≤0.3. The objectiveistoshowthe proposed procedures to find asuitablerobustcontroller gainKtoensurethe admissibilityofthe tube-based predictivecontroller, i.e. ensuring thatX⊖Zand U⊖KZarenotemptysets. Totest the proposed synthesismethod, wedesigntwo matrices,KLQRand Kop.The firstmatrixisobtained as the LQR gainforQ=I2and R=10I2.The second matrix isdesigned using the proposed methodwithavalue of ρ=0.48 inorder to get the sameset KZasinthe previous case. Figure1showsthe approximated RPIforbothgains. Itcan beseen that the proposed methodprovides asmaller set Z. −0.8 −0.6 −0.4 −0.2 00.2 0.4 0.6 0.8 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 x1 x2 Comparation of the minimal robust invariant sets −0.3 −0.2 −0.1 00.1 0.2 0.3 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 u1 u2 Control constraints KZLQR KZop U _ _ ULQR Uop Zop ZLQR Fig.1.Comparativeofthe admissibleminimalrobust invariantsets Inorder todemonstratethe roleofthe parameter ρ, let us trytocalculatethe minimum robustinvariantset with the minimum size, forthat,pickaρ=1 and consider a LQR gainwithQ=1000 ∗I2and R=1.Figure2shows the RPIderived fromLQR and the proposed method. It can beseen howthe set KZLQRisnotfeasibledue to violates the controlconstraints(¯ ULQR=U⊖KZLQRis anemptyset).The proposed methodprovides afeasible RPIbut,since ρ=1,the set ¯ Uop=U⊖KZopisquite small, sosmaller isthe parameter ρbigger isthe robust minimal invariantset (worstdisturbance rejection) but bigger is¯ Uop(The nominalsystem hasalessrestrictive controlconstraints,sothe evolutionofthe nominalsystem isfaster).Thus,withthe value ofρitispossibleto choosebetween disturbance rejectionand performance, as bigger isρbetter disturbance rejectionbut worseisthe performance. 6.CONCLUSIONS Thispaper dealswiththe designstep ofthe robust predictivecontroller fortracking (Alvarado et al. [2007]. Thiscontroller can bethoughtasanaturalextensionof predictivecontrollersto achieveset point tracking and has −0.2 −0.15 −0.1 −0.05 00.05 0.1 0.15 0.2 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 x1 x2 Comparation of the minimal robust invariant sets −0.3 −0.2 −0.1 00.1 0.2 0.3 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 u1 u2 Control constraints ZLQR KZop Zop U Uop _ KZLQR Fig.2.Comparativeofthe minimalrobustinvariantsets someextraparametersthatcan betuned to achievethe following objectives: •Offset minimization •Disturbance rejection byminimizing the RPI •Guarantee of feasibilityofthe tube-based MPCand enlargementofthe domainofattraction. The firstone depends onthe tracking errorcostweighting matrixT,whichcan befreelychosen (under amild assumption) topenalize moreheavilysomeoutputsin order tominimize theiroffset.The last two objectives are achieved bymeans aLMIwhichcan beefficientlysolved. 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