MATLAB外文文献

Automatica42(2006)2169–2174
P.D.Couchman∗,M.Cannon,B.KouvaritakisDepartmentofEngineeringScience,UniversityofOxford,ParksRoad,OxfordOX13PJ,UKReceived21July2005;receivedinrevisedform23June2006;accepted4July2006Availableonline17October2006AbstractStochasticuncertaintyisacommonfeatureofmanycontrolengineeringproblems,andisalsopresentinawiderclassofapplications,e.g.financeandsustainabledevelopment.RecentworkproposedaconstrainedMPCapproachthattookexplicitaccountofthedistributionsofuncertainmodelparametersbutusedterminalequalityconstraintstoensurestability.Thepresentpaperreformulatestheprobleminordertorelaxthestabilityconstraintsbyinvokingappropriateterminalinequalities.Theapplicationoftheproposedstrategyanditsadvantagesoverearlierworkareillustratedbymeansofanumericalexample.2006ElsevierLtd.Allrightsreserved.Keywords:Constrainedcontrol;Stochasticsystems;Modelpredictivecontrol1.IntroductionThereexistsaconsiderablebodyofworkthatconsidersconstrainedmodelpredictivecontrol(MPC)inthepresenceofuncertainty,bothwithadditivedisturbancesanduncertainmultiplicativemodelparameters.Implicitinthisworkistheassumptionthattheuncertaintyisbounded(e.g.polytopic)andinformationontheprobabilisticdistributionofuncertaintyisnotused.Thisappliestoboththedefinitionofpredictedcostandtoconstrainthandling,whetheropen-orclosed-looppredic-tionsareemployed(e.g.Bemporad,Borrelli,&Morari,2003).Inmanyapplications(e.g.sustainabledevelopment),distri-butionscanbequantifiedfortheuncertainty,andsomecon-straintsareprobabilisticinnature.Ignoringthisinformation(bydefiningworst-casecostsandinvokingconstraintsoveralluncertaintyrealizations)canleadtoconservativeresults,andtheneedforastochasticextensiontoconstrainedMPCisclear.Takingexpectedvaluesofthecostprovidesanobviouswaytoutiliseprobabilisticinformation.Forexample,LeeandCooley(1998)performanoptimisationoveropen-loop
ThispaperwasnotpresentedatanyIFACmeeting.ThispaperwasrecommendedforpublicationinrevisedformbyAssociateEditorMartinGuayunderthedirectionofEditorFrankAllgöwer.∗Correspondingauthor.E-mailaddress:[e-mail](P.D.Couchman).0005-1098/$-seefrontmatter2006ElsevierLtd.Allrightsreserved.doi:10.1016/j.automatica.2006.07.006predictionswhileBatina,Stoorvogel,andWeiland(2002)useMonteCarlosimulationtechniquestoderiveaschemeforoptimisingoverclosed-looppredictions,bothwithhardinputconstraints.However,deploymentofexpectedvalueslimitsflexibilityinthecostbyfixingtherelativeimportanceofmeanandvarianceterms.Aprobabilisticformulationofcostcanbeusedtodecoupletheseterms(Kouvaritakis,Cannon,&Couchman,2006).Equally,constraintsoftenadmitaproba-bilisticformulation,e.g.avariableshouldnotexceedacertainboundwithagivenprobability.Thecontrolofwindfarmsprovidesanexample;tomaximisepoweroutputitisdesirabletoexceedtolerancelimits,providedthisisdonewithagivenprobabilitysothatfatiguedamageiswithinacceptablelevels.MPCwithprobabilisticconstraintsisconsideredinvanHessem,Scherer,andBosgra(2001)andYanandBitmead(2005),theformerusingstatemeasurementandthelatterstateestimation.Forboththecostisbasedontheexpectedvalueofalinearfunctionofstate,andtheimplementationofprobabilis-ticconstraintscanbeconservativeduetotheuseofstatisticalconfidenceellipsoidalapproximations.AlsothepresenceofGaussianprocessnoiseinthesystemsconsideredinbothre-sultsinalackofaguaranteeofrecursivefeasibilityoftheprobabilisticconstraints.InLobo,Vandenberghe,Boyd,andLebret(1998)itisshownthatprobabilisticlinearconstraintscanbewrittenassecond-ordercone(SOC)constraints(andareconvex)providedtheprobabilityinvolvedisgreaterthan0.5.2170P.D.Couchmanetal./Automatica42(2006)2169–2174ProbabilisticconstraintsareintroducedinSchwarmandNikolaou(1998)andseparatelyinLi,Wendt,andWozny(2002),butbothconfineanalysistoopenloopoptimisation,andthusstabilityandrecursivefeasibilityissuesarenotconsidered.InKouvaritakisetal.(2006)astochasticMPCstrategyincorporatingaprobabilisticcostandconstraintswasproposedtosolveasustainabledevelopmentpolicyassessmentproblem.Itwasshownthatamovingaverage(MA)modelwithrandomcoefficientswassuitablefortheproblemandthecontrollerwasdesignedaroundthis.Thenatureofthemodelallowedfortheuseofaterminalequalityconstrainttoguaranteerecursivefeasibilityandstabilityofthecontroller.AnefficientonlineoptimisationwasderivedusingSOCconstraints.How-ever,equalitystabilityconstraintscanmaketheactionofthecontrolleroverlyconservative.AlsotheMAmodelassump-tion,thoughsuitableforthesustainabledevelopmentproblemconsidered,canberestrictive.ThecurrentpaperextendsthemethodologyofKouvaritakisetal.(2006)toawiderclassofmodel,namelystatespacemodelswithstochasticuncertaintyintheoutputmap.Inthemostgeneralcase,uncertaintywouldaffectbothstateandinputmaps,however,therestrictiontoout-putmapuncertaintyiswell-suitedtopolicyassessmentprob-lems(Couchman,Cannon,&Kouvaritakis,2006,Couchman,Kouvaritakis,&Cannon,2006;Kouvaritakis,2000;Kouvaritakisetal.,2006)andthisformsthemotivationforthechoiceofmodel.Asecondaimofthispaperistogai