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1、Automatica37(2001)781}786TechnicalCommuniqueAsymptoticvarianceexpressionsforclosed-loopidenti"cationMichelGevers,LennartLjung,PaulVandenHof*CESAME,BaLtimentEuler,LouvainUniversity,B-1348Louvain-la-Neuve,BelgiumDepartmentofElectricalEngineering,Linko(pingUni
2、versity,S-58183Linko(ping,SwedenSignals,SystemsandControlGroup,DepartmentofAppliedPhysics,DelftUniversityofTechnology,Lorentzweg1,2628CJDelft,NetherlandsReceived12May2000;receivedinrevisedform15November2000AbstractAsymptoticvarianceexpressionsareanalyzedformodelsthat
3、areidenti"edonthebasisofclosed-loopdata.Theconsideredmethodscomprisetheclassical&direct'method,aswellasthemorerecentlydevelopedindirectmethods,employingcoprimefactorizedmodels,dualYoula/Kuceraparametrizationsandthetwo-stageapproach.Thevarianceexpressionsarecomparedwit
4、htheopen-loopsituation,andevaluatedintermsoftheirrelevanceforsubsequentmodel-basedcontroldesign.2001ElsevierScienceLtd.Allrightsreserved.Keywords:Systemidenti"cation;Closed-loopidenti"cation;Asymptoticvarianceexpressions,Predictionerrormethods;Model-basedcontroldesig
5、n1.IntroductionHof&Schrama,1995).LaterithasbeenshowninHjal-marsson,Gevers,andDeBruyne(1996)that,foraWhenidentifyingdynamicmodelsforthespeci"cpur-particularclassofcontroldesignmethods,alsofromposeofsubsequentmodel-basedcontroldesign,itisavariancepointofviewclosed-loope
6、xperimentsarearguedthataclosed-loopexperimentalsetupduringthepreferredoveropen-loopones.identi"cationexperimentssupportstheconstructionofInthistechnicalnote,asymptoticvarianceexpressionsanidenti"edmodelthatisparticularlyaccurateinthatwillbepresentedforidenti"edmodelsb
7、asedonseveralfrequencyregionthatisrelevantforthecontroldesign.di!erentclosed-loopidenti"cationmethods,includingThismechanismwhichplaysamajorroleinmanycon-therecentlyintroducedindirectmethodsusingacoprimetributionsintheareaof`identi"cationforcontrola,ori-factormodelrep
8、resentation(Schrama,1992;VandenginallywasmotivatedmainlyonthebasisofbiasHof,Schrama,deCallafon,&Bosgra,1995),themethodconsid
