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1、MathematicalProgrammingStudy5(1976)169-190.North-HollandPublishingCompanyONCONSISTENCYANDIDENTIFIABILITY*LennartLJUNGLundInstituteofTechnology,Lurid,Sweden**Received16May1975Revisedmanuscriptreceived31July1975Theconvergencewithprobabilityone,oftheparameterestimatesobtainedfrompredict
2、ionerroridentificationmethods,suchasthemaximumlikelihoodmethod,isanalysedinthispaper.Itisshownthatunderquiteweakassumptionsontheactualsystemthathasgeneratedthedata,theexpectedvalueoftheidentificationcriterioncanbeusedfortheasymptoticanalysisoftheestimates.Inparticular,thetruesystemdo
3、esnothavetobelongtothesetofmodelsoverwhichthesearchforoptimumismade.Theimplicationsofthisresultforconsistencyanalysisandforquestionsofidentifiability,aswellasforotherrelatedproblemsarediscussed.1.IntroductionTheidentificationproblemistodetermineadynamicmodelthat(accord-ingtosomecrite
4、rion)describedtheinput-outputdatameasuredfromsomeprocessaswellaspossible.Onceacertainmethodtosolvethisproblemhasbeendeviseditisnaturaltotestitsperformanceinvariousways.Thetestscanbenumerical,likewhenthemethodisappliedtodatasimulatedonacomputer.Aparticularlycommonanalyticaltestistostu
5、dytheasymptoticbehaviourofthemethod(andoftheestimatesthatitproduces)asthenumberofmeasureddatatendstoinfinity.Sincethedataoftenareconsideredtoberandomprocesses,suchanalysishastobeperformedusingprobabilisticmethods.Theconceptsofconsistencyandidentifiabilityarecloselyrelatedtosuchanalys
6、isandtothelimitsoftheestimates(iftheyexist),asisfurtherexplainedinSection3.*ThisworkwassupportedinpartbytheAirForceOfficeofScientificResearch,AFSystemsCommand,undercontractAF44-620-69-C-0101,theJointServicesElectronicsProgramundercontractN-00014-67-A-0112-0044,andtheIndustrialAffilia
7、tesProgramattheInfor-mationSystemsLaboratory,StanfordUniversity.**Atpresent:StanfordUniversity,Stanford,Calif.U.S.A.169I70L.Ljung/ConsistencyandidentifiabilityTheclassofidentificationmethodstobestudiedherearedefinedasproceduresthatminimizethepredictionerrorofthemodel,whenappliedtothe
8、recordeddata
