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1、SystemIdentificationRikPintelonVrijeUniversiteitBrussels,dept.ELECVersion7June20131/284TableofContents1.PreliminaryExample32.NonparametricmodelsofLTIsystems293.ParametricmodelsofLTIsystems964.ImprovednonparametricmodelsforLTIsystems1385.Estimationparametricplantmodelwithknownnoise
2、model1866.Estimationparametricplantmodelwithestimatednonparametricnoisemodel2147.Estimationparametricplantand/ornoisemodels2418.Guidelines2739.TakeHomeMessages28310.Booksonsystemidentification2842/2841.PreliminaryExample1.aMeasurementofthevalueofaresistor41.bLeastsquaresestimatoro
3、ftheoutputerrorproblem61.cMaximumlikelihoodestimatoroftheerrors-in-variablesproblem71.dPropertiesestimators101.eStochasticconvergenceestimators111.fAsymptoticvarianceestimators121.gAsymptoticnormalityestimators151.hAsymptoticefficiencyestimators171.iRobustnessestimators191.jSummar
4、yproperties211.kExtensiontoarbitrarycurrents231.lReferences283/2841.PreliminaryExample1.aMeasurementofthevalueofaresistorR0=1?Rk()=uk()?ik()5)ANik()Vuk()Measuredresistance(ik()=i0+ni()k0050100uk()=u+n()kk0uuk()ik()n()k?N()0??222iinu()k?N()0??u2Voltage(V)Current(A)00050100050100kk4
5、/2841.PreliminaryExample1.aMeasurementofthevalueofaresistor(cont’d)2NN11uk()R?SA()N==----?Rk()----?---------1NNik()k=1k=1EstimatedvalueR0110100100010000NumberofmeasurementsObservation:theestimateseemsnottoconvergeasN??Question:whatisgoingwrong?5/2841.PreliminaryExample1.bLeastsqua
6、resestimatoroftheoutputerrorproblem2NVLS()Rz?=???uk()–Rik()2k=11N?uk()ik()k=1R?LS()N==argminVLS()Rz?---------------------------------Ni2()kEstimatedvalueRR?k=10110100100010000NumberofmeasurementsObservation:theestimateconvergestoatoosmallvalueasN??Question:whatisgoingwrong?6/2841.
7、PreliminaryExample1.cMaximumlikelihoodestimatoroftheerrors-in-variablesproblemCallthevectorofthemeasurementszzu=??()1???????u()2?uN()i()1i()2?iN()TGaussianiidrandomvariables?probabilitydensityfunctionofthemeasurementszNfz()zRu??pip=?k=1fuk()()uk()Ru??pipfik()()ik()Ru??pipN??uk()–u
8、2??ik()–i211pp=------------------
