SystemIdentification

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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()RˆSA()N==----Rk()-------------1NNik()k=1k=1EstimatedvalueR0110100100010000NumberofmeasurementsObservation:theestimateseemsnottoconvergeasNQuestion:whatisgoingwrong?5/2841.PreliminaryExample1.bLeastsqua

6、resestimatoroftheoutputerrorproblem2NVLS()Rz=uk()–Rik()2k=11Nuk()ik()k=1RˆLS()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=------------------