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MassachusettsInstituteofTechnology6.435SystemIdentificationProf.M.A.DahlehOut2/23/1994ProblemSetNo.2Due3/07/1994Reading:Chapter4.Problem1:DothefollowingproblemsfromLjung'sbook:a.6G.3b.6G.4c.6E.1d.6E.2e.6E.3Also,computetheminimalvariance.Whatcanyousayaboutestimates(6.69)and(6.70),p.158,ofthetextbook?f.6T.2Applytheresultofthisexercisetolemma6.1.Howrestrictiveisthetechnicalassumptiononthecorrelationfunction?Problem2:Considerthefollowingsystem:y=Gu+v,whereGisanunknownplanttobeidentifiedfromthedatarecordu,y.Apracticalsituationthatoftenarisesiswhenboththeinputprocessuandtheoutputnoiseprocessvarecorruptedbyaperiodicsignal,forinstancea60Hzhum.Aftertaking6.435,youhavedecidedtouseacorrelationmethodtoestimatetheplantG.Analysetheperformanceofyourestimateandcommentonyourdecision.Inordertosimplifytheanalysis,youcanassumethatbothuandvareperiodicsequenceswithexactlyonecommonfrequency.Problem3:TheobjectiveofthisproblemistoenhanceyourunderstandingofthepropertiesoftheETFEestimateofthetransferfunctionandtheroleofsmoothinginimprovingtheETFEestimate.Giventhefollowingprocess:q-11+.5q-1'y=Gu+HeG=H=-Y=Gu1HeG=-.6q-11-.2q-'Lettheinputbegivenbytheperiodicsignalu=Ecos(k)k=landletebewhitegaussiannoisewithmean0andvariance1.a.GenerateMdatarecordsoflengthsN1N2...NMforthissystem.Thenumberandlengthsoftheserecordsarelefttoyourjudgement,stamina,andpatience.b.Foreachrecord,useETFEtoobtainanestimateofG.Computeanestimateofthevarianceatonefrequencyofyourchoice,andanestimateofthecorrelationofanytwofrequencies.ShowaplotofthesequantitiesasafunctionofNk,1kM.DotheresultsconfirmwiththetheoreticalpropertiesofETFE?c.UseSPAwithawindowoflength20toobtainaspectralestimateofG.Foronefrequencyofyourchoice,repeatthequestionsinpartb.Commentontheobserveddifferencesbetweentheplotsinbandc.AlsoCommentonthetradeoffbetweenthewindowlengthandthelengthofthedatarecord.Problem4:Theobjectiveofthisproblemistostudynonparametricidentificationinthepresenceofoutputfeedback.Considertheclosedloopsystemshownbelow.ThesignalristhereferenceinputassumedtobeaWGNofmean0andvariance1.Thesignalvistheprocessnoise,withapowerspectrumgivenbyP,,=IH(eiW)12whereHisgivenasinproblem2.Thesignalsrandvareuncorrelated.TheplantGandthecontrollerKarefixedtobe:q-1G-K=.51-.5q-1Assumethatyoucanmeasurebothuandyasshowninthediagram.a.GenerateadatarecordoflengthN=1000fortheabovesystem.b.Usethecorrelationmethoddirectlyony,utoestimatetheplantG.Hint:computa-tionally,use(6.51).Estimatethenoisespectrum.c.Explainyourresultstheoretically.Hint:derivetheequationssatsifiedbythedifferentcorrelationfunctions.Youarenotaskedtoperformanyasymptoticanalysis.d.Comeupwithaschemethatgetsridofthebiasintheaboveresults(Hint:Userdirectly)ILAYC~-IF~~~I�