chapter 4. optimal linear reconstruction of the state

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1、4OPTIMALLINEARRECONSTRUCTIONOFTHESTATE4.1INTRODUCTIONAIItheversionsoftheregulatorandtrackingproblemssolvedinChapter3havethefollowingbasicassumptionincommon:theco~ilpletestatevectorcar1bemeasitredacc~rrately.Thisassumptionisortenunrealistic.Themostfrequentsituationi

2、sthatforagivensystemx(t)=A(t)x(t)+B(t)i!(f),x(t0)=xu,4-1onlycertainlinearcombinationsofthestate,denotedbyy,canbemeasured:y(t)=C(t)x(t).4-2Thequantity?I,whichisassumedtobeanI-dimensionalvector,withIusuallylessthanthedimensionrrofthestatex,willbereferredtoastheobseru

3、edvariable.Thepurposeofthischapteristopresentmethodsofreconstructingthestatevector,orfindingapproximationstothestatevector,fromtheobservedvariable.Inparticular,wewishtofindafunctionalF,xl(t)=F[~/(T),to<7

4、rrctedstate.Here1,istheinitialtimeoftheobservations.NotethatF[y(r),to27

5、ter3,whichassumeknowledgeofthecompletestatevector,byre-placingtheoctiralstatewiththereconsirisiedstate.InSection4.2weintroducetheobserver,whichisadynamicsystemwhoseoutputapproaches,astimeincreases,thestatethatmuslbereconstructed.Althoughthisapproachdoesnotexplicitl

6、ytakeintoaccountthedifficultiesthatarisebecauseofthepresenceofnoise,itseeksmethodsofrecon-structingthestatethatimplicitlyinvolveacertaindegreeoffilteringofthenoise.3284.2Observers329InSection4.3weintroduceallthestochasticphenomenaassociatedwiththeproblemexplicitlya

7、ndquantitativelyandfindtheoptimalobseruer,alsoreferredtoastheKa/nta11-Blrcj~filter.Thederivationoftheoptimalobserverisbasedupinthefactthattheoptimalobserverproblemis"dual"totheoptimalregulatorproblempresentedinChapter3.Finally,inSection4.4thesteady-stateandasymptot

8、icpropertiesoftheKalman-Bucyfilterarestudied.Theseresultsareeasilyobtainedfromoptimalregulatortheoryusingthedualityoftheoptimalregulatorandobserv