application to systems and control

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1、Chapter6APPLICATIONTOSYSTEMSANDCONTROLAssumeacontrolsystemdependsonaparameterandthesystemoperationreachesitsidealstatuswhentheparameterequalssomeSinceisunknown,wehavetoestimateitduringtheoperationofthesystem,which,therefore,canworkonlyontheestimateofInotherwords

2、,therealsystemisnotundertheidealparameterandtheproblemistoon-lineestimateandtomakethesystemasymptot-icallyoperatingintheidealstatus.ItisclearthatthiskindofsystemparameteridentificationcanbedealtwithbySAmethods.Adaptivecontrolforlinearstochasticsystemsisatypicale

3、xampleforthesituationdescribedabove.Ifthesystemcoefficientsareknown,thentheoptimalstochasticcontrolmaybeafeedbackcontrolofthesystemstate.Thecorrespondingfeedbackgaincanbeviewedastheidealparameterwhichdependsonthesystemcoefficients.Inthesetupofadaptivecontrol,sys

4、temcoefficientsareunknown,andhenceisunknown.Theproblemistoestimateandtoprovethattheresultingadaptivecontrolsystembyusingtheestimateasthefeedbackgainisasymptoticallyoptimalastendstoinfinity.InSection6.1theidealparameterisidentifiedbySAmethodsforsystemsinagenerals

5、etting,andtheresultsareappliedtosolvingtheadaptivequadraticcontrolproblem.TheadaptivestabilizationproblemissolvedforstochasticsystemsinSection6.2,whiletheadaptiveexactpoleassignmentisdiscussedinSection6.3.Anadaptiveregulationproblemfornonlinearandnonparametricsy

6、stemsisconsideredisSection6.4.289290STOCHASTICAPPROXIMATIONANDITSAPPLICATIONS6.1.ApplicationtoIdentificationandAdaptiveControlConsiderthefollowinglinearstochasticsystemdependingonparam-eterwhereandareunknown.TheidealparameterforSystem(6.1.1)isarootofanunknownfun

7、ctionThesystemactuallyoperateswithequaltosomeestimatefor,i.e.,therealsystemisasfollows:Forthenotationalsimplicity,wesuppressthedependenceonthestateandrewrite(6.1.3)asTheobservationattimeiswhereisanoiseprocess.From(6.1.5)itisseenthatthefunctionisnotdirectlyobserv

8、ed,butitisconnectedwithasfollows:whereisgeneratedby(6.1.1).Letbeasequenceofpositivenumbersincreasinglydivergingtoinfinityandletbeafixedpoint.Fixinganinitialvaluewerec