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1、Vol.33,No.10ACTAAUTOMATICASINICAOctober,2007Discrete-timeDirectModelReferenceAdaptiveControl:ASystematicApproach11;2LIJun-LingXIEXue-JunAbstractForaclassofdiscrete-timesystems,thedesignandanalysisofdirectmodelreferenceadaptivecontrol(MRAC)withnormalizedadaptivelawareinvestigated.Wereprovet
2、hediscrete-timeconclusionsontheLpandL2±relationshippropertiesbetweentheinputandtheoutput,andthediscrete-timeswappingLemmas1and2.Wealsoestablishthepropertiesofdiscrete-timeadaptivelaw,deˉnethenormalizingsignal,andrelatethesignalwithallthesignalsintheclosed-loopsystem.Thus,thestabilityandcon
3、vergencepropertiesofthediscrete-timeMRACschemeareanalyzedrigorouslyinasystematicfashionasinthecontinuous-timecase.KeywordsDiscrete-timesystems,swappinglemma,L2±-norm,thenormalizingsignal,modelreferenceadaptivecontrol1Introductionpropertiesofthediscrete-timeMRACschemeareanalyzedrigorouslyin
4、asystematicfashionasinthecontinuous-timeDuringthepasttwodecades,forlinearcontinuous-timecase.systems,thecertaintyequivalence"adaptivecontrollerswithnormalizedadaptivelawshavedominatedtheliter-2Problemstatementatureofadaptivecontrolduetothesimplicityofthede-signaswellastherobustnesspropert
5、iesinthepresenceofLetusconsiderthediscrete-timelineartime-invariant[1?5]plantmodelingerrors.Animportantfeatureofthisclassofadaptivecontrollersistheuseoferrornormalization,whichRp(z)y(t)=kpZp(z)u(t);t=0;1;2;￠￠￠(1)allowsthecompleteseparationoftheadaptiveandcontrollawsdesign.Byusingthepropert
6、iesofL2±-norm,swappingwhereu(t)andy(t)2Raretheinputandoutput,respec-PPlemmas,andBellman-GronwallLemma,amoreelaboratetively,Rnn?1imm?1jp(z)=z+aiz,Zp(z)=z+bjzijbutyetmoresystematicmethodisgivenintheanalysisofwithunknownconstantskp;ai,andbj.zisusedtode-adaptivecontrolschemes.notetheZ-transfor
7、mvariableortime-advanceoperatorAsweknow,theˉrstanalogoustheoreticalresultforthezx(t)=x(t+1),i.e.,z?1isthetime-delayoperatordiscrete-timesystemsseemstobeduetoYdstie[6],whoz?1x(t)=x(t?1).usedtheinternalmodelcontrol(IMC)implementationforThereferencemodelischosena
