##Multiple Lyapunov functions and other analysis tools for switched and hybrid systems.pdf

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1、IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.43,NO.4,APRIL1998475MultipleLyapunovFunctionsandOtherAnalysisToolsforSwitchedandHybridSystemsMichaelS.Branicky,Member,IEEEAbstractÐInthispaper,weintroducesomeanalysistoolsforModelslike(1)havebeenstudiedforstability[1]±[

2、3].switchedandhybridsystems.WefirstpresentworkonstabilityHowever,thosepaperswerepredominantlyconcernedwithanalysis.WeintroducemultipleLyapunovfunctionsasatoolforthecasewhereallthearelinear.Becauseofthis,however,analyzingLyapunovstabilityanduseiteratedfunction

3、systemsPeletiesandhiscoworkerswereabletoderiveswitchinglaws(IFS)theoryasatoolforLagrangestability.Wealsodiscussthecasewheretheswitchedsystemsareindexedbyanarbitrarybasedongeneralizedeigenvalueanalysisoftheconstituentcompactset.Finally,weextendBendixson'stheo

4、remtothecasematrices.Wediscussthegeneralcasesbelow.ofLipschitzcontinuousvectorfields,allowinglimitcycleanalysisWealsodiscussdifferenceequationsofaclassofªcontinuousswitchedºsystems.IndexTermsÐHybridsystems,limitcycles,Lyapunovmethods,(2)nonlinearsystems,stabi

5、lity,switchedsystems,variable-structuresystems.where.Here,weonlyaddtheassumptionthateachisgloballyLipschitzcontinuous.Again,theseequationsI.INTRODUCTIONcanbethoughtofastheªcontinuousºportionofthedynamicsHISPAPERdevelopssomeanalysistoolsapplicableofhybridsyst

6、emscombiningdifferenceequationsandfiniteTtothestudyofswitchedandhybridsystems.Below,automata[4].weintroducethesystemsunderstudyandsummarizeourFinally,wealsostudycontinuousswitchedsystems.Acontributions.continuousswitchedsystemisaswitchingsystemwiththeaddition

7、alconstraintthattheswitchedsubsystemsagreeattheA.SwitchedSystemsswitchingtime.Morespecifically,consider(1)andsupposethatattimesthereisaswitchfromWehaveinmindthefollowingmodelasaprototypicalto.Thenwerequire.Thatexampleofaswitchedsystemis,werequirethatthevector

8、fieldiscontinuousovertime.Throughout,denotethereals,nonnegative(1)reals,integers,andnonnegativeintegers,respectively.where.Weaddthefollowingswitchingrules.·EachisgloballyLipschitzcontinuous.B.Hyb