Basic problems in stability and design of switched

ToappearinIEEEControlSystemsMagazine1BasicProblemsinStabilityandDesignofSwitchedSystemsDanielLiberzonandA.StephenMorseDepartmentofElectricalEngineeringYaleUniversityNewHaven,CT06520-8267fliberzon,[e-mail],wemeanahybriddynamicalsystemconsistingofafamilyofcontinuous-timesubsystemsandarulethatorchestratestheswitchingbetweenthem.Thisarticlesurveysrecentdevel-opmentsinthreebasicproblemsregardingstabilityanddesignofswitchedsystems.Theseproblemsare:stabilityforarbitraryswitchingsequences,stabilityforcertainusefulclassesofswitchingsequences,andconstructionofstabilizingswitchingsequences.Wealsoprovidemotivationforstudyingtheseproblemsbydiscussinghowtheyariseinconnectionwithvariousquestionsofinterestincontroltheoryandapplications.ProblemStatementandMotivationManysystemsencounteredinpracticeexhibitswitchingbetweenseveralsubsystemsthatisdependentonvariousenvironmentalfactors.Someexamplesofsuchsystemsarediscussedin[1-3].Anothersourceofmotivationforstudyingswitchedsystemscomesfromtherapidlydevelopingareaofswitchingcontrol.Controltechniquesbasedonswitchingbetweendierentcontrollershavebeenappliedextensivelyinrecentyears,particularlyintheadaptivecontext,wheretheyhavebeenshowntoachievestabilityandimprovetransientresponse(see,amongmanyreferences,[4-6]).Theimportanceofsuchcontrolmethodsalsostemsinpartfromtheexistenceofsystemsthatcannotbeasymptoticallystabilizedbyasinglecontinuousfeedbackcontrollaw[7].Switchedsystemshavenumerousapplicationsincontrolofmechanicalsystems,automotiveindustry,aircraftandairtraccontrol,switchingpowerconverters,andmanyotherelds.Thebook[8]containsreportsonvariousdevelopmentsinsomeoftheseareas.Inthelastfewyears,everymajorcontrolconferencehashadseveralregularandinvitedsessionsonswitchingsystemsandcontrol.Moreover,workshopsandsymposiadevotedspecicallytothesetopicsareregularlytakingplace.Almosteverymajortechnicalcontroljournalhashadorisplanningtohaveaspecialissueonswitchedandhybridsystems.Thesesourcescanbeconsultedforfurtherreferences.Mathematically,aswitchedsystemcanbedescribedbyadierentialequationoftheform_x=f(x);(1)whereffp:p2PgisafamilyofsucientlyregularfunctionsfromRntoRnthatisparameterizedbysomeindexsetP,and:[0;1)!Pisapiecewiseconstantfunctionoftime,calledaswitchingsignal.Inspecicsituations,thevalueofatagiventimetmightjustdependontorx(t),orboth,ormaybegeneratedusingmoresophisticatedtechniquessuchashybridfeedbackwithmemoryintheloop.Weassumethatthestateof(1)doesnotjumpattheswitchinginstants,i.e.,thesolutionx()iseverywherecontinuous.Notethatthecaseofinnitelyfastswitching(chattering),whichcallsforaThisresearchwassupportedbyARODAAH04-95-1-0114,NSFECS9634146,andAFOSRF49620-97-1-0108.conceptofgeneralizedsolution,isnotconsideredinthisarticle.ThesetPistypicallyacompact(oftennite)subsetofanite-dimensionallinearvectorspace.Intheparticularcasewherealltheindividualsubsystemsarelinear,weobtainaswitchedlinearsystem_x=Ax:(2)Thisclassofsystemsistheonemostcommonlytreatedintheliterature.Inthisarticle,wheneverpossible,problemswillbeformulatedanddiscussedinthemoregeneralcontextoftheswitchedsystem(1).Therstbasicproblemthatwewillconsidercanbeformulatedasfollows.ProblemA.Findconditionsthatguaranteethattheswitchedsystem(1)isasymptoticallystableforanyswitchingsignal.OnesituationinwhichProblemAisofgreatimportanceiswhenagivenplantisbeingcontrolledbymeansofswitchingamongafamilyofstabilizingcontrollers,eachofwhichisdesignedforaspecictask.Theprototypicalarchitectureforsuchamulti-controllerswitchedsystemisshowninFig.1.Ahigh-leveldecisionmaker(supervisor)determineswhichcontrolleristobeconnectedinclosedloopwiththeplantateachinstantoftime.Stabilityoftheswitchedsystemcanusuallybeensuredbykeepingeachcontrollerintheloopforalongenoughtime,toallowthetransienteectstodissipate.However,moderncomputer-controlledsystemsarecapableofveryfastswitchingrates,whichcreatestheneedtobeabletoteststabilityoftheswitchedsystemforarbitrarilyfastswitchingsignals....Controller1Controller2ControllermPlantEnvironmentDecisionMakerSwitchingSignalu1u2yuumFigure1:Multi-controllerarchitectureWeareassumingherethattheindividualsubsystemshavetheoriginasacommonequilibriumpoint:fp(0)=0,p2P.Clearly,anecessaryconditionfor(asymptotic)stabilityunderarbitraryswitchingisthatalloftheindividualsubsystemsare(asymptotically)stable.Indeed,ifthep-thsystemisunstable,theswitchedsystemwillbeunstableifweset(t)p.Toseethatstabilityofalltheindividualsubsystemsisnotsucient,considertwosecond-orderasymptoticallystablesystemswhosetrajectoriesaresketchedinthetoprowofFig.2.Dependingonaparticularswitchingsignal,thetrajectoriesoftheswitchedsystemmightlookasshowninthebottomleftcorner(asymptoticallystable)orasshowninthebottomrightcorner(unstable).TheaboveexampleshowsthatProblemAisnottrivialinthesensethatitispossibletogetinstabilitybyswitchingbetweenasymptoticallystablesystems.1Ifthishappens,onemayaskwhethertheswitchedsystemwillbeasymptoticallystableforcertainusefulclassesofswitchingsignals.Thisleadstothefollowingproblem.1However,therearecertainli