7.DesignofcontrolLyapunovfunctionsforJurdjevic-QuinnsystemsLudovicFaubourgandJean-BaptistePometI.N.R.I.AB.P.93,06902SophiaAntipoliscedexFranceE-mail:Ludovic.Faubourg@sophia.inria.fr,Jean-Baptiste.Pomet@sophia.inria.frUrl:yamethodtodesignexplicitcontrolLya-punovfunctionsforcontrolsystemsthatsatisfytheso-calledJurdjevic-Quinnconditions,i.e.possesanenergy-likefunctionthatisnaturallynon-increasingfortheun-forcedsystem.Theresultswithproofwillap-pearinafuturepaper.Thepresentnoteratherfocusesonthemethod,andonitsapplicationtothemodelofamechanicalsystem,thetransla-tionaloscillatorwithrotationactuator(TORA)(alsoknownasRTAC).7.1IntroductionFordierentiabledynamicalsystems(withoutcontrol),aLyapunovfunctionisaconvenienttooltoanalyzetheasymptoticstabilityofanequilibrium.See[12]forinstance.Itisofcoursenottheonlyone,anditsmaindrawbackisthatthereisnosystematicwaytondaLyapunovfunctioningeneral,eventhoughconverseLyapunovtheorems(see[12,15])tellusthatexistenceofaLyapunovfunctionisequivalenttoasymptoticstability.Forcontrolsystemsandthestabilizationproblem,Lyapunovfunctionshavealsobeenusedextensively.Forlinearsystems,thathavetheniceprop-ertythatLyapunovfunctionsmayalwaysbetakenquadratic,optimizingaquadraticcriteriaorassigningaquadraticLyapunovfunctionaremoreorlesssynonymous(seeforinstance[4]).Fornonlinearsystems,theso-calledLyapunovdesign(see[2,6])consistsindesigningaLyapunovfunctionto-getherwithacontrolthatmakesitdecrease,i.e.thatassignsthisfunctiontobeaLyapunovfunctionfortheclosed-loopsystem.Artstein’stheorem[1]1387.DesignofcontrolLyapunovfunctionsforJurdjevic-QuinnsystemsmakesthismethodtheoreticallyconsistentbycharacterizingthefunctionsthatmaybeassignedtobeLyapunovfunctionsviaasuitablecontinuousfeedbackcontrol.Letusrecallitbriey:consideracontrolsystem_x=f0(x)+mXk=1ukfk(x)(7.1)withstatex2IRnandcontrol(u1;:::;um)=u2IRm,thefk’sbeingsmoothvectoreldsinIRn.Theorem1(Artstein’stheorem)AdierentiablefunctionVthatispos-itivedenite(zeroattheorigin,positiveelsewhere)andinniteatinnitycanbeassignedtobeaLyapunovfunctionfortheclosed-loopsystemviaacontinuousfeedbackin(7.1)ifandonlyif1.itisacontrolLyapunovfunction(CLF):8x2IRnnf0g;Lf1V(x)=0...LfmV(x)=09=;=)Lf0V(x)0;(7.2)2.itsatisesthesocalledsmallcontrolproperty(SCP):forany0,thereexistsa0suchthat,forx2IRn,x6=0kxk9ukukLf0+Pmk=1ukfkV(x)0:(7.3)Thisisonlyanexistenceresult,butSontaggavein[20]anexplicitformulathatgivesasystematicwaytoobtainastabilizingcontrolcorrespondingtoagivenCLF.Onpracticalexamples,thereareofcoursemanyotherpossi-bilities.OthermethodsthanbuildingacontrolLyapunovfunctionexisttodesignstabilizingcontrollaws.However,itiswellknow(seeforinstance[18,19,6])thatacontrolLyapunovfunction,whenavailable,isaveryconvenienttooltoanalyzestability,anditsrobustnesstoperturbations,oreventomodifythedesigntoenhancerobustnessorperformances.Forthesereasons,itisinterestingtoobtaincontrolLyapunovfunctionsforsystemsthatmaybestabilizedbyothermethods.Thisisthetopicofthepresentpaper,atleastforthesituationwherestabilizationhasbeenobtainedusingtheJurdjevic-Quinnmethod,alsocalleddampingcontrol.Letuscall(7.2)-(7.3)Artstein’sequation,anddrawaparallelwithoptimalcontrolandHamilton-Jacobi-Bellman’sequation.Optimalcontrol(forin-stanceminimumtime)isaquantitativeproblem,whosesolutionisunique,andoftenverydiculttodescribe.Asymptoticstabilization,aqualitative7.2Jurdjevic-Quinnsystems139problem,whosesolutionishighlynon-unique,therequirementbeingsome-howweaker.Artstein’sequationsplay,forstabilization,theroleofHamilton-Jacobi-Bellman’sequationforoptimalcontrol:thedynamicprogrammingprincipleassertsthatasolutiontotheHJB’sequationyields(atleastwhenitisdierentiable!)theoptimalsynthesis,andinthesameway,Artstein’stheoremallowsonetoderiveastabilizingcontrollawfromasolutiontoArtstein’sequations,withsomeuniversalformulasavailable.HoweveritisknownthatthesolutionstoHJB’sequationareingeneralnon-smooth,andtheanalysisofthisequationhasgivenrisetoalotofmathematicaldevelop-mentstohandlethisproblem.Artstein’sequationsaremuchlessconstrained,andforthisreason,theyhavesmoothsolutions,farfrombeingunique:forinstanceonemayfreelychangeVoutsidetheoriginandthepointswhereitsderivativealongthecontrolvectoreldsiszero,atleastifthechangesarenotbigenough(inC1topology)forthederivativeofthedeformedfunctionalongthecontrolvectoreldstovanish.WedonotgiveabibliographyonHJB’sequationanddynamicprogramming,butthereadermayndin[11,9,18,19]someinvestigationsonthelinkbetweenoptimality,controlLyapunovfunc-tionsandrobustness.Weshallnotelaboratemorealongtheseline,butwedothinkthatArtstein’sequationsdeserveadeeperanalysis.Thispaperisacontributiontothisanalysisinaparticularcase.Thispaperisorganizedasfollows.Insection7.2,wepresentthetypeofsystemsweareinterestedin,andreviewverybrieythepopularmethodtostabilizethem(withoutobtainingaCLF).Themethodisoutlinedinsection7.3andgivessomegeneralresults.Wethenapply,insection7.4themethodtoamechanicalsystemusedasabenchmarkin[19],seealso[3].Notethatthismodeldoe
本文标题:7. Design of control Lyapunov functions for iJurdj
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