Lyapunov Functions and Feedback in Nonlinear Control

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Niveau: Supérieur, Doctorat, Bac+8
Lyapunov Functions and Feedback in Nonlinear Control Francis Clarke Professeur a l'Institut Universitaire de France. Institut Desargues, Universite de Lyon 1, 69622 Villeurbanne, France. Summary. The method of Lyapunov functions plays a central role in the study of the controllability and stabilizability of control systems. For nonlinear systems, it turns out to be essential to consider nonsmooth Lyapunov functions, even if the underlying control dynamics are themselves smooth. We synthesize in this article a number of recent developments bearing upon the regularity properties of Lyapunov functions. A novel feature of our approach is that the guidability and stability is- sues are decoupled. For each of these issues, we identify various regularity classes of Lyapunov functions and the system properties to which they correspond. We show how such regularity properties are relevant to the construction of stabilizing feed- backs. Such feedbacks, which must be discontinuous in general, are implemented in the sample-and-hold sense. We discuss the equivalence between open-loop con- trollability, feedback stabilizability, and the existence of Lyapunov functions with appropriate regularity properties. The extent of the equivalence confirms the cogency of the new approach summarized here. 1 Introduction We consider a system governed by the standard control dynamics x˙(t) = f(x(t), u(t)) a.

  • stable controllability

  • proximal weak

  • asymptotically guidable

  • interest when

  • stabilizing feedback design

  • valued

  • lyapunov functions

  • nonlinear control


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LyapunovFunctionsandFeedbackinNonlinearControlFrancisClarkeProfesseura`l’InstitutUniversitairedeFrance.InstitutDesargues,Universite´deLyon1,69622Villeurbanne,France.clarke@igd.univ-lyon1.frSummary.ThemethodofLyapunovfunctionsplaysacentralroleinthestudyofthecontrollabilityandstabilizabilityofcontrolsystems.Fornonlinearsystems,itturnsouttobeessentialtoconsidernonsmoothLyapunovfunctions,eveniftheunderlyingcontroldynamicsarethemselvessmooth.WesynthesizeinthisarticleanumberofrecentdevelopmentsbearingupontheregularitypropertiesofLyapunovfunctions.Anovelfeatureofourapproachisthattheguidabilityandstabilityis-suesaredecoupled.Foreachoftheseissues,weidentifyvariousregularityclassesofLyapunovfunctionsandthesystempropertiestowhichtheycorrespond.Weshowhowsuchregularitypropertiesarerelevanttotheconstructionofstabilizingfeed-backs.Suchfeedbacks,whichmustbediscontinuousingeneral,areimplementedinthesample-and-holdsense.Wediscusstheequivalencebetweenopen-loopcon-trollability,feedbackstabilizability,andtheexistenceofLyapunovfunctionswithappropriateregularityproperties.Theextentoftheequivalenceconfirmsthecogencyofthenewapproachsummarizedhere.1IntroductionWeconsiderasystemgovernedbythestandardcontroldynamicsx˙(t)=f(x(t),u(t))a.e.,u(t)∈Ua.e.orequivalently(undermildconditions)bythedifferentialinclusionx˙(t)F(x(t))a.e.Theissueunderconsiderationisthatofguidingthestatextotheorigin.(Theuseofmoregeneraltargetsetspresentsnodifficultiesintheresultspresentedhere.)Acenturyago,fortheuncontrolledcaseinwhichthemultifunctionFisgivenbya(smooth)single-valuedfunction(thatis,F(x)={f(x)}),Lyapunovintroducedacriterionforthestabilityofthesystem,apropertywherebyallthetrajectoriesx(t)ofthesystemtendtotheorigin(inacertainsense