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Lecture12:StabilityII Lecture12:StabilityII • StabilityofHybridSystems • SwitchingLinearSystems • PiecewiseLinearSystems: • globallyquadraticLyapunovfunction • piecewisequadraticLyapunovfunction LyapunovStability Theorem1(Lyapunov StabilityTheorem) ConsiderahybridautomatonHwith ,anequilibriumpointand.Assumethatthere existsanopensetwith Letbeafunctioninxsuchthatforall: Ifforallstartingatwhere,and all,thesequenceisnon-increasing (orempty),thenisastable equilibriumofH. LyapunovStability Sketchoftheidea Example:SwitchingLinearSystem stableswitching Example:SwitchingLinearSystem Sketchofproof: x =0isanequilibrium: ConsiderthecandidateLyapunovfunction: Ineachdiscrete,thecontinuoussystemisstable! Example:SwitchingLinearSystem Testfornon-increasingsequencecondition: LyapunovStability • SuchV iscalleda“Lyapunov-like”function. • AdrawbackofthisTheoremisthatthesequence mustbeevaluated whichmayrequireintegratingthe continuousdynamics. • Ingeneral,itishardtofindsuchV. LyapunovStability Theorem2(MoreRestrictiveLyapunovStabilityTheorem) Considerahybrid automatonHwithanequilibriumpoint,and Assumethatthereexistsanopensetwith Letbeafunctioninxsuchthatforall: Thenisastable equilibriumofH. Proof: Define AndapplyTheorem1. PiecewiseLinearSystems Withthefollowingproperties: PiecewiseLinearSystems Theorem3 (GloballyQuadraticLyapunovFunction) Ifthereexistssuchthat ofisasymptoticallystable. LinearMatrixInequality(LMI) condition! LinearMatrixInequality(LMI) Linearinequalities Linearequalitiesandinequalities PiecewiseLinearSystems Proof (GloballyQuadraticLyapunovFunction)Firstnotethatthereexists suchthat withthegivenassumptionsthereexistsaunique,infinite,andnon-Zeno executionforeveryinitialcondition.Thecontinuousevolutionsatisfiesthe followingdifferentialequation: whereisafunctions.t.for Let Thesystemresidesinone discretestateatatime. PiecewiseLinearSystems From,wehave wherearethesmallestandlargesteigenvaluesofP respectively. goestozeroexponentiallyas,whichimpliesthatthe equilibriumpointisasymptotically(actuallyexponentially) stable. Example:PiecewiseLinearSystems AsymptoticallySTABLE! Example:PiecewiseLinearSystems P doesnotexist! Asymptoticallystable! Theabovecondition maybetoostrong? PiecewisequadraticV MATLAB:LMIToolbox LyapunovStabilityTest LMI • RobustControlToolbox:LMIsolvers • DetaileddescriptionsoftheLMItoolsareinLMIunderRobust ControlToolbox inthe“Help”menu. • Demo:lmidem • GUI: lmiedit • References • S.Boyd,L.ElGhaoui,E.Feron,andV.Balakrishnan.“Linear MatrixInequalitiesinSystemandControlTheory”.SIAM, Philadelphia,1994. PiecewiseLinearSystems Withthefollowingproperties: PiecewiseLinearSystems Forstability,fromTheorem3 maybetoostrong! Piecewisequadratic!! hasnon-negativeelements. unknowns PiecewiseQuadraticLyapunovFunction Theorem 4(PiecewiseQuadraticLyapunovFunction)Ifthereexist suchthat satisfies: :continuousacrosstheboundary :Positivedefinite wherearenon-negative,theequilibriumpointisasymptotically stable. LMIproblem! M.JohanssonandA.Rantzer.ComputationofpiecewisequadraticLyapunov functionsforhybridsystems,IEEETr.AutomaticControl,1998 PiecewiseQuadraticLyapunovFunction Ontheboundary continuousacrosstheboundary Example:PiecewiseLinearSystems Example:PiecewiseLinearSystems asymptoticallystable Example:PiecewiseLinearSystems asymptoticallystable MATLAB:LMIToolbox LyapunovStabilityTest(Theorem4) LMI • RobustControlToolbox:LMIsolvers • DetaileddescriptionsoftheLMItoolsareinLMIunderRobust ControlToolbox inthe“Help”menu. •References • S.Boyd,L.ElGhaoui,E.Feron,andV.Balakrishnan.“Linear MatrixInequalitiesinSystemandControlTheory”.SIAM, Philadelphia,1994.
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