On Several Composite Quadratic Lyapunov Functions for Switched Systems.pdf

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1、OnSeveralCompositeQuadraticLyapunovFunctionsforSwitchedSystemsTingshuHu¤,LiqiangMa¤andZongliLinyAbstract—ThreetypesofcompositequadraticLyapunovStabilityandstabilizationofboththeLDIandtheswitchedfuntionsareusedforderivingconditionsofstabilizationandsys

2、temhavebeenextensivelystudiedinrecentyears(see,forconstructingswitchinglawsforswitchedsystems.Thee.g.,[1],[4],[14],[15],[16]forsomerecentsurveys).Forthreetypesoffunctionsare,themaxofquadratics,themintheLDI(2),eventhoughthevectorx_canbediscontinuousofq

3、uadraticsandtheconvexhullofquadratics.DirectionalderivativesoftheLyapunovfunctionsareusedforthecharac-ateachtimeinstant,theset-valuedmapfromxtofAix:terizationofconvergencerate.Stabilityresultsareestablishedi=1;2;¢¢¢;Ngiscontinuousandlinear.Becauseofwi

4、thcarefulconsiderationoftheexistenceofslidingmodeandthelinearity,itwasshownin[3]thatasymptoticstabilitytheconvergenceratealongtheslidingmode.Dualstabilizationisequivalenttotheexistenceofacommonconvexandresultisestablishedwithrespecttothepairofconjugat

5、ehomogeneousLyapunovfunctionforeverymembersystemLyapunovfunctions:themaxofquadraticsandtheconvexhullofquadratics.Itisobservedthattheminofquadratics,whichx_=Aix.SincequadraticLyapunovfunctionsareknowntoisnondifferentiableandnonconvex,maybeamoreconvenie

6、ntbeconservativeforLDIs,recenteffortshavebeendedicatedtoolthantheothertwotypesoffunctionswhichareconvextotheconstructionofnumericallytractablenonquadraticand/ordifferentiable.Lyapunovfunctions(e.g.,[2],[7],[22]).keywords:switchedsystem,compositequadra

7、ticfunctions,Thesituationwiththeswitchedsystem(3)isquitedif-stabilization,BMI,slidingmode.ferent.Withagivenswitchingstrategy¾(x),theresultingsystemisintrinsicallynonlinearanddiscontinuous.ThereareI.INTRODUCTIONbasicallytwoapproachestostabilityanalysis

8、andstabiliza-Givenafamilyoflinearsystems,tionforswitchedsystems:onebasedondetailedanalysisofthevectorfieldandtheotheronebasedonLyapunovtheory.x_=Aix;i=1;2;¢¢¢;N;(1)Forsecondordersystems,somenecessaryandsufficienttwotypesofswitchingsystemscanbede