JournalofMathematicalSystems,Estimation,andControlc1995Birkhauser-BostonVol.5,No.4,1995,pp.1{22RegionsofAttractionofClosed-LoopLinearSystemswithSaturatedLinearFeedbackRodolfoSuarezJoseAlvarezJesusAlvarezAbstractInthisarticleweaddresstheproblemofdeterminingtheregionofattraction(RA)ofaclosed-loopsingle-inputlinearsystemwithasaturatedstabilizinglinearfeedback.Itisshownthattheshapeoftheregionofattractiondependsstronglyonthenumbernuofeigen-valueswithpositiverealpartoftheopen-loopsystem.Inparticular,undercertainconditionsonthecontrolfunction,iftheopen-loopsystemhaseigenvalueswithstrictlynon-zerorealpart,thecorre-spondingRAishomeomorphictothecylinderRnnuBnu.Keywords:saturatedlinearfeedback,non-lineardynamicalsystems,regionsofattractionAMSSubjectClassications:93C10,93D200BasicNotationBn(r)n-dimensionalopenballofradiusr.Snn-dimensionalunitarysphere.ns;nunumberofeigenvaluesofthematrixAwithnegativeandpositiverealpart,respectively.Ws();Wu()stableandunstablemanifoldsoftheinvariantset.@S;cl(S)boundaryandclousureofthesetS.(0)regionofattractionoftheorigin.(A)thesetofeigenvalues(spectrum)ofthematrixA.C+;C;C0complexnumberswithpositive,negativeandzerorealpart.ReceivedFebruary28,1994;receivedinnalformJuly6,1994.SummaryappearedinVolume5,Number4,1995.1R.SUAREZ,J.ALVAREZ,ANDJ.ALVAREZ1IntroductionConsiderthelinearcontrollablesystem_x=Ax+bu(1:1)wherex2Rn;u2R.Fromthecontrollabilitypropertyofthepair(A;b),thereexistsalinearfeedbacku(x)=kTx;k2Rn(1:2)whichstabilizessystem(1.1)(i.e.,(A+bkT)C).Letu0beapositiverealnumber.Iftheinputuisrestrictedtotakevaluesintheinterval[u0;u0],asaturatedfeedbackandaclosed-loopnon-linearvectoreldareobtained.Denition1.1Letu(x)beastatefeedbackdenedasin(1.2).Thesaturatedlinearfeedbackusat(x)isgivenbyusat(x)=8:u0ifu(x)u0u(x)ifu0u(x)u0u0ifu(x)u0(1:3)Thesaturatedvectoreld,Ax+busat(x),willbedenotedfsat(x).Earlierstudiesonthesaturatedlinearfeedbackproblemhavefocusedonthederivationofsucientconditionsfortheglobalasymptoticstabilityof(1.1)-(1.3)whenAisamarginallystablematrix(seeforinstance[9,10]).Inarecentpaper[1],forthetwo-dimensionalcase,weusedqualitativemeth-odstotopologicallycharacterizetheregionofattraction(RA)of(1.1)-(1.3)anditsbifurcations.Alongthismethodologicalline,inthisworkwestudythecharacterizationoftheRAforsingle-inputn-dimensionalhyperbolic((A)\C0=;)controllablelinearsystems(1.1)withasaturatedlinearfeedback(1.3).Specically,weprovethat,forasystemwhoseopen-loopeigenvalueshavenon-positiverealpart,theRAisunbounded.Forcompletelyun-stableplants((A)C+),itisprovedthattheRAisboundedandhomeomorphictothen-dimensionalball.Forstableopen-loopsystems,itisprovedthatalltrajectorieseventuallytendtowardssomecompactsetofzerovolume.Forthecaseofsystemswhoseeigenvalueshavepositiveandnegativerealparts,itisfoundthatafeedbackwhichonlyrelocatestheeigenvalueswithpositiverealpart,makestheRAhomeomorphictotheproductoftheRAsassociatedtothestableandstabilizedparts.Conse-quently,theRAoftheclosed-loopsystemishomeomorphictothecylinderRnnuBnu.Fornu=1,andkeepingxedtherelocatedeigenvalues,thecylindricstructureoftheRAisretainedundersmallchangesintheloca-tionsoftheopen-loopstableeigenvalues.ToestimatetheRAweprove2CLOSED-LOOPLINEARSYSTEMSthat@(0)=[iWs(j)whenthecriticalelementsjin@(0)arecon-nectedtotheorigin.Threeaplicationexamplesarepresentedtoillustratehowthetheoreticalresultsimprovetheunderstandingoftheproblemandassistcontroldesignprocedures.2TheSaturatedClosed-LoopSystemThefeedbackusat(x)inducesapartitionofRnintothreeregions(S+;SandS0):S+()=fx2Rn:usat(x)=+()u0g;S0=fx2Rn:ju(x)ju0g:S0isanopenset,andS+[S0[S=Rn.S=S+[SandS0arereferredtoassaturationandnon-saturationregions,respectively.NotethattheboundariesofthesaturationregionsS+andSarethe(n1)-dimensionalhyperplaneskTx=u0.OneachoftheregionsS+;S0andS,system(1.1)-(1.3)islinear.OnRn,thesystemispiecewiselinearandcontinuous.Since(A+bkT)C,theoriginisalocallyasymptoticallystable(possibly,non-unique)equilibriumpointof(1.1)-(1.3),anditistheonlyequilibriumpointinS0.Ourmainproblemistoestimatetheregionofattraction(0)oftheorigin.Inthenextproposition,itisprovedthat(0)containspointsintheinteriorofthesaturationregions(S+andS).Proposition2.1(0)\[Rnncl(S0)]6=;.Proof:Assumethat(0)cl(S0).Then,@(0)cl(S0)anddist(0;@(0))0.Theinvarianceof@(0)impliestheexistenceofatleastonetrajectory(t)containedin@(0).Becauseincl(S0)thesaturatedlinearfeedbackcoincideswiththelinearclosed-loopfeedback,satises:_(t)=(A+bkT)(t)forallt2R.ThestabilityofA+bkTimpliesthat(t)!0whent!1.Thisisacontradictiontothefactthatdist(0;@(0))0.Therefore,(0)isnotasubsetofcl(S0)andthepropositionisproved.Inthenextsections,itwillbeshownthatthe\sizeoftheintersection(0)\[Rnncl(S0)]is,ingeneral,ratherlarge.2.1EquilibriumpointsAsapointofdeparturetostudythetopologicalcongurationoftheRA,inthissectionweobtainthesetofequilibriumpointsofthesaturatedsystem.Considersystem(1.1)-(1-3)withthematrixAinvertible.Inadditionto3R.SUAREZ,J.ALVAREZ,ANDJ.ALVAREZtheorigin,theequilibriumpointse=A1bu0(2:1)o
本文标题:Regions of Attraction of Closed-Loop Linear System
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