离散时间切换系统输入输出有限时间稳定性.pdf 22页

'˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn离散时间切换系统输入输出有限时间稳定性张光晨，王为群，高靖波南京理工大学大学理学院，南京　210094摘要：本文讨论具有任意切换和限制切换的切换系统输入输出有限时间问题。任意切换情形，分别对于输入信号W2和W1给出了切换系统有限时间稳定的充分条件。受限切换情形，讨论了已知切换信号和平均驻留时间两种情况下的切换系统输入输出有限时间稳定性问题；相应的充分条件通过线性矩阵不等式给出。最后，所得结论的有效性由一些数值例子进行了验证。关键词：切换系统，有限时间稳定性，输入输出有限时间稳定性，已知切换数值，任意切换，平均驻留时间中图分类号：TP273SomenotesonInput-outputFinite-timeStabilityofDiscrete-timeSwitchedSystemsZHANGGuang-Chen,WANGWei-Qun,GAOJing-BoDepartmentofScience,NanjingUniversityofScienceandTechnology,Nanjing,210094Abstract:Thispapercontainsadiscussiononinput-outputfinite-timestability(I-OFTS)issuesrelatedtoswitchedsystemsfeaturingarbitraryswitchingandrestrictedswitching.Forswitchedsystemsunderarbitraryswitching,sufficientconditionsareseparatelyobtainedforinputsignalsW2andW1,respectively.TheI-OFTSissuesforswitchedsystemsunderrestrictedswitchingarethenaddressed,includingcasesofswitchingsignalswithknowninstantsandaveragedwelltime;correspondingsufficientconditionsareproposedviasolvableLMIsconstraints.Toconcludethepaper,theeffectivenessoftheproposedresultsareillustratedthroughnumericalexamples.Keywords:switchedsystems,finite-timestability,input-outputfinite-timestability,knownswitchinginstants,arbitraryswitching,averagedwelltimeFoundations:SpecializedResearchFundfortheDoctoralProgramofHigherEducation(20133219110040)AuthorIntroduction:Correspondenceauthor：ZhangGuang-Chen（1985-），male，Ph.D.majorresearchdirection：MultidimensionalSystemsTheory.WangWei-Qun（1964-），female，professor，majorresearchdirection：2-Dsystems,fuzzysystemsandsingularsystems.GaoJing-Bo（1988-），male，Ph.D.candidate，majorresearchdirection：Multidimen-sionalSystemsTheory.-1- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn0IntroductionSwitchedsystems,aspecialclassofhybridsystems,consistofanumberofsubsystems(eithercontinuous-timeordiscrete-timeordinarydynamicsystems)andaswitchinglaw,whichgovernssubsystemswitchinganddefinestheactivationduringacertainintervaloftime.Inrecentdecades,switchedsystemshavebeenappliedinawidevarietyoffieldssuchasattractedcommunicationnetworks[1],theaerospaceindustry[2],networkedcontrolsystems[3,4],powerelectronics[5],andflightcontrolsystems[6].TheLyapunovasymptoticstability(LAS)issueinherenttoswitchedsystemsisconsideredaveryimportantresearchsubject[7-11],asitrelatestothebehaviorofthesystemoverasufficientlylengthy(intheory,infinite,)timeinterval.ComparedtoLAS,finite-timestability(FTS)hasacertainadvantageinmanyengineeringfieldsbyallowingthestateofthesystemtofallintoagivenboundduringaspecifiedtimeinterval,whichbetterreflectsthetransientperformanceofthesystem[12-17].Bydefinition,input-outputfinite-timestability(I-OFTS),isfullyconsistentwiththeFTSconcept–generallyspeaking,asystemissaidtobeI-OFTSifforaclassofnorm-boundedinputsignalsdefinedovertimeintervalT,theoutputsofthesystemdonotexceedanassignedthresholdduringthesametimeinterval[18].ItisnecessarytopointoutherethatIO-FTSandclassicLpI-Ostabilityaretwoindependentconcepts,however,whichinterestedreaderscanlearnmoreaboutintheworkscited[19-21].Asofnow,researchershavemadenumerousvaluablecontributionstotheliteratureintermsofI-OFTS.Forexample,studyingtheIO-FTSofdiscrete-timelinearsystems[19],andhybridsystems[20]indetail.OtherstudieshavedefinedthenecessaryandsufficientconditionsforI-OFTSoflinearsystems[21]andfinite-timestabilizationofI-OFTSandI-OstochasticMarkovianjumpsystems[22],aswellasI-OFTSconditionsinsingularlinearsystems[23].Tothebestofourknowledge,theissueofI-OFTSfordiscrete-timeswitchedsystem-shasseenrelativelylittleresearch.Inonerelevantstudy,I-OFTSproblemsofimpulsiveswitchedsystemswithstatedelayswerediscussedindetailunderknownswitchinstants,ar-bitraryswitching,anduncertainswitching[24].Itisimportanttonotethattheseobtainedsufficientconditionsarecloselyrelatedtojumpsystems,however,sowhenconsideringswitchingsystemswithoutjumps,theapproachesandconclusionsarenotdirectlyapplicable.Further,thethreecasesexaminedinthestudy,(knownswitchinginstants,arbitraryswitching,anduncertainswitching,)inevitablyresultinsysteminstabilityduetothefrequentswitchingofsubsystems,whichmaycausetheoutputsignaltosharplyincreaseduringafinitetimeinterval,ultimatelycrashingthesystem.Inefforttosolvethisproblem,weappliedanaveragedwelltimeapproachtoouranalysisofI-OFTSissues.Asdiscussedbelow,weconsideredI-OFTSissuesofswitchedsystemsunderarbitraryswitchingandrestrictedswitching,whichcontainswitchingsignalswithknowninstantsandsatisfyappropriateaveragedwell-timeintervals.Theremainderofthispaperisorganizedasfollows:inthesecondsection,weintroduce-2- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnsomenecessarybackgroundinformation(suchasinputsignalclassesW2andW1anddefini-tionsofrelevantconcepts.)Inthethirdsection,I-OFTSissuesunderarbitraryswitchingarediscussedfirstfollowedbythecorrespondingcriteriaandLMIconditions.WethendiscussI-OFTSissuesunderrestrictedswitching,whichconcernsswitchingsignalswithknowninstantsunderappropriateaveragedwelltime,andreportthecorrespondingsufficientconditionsweidentified.Inthefourthsection,weprovideexamplesthatillustratetheeffectivenessoftheproposedmethods.Thefifthsectionsummarizesandconcludesthepaper.Thenotationsusedinthispaperareshownasfollows:•P>0:asymmetricpositivedefiniterealmatrix;•maxfPgandminfPg:themaximumandminimumeigenvaluesofamatrixP;•AT:thetransposeofarealvectorormatrix.1BackgroundInformationInthispaper,wewillconsiderthefollowingdiscrete-timeswitchedsystem:x(k+1)=A(k)x(k)+G(k)w(k)(1)y(k)=C(k)x(k)+D(k)w(k);k∈Ω=[0;N];withswitchingsignal(k):Ω=[0;N]→M={1;···;M},wherex(k)∈Rnisthestatevector,w(k)∈Wisexogenousdisturbance,andy(k)∈Rmistheoutput.A;G;C(k)(k)(k)andD(k)aresystemmatriceswithcompatibledimensions.ThefollowingdefinitionisnecessaryinordertostudytheI-OFTSofsystem(1),thefollowingdefinitionisneededtobeintroducedfirstly.Defnition1.GiventhesetΩ,inputsignalsWdefinedoverΩ,aswitchingsignal(k),andapositivedefinitematrixR,theswitchedsystem(1)issaidtobeinput-outputfinite-timestable(I-OFTS)withrespectto(W;R;;N),ifTw(k)∈W⇒y(k)Ry(k)<1;(2)foranyk∈Ω:WeconsidertheinputsignalWtofallintooneoftwoclasses,W2andW1[14]underthefollowingsets.ForthegivensetΩ=[0;N],w(k)∈Lp;Ω(i.e.,therestrictionofLponΩ,)can-3- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnbedescribedasfollows:()1∑pp∥w(k)∥<∞:ΩTherestrictionofL1onΩ,ismarkedbyL1;Ω,thatis,anyw(k)∈L1satisfiesthefollowing:∥w(k)∥1<∞;k∈Ω:Aftertheabovepreparations,forthegivensymmetricpositivedefinitematrixQ,inputsignalsclassesW2andW1canbedescribedasfollows:(i)∑NTW2={w(k)∈L2;Ω:w(t)Qw(k)≤1};k=1(ii)TW1={w(k)∈L1;Ω:w(k)Qw(k)≤1}:Forswitchingsignal(t),wesupposethattheswitchinginstantsof(t)areknownanddenotethembyki;i=1;2···;lIfk∈[ki;ki+1);thei-thsubsystemisactivated.LetΩi=∪l[ki;ki+1);andofcourse,Ω=[0;N]=i=1Ωi:WeadoptanaveragedwelltimeapproachtostudyI-OFTS,whichisdefinedasfollows.Defnition2.Forswitchingsignal(k)andanyK≥k≥K0;letN(K;k)denotethenumberofswitchingsover[k;K]:IfK−kN(K;k)≤N0+;(3)tholdsfort>0andanintegerN0,thentiscalledanaveragedwell-timeandN0iscalledthechatterbound.Withoutlossofgenerality,wechooseN0=0:2Input-outputFinite-timeStabilityAnalysisThefollowingsectionfocusesonI-OFTSanalysisofswitchedsystem(1)underarbitraryswitchingandrestrictedswitching,whichincludesswitchingsignalswithknownswitchinginstantsandaveragedwelltime.-4- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn2.1ArbitrarySwitchingCaseBelow,weexamineacaseofarbitraryswitchinginwhichthesufficientconditionfallsunderthefollowingtheorem.Theorem1.Considertheswitchedsystem(1)andtheswitchingsignal(k)witharbitraryswitching,whereforanyi∈M,ifthereexistsymmetricmatricesPi>0,andscalarsi>1;0<i<1,suchthat(i)[]ATPA−PATPGijiiiji<0;∀i;j∈M(4)GTPAGTPG−Qijiiiii(ii)For(k)∈Mandanyk∈[0;N];T1C(k)RC(k)0,defineLyapunov-likefunctionsasfollows:TV(k)=x(k)P(k)x(k);(7)thenforanyk∈[0;N];T∆V(x(k))−(k)w(k)Qw(k)T=V(k+1)(k+1)−V(k)(k)−(k)w(k)Qw(k)TTT=[A(k)x(k)+G(k)w(k)]P(k+1)[A(k)x(k)+G(k)w(k)]−x(k)P(k)x(k)−(k)w(k)Qw(k)[][]ATPA−PATPGx(k)=[xT(k);wT(k)](k)(k+1)(k)(k)(k)(k+1)(k);GTPAGTPG−Qw(k)(k)(k+1)(k)(k)(k+1)(k)(k)(8)Notethatpart(i)ofcondition(4)impliesthefollowing:TT∆V(k)−(k)w(k)Qw(k)=V(k+1)−V(k)−(k)w(k)Qw(k)<0;(9)orequivalently,TV(k+1)≤V(k)+(k)w(k)Qw(k)<0;k∈[0;N]:(10)Thus,byworkingiterativelythroughEq.(10)andtakingintoaccountthatx(0)=0,∑k1TTx(k)P(k)x(k)=V(k)≤V(0)(0)+(k)w(s)Qw(s)s=0∑NT≤w(s)Qw(s)≤1:(11)s=0-5- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnForsystem(1),TTy(k)Ry(k)=[C(k)x(k)+D(k)w(k)]R[C(k)x(k)+D(k)w(k)]TTTT=x(k)C(k)RC(k)x(k)+x(k)C(k)RD(k)w(k)TTTT+w(k)D(k)RC(k)x(k)+w(k)D(k)RD(k)w(k):(12)NoticethatRisapositivedefinitematrix,sothereexistsaninvertiblematrixR1withsuitabledimensionssuchthatR=RTR,and11TTTTx(k)C(k)RD(k)w(k)+w(k)D(k)RC(k)x(k)TTTTTT=x(k)C(k)R1R1D(k)w(k)+w(k)D(k)R1R1C(k)x(k)TTTTTT≤x(k)C(k)R1R1C(k)x(k)+w(k)D(k)R1R1R1D(k)w(k)TTTT=x(k)C(k)RC(k)x(k)+w(k)D(k)RD(k)w(k):(13)whichimpliesthatTTTTTy(k)Ry(k)≤2[x(k)C(k)RC(k)x(k)+w(k)D(k)RD(k)w(k)](14)Furthermore,underconditions(5)and(6)part(ii)andxT(k)Px(k)≤1incondition(11),(k)T1T(k)−1Ty(k)Ry(k)≤2[x(k)P(k)x(k)+w(k)Qw(k)]<1(15)2(k)2(k)foranyk∈[0;N]andthiscompletestheproof.Remark1TheinequalityinEq.(13)isestablishedbecauseforarbitrarymatricesX;YandP>0withsuitabledimensionsTTTT1XY+YX≤XPX+YPY:Remark2AccordingtoTheorem3.1,conditions(4)to(6)seemtobeindependentalongthelengthoffinite-timeintervalN.However,everyconditiondependsassuchonN,bywhichdecidestheactivesubsystematinstantNtoreflectoutputinformation.Additionally,suchfiniteinstantsk=0;1;···;Nturnconditions(4)to(6)intoequationsusefulforverifyingthesystem.IfNisinfinite,though,conditions(4)to(6)with(k)dependentonNarequitedifficulttorealizeaccurately.Remark3Comparedtoacasewitharbitraryswitching[19],thesufficientconditioninThe-orem3.1abovemakesfulluseofsystemmatricesonswitchinginstantsandthusshowslowerconservatism.IftheinputsignalisconsideredtobeW1;,thecorrespondingresultcanbeformulatedasthefollowingtheorem.-6- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnTheorem2.Considersystem(1)andtheswitchingsignal(k)witharbitraryswitching.Foranyi∈M,ifthereexistsymmetricmatricesTi,Pi>0,andscalarsi>1;0<i<1,suchthat(i)conditions(4)and(6)aresatisfied;(ii)Whenswitchingsignal(k)∈Mandk∈Ω;T1C(k)RC(k)0,and-7- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnscalarsi>1;0<i<1,suchthat(i)[]ATPA−PATPGiiiiiii<0;i∈M(20)GTPA−Tiiii[]ATPA−PATPGjjjijjj<0;i̸=j;i;j∈M(21)GTPAGTPG−Qjjjjjji(ii)TGiPiGi+Ti<iQ;(22)(iii)For(k)∈Mandk∈Ω;T1C(k)RC(k)0;selectthefollowingLyapunov-likefunctionsTV(k)=x(k)P(k)x(k);(25)andifk∈Ωi;P(k)=Pi:Foranyk∈Ωi,then(k)=(k+1)=i(i.e.k+1∈Ωi),∆Vi(k)=Vi(k+1)−Vi(k)TT=[Aix(k)+Giw(k)]Pi[Aix(k)+Giw(k)]−x(k)Pix(k)TTTTTT=x(k)AiPiAix(k)+x(k)AiPiGiw(k)+w(k)GiPiAix(k)TTT+w(k)GGiPiGiw(k)−x(k)Pix(k)(26)ForthegivenTi>0;denote11(k)=T2w(k)−T2GTPAx(k);(27)iiiiiitoobtainTi(k)i(k)TTT1TTTTT=w(k)Tiw(k)+x(k)AiPiGiTiGiPiAix(k)−x(k)AiPiGiw(k)−w(k)GiPiAix(k)(28)-8- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnTogetherwithEqs.(26)and(28),T∆Vi(k)+i(k)i(k)TTTTT=x(k)AiPiAix(k)+w(k)GiPiGiw(k)−x(k)Pix(k)TTT1T+w(k)Tiw(k)+x(k)AiPiGiTiGiPiAix(k)TTT1T=x(k)[AiPiAi+AiPiGiTiGiPiAi−Pi]x(k)TT+w(k)[GiPiGi+Ti(k)]w(k):(29)ByusingtheSchurcomplementlemma,Condition(20)part(i)canberewrittenequivalentlyasfollows:TT1TAiPiAi+AiPiGiTiGiPiAi−Pi<0;(30)andthen,byvirtueofconditions(22)inpart(ii)and(30),TT∆Vi(k)+i(k)i(k)<iw(k)Qw(k);(31)thatisT∆V(k)=Vi(k+1)−Vi(k)<iw(k)Qw(k):(32)Conversely,if(k)=i̸=(k+1)=j;thatisk∈Ωiandk+1∈Ωj,thenT∆V(x(k))−iw(k)Qw(k)T=Vj(k+1)−Vi(k)−iw(k)Qw(k)TTT=[Ajx(k)+Gjw(k)]Pj[Ajx(k)+Gjw(k)]−x(k)Pix(k)−iw(k)Qw(k)[][]ATPA−PATPGx(k)=[xT(k);wT(k)]jjjijjj:(33)GTPAGTPG−Qw(k)jjjjjjiAccordingtocondition(21)inpart(i)and(33),thusT∆V(k)=Vj(k+1)−Vi(k)<iw(k)Qw(k);(34)andunderConditions(32)and(34),foranyk∈Ω=[0;N];∑k1∑k1T∆V(k)=V(k)−V(0)(0)−iw(s)Qw(s)<0;(35)k=0s=0Bytakingintoaccountx(0)=0,thenforanyk∈Ω,itisderived∑k1∑NTTTx(k)P(k)x(k)=V(k)≤V(0)(0)+iw(s)Qw(s)≤iw(s)Qw(s)≤1(36)s=0s=0-9- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnandfinally,similarlytoTheorem3.1,fork∈Ω;T1T(k)−1TyRy(k)≤2[x(k)P(k)x(k)+w(k)Qw(k)]<1:(37)2(k)2(k)Thiscompletestheproof.Remark4Notethatinthepreviousstudywereferenced[19],theconstraintsonswitchinginstantswereestablishedbyimpulsivesystemmatrices.ButunderTheorem3.3,thecon-straintconditionsontheseswitchinginstantsareprovidedbasedontheconsecutiveswitchedsubsystems,andassucharemoreeasilyverified.IftheinputsignalisconsideredtobeW1,thecorrespondingresultcanbeformulatedasfollows.Theorem4.Considersystem(1)andaswitchingsignal(k)withtheknownswitchinginstantski:Foranyi∈M,ifthereexistsymmetricmatricesTi,Pi>0,andscalarsi>1;0<i≤1,,suchthat(i)[]ATPA−PATPGiiiiiii<0;i∈M(38)GTPA−Tiiii[]ATPA−PATPGjjjijjj<0;i̸=j;i;j∈M(39)GTPAGTPG−Qjjjjjji(ii)TGiPiGi+Ti<iQ;(40)(iii)For(k)∈Mandk∈Ω,T1C(k)RC(k)0,andscalarsi>1,0<i<1,suchthat(i)[]ATPA−PATPGiiiiiii<0;i∈M(45)GTPA−Tiiii(ii)TGiPiGi+Ti<iQ;(46)(iii)For(k)∈Mandk∈[0;N];T1C(k)RC(k)0,considerthefollowingLyapunov-likefunctions:TV(k)=V(k)(k)=x(k)P(k)x(k);(50)andfor(k)=i,P(k)=Pi.Assumethatkiareswitchinginstantsandfork∈[ki;ki+1),thei-thsubsystemisactivated.Accordingtoswitchedsystem(1),then∆Vi(k)=V(k+1)−V(k)TT=[Aix(k)+Giw(k)]Pi[Aix(k)+Giw(k)]−x(k)Pix(k)TTTTTT=x(k)AiPiAix(k)+x(k)AiPiGiw(k)+w(k)GiPiAix(k)TTT+w(k)GiPiGiw(k)−x(k)Pix(k)(51)Denote11(k)=T2w(k)−T2GTPAx(k);(52)iiiiiiSimilarlytoTheorem3.1,accordingtoconditions(45)and(46),TT∆Vi(k)+i(k)i(k)<iw(k)Qw(k);(53)therefore,TVi(k+1)−Vi(k)<iw(k)Qw(k);(54)whichimplies∑k1TVi(k)0,andscalarsi>1,0<i<1,suchthat(i)conditions(45),(46)and(48)arefulfilled;(ii)forany(k)∈Mandk∈[0;N];thefollowingLMIholds,T1C(k)RC(k)