Chapter7:Input-to-StateStability1 MotivationConsideragainthesystemX=f(x.u)(1)Assumingthatx=0)hasauniformlyasymptoticallystableequilibriumpointattheorigin,westudywhathappenswhenu启0・Example1:Considerthe.followingfirst-ordernonlinearsystem:龙=—2+(亿+x3)u.Ifu=0;weobtainx=—x,(asymptoticallystableeqmhbnumpoint)・Whenu(t)=1,however,weobtainx=x3,whichresultsznanunboundedtrajectoryforanyinitialcondition・口=>Asymptoticstabilityofz=0doesnotsaymuchabouttheforcedsystem.2 DefinitionsThenotionofinput-to-statestability(ISS)atteptstocapturethenotionof^boundedinput-boundedstate”・Definition1:Thesystem(1)诒saidtobelocallyznpvt-to-state-stable(ISS)zfthereexistaICCfunctionfl,aclass7Cfunction7andconstants畑€R+suchthat11^(011<0(闷I")+7(hr()hJno.00||uj(t)||=llurllr^0.\\xo\\0andtrajectoriesapproachtheballofradius丁(6),i-e.,题问⑴||=LL<7(6)7(Jiscalledtheultimateboundofthesystem(1).• AlternativeDefimhon:AvariationofDefinition1istoreplaceequation(2)withthefollowingequation:11^(0110,00(4)-u)S—a'3(||^||)uEDu.||^dL>/V(||u||).⑸VissaidtobeanISSGlobalLyapunovfunctionifIj=Rn,Du=Rm,andQ],&2；a3€兀oo・Remarks:thismeansthatVisanISSLyapunovfunctionif(a) ItispositivedefiiiiteinD・(b) Itisnegativedefinitealongthetrajectodiesof(1)wheneverthetrajectoriesareovisideoftheballdefinedby\\x^\\=zt(||u||)・3Input-to-StateStability(ISS)TheoremsTheorem1:(LocalISSTheorem)Considerthesystem(1)andletV:O—>RbeanISSLyapunovfunctionforthissystem・Then(1)isinput-to-state-stableaccordingtoDefimtion1with\J/\J/\)/\J/67897(\7/vz(\z(\7=a”o°ki=aj(ai(”)k2=尢T(min{h尢(q)}).z.e||a?||0.Thiswillbethecase,providedthata^|x|3>\u\or,equivalently圍＞##仙=(5)1/3-Itfollowsthatthesystemzsgloballyinput-to-state-stablewith#Example3:Nowconsiderthe.followingsystem,whichisaslightlymodifiedversionoftheoneinExample2:oqx=—ax+x^ua>0UsingthesameISSLyapunovfunctioncandidateusedznExample2,wehavethatV=—ax4+x3u=—ax4+aOx4—a0xA+x3u0<0<1=—a(l—0)z4—x3(a0x—u)<—a(l—&)/〉providedx3(a0x—u)>0or.岡>热Thus,thesystemisgloballyinput-to-statestablewith7(u)=兽.#4 Input-to-StateStabilityRevisitedTheorem3:AcontinuousfunctionV：DRisanISSLyapunovfunctiononDforthesystem(1)ifandonlyifthereexistclass7CfunctionsQi・Q2：Q3‘ctnd(7suchthatthefollowingtwoconditionsaresatisfied:(11)(12)Ql(lkll)
训)eD.ueDuV2sanISSGlobalLyapunovfunctionifD=Rn,Du=Rm,andai.cvo.Q3,and(7G/Coo・Remarks:Noticethat，givenru>0,thereexistpointsx6Rnsuchthat他(闻|)"(%)•Thisimpliesthat2dGsuchthata3(d)=dandanyU:II训£oo<血：薯危卫)<_q(胡|)+叩|训)<_a(||d||)+列|训心).Thus,thetrajectoty无(t)resultingfromaninputu(t):|也||匕8
