Asynchronous

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1、AsynchronoushybridsystemswithjumpsÐanalysisandsynthesismethodsGeirE.DullerudSanjayG.LallDepartmentofAppliedMathematicsMailCode116-81UniversityofWaterlooControlandDynamicalSystemsWaterloo,OntarioCaliforniaInstituteofTechnologyCANADAN2L3G1PasadenaCA91125,USAAugust26,1997;submittedtoSyste

2、ms&ControlLettersAbstractInthispaperweshowthattheH∞synthesisproblemforaclassoflinearsystemswithasynchronousjumpscanbereducedtoapurelydiscretetimesynthesisproblem.Thesytemclassconsideredincludescontinuous-timesystemswithdiscretejumps,ordiscontinuities,inthestate.Newtechniquesaredevelope

3、dfortheanalysisofasyn-chronoustimevaryinghybridsystemswhichallowaparticularlysimpletreatment,andprovideanelementaryproofforthesampled-dataH∞problem.1IntroductionInthispaperweconstructasolutiontotheH∞synthesisproblemforlinearsystemswithjumps.Ajumpsystemisasystemthatsatisfiestheusualconti

4、nuoustimestatespaceequationsx(t)˙=A(t)x(t)+B(t)u(t)y(t)=C(t)x(t)withtheprovisothatatparticulartimestk,thestateofthesystemundergoesa‘jump’accordingtotherule+x(tk)=Ad(tk)x(tk)+Bd(tk)ud(tk)+wherex(tk)meansthelimitofx(t)asttendstotkfromabove.Thusthesystemisahybridsystem,withbothdiscreteand

5、continuousdynamicsanddiscreteandcontinu-ousinputsandoutputs.Jumpsystemsareanaturalgeneralizationofthepurely-discreteandpurely-continuoussystemstypicallystudiedincontrol,andtheirgreatergeneralitymakesthemparticularlyusefulinthestudyofsampled-datasystems;itisstraightfor-wardtorewriteacon

6、tinuouslinearsystemwith(possiblygeneralized)sampleandholddevicesasajumpsystem.MuchworkhasbeencarriedoutintheareaofH∞controlofsampled-datasystems.BamiehandPearson[2]giveanelegantsolutiontotheproblemofH∞synthesisforlin-eartimeinvariantsystemsundersampled-datacontrol,showingthattheproblem

7、canbeconvertedtoanequivalentH∞synthesisproblemfordiscretetimesystems.Simi-larresultsarealsoderivedbyYamamoto[15],HaraandKabamba[8],andToivonen[13].OtherresultsaregivenbySunetal[12]andSivashankarandKhargonekar[11]foraclassofsynchronousjumpsystems;thesearesystemsinwhichthejumpsoccurper