Chaos Synchronization of Two Chaotic

ChaosSynchronizationofTwoChaoticNonlinearGyrosusingBacksteppingDesign,Ph.D:LOEMBE-SOUAMYProfessor:Guo-PingJiangCo-Author:Chun-XiaFanXin-WeiWang,1,2020/6/6,2,BackgroundandSignificanceProblemFormulationBacksteppingControlSimulationResultsSummaryandOutlook,Outline,2020/6/6,3,BackgroundandSignificance,SincethepioneeringworkofsynchronizationofchaoticsystemsbyPecoraandCaroll1,varioussynchronizationschemessuchasactivecontrol2,adaptivecontrol3andbacksteppingcontrol,havebeensuccessfullyemployedforchaossynchronization.,1L.M.Pecora,T.L.Carroll,Synchronizationinchaoticsystems,PhysicalReviewLetters64(1990)821824.2M.C.Ho,Y.C.Hung,Synchronizationoftwodifferentsystemsbyusinggeneralizedactivecontrol,PhysicsLettersA,Vol.301,No.5-6,424-428,2002.3S.H.Chen,J.H.Lu,Synchronizationofanuncertainunifiedchaoticsystemviaadaptivecontrol,Chaos,Solitons&Fractals,Vol.14,No.4,643-647,2002.,BackgroundandSignificance,Inrecentyears,backsteppingtechnique4ofchaossynchronizationhasattractedmanyresearchersattentionbecauseofitsadvantages.Backsteppingdesigntechniqueisakindofsynthetictechniquetocontroller,whichrecursivelyinterlacesthechoiceofaLyapunovfunctionwiththedesignoffeedbackcontrol.,4,2020/6/6,4X.H.Tan,J.Y.Zhang,Y.R.Yang,Synchronizingchaoticsystemsusingbacksteppingdesign,Chaos,Solitons&Fractals,Vol.16,No.1,37-45,2003,2020/6/6,2020/6/6,5,Fig.1:Aschematicdiagramofasymmetricgyroscope.,BackgroundandSignificance,2020/6/6,6,BackgroundandSignificance,Thesystem(1)canbetransformedintothefollowingnominalform(2):,2020/6/6,2020/6/6,7,BackgroundandSignificance,Fig.2:Phaseplanetrajectoryofthechaoticgyro.,.,2020/6/6,2020/6/6,8,ProblemFormulation,.,Wetakethegyroscope(2)asthemastersystem.Byusingthebacksteppingtechnique,wedesignacontrollerinthefollowingslavesystem(3):,2020/6/6,2020/6/6,9,ProblemFormulation,Thegoalofthecurrentcontrolproblemistodesignthecontroller,sothatforanyinitialconditionsofthetwosystems(2)and(3),thebehavioroftheslaveconvergestothatofthemaster,i.e.,Wedefinetheerrorstatesbetweenthemastersystemandslavesystemas:,2020/6/6,2020/6/6,10,ProblemFormulation,TheerrordynamicsofEq.(4)canbeobtaineddirectlybysubtractingEq.(3)fromEq.(2).,2020/6/6,2020/6/6,11,BacksteppingControl,Comparedwithothercontrollers,wearegoingtodesignamoresimplecontroller.Basedonthebacksteppingmethod,aerrorvariableneedstobedefinedas:,2020/6/6,2020/6/6,12,BacksteppingControl,Wedesignacontrollerasfollows:,2020/6/6,2020/6/6,13,BacksteppingControl,Theorem1.Considerthemaster-slavesystemgiveninEqs.(2)and(3).ThetwosystemscanbegloballyasymptoticallysynchronizedbythecontrollerdefinedinEq.(6).Thatis,theerrordynamicalsystem(4)isgloballyexponentiallystableabouttheorigin.,2020/6/6,2020/6/6,14,BacksteppingControl,Proof:ChoosetheLyapunovfunction:,ThederivativeofEq.(7)is:,2020/6/6,2020/6/6,15,BacksteppingControl,Assumption1:,FromAssumption1,weobtain:,2020/6/6,2020/6/6,16,BacksteppingControl,2020/6/6,2020/6/6,17,BacksteppingControl,Therefore,thesynchronizationerrorisverifiedtobeasymptoticallystable.,2020/6/6,2020/6/6,18,SimulationResults,Theparametersofthenonlineargyrosystemsarespecifiedasfollows:,Theinitialconditionsaredefinedas:,2020/6/6,2020/6/6,19,SimulationResults,Fig.3Timeresponsesofstatesofthecontrolledchaoticgyro.,2020/6/6,2020/6/6,20,SimulationResults,Fig.4Timeresponsesofstatesofthecontrolledchaoticgyro.,2020/6/6,2020/6/6,21,SimulationResults,Fig.5Timeresponsesoferrorstatesofthecontrolledchaoticgyro.,2020/6/6,2020/6/6,22,SummaryandOutlook,BasedonLyapunovstabilitytheory,acontrolleremployedthebacksteppingapproachhasbeendesignedforthesynchronizationoftwochaoticnonlineargyros.Thesimulationresultsshowthatthesynchronizationschem