控制一类超混沌系统到达任意目标

1ControloneclassofhyperchaoticsystemtoreacharbitrarydesiredtargetMaJun1PuZhongsheng1TangGuoning2WangChunni1(1Departmentofphysics,schoolofscience,LanZhouUniversityofTechnology,China,GansuLanZhou.730050)(2CollegeofphysicsandinformationTechnology,GuangxiNormalUniversity,China,GuangxiGuilin541004)Abstract:Basingonthestabilitytheory,itgetstherightformofcontrolleranalytically.Usingtherelevantcontrollerproposedinthispaper,itcancontrolthesystemtoreacharbitrarystablepoint,limitcircle,differentperiodorbitsofthesystemandoutside,thenumericalresultisconsistentwiththetheoryanalysis.Keywords:L.Coscillator;stability;Lyapunovfunction；hyperchaosPACC:0545IntroductionInrecentyears,synchronizationandcontrolofchaosattractedmoreandmoreattentionamongpeoples,manyrelevantmethodswereproposedtosynchronizeandcontrolchaosfordifferentchaoticsystems[1-6],butitstillencountersomeproblemstofindtherightrequirementforcontrollerswhichcancontrolthechaoticsystemtoreachthedesiredtargetanalytically,Inthispaper,itproposedmethodstocontrolthe4DL.Coscillator,basingonthestabilitytheoryandmatrixtheory,itanalyticallygottherequirementsformofcontrollerstocontrolthehyperchaoticsystemtoreacharbitrarystablepoint,limitcircle,differentperiodorbitsofthesystem(2-period,3-period,etal)andoutside,thenumericalresultisconsistentwiththetheoryanalysis,thetheoryandanalyticmethodcanbeputintousetocontrolotherchaoticandhypercahoticsystems.1ControlprincipleDynamicequationofthecontrolledL.Coscillatorhyperchaoticsystem[1-2,7]：()()⎪⎪⎩⎪⎪⎨⎧+---=--=-=+--=4443443132121321111uxHxdxxxcxxxbxxxuxxaxx&&&&em(1)u1andu4arecontrollers,whenu1=0andu4=0,itishyperchaotic[7].Partone:UsingoutsidestandardsignaltocontrolthesystemSupposingr(t)isstandardsignal,itcanconstructLyapunovfunctionasfollowing[6]：()()()()()()()()()222221txtrtxtrtrtxtV&&-+-+-=(2)dtdVV/=&=V2-+()222xrxr&&-+-()[22122xbrrr--++&&&()]1312uxxb-++--a(3)Evidently,setthecontrollerasthefollowingform(4),itwillmakeequation(5).1SupportedbyNationalsciencefund(number10147101).E-mil:[e-mail]_______________________________________________________________________________()()312212122xxbxbrrru++----++=a&&&（4）dtdVV/=&=V2-（5）Certainly,theLyapunovfunctionVisconvergentstably,asaresult,when∞→t,itresultsin:()()trtx→2（6）Supposing)(tr′isanotherstandardsignal,itcanconstructLyapunovfunctionsamely.()()()()()()()()()233232txtrtxtrtrtxtV&&-′+-′+′-=（7）Evidently,setthecontrollerasthefollowingform:()()()()()221412/)2(22xbxbcrrru-+----+++=eeememee&&&)22(rrr′+′+′-&&&me()()()()11/21/2244432--+---+-+xHxdxcxccmemeeme（8）undertheactionofcontroller（8）,when∞→t，itmakes()()trtx′→3.Additionally,r(t)and()tr′cancomefromanothersystemorsignalgeneratoroutside.Partone:Usingstandardsignaloutsidetocontrolsystem1.1ControlthesystemtoreacharbitrarystablepointTocontrolthesystemtoreachthepoint(bx02,x02,0,bx02)，onecontrolleru1（4）isgoodenough.Undertheactionofu1（4），selectingr(t)=x02，fromthetheoryanalysisabove,()()022xtrtxt=──→─∞→（9）Consideringthesecondpartequationofthesystem(1),itiseasytofind221bxxx+=&020bxt+──→─∞→=02bx(10)Thefrondosecontroller()()3122021212xxbxbxu++----=a(11)Putting(11)and(9)intothefirstpartequationofthesystem(1),itiseasytofindthevariablex3decreasestozeroagainsttime，consideringthethirdpartequationofthesystem,itcanfindthatthevariablex4willreachthevaluebx02Tocontrolthesystemtoreachthepoint(bx02,x02,x03,bx02-cx03)，twocontrollersu1andu4arenecessary.Undertheactionofu1（4）andu4（8）:()()022xtrtxt=──→─∞→，021bxxt──→─∞→,()033xtrxt=′──→─∞→forwardly,itresultsin:03023314cxbxxcxxxt-──→─--=∞→&m（12）1.2Controlthesystemtoreacharbitraryperiodorbit中国科技论文在线_______________________________________________________________________________，making()tktrwsin1=，wisconstant，1kissignalextent，usingonesinglecontrolleru1:()()312212122xxbxbrrru++----++=a&&&3122211)2()1(sincos2sin2xxabxbtktktk++-----+=（13）when∞→t，itwillmake：()()tktrtxwsin12=→（14）221bxxx+=&wtbkwtktsincos11+──→─∞→w=)sincos(1tbtk=()fww++tbksin221（15）)/arccos(22wf+=bb，thevariable1xisperiodsignal，3xand4xarealsoperiodsignal.Parttwo:NousingstandardsignaloutsidetocontrolsystemTocontrolthesystemtoreachlimitcircle,selectingcontrollerasfollowing：3322111xkxkxku++=(16)⎩⎨⎧--≥+-=1)(14444444xxkdxxkdu（17）Puttingcontroller(16)and（17）intosystem（1）,itwillmake()()TTxxxxAxxxx43214321,,,,,,=&&&&（18）()⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡-------+=eemmm//100/1/0/10010114321kdcbkkkaA（19）Settingrightvalueforki,i=1,2,3,4，twoeigenvaluesofequation（18）arenegative，thelastpairisimaginarynumber，itcancontrolthesystemtoreachstablelimitcircle.forexample:k1=-0.181653458,k2=k3=0,k4=3.0,UsingsoftwareofMathCADtocomputetheeigenvaluesofequation（19）:eigenvals0.418346542110.301-0.05-001-00.05-10.33001-0.37-0.33⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎟⎟⎠2.047i2.047i-20.727-0.166-⎛⎜⎜⎜⎜⎝⎞⎟⎟⎟⎟⎠=（20）2Numericalsimulation:2.1.1Controlsystemtoreach(bx02,x02,0,bx02)Settingx02=2.0,undertheactionofcontroller（4），with4-orderR-Kmethodtocomputeequationsystem(1),stepsizeh=0.001，thefollowingGraph1showthenumericalsimulationresult,thevariablexi,i=1,2,3,4,werecontrolledto(0.1,2