delayed feedback control of chaos

DelayedfeedbackcontrolofchaosBYKESTUTISPYRAGAS*T&TSemiconductorPhysicsInstitute,11A.Gosˇtauto,011088Vilnius,LithuaniaTime-delayedfeedbackcontroliswellknownasapracticalmethodforstabilizingunstableperiodicorbitsembeddedinchaoticattractors.Themethodisbasedonapplyingfeedbackperturbationproportionaltothedeviationofthecurrentstateofthesystemfromitsstateoneperiodinthepast,sothatthecontrolsignalvanisheswhenthestabilizationofthetargetorbitisattained.Abriefreviewonexperimentalimplementations,applicationsfortheoreticalmodelsandmostimportantmodificationsofthemethodispresented.Recentadvancementsinthetheory,aswellasanideaofusinganunstabledegreeoffreedominafeedbacklooptoavoidawell-knowntopologicallimitationofthemethod,aredescribedindetail.Keywords:dynamicalsystem;controllingchaos;delayedfeedback;time-delayautosynchronization;Pyragasmethod;unstableperiodicorbit1.IntroductionControltheoryisoneofthecentralsubjectsinengineeringscience.Despitethefactthatengineersandappliedmathematicianshavebeendealingwithcontrolproblemsforalongtime,anideaofcontrollingchaoshasbeenintroducedrelativelyrecently(Ottetal.1990).Thisnewideahasattractedgreatinterestamongphysicistsandboostedanenormousamountofworkoncontrolproblems.Whyarechaoticsystemsinterestingsubjectsforcontroltheoryandapplications?Themajoringredientforthecontrolofchaosistheobservationthatachaoticset,onwhichthetrajectoryofthechaoticprocesslives,hasembeddedwithinitalargenumberofunstableperiodicorbits(UPOs).Inaddition,owingtoergodicity,thetrajectoryvisitsoraccessestheneighbourhoodofeachoftheseperiodicorbits.Someoftheseperiodicorbitsmaycorrespondtoadesiredsystem’sperformanceaccordingtosomecriterion.Thesecondingredientistherealizationthatchaos,whilesignifyingsensitivedependenceonsmallchangestothecurrentstateandhenceforthrenderingthesystemstateunpredictableinthelongterm,alsoimpliesthatthesystem’sbehaviourcanbealteredbyusingsmallperturbations.Then,theaccessibilityofthechaoticsystemtomanydifferentperiodicorbits,combinedwithitssensitivitytosmallperturbations,allowsforthecontrolandthemanipulationofthechaoticprocess.TheseideasstimulatedPhil.Trans.R.Soc.A(2006)364,2309–2334doi:10.1098/rsta.2006.1827Publishedonline27July2006Onecontributionof15toaThemeIssue‘Exploitingchaoticpropertiesofdynamicalsystemsfortheircontrol’.*pyragas@pfi.lt2309q2006TheRoyalSocietythedevelopmentofarichvarietyofnewchaos-controltechniques(seeShinbrotetal.1993;Kapitaniak1996;Shuster1999;Chen&Raton2000;Boccalettietal.2000;Chen&Yu2003forreview),amongwhichthedelayedfeedbackcontrol(DFC)methodproposedbyPyragas(1992)hasgainedwidespreadacceptance.TheDFCmethodisreference-freeandmakesuseofacontrolsignalobtainedfromthedifferencebetweenthecurrentstateofthesystemandthestateofthesystemdelayedbyoneperiodoftheUPO.Theblockdiagramofthemethodispresentedinfigure1.Alternatively,theDFCmethodisreferredtoasamethodoftime-delayautosynchronization,sincethestabilizationofthetargetorbitmanifestsitselfasasynchronizationofthecurrentstateofthesystemwithitsdelayedstate.Themethodallowsustotreatthecontrolledsystemasablackbox;noexactknowledgeofeithertheformoftheperiodicorbitorthesystemofequationsisneeded.Takingintoaccountonlytheperiodoftheunstableorbit,thesystemundercontrolautomaticallysettlesonthetargetperiodicmotion,andthestabilityofthismotionismaintainedwithonlysmallperturbations.TheDFCalgorithmisespeciallysuperiorforfastdynamicalsystems,sinceitdoesnotrequireanyreal-timecomputerprocessing.Experimentalimplementations,applicationsfortheoreticalmodelsandmostimportantmodificationsoftheDFCmethodarebrieflylistedin§1a–c.(a)ExperimentalimplementationsThetime-delayedfeedbackcontrolhasbeensuccessfullyusedinexperimentalcontexts,whicharequitediverse.Pyragas&Tamasˇevicˇius(1993),Kitteletal.(1994),Gauthieretal.(1994)andCelka(1994)verifieditforelectronicchaososcillators.Hikihara&Kawagoshi(1996)andChristinietal.(1997)stabilizedUPOsinmechanicalpendulums.Bielawskietal.(1994),Bassoetal.(1997a,b)andLuetal.(1998)appliedtheDFCtolasersystems.Pierreetal.(1996)investigatedtheDFCtoagasdischargesystem,Mausbachetal.(1997)stabilizedionizationwavechaos,Fukuyamaetal.(2002)controlledchaoscausedbythecurrent-drivenionacousticinstabilityandGravieretal.(2000)stabilizeddriftwavesinamagnetizedlaboratoryplasma.AnapplicationoftheDFCtoahydrodynamicsystem,namely,achaoticTaylor–Couetteflow,wasconsideredbyLu¨thjeetal.(2001).Parmanandaetal.(1999)andGuderianetal.(1998)usedtheDFCtocontrolelectrochemicalsystems.Benner&Just(2002)delaydynamicalsystemy(t)y(t)y(t)y(t–t)––Kp=p0–K[y(t)–y(t–t)]K[y(t)–y(t–t)]p0Figure1.Blockdiagramofthedelayedfeedbackcontrolmethod.yðtÞ,isanoutputvariable;p,acontrolparameter;p0,itsvalueatwhichthedynamicalsystemhasanunstableperiodicorbitwithaperiodt;andK,thefeedbackgain.K.Pyragas2310Phil.Trans.R.Soc.A(2006)stabilizedaUPOinahigh-powerferromagneticresonancesystem.ApossibilityofusingtheDFCtocontrolhelicopterrotorbladeswasconsideredbyKrodkiewski&Faragher(2000).Sugimoto&Osuka(2002)appliedtheDFCtowalkingcontrolofarobot.Lastly,Halletal.(1997)usedtheDFCforacardiacsystem.(b)Applicationsfortheoreti