Control of Chaos Methods and Applications Applicat

AutomationandRemoteControl,Vol.65,No.4,2004,pp.505{533.TranslatedfromAvtomatikaiTelemekhanika,No.4,2004,pp.3{34.OriginalRussianTextCopyrightc2004byAndrievskii,Fradkov.REVIEWSControlofChaos:MethodsandApplications.II.Applications1B.R.AndrievskiiandA.L.FradkovInstituteofProblemsofScienceofMachines,RussianAcademyofSciences,St.Petersburg,RussiaReceivedNovember4,2003Abstract|Reviewedweretheproblemsandmethodsforcontrolofchaos,whichinthelastdecadewasthesubjectofintensivestudies.Considerationwasgiventotheirapplicationinvar-iousscienticeldssuchasmechanics(controlofpendulums,beams,plates,friction),physics(controlofturbulence,lasers,chaosinplasma,andpropagationofthedipoledomains),chem-istry,biology,ecology,economics,andmedicine,aswellasinvariousbranchesofengineeringsuchasmechanicalsystems(controlofvibroformers,microcantilevers,cranes,andvessels),spacecraft,electricalandelectronicsystems,communicationsystems,informationsystems,andchemicalandprocessingindustries(stirringofuidowsandprocessingoffree-owingmate-rials)).1.INTRODUCTIONIntherstyearsafterthepenetrationoftheconceptofdeterministicchaosintothescienticliterature,chaoticbehaviorwasregardedasanexoticphenomenonwhichmightbeofinterestonlyasamathematicalspeculationandwouldneverbeencounteredinpractice.Lateron,however,thepossibilityofchaoticdynamicswasdiscoveredinnumeroussystemsinmechanics,communication,laserandradiophysics[10,12,16,18,19],chemistryandbiochemistry[46],biology[55],economics[47,124,144],andmedicine.Yetfurtherdevelopmenthighlightedanumberofapplicationswherechaoticmodesmayappear|sometimesasharmful,sometimesasuseful.Moreover,entireclassesofproblemsthatareofpracti-calimportancearosewhereonehastocontrolanonlinearsystembyreducingor,onthecontrary,increasingthedegreeofitschaoticity.Methodsforsolvingtheseproblemsalsowereactivelyde-veloped.Themainofthemweredescribedintherstpart[6]ofthepresentreviewwhosesecondpartisdevotedtotheirapplications.Morethan300papersdevotedtovariousapplicationsofthemethodsforcontrolofchaoticprocesseswerepublishedinthepeer-reviewedjournalsbetween1997and2002.Thequestionsofchaoscontrolareactivelydiscussedinscienticandtechnicaleldssuchasphysicsofturbulentprocesses,laserphysicsandoptics,physicsofplasma,molecularandquantumphysics,mechanics,chemistryandelectrochemistry,biologyandecology,economicsandnances,medicine,mechanicalengineering,electricalengineeringandchemicalindustry,traccontrol,orcommunicationandinformationsystems.Itisappropriatetodecomposetheappliedworksonchaosintoscienticandtechnical(engineering)applications.Theworksonengineeringapplicationsdemonstratetheuseofchaosandthemethodsforcontrolofchaoticsystemsinparticularpracticalproblemsoratleastshowtheirfeasibility.Thescientic1ThisworkwassupportedbytheRussianFoundationforBasicResearch,projectno.02-01-00765,theScienticProgram19ofthePresidiumoftheRussianAcademyofSciences,projectno.1.4,andtheFederalProgram\Integration.TheauthorsarealsogratefultoProf.R.J.EvansofUniversityofMelbourneforsupport.0005-1179/04/6504-0505c2004MAIK\Nauka/Interperiodica506ANDRIEVSKII,FRADKOVapplications(inphysics,chemistry,orbiology),onthecontrary,arevectoredtothecontroltheoryandmethodsfordiscoveringnewpropertiesandregularitiesinbehaviorofphysical(chemical,biological)systems,ratherthantoparticularapplications.Theyoftenmakeuseofsimplisticmodeldescriptionsofthesystemsunderstudy.Atthesametime,thesmallnessrequirementorotherconstraintsontheclassofadmissiblecontrolactionsplayanimportantrole.Introductionofthese(explicitorimplicit)constraintsisaimedatelucidatingtheinternalpropertiesinherentinthesystemitselfandnotforcedonitbyastrongcontrolaction.Thescienticandtechnicalapplicationsarediscussedbelow,respectively,inSections2and3.ThemiscellaneousapplicationsarediscussedinSection4.2.SCIENTIFICAPPLICATIONS2.1.MechanicsControlofpendulums,beams,andplates.Thependulumrepresentsthesimplestclassofme-chanicalsystemsfeaturingcomplexdynamics.Thependulumsystemscanmanifestessentially\nonlinearbehaviorsuchasmultistability,bifurcations,orchaos.Simplicityandobviousnessofphysicalexperimentsmakethependulumattractivebothforresearchandtutorialpurposes[33,98,118,169].Motionofthesimplependulumwithfriction,aswellasofmanyothermodelsofnonlinearoscillatorswithonedegreeoffreedom,isknowntobeabletobecomechaoticifexcitedbyaharmonicforceofasucientamplitude.SomeworksexaminedtheproblemposedbyS.W.Shawin1989:\Howmuchdoesincreasetheamplitudeofaperiodicexcitingforceforwhichchaosisnotyetobserved,providedthattheformoftheexcitingforcecanbevaried?Aninversependulumprovidedwithstopsandexcitedbyhorizontaloscillationsofthesuspensionaxiswasstudiedin[118].Theoptimalproleoftheexcitingforceguaranteeing(bytheMel'nikovcriterion)theabsenceofchaosforthegreatestamplitudewasdeterminedanalytically,andthegreatestamplitudewasshowntoexceedthecorrespondingamplitudeofforharmonicexcitationbythefactoroftwo.Introductionofafeedbackresultsinfurtherextensionofthezoneofnonchaoticmodes[120].SimilarresultswereobtainedfortheDungandHelmholtzoscillators[119].H.K.Chen[50]investigatedthedynamic