Cooperative Control of Dynamical Systems With Appl

894IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.53,NO.4,MAY2008CooperativeControlofDynamicalSystemsWithApplicationtoAutonomousVehiclesZhihuaQu,SeniorMember,IEEE,JingWang,Member,IEEE,andRichardA.Hull,Member,IEEEAbstract—Inthispaper,anewframeworkbasedonmatrixthe-oryisproposedtoanalyzeanddesigncooperativecontrolsforagroupofindividualdynamicalsystemswhoseoutputsaresensedbyorcommunicatedtoothersinanintermittent,dynamicallychang-ing,andlocalmanner.Intheframework,sensing/communicationisdescribedmathematicallybyatime-varyingmatrixwhosedi-mensionisequaltothenumberofdynamicalsystemsinthegroupandwhoseelementsassumepiecewise-constantandbinaryvalues.Dynamicalsystemsaregenerallyheterogeneousandcanbetrans-formedintoacanonicalformofdifferent,arbitrary,butfiniterela-tivedegrees.Utilizingasetofnewresultsonaugmentationofirre-duciblematricesandonlowertriangulationofreduciblematrices,theframeworkallowsadesignertostudyhowagenerallocal-and-output-feedbackcooperativecontrolcandeterminegroupbehaviorsofthedynamicalsystemsandtoseehowchangesofsens-ing/communicationwouldimpactthegroupbehaviorsovertime.Anecessaryandsufficientconditiononconvergenceofamulti-plicativesequenceofreduciblerow-stochastic(diagonallypositive)matricesisexplicitlyderived,andthroughsimplechoicesofagainmatrixinthecooperativecontrollaw,theoverallclosed-loopsys-temisshowntoexhibitcooperativebehaviors(suchassinglegroupbehavior,multiplegroupbehaviors,adaptivecooperativebehaviorforthegroup,andcooperativeformationincludingindividualbe-haviors).Examples,includingformationcontrolofnonholonomicsystemsinthechainedform,areusedtoillustratetheproposedframework.IndexTerms—Consensus,cooperativecontrol,cooperativecontrollability,formationcontrol,high-orderdynamicalsys-tems,matrixtheory,networkedsystems,time-varyingsens-ing/communication.I.INTRODUCTIONTHISpaperproposesamatrix-theory-basedframeworkofanalysisandcooperativecontroldesignsforagroupofindividualbutheterogeneousdynamicalsystemsandseeksfortheleastrestrictiverequirementonsensingandcommunicationamongthesystems.Asanexample,agroupofunmannedau-tonomousvehiclesarecommandedtoperformasetoftasksasagroup,andindividualrobotsofdifferentcapabilitiesaretoexhibitnotonlycertaingroupbehaviorbutalsotheirindi-vidualbehaviors.Inthegeneralcase,thedynamicalsystemsoperateinadynamicallychanginganduncertainenvironment.Assuch,sensingandcommunicationamongthesystemsareintermittentandlocal,andtheirchangesarenotknownaprioriManuscriptreceivedMay19,2006;revisedOctober25,2006andApril30,2007.RecommendedbyAssociateEditorF.Bullo.ThisworkwassupportedinpartbyCorporateGrantsfromLockheedMartinCorporation.Z.QuandJ.WangarewiththeSchoolofElectricalEngineeringandCom-puterScience(EECS),UniversityofCentralFlorida,Orlando,FL32816USA(e-mail:[e-mail];[e-mail]).R.A.HulliswiththeScienceApplicationsInternationalCorporation(SAIC),Orlando,FL32801USA(e-mail:[e-mail]).DigitalObjectIdentifier10.1109/TAC.2008.920232orpredictablebyeitherdeterministicorprobabilisticmodels.Thefundamentalquestionsareasfollows:whatisthenecessaryandsufficientconditionforsensing/communicationandhowtodesigncooperativecontroltoachieveaguaranteedperformance.Therehavebeenmanyearlierresultsondistributedrobotics,andtheseresultsareobtainedusingheuristicapproaches.Forexample,artificialintelligencemethods[1]havebeenexten-sivelyusedtoexplorethearchitecture,taskallocation,mappingbuilding,coordination,andcontrolalgorithmsinmultirobotmo-tionsystems[2]–[5].Multirobotlocalizationandexplorationarestudiedin[6]usingaprobabilisticapproach.Pathplanningandformationcontrolareinvestigatedin[7]usingbehavior-basedcontrolparadigm[8],wheretherule-basedformationbehaviorshavebeendefinedandevaluatedthroughsimulations.Asimpleheuristicdistributedalgorithmisproposedin[9]foridenticalmobilerobotstoformacircleofagivenradius,whereeachrobotupdatesitspositionaccordingtoasetofrules.Inmanycases,cooperativerulesarechosentomimicanimalbehav-iors[10].Thebasiccohesion,separation,andalignmentrulesareextractedbyobservingtheanimalflockingandsimulatedthroughcomputeranimation[11].Thealignmentproblemisre-centlystudiedin[12],andtheso-callednearest-neighborruleisderivedexperimentally.Thatis,alltheparticlesofpointmassmoveintheplanewiththesamespeed,andtheirheadingsareupdatedindividuallybythesamediscreteandlocalruleofav-eragingitsownheadingandtheheadingsofitsneighbors.Thegroupflockingbehaviorssuchasavoidance,aggregation,anddispersionhavealsobeenexplored[13].Whileheuristicandbioinspiredapproacheshaveproducedmanyinterestingandveryusefulresults,therewasalackoftheoreticalframeworksforbothanalysisandcontroldesign.Undertheassumptionthatsensing/communicationistime-invariant,analysisandcontrolofmultivehiclesystemscanbedoneusingvariousstandardapproachesincontroltheory,forinstance,[14]–[20].However,cooperativecontrolofdynami-calsystemsofteninvolvesintermittent,local,anddynamicallychangingcommunication/sensing.Thus,thecentralanddiffi-cultquestionistwofold:whatisthenecessaryandsufficientconditiononsensing/communicationtoensurecooperativecon-trollabilityandhowto