High-Order-Consensus-Algorithms-in-Cooperative-Veh

High-OrderConsensusAlgorithmsinCooperativeVehicleSystemsWeiRen,Member,IEEE;KevinMoore,SeniorMember,IEEE;YangQuanChen,SeniorMember,IEEEAbstract—Inthispaperwestudy`thorder(`=3)consensusalgorithms,whichgeneralizetheexistingfirst-orderandsecond-orderconsensusalgorithmsintheliterature.Wewillshowsufficientconditionsunderwhicheachinformationvariableandtheirhigher-orderderivativesconvergetocommonvalues.Wewillpresenttheideaofhigher-orderconsensuswithaleaderandintroducetheconceptofan`thordermodel-referenceconsensusproblem,whereeachinformationvariableandtheirhigh-orderderivativesnotonlyreachconsensusbutalsoconvergetothesolutionofaprescribeddynamicmodel.Theeffectivenessofthesealgorithmsaredemonstratedthroughsimulationsandamulti-vehiclecooperativecontrolapplicationwhichmimicsflockingbehaviorinbirds.I.INTRODUCTIONInplainlanguage,whenseveralentitiesoragentsagreeonacommonvalueofavariableofinteresttheyaresaidtohavecometo“consensus.”Inagroupofnetworkedmobileagentswithacommonmissionortask,theconsensusproblemcanplayapivotalrole,particularlywhenthecommunica-tioncapabilityforeachagentislimitedand/orpurposelyconstrained.Forexample,whenthedynamicenvironmentchanges,theagentsintheteammustbeinagreementastowhatchangeshavetakenplace,evenwheneveryagentcannottalkdirectlytoeveryotheragent.Toachieveconsensus,theremustbeasharedvariableofinterest(calledthecoordinationdataorvariable)aswellasappropriatealgorithmicmethodsfornegotiatingtoconsensusaboutthevalueofthatvariable(calledaconsensusalgorithmorprotocol).Cooperativecontrolformulti-agentsystemshaveprimarilybeenappliedtoformationcontrolproblemswithapplicationstomobilerobots,unmannedairvehicles(UAVs),autonomousunderwatervehicles(AUVs),satellites,aircrafts,spacecrafts,andautomatedhighwaysystems[1],butalsoincludenon-formationcooperativecontrolproblemssuchastaskassign-ment,payloadtransport,roleassignment,airtrafficcontrol,timing,andsearch[2].Aspointedoutin[1],forcooperativecontrolstrategiestobesuccessful,numerousissuesmustbeaddressed,includingthedefinitionandmanagementofsharedinformationamongagroupofagentstofacilitatethecoordinationoftheseagents.Informationnecessaryforcooperationmaybesharedinavarietyofways.However,thequestion“consensustowhat?”hasnotbeenexplicitlyansweredintheliterature,althoughasimilarquestion“for-mationtowhatformorshape?”wasaskedin[3]withinthecontextofmobileactuatorandsensornetworks[4].W.RenandY.-Q.ChenarewiththeDepartmentofElectricalandComputerEngineering,UtahStateUniversity,Logan,UT84322-4120,USAemail:[e-mail],[e-mail]).K.L.MooreiswithDivisionofEngineering,ColoradoSchoolofMines,1610IllinoisStreet,Golden,CO80401,USA(email:[e-mail]).Inthispaperwegeneralizethefirst-orderandsecond-orderconsensusalgorithmsintheliterature(see[5]andthereferencestherein).Weshowsufficientconditionsunderwhichtheconsensusinformationvariableanditshigher-orderderivativesconvergetocommonvalues.Onemotiva-tionforstudyinghigher-orderconsensuscomesfromobserv-ingthebehaviorofflocksofbirds.Itisoftennotedthatsuchflocksflyinsomewhatofaformation,maintaininganominalseparationfromeachother,buteachtravelingwiththesamevelocityvector.In[6]itwasshownhowsecond-orderconsensuscanproducethebehaviorofaseparationandcom-monvelocityunderdirectedinformationexchange.However,sometimesabirdflockabruptlychangesdirection,perhapswhenoneofthemsuddenlyperceivesasourceofdangerorfood.Clearlythebirdsinthissettingneedtobuildconsensusonnotonlytheirrelativepositionandtheirvelocity,butalsoonacceleration.Thismotivatestheideaofhigher-orderconsensus.HigherorderconsensusmakesobvioussenseforcooperativecontrolofateamofUAVswhenconfrontinganotherteamof“hostileintelligentgaming(HIG)UAVs”.Withthequestion“consensustowhat”inmind,wearealsomotivatedtoconsideran`thordermodel-referenceconsensusproblem,whereeachinformationvariableandtheirhigh-orderderivativesnotonlyreachconsensusbutalsoconvergetothesolutionofaprescribeddynamicmodel.Weintroducethismodel-referenceconsensusproblemandestablishsufficientconditionsforconsensusconvergence.Theremainderofthepaperisorganizedasfollows.Sec-tionIIpresentssomebackgroundmaterialsandsomemathe-maticalpreliminariesforourlaterdevelopment.Higher-orderconsensusalgorithmsaregiveninSec.III,includingthestandard,unforcedcaseandthecasesofsetpointtracking,whichleadtotheideaofconsensuswithaleader,andmodel-referenceconsensus.Theeffectivenessoftheproposedal-gorithmsisillustratedthroughoutbysimulations,includinganexampleofflockingbehaviorinSectionIV.SectionVconcludesthepaper.II.BACKGROUNDANDPRELIMINARIESItisnaturaltomodelinformationexchangebetweenvehiclesbydirected/undirectedgraphs.Adigraph(directedgraph)consistsofapair(N,E),whereNisafinitenonemptysetofnodesandE∈N2isasetoforderedpairsofnodes,callededges.Asacomparison,thepairsofnodesinanundirectedgraphareunordered.Ifthereisadirectededgefromnodevitonodevj,thenviisdefinedastheparentnodeandvjisdefinedasthechildnode.Adirectedpathisasequenceoforderededgesoftheform(vi1,vi2),(vi2,vi3),···,wherevij∈N,inadigraph.Anundirectedpathinanundirectedgraphisdefinedaccordingly.457Adigraphiscal