河南理工大学.docx

河南理工大学.docx

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河南理工大学

河南理工大学

HENANPOLYTECHNICUNIVERSITY

英文文献翻译

Englishliteraturetranslation

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2014年6月2日

ConsensusandCooperationin

NetworkedMulti-AgentSystems

Algorithmsthatproviderapidagreementandteamworkbetweenallparticipants

alloweffectivetaskperformancebyself-organizingnetworkedsystems.

Thispaperprovidesatheoreticalframeworkforanalysisofconsensusalgorithmsformulti-agentnetworkedsystemswithanemphasisontheroleofdirectedinformationflow,robustnesstochangesinnetworktopologyduetolink/nodefailures,time-delays,andperformanceguarantees.Anoverviewofbasicconceptsofinformationconsensusinnetworksandmethodsofconvergenceandperformanceanalysisforthealgorithmsareprovided.Ouranalysisframe-workisbasedontoolsfrommatrixtheory,algebraicgraphtheory,andcontroltheory.Wediscusstheconnectionsbetweenconsensusproblemsinnetworkeddynamicsystemsanddiverseapplicationsincludingsynchronizationofcoupledoscillators,flocking,formationcontrol,fastconsensusinsmall-worldnetworks,Markovprocessesandgossip-basedalgorithms,loadbalancinginnetworks,rendezvousinspace,distributedsensorfusioninsensornetworks,andbeliefpropagation.Weestablishdirectconnectionsbetweenspectralandstructuralpropertiesofcomplexnetworksandthespeedofinformationdiffusionofconsensusalgorithms.Abriefintroductionisprovidedonnetworkedsystemswithnonlocalinformationflowthatareconsiderablyfasterthandistributedsystemswithlattice-typenearestneighborinteractions.Simulationresultsarepresentedthatdemonstratetheroleofsmall-worldeffectsonthespeedofconsensusalgorithmsandcooperativecontrolofmultivehicleformations.

I.INTRODUCTION

Consensusproblemshavealonghistoryincomputerscienceandformthefoundationofthefieldofdistributedcomputing.Formalstudyofconsensusproblemsingroupsofexpertsoriginatedinmanagementscienceandstatisticsin1960s（seeDeGrootandreferencestherein）.TheideasofstatisticalconsensustheorybyDeGrootreappearedtwodecadeslaterinaggregationofinformationwithuncertaintyobtainedfrommultiplesensorsandmedicalexperts.

DistributedcomputationovernetworkshasatraditioninsystemsandcontroltheorystartingwiththepioneeringworkofBorkarandVaraiyaandTsitsiklisandTsitsiklis,Bertsekas,andAthansonasynchronousasymptoticagreementproblemfordistributeddecision-makingsystemsandparallelcomputing.

Innetworksofagents（ordynamicsystems）,"consensus"meanstoreachanagreementregardingacertainquantityofinterestthatdependsonthestateofallagents.A"consensusalgorithm"（orprotocol）isaninteractionrulethatspecifiestheinformationexchangebetweenanagentandallofitsneighborsonthenetwork.

ThetheoreticalframeworkforposingandsolvingconsensusproblemsfornetworkeddynamicsystemswasintroducedbyOlfati-SaberandMurrayin[9]and[10]buildingontheearlierworkofFaxandMurray.Thestudyofthealignmentprobleminvolvingreachingan

Agreement——withoutcomputinganyobjectivefunctions——appearedintheworkofJadbabaieetal..Furthertheoreticalextensionsofthisworkwerepresentedin[14]and[15]withalooktowardtreatmentofdirectedinformationflowinnetworksasshowninFig.1（a）.

Fig.1.Twoequivalentformsofconsensusalgorithms:

（a）anetworkofintegratoragentsinwhichagentireceivesthestate

ofitsneighbor,agentj,ifthereisalink（i,j）connectingthetwonodes;and（b）theblockdiagramforanetworkofinterconnecteddynamicsystemsallwithidenticaltransferfunctions

.Thecollectivenetworkedsystemhasadiagonaltransferfunctionandisamultiple-inputmultiple-output（MIMO）linearsystem.

Thecommonmotivationbehindtheworkin[5],[6],and[10]istherichhistoryofconsensusprotocolsincomputerscience[1],whereasJadbabaieetal.[13]attemptedtoprovideaformalanalysisofemergenceofalignmentinthesimplifiedmodelofflockingbyVicseketal.[16].Thesetupin[10]wasoriginallycreatedwiththevisionofdesigningagent-basedamorphouscomputers[17],[18]forcollaborativeinformationprocessinginnetworks.Later,[10]wasusedindevelopmentofflockingalgorithmswithguaranteedconvergenceandthecapabilitytodealwithobstaclesandadversarialagents[19].

GraphLaplaciansandtheirspectralproperties[20]–[23]areimportantgraph-relatedmatricesthatplayacrucialroleinconvergenceanalysisofconsensusandalignmentalgorithms.GraphLaplaciansareanimportantpointoffocusofthispaper.ItisworthmentioningthatthesecondsmallesteigenvalueofgraphLaplacianscalledalgebraicconnectivityquantifiesthespeedofconvergenceofconsensusalgorithms.Thenotionofalgebraicconnectivityofgraphshasappearedinavarietyofotherareasincludinglow-densityparity-checkcodes（LDPC）ininformationtheoryandcommunications[24],Ramanujangraphs[25]innumbertheoryandquantumchaos,andcombinatorialoptimizationproblemssuchasthemax-cutproblem[21].

Morerecently,therehasbeenatremendoussurgeofinterest——amongresearchersfromvariousdisciplinesofengineeringandscience——inproblemsrelatedtomultiagentnetworkedsystemswithclosetiestoconsensusproblems.Thisincludessubjectssuchasconsensus[26]–[32],collectivebehaviorofflocksandswarms[19],[33]–[37],sensorfusion[38]–[40],randomnetworks[41],[42],synchronizationofcoupledoscillators[42]–[46],algebraicconnectivityofcomplexnetworks[47]–[49],asynchronousdistributedalgorithms[30],[50],formationcontrolformultirobotsystems[51]–[59],optimization-basedco-operativecontrol[60]–[63],dynamicgraphs[64]–[67],complexityofcoordinatedtasks[68]–[71],andconsensus-basedbeliefpropagationinBayesiannetworks[72],[73].Adetaileddiscussionofselectedapplicationswillbepresentedshortly.

Inthispaper,wefocusontheworkdescribedinfivekeypapers——namely,Jadbabaie,Lin,andMorse[13],Olfati-SaberandMurray[10],FaxandMurray[12],Moreau[14],andRenandBeard[15]Vthathavebeeninstrumentalinpavingthewayformorerecentadvancesinstudyofself-organizingnetworkedsystems,orswarms.Thesenetworkedsystemsarecomprisedoflocallyinteractingmobile/staticagentsequippedwithdedicatedsensing,computing,andcommunicationdevices.Asaresult,wenowhaveabetterunderstandingofcomplexphenomenasuchasflocking[19],ordesignofnovelinformationfusionalgorithmsforsensornetworksthatarerobusttonodeandlinkfailures[38],[72]–[76].

Gossip-basedalgorithmssuchasthepush-sumprotocol[77]areimportantalternativesincomputersciencetoLaplacian-basedconsensusalgorithmsinthispaper.Markovprocessesestablishaninterestingconnectionbetweentheinformationpropagationspeedinthesetwocategoriesofalgorithmsproposedbycomputerscientistsandcontroltheorists[78].

Thecontributionofthispaperistopresentacohesiveoverviewofthekeyresultsontheoryandapplicationsofconsensusproblemsinnetworkedsystemsinaunified

framework.Thisincludesbasicnotionsininformationconsensusandcontroltheoreticmethodsforconvergenceandperformanceanalysisofconsensusprotocolsthatheavilyrelyonmatrixtheoryandspectralgraphtheory.Abyproductofthisframeworkistodemonstratethatseeminglydifferentconsensusalgorithmsintheliterature[10],[12]–[15]arecloselyrelated.Applicationsofconsensusproblemsinareasofinteresttoresearchersincomputerscience,physics,biology,mathematics,robotics,andcontroltheoryarediscussedinthisintroduction.

A.ConsensusinNetworks

TheinteractiontopologyofanetworkofagentsisrepresentedusingadirectedgraphG=（V,E）withthesetofnodesV={1,2,...,n}andedges

.Theneighborsofagentiaredenotedby

.Accordingto[10],asimpleconsensusalgorithmtoreachanagreementregardingthestateofnintegratoragentswithdynamics

canbeexpressedasannth-orderlinearsystemonagraph

Thecollectivedynamicsofthegroupofagentsfollowingprotocol

（1）canbewrittenas

（2）

where

isthegraphLaplacianofthenetworkanditselementsaredefinedasfollows:

Here,

denotesthenumberofneighborsofnodei（orout-degreeofnodei）.Fig.1showstwoequivalentformsoftheconsensusalgorithmin

（1）and

（2）foragentswithascalarstate.TheroleoftheinputbiasbinFig.1（b）isdefinedlater.

AccordingtothedefinitionofgraphLaplacianin（3）,allrow-sumsofLarezerobecauseof

Therefore,Lalwayshasazeroeigenvalue

.Thiszeroeigenvaluescorrespondstotheeigenvector

becausebelongstothenull-spaceof

.Inotherwords,anequilibriumofsystem

（2）isastateintheform

whereallnodesagree.Basedonanalyticaltoolsfromalgebraicgraphtheory[23],welatershowthatxisauniqueequilibriumof

（2）（uptoaconstantmultiplicativefactor）forconnectedgraphs.

Onecanshowthatforaconnectednetwork,theequilibrium

isgloballyexponentiallystable.Moreover,theconsensusvalueis

thatisequaltotheaverageoftheinitialvalues.Thisimpliesthatirrespectiveoftheinitialvalueofthestateofeachagent,allagentsreachanasymptoticconsensusregardingthevalueofthefunction

.

Whilethecalculationoff（z）issimpleforsmallnetworks,itsimplicationsforverylargenetworksismoreinteresting.Forexample,ifanetworkhas

nodesandeachnodecanonlytalkto

neighbors,findingtheaveragevalueoftheinitialconditionsofthenodesismorecomplicated.Theroleofprotocol

（1）istoprovideasystematicconsensusmechanisminsuchalargenetworktocomputetheaverage.Thereareavarietyoffunctionsthatcanbecomputedinasimilarfashionusin