河南理工大学.docx
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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
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