复杂网络的同步与多主体系统的一致性

SynchronizationofComplexNetworksandConsensusofMulti-AgentSystems:SynchronizedRegionandConsensusRegionZhishengDuanPekingUniversityMay25,2010DuanZ.S.(PekingUniversity)2010-5-251/40Outline1ConnectionsBetweenSynchronizationandConsensus2CharacteristicsofSynchronizedRegion3H2orH1Performance4OtherRelatedProblemsDuanZ.S.(PekingUniversity)2010-5-252/40Outline1ConnectionsBetweenSynchronizationandConsensus2CharacteristicsofSynchronizedRegion3H2orH1Performance4OtherRelatedProblemsDuanZ.S.(PekingUniversity)2010-5-253/401.1SynchronizationofcomplexnetworksNetworkmodel_xi=f(xi)+cNXj=1lijH(xj);i=1;2;


;N:(1)Networknode_xi=f(xi);Inner-linkingfunctionH(
),thecouplingstrengthc,LaplacematrixL=(lij).EigenvaluesofLaplacematrixL=(lij)(symmetrical):0=123


N:(2)Synchronization:x1(t)!x2(t)!


!xN(t);ast!1:(3)Diusivecoupling!x1(t)!x2(t)!


!xN(t)!s(t);_s(t)=f(s(t)):DuanZ.S.(PekingUniversity)2010-5-254/40Synchronizedregion(Masterstabilityfunctionmethod)Linearizedequation:_i=Df(s(t))i+cPNj=1lijDH(s(t))Masterstabilityequation:_!=[Df(s(t))+DH(s(t))]!:(4)SynchronizedregionS:theregionofsuchthatthelargestLyapunovexponentof(4)Lmax0.Astabilityconditionofthesynchronousstates(t):ck2S;k=2;3;


;N:(5)Ifthesynchronousstates(t)isanequilibriumpoint,themasterstabilityequationbecomes:_!=[F+H]!:(6)ThesynchronizedregionSbecomesthestableregionofF+Hwithrespecttoparameter.DuanZ.S.(PekingUniversity)2010-5-255/401.2Consensusofmulti-agentsystemsAgentsystems(linearmodel)µ_xi=Axi+Bui;yi=Cxi;i=1;


;N:(7)Anobserver-baseddynamicprotocolµ_vi=(A+BK)vi+cV0@NXj=1lijC(vivj)NXj=1lij(yiyj)1A;ui=Kvi;i=1;


;N;(8)wherec0isthecouplingstrength,L=(lij)isthetopologicalmatrix,VandKarefeedbackgainmatrices.DuanZ.S.(PekingUniversity)2010-5-256/40Multi-agentnetworkConnecting(7)and(8)gives_i=Fi+cNXj=1lijHj;i=1;


;N:(9)wherei=xivi;F=ABK0A+BK;H=00VCVC:Node_i=Fi;inner-linkingmatrixH.ConsensusGivenagentsystems(7)§itiscalledthattheprotocol(8)solvestheconsensus,ifthemulti-agentnetwork(9)satiseslimt!1kxi(t)xj(t)k=0;limt!1vi=0;8i;j=1;2;


;N:(10)DuanZ.S.(PekingUniversity)2010-5-257/40ConsensusconditionTheorem1Givenagentsystems(7).SupposethatthecommunicationtopologyLhasaspanningtree,theprotocol(8)solvestheconsensusproblemifandonlyifA+BKandA+ciVC,i=2;


;NareHurwitzstable§wherei;i=2;


;Narethenon-zeroeigenvaluesoftheLaplacematrixL.Remark1µTheprotocol(8)canbeviewedasageneralizationoftheobserver-basedcontrollerforthetraditionalcontrolsystems.Theseparationprinciplestillholds.Byintroducinganewparameterc,onecanintroduceanewconcept\Consensusregionsimilartothesynchronizedregioninthecomplexnetworks.DuanZ.S.(PekingUniversity)2010-5-258/40ConsensusregionByTheorem1§thestabilityofA+ciVCisveryimportantforconsensus.ViewingciasasingleparameterleadstoDenition(Consensusregion)TheregionofsuchthatA+VCisstableiscalledtheconsensusregion.ByTheorem1§theconditionfornework(9)achievingconsensusisthatA+BKisstableandck2S;k=2;3;


;N:Forundirectedtopology,theconsensusregionisontherealaxis;fordirectedtopology,theconsensusregionisinthecomplexplane.Theregioncanbebounded,unbounded,orasetofseveralboundedregions,etc.DuanZ.S.(PekingUniversity)2010-5-259/40AnexampleGivenmatricesA=241:430512:51423:37591110:39115:58452:336935;B=2410010035;C=0:89:62:60:351;V=240:89:6000:3535;K=0:172013:34423:54640:91021:09540:7396:Forundirectedtopology:theconsensusregionS=(0;1:4298)[(2:2139;7:386).DuanZ.S.(PekingUniversity)2010-5-2510/40RemarkFromtheabovediscussion,onecanseethattheproblemsinsynchronizationofcomplexnetworksandconsensusofmulti-agentsystemscanbestudiedinauniedframework.Theconcept\synchronizedregionor\consensusregionisveryimportant.Thelargertheconsensusregion,theeasiertheconsensus.Theconsensusregionshowstherobustnessofconsensus.Actually,thetopicsin\synchronizationofcomplexnetworksisverypopularinthecommunityofphysics.Theconceptofsynchronizationappearedveryearly.Thephysicistspaymuchattentiononsynchronizationphenomena,andfactorsofinuencingsynchronization.Thetopicsin\consensusofmulti-agentsystemsarepopularintheareaofdynamicsandcontrol.Thecontrolscientistspaymuchattentiononthedesignofprotocolstoachieveconsensus.Thesetwoconceptsarecloselyrelatedtoeachother.DuanZ.S.(PekingUniversity)2010-5-2511/40Outline1ConnectionsBetweenSynchronizationandConsensus2CharacteristicsofSynchronizedRegion3H2orH1Performance4OtherRelatedProblemsDuanZ.S.(PekingUniversity)2010-5-2512/402.Characteristicsofsynchronizedregion2.1DisconnectedcharacteristicsTheexistenceofndisconnectedstableregionsforF+H.Theorem2Foranynaturalnumbern,therearematricesFandHofordernsuchthatF+Hhasatleast[n=2]+1disconnectedstableregionswithrespecttoparameter.MainideaµIfarealpolynomialisstable,thenallitscoecientsarepositive.ConstructFandHsuchthattheconstanttermofcharacteristicpolynomialofF+Hisapolynomialwithvariableofordern§theothercoecientsareconstants.DuanZ.S.(PekingUniversity)2010-5-2513/40ConstructingFandHH=0BBBB@01


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