Convergence in multiagent coordination consensus and flocking - PDF document

Equalneighbormodel:Here,aij(t)=1=ni(t);ifj2Ni(t),0;ifj=2Ni(t),whereNi(t)=fjj(j;i)2E(t)gisthesetofagentsjwhosevalueistakenintoaccountbyiattimet,andni(t)isitscardinality.ThismodelisalinearversionofamodelconsideredbyVicseketal.[15].NotethatheretheconstantofAssumption1isequalto1=n.Pairwiseaveragingmodel([5]):Thisisthespecialcaseofboththesymmetricmodelandoftheequalneighbormodelinwhich,ateachtime,thereisasetofdisjointpairsofagentswhocommunicate(bidirectionally)witheachother.Ificommunicateswithj,thenxi(t+1)=xj(t+1)=(xi(t)+xj(t))=2.Notethatthesumx1(t)+


+xn(t)isconserved;therefore,ifconsensusisreached,ithastobeontheaverageoftheinitialvaluesofthenodes.Theassumptionbelowisreferredtoas“partialasynchro-nism”in[3].Wewillseethatitissometimesnecessaryforconvergence.Assumption3.(Boundedintercommunicationintervals)Ificommunicatestojaninnitenumberoftimes[thatis,if(i;j)2E(t)innitelyoften],thenthereissomeBsuchthat,forallt,(i;j)2E(t)[E(t+1)[


[E(t+B�1).III.CONVERGENCERESULTSINTHEABSENCEOFDELAYS.Wesaythattheagreementalgorithmguaranteesasymp-toticconsensusifthefollowingholds:foreveryx(0),andforeverysequencefA(t)gallowedbywhateveras-sumptionshavebeenplaced,thereexistssomecsuchthatlimt!1xi(t)=c,foralli.Theorem1.UnderAssumptions1,2(connectivity),and3(boundedintercommunicationintervals),theagreementalgorithmguaranteesasymptoticconsensus.Theorem1subsumesthespecialcasesofsymmetryoroftheequalneighbormodel,andthereforesubsequentconver-genceresultsandproofsforthosecases.Theorem1ispresentedin[14]andisprovedin[13];asimpliedproof,forthespecialcaseofxedcoefcientscanbefoundin[3].Themainidea,whichappliestomostresultsofthistype,isasfollows.Letm(t)=minixi(t)andM(t)=maxixi(t).SinceeachA(t)isstochastic,itisstraightforwardtoverifythatm(t)andM(t)arenondecreas-ingandnonincreasing,respectively.ItthensufcestoverifythatthedifferenceM(t)�m(t)isreducedbyaconstantfactoroverasufcientlylargetimeinterval;theintervalischosensothateveryagentgetstoinuence(indirectly)everyotheragent;bytracingthechainofsuchinuences,andusingtheassumptionthateachinuencehasanontrivial“strength”(ourassumptionthatwheneveraij(t)isnonzero,itisboundedbelowby�0),theresultfollows.Intheabsenceoftheboundedintercommunicationintervalassumption,thealgorithmdoesnotguaranteeasymptotic consensus,asshownbyExample1below(Exercise3.1,inp.517of[3]).Inparticular,convergencetoconsensusfailseveninthespecialcaseoftheequalneighbormodel.Themainideaisthattheagreementalgorithmcancloselyemulateanonconvergentalgorithmthatkeepsexecutingthethreeinstructionsx1:=x3,x3:=x2,x2:=x1,oneaftertheother.Example1.Letn=3,andsupposethatx(0)=(0;0;1).Let1beasmallpositiveconstant.Considerthefollowingsequenceofevents.Agent3communicatestoagent1;agent1formstheaverageofitsownvalueandthereceivedvalue.Thisisrepeatedt1times,wheret1islargeenoughsothatx1(t1)1�1.Thus,x(t1)(1;0;1).Wenowletagent2communicatestoagent3,t2times,wheret2islargeenoughsothatx3(t1+t2)1.Inparticular,x(t1+t2)(1;0;0).Wenowrepeattheabovetwoprocesses,innitelymanytimes.Duringthekthrepetition,1isreplacedbyk(andt1;t2getadjustedaccordingly).Furthermore,bypermutingtheagentsateachrepetition,wecanensurethatAssumption2issatised.Afterkrepetitions,itcanbecheckedthatx(t)willbewithin1�1�


�kofaunitvector.thus,ifwechoosetheksothatP1k=1k1=2,asymptoticconsensuswillnotbeobtained.Ontheotherhand,inthepresenceofsymmetry,theboundedintercommunicationintervalassumptionisunnec-essary.Thisresultisprovedin[9]and[4]forthespecialcaseofthesymmetricequalneighbormodelandin[11],[7],forthemoregeneralsymmetricmodel.AmoregeneralresultwillbeestablishedinTheorem4below.Theorem2.UnderAssumptions1and2,andforthesym-metricmodel,theagreementalgorithmguaranteesasymp-toticconsensus.IV.PRODUCTSOFSTOCHASTICMATRICESANDCONVERGENCERATETheorem1and2canbereformulatedasresultsontheconvergenceofproductsofstochasticmatrices.Corollary1.ConsideraninnitesequenceofstochasticmatricesA(0);A(1);A(2);:::,thatsatisesAssumptions1and2.IfeitherAssumption3(boundedintercommunicationintervals)issatised,orifwehaveasymmetricmodel,thenthereexistsanonnegativevectordsuchthatlimt!1A(t)A(t�1)


A(1)A(0)=1dT:(Here,1isacolumnvectorwhoseelementsareallequaltoone.)AccordingtoWolfowitz'sTheorem([16])convergenceoccurswheneverthematricesarealltakenfromanitesetofergodicmatrices,andthenitesetissuchthatanyniteproductofmatricesinthatsetisagainergodic.Corollary1extendsWolfowitz'theorembynotrequiringthematricesA(t)tobeergodic,thoughitislimitedtomatriceswithpositivediagonalentries. increasingsequenceoftimes,witht0=0andtk+1�tk!1.Iftkttk+1,theagentsupdateaccordingtox1(t+1)=(x1(t)+x2(tk))=2;x2(t+1)=(x1(tk)+x2(t))=2:Wewillthenhavex1(t1)=1�1andx2(t1)=1,where1&#x-277;0canbemadearbitrarilysmall,bychoosingt1largeenough.Moregenerally,betweentimetkandtk+1,theabsolutedifferencejx1(t)�x2(t)jcontractsbyafactorof1�2k,wherethecorrespondingcontractionfactors1�2kapproach1.IfthekarechosensothatPkk1,thenQ1k=1(1�2k)&#x]TJ/;༔ ; .96; Tf;&#x 11.;ɸ ;� Td;&#x[000;0,andthedisagreementjx1(t)�x2(t)jdoesnotconvergetozero.Accordingtotheprecedingexample,theassumptionofboundeddelayscannotberelaxed.Ontheotherhand,theassumptionofboundedintercommunicationintervalscanberelaxed,inthepresenceofsymmetry,leadingtothefollowinggeneralizationofTheorem2,whichisanewresult.Theorem4.UnderAssumptions1,2(connectivity),and4(boundeddelays),andforthesymmetricmodel,theagreementalgorithmwithdelays[cf.Eq.(1)]guaranteesasymptoticconsensus.Proof.LetMi(t)=maxfxi(t);xi(t�1);:::;xi(t�B+1)g;M(t)=maxiMi(t);mi(t)=minfxi(t);xi(t�1);:::;xi(t�B+1)g;m(t)=minimi(t):Aneasyinductiveargument,asinp.512of[3],showsthatthesequencesm(t)andM(t)arenondecreasingandnonincreasing,respectively.Theconvergenceproofrestsonthefollowinglemma.Lemma1:Ifm(t)=0andM(t)=1,thenthereexistsatimetsuchthatM()�m()1�nB.GivenLemma1,theconvergenceproofiscompletedasfollows.Usingthelinearityofthealgorithm,thereexistsatime1suchthatM(1)�m(1)(1�nB)(M(0)�m(0)).ByapplyingLemma1,withtreplacedbyk�1,andusinginduction,weseethatforeverykthereexistsatimeksuchthatM(k)�m(k)(1�nB)k(M(0)�m(0)),whichconvergestozero.This,togetherwiththemonotonic-itypropertiesofm(t)andM(t),impliesthatm(t)andM(t)convergetoacommonlimit,whichisequivalenttoasymptoticconsensus.q.e.d.ProofofLemma1:Fork=1;:::;n,wesaythat“PropertyPkholdsattimet”ifthereexistatleastkindicesiforwhichmi(t)kB.Weassume,withoutlossofgenerality,thatm(0)=0andM(0)=1.Then,m(t)0forallt,becauseofthemonotonicityofm(t).Furthermore,thereexistssomeiandsome2f�B+1;�B+2;:::;0gsuchthatxi()=1.Usingtheinequalityxi(t+1)xi(t),weobtainmi(+ B)B.ThisshowsthatthereexistsatimeatwhichpropertyP1holds.Wecontinueinductively.SupposethatknandthatPropertyPkholdsatsometimet.LetSbeasetofcardinalitykcontainingindicesiforwhichmi(t)kB,andletScbethecomplementofS.Letbethersttime,greaterthanorequaltot,atwhichaij()6=0,forsomej2Sandi2Sc(i.e.,anagentjinSgetstoinuencethevalueofanagentiinSc).Suchatimeexistsbytheconnectivityassumption(Assumption2).Notethatbetweentimestand,theagents`inthesetSonlyformconvexcombinationsbetweenthevaluesoftheagentsinthesetS(thisisaconsequenceofthesymmetryassumption).SinceallofthesevaluesareboundedbelowbykB,itfollowsthatthislowerboundremainsineffect,andthatm`()kB,forall`2S.Fortimess,andforevery`2S,wehavex`(s+1)x`(s),whichimpliesthatx`(s)kBB,fors2f+1;:::;+Bg.Therefore,m`(+B)(k+1)B,forall`2S.Considernowanagenti2Scforwhichaij()6=0.Wehavexi(+1)aij()xj(ij())mi()kB+1:Usingalsothefactxi(s+1)xi(s),weobtainthatmi(+B)(k+1)B.Therefore,attime+B,wehavek+1agentswithm`(+B)(k+1)B(namely,theagentsinS,togetherwithagenti).ItfollowsthatPropertyPk+1issatisedattime+B.ThisinductiveargumentshowsthatthereisatimeatwhichPropertyPnissatised.Atthattimemi()nBforalli,whichimpliesthatm()nB.Ontheotherhand,M()M(0)=1,whichprovesthatM()�m()1�nB.q.e.d.Thesymmetrycondition[(i;j)2E(t)iff(j;i)2E(t)]usedinTheorem4issomewhatunnaturalinthepresenceofcommunicationdelays,asitrequiresperfectsynchronizationoftheupdatetimes.Alooserandmorenaturalassumptionisthefollowing.Assumption5.ThereexistssomeB&#x-521;0suchthatwhenever(i;j)2E(t),thenthereexistssomethatsatisesjt�jBand(j;i)2E().Assumption5allowsforprotocolssuchasthefollowing.Agentisendsitsvaluetoagentj.Agentjrespondsbysendingitsownvaluetoagenti.Bothagentsupdatetheirvalues(takingintoaccountthereceivedmessages),withinaboundedtimefromreceivingotheragent'svalue.Inarealisticsetting,withunreliablecommunications,eventhisloosesymmetryconditionmaybeimpossibletoenforcewithabsolutecertainty.Onecanimaginemorecomplicatedprotocolsbasedonanexchangeofacknowledgments,butfundamentalobstaclesremain(seethediscussionofthe“two-armyproblem”inpp.32-34of[2]).Amorerealisticmodelwouldintroduceapositiveprobabilitythatsomeoftheupdatesarenevercarriedout.(Asimplepossibilityisto