初始状态学习条件下的不确定通讯拓扑结构的多智能体一致性的分布式模糊自适应迭代学习控制.pdf 15页

'中国科技论文在线http://www.paper.edu.cnDistributedfuzzyadaptiveiterativelearningcontrolwithinitial-statelearningforconsensusofmulti-agentsystems#5withuncertaincommunicationtopologystructure**WUHui,LIJunmin(SchoolofMathematicsandStatistics,XidianUniversity,Xian710126)Abstract:Distributedconsensusproblemisaddressedinthispaperforlinearlyparameterizedmulti-agentsystemswithuncertaincommunicationtopologystructureunderinitial-statelearning10condition.T-Sfuzzymodelsarepresentedtodescribetheuncertaincommunicationtopologystructure,andadistributediterativelearningcontrolprotocolisproposedwithoutusinganyglobalinformationfortheconsensusproblem.TheAILCprotocolsaredesignedwithdistributedinitial-learninganditisnotessentialtofixtheinitialvalueatthestartofeachiteration.Itisprovedthattheproposedprotocolensuresalltheinternalsignalsinthemulti-agntsystemareboundedandthefolloweragentstrackthe15leaderexactlyon[0,T].Sufficientconditionsofperfectlyconsensusformulti-agentsystemsareobtainedbyappropriatelyconstructingLyapunovfunction.Theformationcontrolproblemisalsostudiedbyconvertingtotheconsensusproblem.Finally,thesimulationexamplesaregiventoverifytheefficacyofthetheoreticalanalysis.Keywords:Multi-agentsystem;AILC;T-Sfuzzymodel;Uncertaincommunicationtopology20structure0IntroductionRecentyearshavewitnessedthatthemulti-agentsystems(MAS)receivesincreasingattentioninmanyfieldsincludingphysics,biologyandengineering.Consensusofmulti-agent25systemisoneofthecriticalresearchproblems.Moreover,theobjectiveofitistofinddistributedcontrollawsthatenablemultipleagentstoreachanagreementoncertaincriteriausinglocalneighboringinformation.Itishighlyappliedinformationcontrol,synchronizationofcomplexnetworksanddistributedsensornetworkdesign[1-6].Todate,numerousresultshavebeenreportedduringthelastdecadeonconsensusforbothfirst-order,second-orderandhigh-order30multi-agentsystems[7-11].Variouscontrolmethodshavebeenemployed,suchasadaptivecontrol[12-13],iterativelearningcontrol[14-15],pinningcontrol[16]androbustcontrol[17]andsoon.Aswellknown,iterativelearningcontrol(ILC)isawellformativelearningcontroltactics,whichsufficientlyutilizesthepreviouscontrolexperienceandtrackingerrortoenhancethe35currenttrackingperformance.TheapplicationofILCtoMASisarelativelynewfieldandafewresearchresultshavereportedintheliteratureatpresent.Thesynchronizationofmulti-agentsystembasedonILCmethodwasfirstlyappearedin[18].In[19],theformationcontrolofdiscrete-timemulti-agentsystemwasstudiedbyusingtheiterativelearningcontrolmethod.AdistributedDtypeiterativelearninglawwasdesignedtolearntheformationcontrolfora40leaderlessmulti-agentsystemin[20].In[21],thefinite-timeoutputconsensusproblemofMASwasconsideredwithILCschemesbasedonthecontractionmappingapproach.Butinrealworld,practicalsystemsusuallyhavecomplexandnonlineardynamics,theadaptivecontrolwasconsideredasanusefulandeffectivemethodforsystemwithsystemFoundations:TheNationalNatureScienceFoundationofChinaunderGrant61573013,61603286andbyPh.D.PragramsFoundationofMinistryofEducationofChinaunderGrant20130203110021Briefauthorintroduction:WUHui（1994-），famale，master，majorresearchdirection：adaptiveiterativelearningcontrol,multi-agentsystemCorrespondanceauthor:LIJunmin(1964-)，male，professor，majorresearchdirection：adaptivecontrol,learningcontrolofMAS,hybridsystemcontroltheoryandthenetworkedcontrolsystems,etc.E-mail:jmli@mail.xidian.edu.cn-1- 中国科技论文在线http://www.paper.edu.cnuncertainties[22-23].ManyscholarscombinedtheadaptivecontrolmethodandILC,and45formulatedanewexplorationmethodnamelyadaptiveiterativelearningcontrol(AILC)forsystemuncertainties.Recently,severalworkshavebeenappliedAILCmethodtomulti-agentsystems.TheconsensustrackingofMASwasstudiedbasedonAILCundereitherresettingconditionoralignmentconditionin[24].Itisknownthatinitialconditionsplayafundamentalroleinallkindsofiterativelearningcontrolmethods.Theapplicationoccasionandlimitationsoffive50differentinitialconditionswerestudieddetailedlyin[25].Amongtheseinitialconditions,theidenticalinitialconditionusuallyrequirestheinitialstateofallfolloweragentsresettotheinitialstateoftheleaderatthestartofeachiteration,whichrestrictsthepracticalapplicationofILC.Therefore,numerouseffortsaretakentoavoidthislimitation.AILCforMASconsensustrackingunderalignmentinitialconditionhavebeenstudiedin[26-27].Moreover,In[28],theinitial-state55learningconditionwasusedtostudythecoordinationcontrolofleader-followingmulti-agentsystems.IntheexistingliteratureontheAILCfortheconsensusproblemofMAS,itiscommonlyassumedthatthecommunicationtopologyisfixed,time-varyingorswitching.However,thisassumptionmaybeunrealisticinpracticalapplicationastheconnectionsbetweenthenodesmay60beinfluencedbysomeinherentandexternalfactorssuchastime,temperature,andexternalenvironment.Thustheconnectionsbetweenthenodesarevariable,evenareuncertain.ItiswidelyacceptedthattheT–Sfuzzymodelsisaneffectivemethodfortheirpowerfulcapacitytoapproximateuncertainnonlinearsystems[29].In[30],T-SfuzzymodelswerefirstpresentedtodescribevariablestructureforleaderlessMASandstudiedtheconsensusproblemoffirst-order65MAS.Motivatedbythisobservationandbasedontheaforementioneddiscussions,thispapermakesfurthereffortstoconsidertheconsensusproblemformulti-agentsystemwithuncertaincommunicationtopologystructurebasedonT-Sfuzzymodels.Moreover,weproposeadistributedfuzzyadaptiveiterativelearningcontrolprotocolforconsensusproblemofMASwithuncertaincommunicationtopologygraphunderinitial-statelearningcondition.Thecontributions70ofthispaperaretwo-ford.Firstly,T-Sfuzzysystemsarepresentedtodescribetheuncertaincommunicationtopologystructureformulti-agentsystemsinthispaper.Asfarasauthorsknown,theapplicationofAILCfortheconsensusproblemofMASwithuncertaintopologygraphunderinitial-statelearningconditionisseldominvestigateduntilnow.Secondly,weuseamoreapplicabledistributedinitial-statelearningconditionandcombinedthecharacteristicsofILCand75multi-agentsystems,andwepresentanewdistributedAILCprotocolforuncertainMAS,whichcanobtaintheperfectlyconsensusresultforuncertainMAS.Thepaperisorganizedasfollows.InSection1,wedescribethesystemmodelandsomepreliminaries,andT-Sfuzzymodelsarepresentedtodescribetheuncertaintopologystructureformulti-agentsystem.Themainresultsforconsensusproblemofmulti-agentbasedonAILC80schemearepresentedinSections2.Moreover,theresultforformationcontrolproblemisextendedinsection3.InSection4,thesimulationsareprovidedtoshowthevalidityoftheproposedalgorithms.Finally,conclusionsaregiveninSection5.1Preliminariesandproblemformulation1.1Graphtheory85LetG(,,)VEAbeanundirectedgraphwithasetofverticesV(,...,),vvasetof1nedgesEVV,andtheweightedadjacencymatrixAa[].Ifthereistheinteractionbetweenijagentiandj,thatis(,)vvE,thenaa0,andotherwiseaa0.Moreover,self-loopsarejiijjiijjinotallowed.N{:(,v)}vvEistheneighborsetofnodev.TheLaplacianmatrixassociatedijjiinwithgraphGisL[]lijDA,whereDdiagd{,...,1dn},dij1aij.TheweightedadjacencymatrixAis90symmetricifthegraphGisundirectedandconnected.-2- 中国科技论文在线http://www.paper.edu.cnLabeltheleaderasnode0andaddedittoG,thenonederivetheoutspreadgraphG.Denotetheconnectionweightbetweenagentiandtheleaderasb0,inwhichb0iftheithfolloweriiagenthasaccesstotheinformationoftheleader,orelseb0.ThenonecaneasilyderivethattheiweightcommunicationmatrixHLBcorrespondingtographGissymmetricwiththeLaplacian95matrixLofgraphGandthediagonalmatrixBdiagb,...,b1nLemma1.(See[31].)IfgraphGisconnected,thenthesymmetricmatrixHassociatedwithGispositivedefinite.1.2Problemformulation100Inthissection,weconsideraleader-followermulti-agentsystemwhosecommunicationtopologygraphisimprecise,andthedynamicsoftheithfolloweragentarerepresentedaskkkkxti()i()tfxtii()uti()dti()(1)wherekrepresentstheiterationnumber,kkxt()Randut()Rarethestateandinputforithiifolloweragent,respectively,k105()tRisanunknowntime-varyingsystemparameterandfxtt((),)isiiaknownnonlinearfunction,andkdt()istheunknownboundedinputdisturbancethatik**satisfiesdt()d,withdanunknownpositiveconstant.iiiDenotethestateoftheleaderasx,whichisacontinuouslydifferentiablefunctionon0,T0andthedynamicsatisfies110xt0()fxtt00(),(2)Theconsensuserroroftheuncertainmulti-agentsystemisdefinedaskkii()txt()xt0()(3)Definition1.ThefuzzygeneralconsensuserrorfortheuncertainMASisdefinedasRulep:115Ifz(t)isMandz(t)isMand...andz(t)isM1p12p2npnThenetkapxtkxtkbpxtxtk(4)i()ijj()i()i0()i()jNiwherei1,2,...,n,z(t)x(t)isthepremisevariable,M,p1,...,,ri1,...,nrepresentthefuzzyiipipsets,risthenumberofif-thenrules,Nistheneighborsoffolloweragenti,a0istheiijconnectionweightbetweenfolloweragentsiandp120jinrulep,andbistheconnectionweightibetweenagentiandtheleadercorrespondingtotheppthrule,b0iftheithfolloweragenthasiaccesstotheinformationoftheleader.Basedontheabovedefinitions,weproposeaT-Sfuzzymulti-agentmodelwhosepthruleisgivenasthefollowingform:125Rulep:Ifz1(t)isMp1andz2(t)isMp2and...andzn(t)isMpnThenkket()H()t(5)pkkkkTkkTwhereet()[(),etet12(),...,etn()]，xt()1nxt0(),1n[1,1,...,1].rrr130whereTandzt()[,,...,],zz12znhpp(())/ztp(()),ztHhztHp(())p,p(())ztMpi(())ztp1p1p1M(())ztrepresentingthegradeofmembershipsofzt()inM.pipiThen,itcanbeseenthatrhztpp(())0，p1,..,,rhzt(())1p1-3- 中国科技论文在线http://www.paper.edu.cnforalltR.135Assumption1.TheweightcommunicationmatrixHinrulepp,1,...rarepositivedefinite.pRemark1.Herewestudytheuncertaincommunicationgraphindifferentrulesbasedontheconnectedandundirectedtopologygraph,thustheAssumption1isrealistic.Assumption2.Theleaderdynamicfxtt((),)isbounded,thatis,thereexistsanunknownreal00constantboundM0suchthatf140f00xtt(),Mft[0,].TAssumption3.Themembershipfunctionhzt()isdifferentiableandhzt(),p1,...,rispp1bounded,thustheelementsofthematricesandarebounded.HHr1Remark2.Fromassumption1andhztp()0,wecanconcludethatHhztHp(())pandHarep1positivedefinite.Sincehzt()isboundedandcombining(6),wecanobtainthattheelementsofp.111145thematricesandarebounded.Here,wedenote()as.HHHHLemma2.LetthegraphGp,1,...,rbeconnected.Thenitfollowsfrom(6)thatpkk()tet()/()(7)H22whererepresentstheEuclideannorm,and()0istheminimumsingularvalueof.2HHkkThuslimet()0canderivethatlim()t0.kiki150Hereweuseamoreapplicablelearningconditioninsteadofthestrictidenticalinitialcondition,andeachfolloweragentcantakeanyarbitraryinitialstate.Thefollowinginitial-statelearningprotocolisproposed:k1kkx(0)x(0)e(0)(8)iiiwhereisadesignparametertobedetermined.155Lemma3.(see[28].)IfI(0)1,theproposedinitial-statelearningcanguaranteethatHklim(0)0.k2Proof:From(8),onehask1kkx(0)x(0)e(0)(9)160From(6),kkkke(0)H(0)(0)H(0)x(0)1nx00(0)H(0)x(0)H(0)1nx(0).Wecanconcludethatk11kkke(0)e(0)H(0)x(0)x(0)(10)Substituting(9)into(10),weobtain165kk1e(0)(IH(0))(0)e(11)Takingnormsonbothsidesof(11)yieldskk1e(0)I(0)e(0)22H2kThustheconditionI(0)1caneasilyobtainthatlime(0)0.MoreoverwehaveHk2klim(0)0.k2170Remark2.From(11),onehaskk110e(0)(I(0))e(0)H(12)kwecanalsoobtainthates(0)foranyiterationfromLemma3withsapositiveconstant.2Theentiredynamicsofthefolloweragentscanbedescribedaskkkkxt()fxt()ut()dt()(13)TkkkkTkkkkTk175where=[,,,...,12n],fxt()[(fxt11()),fxt2(2()),...,fxtn(n())],ut()=[utut1(),2(),...,utn()],dt()=kkkT[dtdt(),(),...,dt()].12n-4- 中国科技论文在线http://www.paper.edu.cnkTherefore,thedynamicoftheerroret()iskkk1kkkket()H()tH()tHHet()Hfx()ud1nfxtt00((),)(14)2Mainresultsk180Theconsensusobjectiveofthispaperistodesignthecontrolinputut()foreachagenttoimakethefolloweragentstracktheleaderexactlyon[0,]Tastheiterationnumberkapproachesinfinity,thatis,klim()t0,i1,2,...,.nkiInordertoachievethisobjective,thecontrolprotocolfortheithagentatthekthiteration185canbedesignedasfollows:kkkkkkkkut()ctet()()ˆf(xt())sgn(())etˆ()t(15)iiiiiiiikkkwherect()isthedesignedtime-varyingparameter,ˆanˆaretheestimationofand,iiiii*respectively,with=.dMiifThevectorformofprotocolscanbewrittenas190utk()Ctetk()()kˆkfxt(())kvtk()whereCtdiagctctk()=((),(),...,()),kkctkˆk[,...,ˆkˆkT],()[(),...,()]vtkvtkvtkTwithvtk()sgn(())etkˆk().t12n1n1niiikkkTheadaptivelawsforˆ(),ttˆ()andct()areproposedasiiiˆkˆk1fekkiiiii(16)ˆ00ˆkk()()itretiikk1(17)(0)()T195iiand2kkcti()ieti()(18)kk1c(0)c()Tii0（ini)0（ini)（ini)（ini)where,randarepositiveparameterstobedesigned,andˆ()t,()ctc,,careiiiiiiiiiconstantvalues.200Theorem1.Considerthemulti-agentsystem(1)underAssumptions1-3andinitial-statelearningcondition(8)withuncertaintopologygraph.Byapplyingthedistributedcontrolprotocol(15)andparameteradaptivelaws(16)-(18),theexactconvergenceoftheconsensuserrorandtheboundnessofalltheinternalsignalsareensuredon[0,]Tastheiterationnumberk,thatis,klim()0,ti1,2,...,.nki205Proof.Thecompositeenergyfunctioncanbechosenasnnk1kT11k1tkTk1k221kEt()et()Het()di()tcti()c0220ii112rii2(21)wherekˆk,kˆk,k[k,...,kT],=diag,,...,,,randcarethepositivedesignediiiiii1i12nii0parameters.Thedifferencebetweenkk1210Et()andEt()isderivedasfollowskkk1Et()Et()E()t1kTT1k1k11k11tkTT1kk11k1et()et()e()te()td2HH220(20)nn112222kk11kki()ti()tcti()c00ci()tcii1122riiNotethat-5- 中国科技论文在线http://www.paper.edu.cn1kT1k1kT1k1ttkT1kkT1ket()Het()=e(0)H(0)(0)+ee()He()de()He()d(21)22200TTk1kk11k1(ˆk1ˆkT)1[2k(ˆkˆk1)](22)215nnn1k221kt1kki()tdi(0)i()()i(23)i122rii1ri0i1rinnn1k221kt1kkcti()c0ci(0)c0ci()cc0i()d(24)i122ii1i0i1ikk1Substituting(21)-(24)into(20),thedifferencebetweenEt()andEt()canbederivedaskkk1Et()Et()E()t1kT1k1k1T1k11ttkT1kkT1k=e(0)H(0)(0)ee()tHe()+te()He()de()He()d22200nnntt(ˆk1ˆkT)1k1k(0)221k1()1k()()kdiitiid00i122rii1rii1rinnn1k221k1t1kkci(0)c0ci()tc0ci()cc0i()di122ii1i0i1iAttimetTandusingupdatinglaws(16)-(18),wehavek1kT1k1k1T1k11TkT1kET()e(0)He(0)e()THe()+Te()He()d2220nTTT1kkkkk2e()C()()edci()c0i(())eid00i1iTTTTek()11ek()d(ˆk1ˆkT)1ek()fx(k)kdHHH00nT1220ek()kk()()d(25)iii0ri1iTkTT11111kk1ke()cI0nHHHHe()de(0)He(0)0221Tkk111e()THe()T2TTT1kkk1ke()Qe()de(0)He(0)021111whereQcI.0nHHHH2TheinequalityQ0isequivalentto1111c.0maxHHHH2Then225TTT1kllk1kET()minQe()e()de(0)He(0)(26)02Accordingto(26),wehavekk1Tk1l1l1lET()ET()+ET()ET()+e(0)He(0)(27)ll222ItcanbededucedfromLemma3and(12)that-6- 中国科技论文在线http://www.paper.edu.cnkk11TTl1llle(0)HH(0)(0)emin((0))e(0)e(0)22ll22k1T02l0min(HH(0))e(0)(I(0))e(0)2l2230k1((0))((I(0)))2le0(0)Te0(0)minHH2l24min(HH(0))((I(0)))00Tee(0)(0)221((IH(0)))Thuswehavek1ET()ET()S(28)where4min(HH(0))((I(0)))00TS=e(0)e(0)221((IH(0)))k1235ItcanbeseenthatthefinitenessofET()dependsonthefinitenessofET().Next,wewillshow1thefinitenessofEt().kFrom(19),thederivativeofEt()canbecomputedasTTTEtk()etk()Ctetk()()ketk()fx(k)ketk()vk1fxtt((),)dtk()n00TT11Tetk()11etk()etk()1etk()k1kHHHH22nn11k()tk()tctk()cctk()iii0iii11riinTTetk()Ctetk()()ketk()fx(k)k+etk()k()tiii1TT11Tetk()11etk()etk()1etk()()t1()tˆk1kHHHH22nn11k()tk()tctk()cctk()iii0iii11riiUsingadaptivelaws(16)-(18),wehavenTTEtk()etk()Ctetk()()+ketk()k()tetk()11etk()iiHHHi111TTTetk()1etk()()t1()[(tˆk1etk()fx(k)]kH22nn112kkkk240i()tretii()cti()c0ieti()ii11rii(29)kT1111ket()cI0nHHHHet()21()ttT1()ˆkk1120（ini)0（ini)Fromˆ()t,()ctcintheadaptivelaws,wehaveiiii0()tˆ0()t,0()tˆ0()t(ini)0.iiiiiiiiiiThenwecanfurtherderivethefollowingequality1(0)ˆ1(0)ˆ0()T,iiiiii1100(0)ˆ(0)ˆ()T，iiiiii10(ini)c(0)cT()ciii245Since()tiscontinuousandboundedover[0,]TandcombinedwithRemark3,itcanbeconcludedi1thatE(0)nn111T111t1T11112211E(0)e(0)H(0)(0)e(0)(0)di(0)ci(0)c0220ii112rii2-7- 中国科技论文在线http://www.paper.edu.cnisbounded.From(29),wecaneasilyderiveitsfirstiterationasfollowsT11T250Et1()et1()cI111et1()()t1()tˆ0110nHHHH22Usingadaptivelaw(16),weobtainT1T1111Et()et()Qet()()t()t(30)21111Choosingc,wehave0maxHHHH2111TEt()max()()t()t2255Since()tiscontinuousandboundedover[0,]T.ThereforethereexistsaconstantLwith11TLmaxmax()()t()ttT0,2Thus,thefollowinginequalitycanbederivedtEt1()E1(0)E1()d0t1(31)E(0)+Ld01E(0)TL11kItcanbeseenthatEt()impliesthatET()isfinite,thusthefinitenessofET()isensured.260Wecanconcludefrom(27)thatkTTk1llET()ET()minQe()e()d+S(32)0l2ThenwecanderivethatthefinitenessofkET()isensuredforanyiteration.Theboundednesskkkkofut(),ˆ(),tˆ()tandct()canbeobtainedfrom(15)-(18).Therefore,alltheinternalsingalsareiiiibounded.Thepositivenessofk1265ET()andtheboundednessofET()implytheconvergenceoftheserieskTTTTllkke()e()d.Moreover,itcanbederivedthatlime()e()d=0.Wecanalsoconclude0k0l2kthatet()isuniformlyboundedon[0,]Tfrom(14),(15)andassumption2-3.Hencewecanderivethatklimet()0exactlyon[0,]TbytheBarbalat-likelemma.Furtherweobtainkklim()0tfromlemma2.Thusitisapprovedthatthefolloweragentscantracktheleaderk270exactlyon[0,]Tastheiterationnumberkapproachesinfinity.3FormationcontrolTheformationcontrolisgivenformulti-agentsysteminthissection.Ifthefolloweragentsreachthedesiredformation,itindicatesthattheMASachievetheleader-followingformationcontrol.275Thepositionerrorforithagentinthekthiterationisdefinedaskkxt()xt()iiiwhereisthedesireddistanceconstantforithagentfromtheleader.ikThegoaloftheformationcontrolistodesignthecontrolinputut()foreachagenttoensureithefolloweragentsretainadesireddistanceconstantfromtheleaderexactlyon[0,]Tasthe280iterationnumberincreases.Similarly,wedefinetheformationerrorforithagentaskk()txt()xii0Furthermore,wedefinethefuzzygeneralformationerrorforithagentas-8- 中国科技论文在线http://www.paper.edu.cnkpkkpketi()aijxtj()xti()bixt0()xti()(33)jNi285HereweinvestigatetheformationproblemsbyconvertingtotheconsensusproblemofMAS.Thenthefollowingtheoremisproposedasinsection3.Theorem2.Withthemulti-agentsystemproposedinTheorem1.Thedistributedcontrolprotocol(15)andparameteradaptivelaws(16)-(18)andwiththegeneralformationerrordefinedas(33)290canassurethatallinternalsignalsremainbounded.Moreover,theexactconvergenceoftheformationerrorisalsoensuredon[0,]Tastheiterationnumberk,thatis,klim()0,ti1,2,...,nki4SimulationexamplesInthissection,theillustrativeexampleswillbegiventoshowtheeffectivenessofthe295distributedfuzzyadaptiveiterativelearningprotocolsoftheorem1andtheorem2.Thefirstexampleisanumericalexampletoverifytheefficacyoftheproposedmethod.Theresultsforformationcontrolaregiveninexample2.Finally,theAILCprotocolisappliedtodealwiththevelocityconsensusformulti-vehiclesysteminexample3.ConsidertheconsensusproblemofMASwiththreefolloweragentsandoneleader.Here,we300presentaT-SfuzzysystemtodescribetheuncertaintopologygraphforMASwiththreeagentsandtheleaderanditsrulesareshownasfollows:Rule1:Ifzt()isMzt(()),then111kkeH1(34)305Rule2:Ifzt()isMzt(()),then121kkeH(35)2whereH1andH2aretheweightcommunicationmatricesinrules1and2.Thecommunication310graphsofRules1and2areassumedasfigures1(a)and1(b)asfollows311110HH12012112101011(a)(b)2013315Fig.1:communicationgraphsExample1.Thedynamicofthethreefolloweragentsaregivenbykkkkkxti()i()txti()sinxti()uti()dti(),i1,2,3Thedynamicoftheleadernodeisgivenasxtcos().022320Selecth(1cos())2,th(sin())2,td0.05,d0.1,d0.07,r0.1,r0.2,r0.3,0.3,0.4,12123123120.2,diag(3,4,7).WiththedistributedAILCprotocol,theconsensusresultsateachnode3areshowninFigs.2-6.-9- 中国科技论文在线http://www.paper.edu.cn325Fig.2:ThestateerrorsofMASFig.3:Thecontroluuu,,Fig.4:Normofˆ,,ˆˆ123123330Fig.5:Normofˆ,,ˆˆFig.6:Normofccc,,123123Fromthesimulationresults,wecanclearlyseethattheproposedalgorithmcanconfirmthe335desiredtrackingperformanceforfolloweragentsandtheboundnessofalltheinternalsignals.Example2.Herewepresenttheformationcontrolforthemulti-agentsystem.ConsidertheMASandthethecommunicationgraphscorrespondingtoRules1and2ofT-Sfuzzymodelaredescribedasexample1.22340Chooseh(1cos())2,th(sin())2,td0.05,d0.1,d0.08,r0.1,r0.2,r0.3,0.3,121231231-10- 中国科技论文在线http://www.paper.edu.cn0.4,0.2,diag(2,4,8).Thedesireddistanceconstantare0.4,0.8,1.2.The23123formationresultareshowninFigs.7-12.345Fig.7:ThestateerrorsofMASFig.8:Thecontroluuu,,Fig.9:Normofˆ,,ˆˆ123123350Fig.10:Normofˆ,,ˆˆFig.11:Normofccc,,123123-11- 中国科技论文在线http://www.paper.edu.cn355Fig.12:Statesoftheagentsinthe400thiterationFromFigs.7-12,wecandrawtheconclusionthattheformationcontrolformulti-agentsystemisachievedwiththedistributedAILCprotocolsproposedinthispaper.360Example3.Inthissimulation,weshowtheconsensusproblemofagroupofvehiclesonlyconsiderthevelocityandignorethedisplacementofeachvehicle.Considerasystemconsistingofthreefollowervehiclesandoneleadervehicle.Theinformationexchangeamongthevehiclesisuncertainandthetwocommunicationgraphscorrespondingtorule1andrule2oftheT-SfuzzymodelareasshowninFig.1(a)and(b).Thedynamicsofeachfollowervehicleisgivenbykkk365vti()1/mii()(())tfvtiiuti(),i1,2,3.wherevisthevelocity,uisthecontrolinput,misthemass,andfisthesumoffriction,airiiiikresistance,andotherforcesforvehiclei.Herewesupposefvt(()isaknownnonlinearfunctioniiandchoosekk2fvt(()(().vtiiiThedynamicsoftheleadervehiclesatisfies0vt()cos.t22370Select1/m1,i1,2,3.h(1cos())2,th(sin())2,tr0.1,r0.2,r0.3,0.3,0.4,i12123120.2,diag(2,5,7).WiththedistributedAILCprotocol,thevelocityconsensusresultsfora3multi-vehiclesystemareshowninFigs.13-17.375Fig.13:Thevelocityerrorsofmulti-vehiclesystem-12- 中国科技论文在线http://www.paper.edu.cnFig.14:Thecontroluuu,,Fig.15:Normofˆ,,ˆˆ123123380Fig.16:Normofˆ,,ˆˆFig.17:Normofccc,,123123385Asshowninfigures13-17,itcanbeconcludedthattheconsensusofallvehiclesisachievedandtheboundnessofalltheinternalsingalsarealsoensured,whichaffirmsthattheproposedalgorithmisalsovalidforthepracticalapplication.5ConclusionThispaperproposesadistributedfuzzyAILCprotocolunderinitial-statelearningcondition390oftheconsensusproblemandformationcontrolproblemforlinearlyparameterizedmulti-agentsystemswithuncertaincommunicationtopologystructure.T-Sfuzzymodelareusedtorepresenttheuncertaintopologygraph.BasedontheproposedAILCprotocol,thedesiredtrackingperformanceforfolloweragentsandtheboundnessofalltheinternalsignalsareensuredonthefiniteinterval[0,]Tforconsensusproblemandformationcontrolproblem.Thentheavailabilityof395theproposedalgorithmisaffirmedbythesimulationresults.Moreover,howtoapplytheproposedalgorithmtotheMASwithswitchingtopologieswillbethefocusofourfuturework.References[1]RENW.Consensusstrategiesforcooperativecontrolofvehicleformations[J].IETControlTheoryAppl.,4002007,505-512[2]Ahn,H.-S,Moore,K.L,CHEN.Y,Trajectory-keepinginsatelliteformationflyingviarobustperiodiclearningcontrol[J].Int.J.RobustNonlinearControl,2010,20(14):1655-1666.[3]RENW,andR.W.Beard.DistributedConsensusinMultvehicleCooperativeControl[M].London,U.K:Springer-Verlag,2008.-13- 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中国科技论文在线http://www.paper.edu.cn吴慧，李俊民（西安电子科技大学数学与统计学院，西安市710126）摘要：本文研究了在初始状态学习条件下的不确定通讯拓扑结构的线性参数化多智能体的分470布式一致性问题。用T-S模糊模型去描述了不确定的通讯拓扑结构，并基于一致性问题提出了不含任何全局信息的分布式自适应迭代学习协议。并用初始状态学习条件去设计自适应迭代学习律从而在每次迭代时不用去修正初始值。可以证明所提出的协议能保证所有多智能体系统的内部信号有界且从节点在[0,T]上能完全跟踪上头结点。通过选择合适的Lyapunov函数可以得到多智能体系统完全一致性的充分条件。同时本文通过转化为一致性问题的方法475也学习了多智能体的编队控制问题。最后，给出了几个仿真例子证实了理论的有效性。关键词：多智能体系统；自适应迭代学习控制；T-S模糊模型；不确定通讯拓扑结构中图分类号：O231.2-15-'