Source document: https://docs-lawep.s3.us-east-2.amazonaws.com/1698831096512.pdf (Sierra Leone)
thelocalinformationcanonlybecollectedoveraﬁniterange,itcanleadtotheformationofmorethanoneﬂock.Thispaperstudiesscenariosinwhichlocalinteractionleadstothefragmentationofgroupsandinvestigateshowtheintroductionofrandomprocessesinanagentcontrollerhelpstoeliminateoralleviatetheproblemoffragmentation.Inthispaper,weconsiderﬂockingintheLyapunovfunc-tionframework.TheLyapunovfunctionforthecontrollerdesignofamulti-agentsystemisbasedontheartiﬁcialpotentialfunctionofagentinteractions[8],[14].Beinginspiredbytheroleplayedbytherandomﬂuctuationindrivingself-organizingprocessesincells,weproposetoincludearandomprocessinthedecentralizedcontrollerforanagentswarm.Consequently,weanalyzetheswarmself-organizationusingthestochasticLyapunovfunction[7].Weshowthatourproposeddecentralizedcontrollerprovidestherobustﬂockingoftheswarmofagents.Thispaperisorganizedasfollows.WeintroduceamodelofswarmingagentsinsectionII.InsectionIII,wedescribeadeterministicdecentralizeddesignofcontrollersprovidingﬂockinginapopulationofagents.Dealingwiththeproblemoflocalminima,wesuggestincludingastochasticprocessintothecontrollerinsectionIV.InsectionV,wederiveemergentbehaviorrelatedtothecenterofthemassoftheﬂock.SectionVIpresentsnumericalsimulationsillustratingtheperformanceofthecontroller,andﬁnallysectionVIIpresentstheconclusionsderivedfromthepresentwork.II.MODELINGOFSWARMSYSTEMANDLYAPUNOVFUNCTIONDEFINITIONThegroupofmobileagentsconsistsofNfullyactuatedagents,eachofwhosedynamicsisgivenbythedoubleintegrator:˙qi=pi(1)˙pi=ui(t)i=1,2,...,Nwhereqi,pianduiarem-dimensionalposition,velocityandcontrolvectorsofagenti,respectively.Thisdoubleintegratorrepresentingparticledynamics[14],[8],[20]isapopularandrealisticmodeltorepresentthemotionofagentsinamulti-agentsystem.Itfacilitatestheimplementationofdecentralizedcontrolalgorithms,andprovidesamechanismtoincludelimitationsduetosensingandcommunicationascomparedtoothermodels,suchascontinuummodel[6],[11],whichrepresentsthecollectivemotionofagentsintheformofparticledensityfunctions.Themodelgiveninequation(1)canbewrittenas:˙x=Ax(t)+Bu(t)(2
)wherevectorx(t)=[q(t)p(t)]Tandq(t)=q1...qN,p(t)=p1...pN,u(t)=u1...uN(3)andmatricesAandBareA(x,t)=[IN×N0N×N]T,B=[0N×NIN×N]T.(4)Inordertocarryoutastabilityanalysisofthecollectivemotionofagents,aLyapunovfunctioncanbechosenasthetotalenergyφ(q,p)=V(q)+12pTp(5)TheLyapunovfunctioniscomposedofthetotalartiﬁcialpotentialenergyV(q)andthekineticenergy,thesecondtermofthesumgiveninequation(5).Wedeﬁnethepotentialenergyasanon-negativefunctionV(q)=12NXi=1Vi=12NXi=1Xj∈NiVij(kqi−qjk)(6)Vij(kqi−qjk)≥0(7)whereViisthetotalartiﬁcialpotentialassociatedwithanagenti.ThisenergydependsonthesetNicomprisedoftheagentsinsidethecommunicationrangeoftheagenti.TheartiﬁcialpotentialfunctionVijdependsupontheEuclediandistancekqi−qjkbetweentheagentiandj,anditisgivenby:Vij=(a(ln(kqi−qjk)+d0kqi−qjk),0≤kqi−qjk≤d1a(ln(d1)+d0d1),kqi−qjk>d1(8)where,aisascalarparameter.Theparametersd0andd1respectivelyrepresenttheinter-agentdistancebelowwhichtheinteractionforceisrepulsive(negative)andabovewhichtheinteractionforceiszero.Figure1showstheinteractionpotentialplottedagainsttheinter-agentdistance.Itcanbeeasilyseenthatthepotentialbecomesminimalwhentheinter-agentdistanceisd0.Theinteractionamongagentshappenswiththehelpofsensingorcommunicationdevices.Theparameterd1,then,canberegardedasasensingoracommunicationrange.05101505101520253035404550Interaction
PotentialInteragent Distance ||qj − qi||Magnitude of Potential d0d1Fig.1.Interactionpotentialversusinter-agentdistanceIII.CONTROLLERDESIGNFORFLOCKINGFlockingisaformofself-organizedbehaviorofagentswarmsinwhichagentsmeetorcometogether.Thecol-lectivedynamicsofthesystemcanbeanalyzedusinga1 2 4
Lyapunovfunction.Wecandifferentiateφ(q,p)withrespecttotime,andusingexpression(5)andmodel(1),onegets:˙φ(q,p)=pT∇V(q)+pT˙p=pT∇V(q)+pTu(t)UsingargumentsbasedontheLyapunovfunctionstabilityanalysis,thestateconﬁgurationp,qwillbestableif˙φ(q,p)=pT∇V(q)+pTu(t)≤0(9)Thecontrollawensuringthistypeofstabilityisdescribedbythefollowinglemmaofwhichadetailedproofisprovidedinreferences[14]and[8].Lemma3.1:ConsiderasystemofNmobileagents.Eachoftheagentsfollowsdynamicsgivenbymodel(1)andwiththefeedbackcontrollawgivenbyui=−∇qiVi+fvi(10)where∇qirepresentsthegradientwithrespecttopositionqiofagenti,c>0isascalargainandfviisgivenby:fvi=cXj∈Ni(pj−pi)(11)Foranyinitialconditionbelongingtothelevelsetofφ(q,p)givenbyΩC={(q,p):φ(q,p)≤C)}withC>0,andwhentheunderlyinggraphofthesystemisconnectedandcohesive,thesystemasymptoticallyconvergestoaninvariantsetΩI⊂ΩCsuchthatthepointsinΩIhaveavelocitythatisboundedandthevelocitiesofallagentsmatch.Toillustratethislemma,itisworthmentioningthatthecontrollaw(10)resultsfromthefactthatitcanbeequivalentlydescribedinthevectorformasu(t)=−∇V(q)−ˆL(q)p(12)whereˆL(q)∈RmN×mNism-dimensionalgraphLaplacian(seereference[14]),whichisapositivesemi-deﬁnitematrix.Obviously,thiscontrolsatisﬁescondition(9).Also,fromLasalle’sInvariancePrinciple,allsolutionsofthesystemstartinginΩCwillconvergetothelargestinvariantsetΩI=n(q,p)∈ΩC:˙φ(q,p)=0)o,andthisoccurswhenthevelocitiesofallagentsmatch.IV.CONTROLLERDESIGNBASEDONTHESTOCHASTICLYAPUNOVFUNCTIONUsingthedeterministicLyapunovfunctioncontrollerde-sign,agentsalwaysmoveinsuchawaythatthecovereddistancesandtheirdirectionsdonotincreasetheLyapunovfunction.Itisultimatelyexpectedthattheagentsreachastableformation(ﬂock)inwhichtheLyapunovfunctionattainsitsextremumvalue.Iftheagentsthatwearedealingwithhavealimitedcommunicationrange,itispossiblethatrobotsreachstableformationswithmorethanonecluster.Inthisconﬁguration,agentsfromoneclusterareoutofthecommunicationrangeofanyagentfromtheotherclusters.ThegraphLaplacianˆL(q)isnolongerpositivesemi-deﬁniteinthiscase,andalocalminimumoftheLyapunovfunctionisreached.Her
e,weintroduceacontrollerwhichincludestherandomprocessterm,providingmeanstoanescapefromlocalminima.UsingthenotationofsectionII,themodelofarobotpopulationcanbewritteninthematrixformas˙x=Ax(t)+Bu(t)+Σξ(t)(13)whereΣ=·000diag(σ1,σ2,...σN)¸2N×2N(14)andthecontrollereffectiscomposedofthepartwhichisassumedtobecomputedbasedonthelocalagentinformationBu(t)andthestochasticpartΣξ(t).UsingtheLyapunovfunctionasbeforeandusingtheˆItoformula,wecanﬁndthetimederivativeoftheLyapunovfunction˙φ=Xipi∂φ∂qi+ui(t)∂φ∂pi+12Xiσ2i∂2φ∂p2i+Xiσi∂φ∂piξi(t)(15)thatresultsin˙φ=Pipi∂Vi(q1,q2,...qN)∂qi+Piui(t)pi+(16)12Piσ2i+Piσipiξi(t)ThelatterexpressionmeansthatinthiscasetheLyapunovfunctionφoftheroboticpopulationisastochasticprocess.Similartothedeterministiccase,ifwewouldliketodesignacontrollerwhichalignstherobotvelocities,i.e.,providesﬂocking,wecandeﬁneuiasui=−∂Vi(q1,q2,...qN)∂qi−[ˆL(q)p]i(17)Underthiscondition˙φ=−pTˆL(q)p+12Xiσ2i+Xiσipiξi(t)(18)whichmeansthatthetotalLyapunovfunctionvalueisastochasticprocess.Thestochasticityprovidesescapefromthelocalminima.Weassume,withoutlosinggenerality,thatσi=σ.Theintensityofthisstochasticprocessisgovernedbytheparameterσwhichneedstobedeterminedbytakingappropriateconsiderationsoffactorsexplainedbelow.Thereisastochasticsteady-state(seereference[7],page50,Theorem6)forthevalueofφinwhichthefollowingconditionissatisﬁed:EnpTˆL(q)po=12Xiσ2i=Nσ22(19)Therearetwolimitsforσthatshouldbeavoided.Oneisσ<σLthatresultsinthedeterministiccontroller.Inthiscasetheswarmdoesnotﬂockrobustly.Thesecondlimitisσ>σH,whenσislargeandleadstoalargeexpectedvalueofpTˆL(q)p.Thismeansthattherobotvelocitiesarepoorlyaligned.Forreasonsoftherobustﬂocking,σshouldbeintherangeofσLandσH.Outofthisrange,theﬂockingdoesnothappenrobustlyduetothesmall(σ<σL)orlarge1
2 5 (σ>σH)intensityoftherandomprocess,respectively.Thevalueofσhastobetunedtakingintoaccounttheconstraintsoftheactuatorthatdrivestherobotsanditisapartofthecontrollerdesign.IntroducedcontrollerdoesnotguaranteetheattainmentofglobalminimumoftheLyapunovfunction,butitcanguaranteethattheformationofthesystemis,mostofthetime,aroundthisminimum.V.EMERGENTBEHAVIOROFASTOCHASTICCONTROLFORFLOCKINGTheclusterisagroupofrobotsinwhicheachrobotisinthecommunicationrangeofatleastonerobot.Motionofeachagentiobeysthefollowingstochasticdifferentialequation˙qi=pi(20)˙pi=ui(t)+σiξi(t).
(21)Thisexpressionincludestwoparts:theﬁrstpartbasedonthelocalinformationandthesecondpartbasedonthestochasticprocessofintensityσi.WeassumeagainthatξistheGaussianwhitenoiseoftheunitintensity.Forreasonsofsimplicity,butwithoutlosinggenerality,wewillconsiderthatσi=σ.Sinceeachrobothasauniquelabeli,theclusterCjcanbedeﬁnedasthesetoflabelsioftherobotsbeingintheclusterj.Assumingthattheclusterexistswithinthetimeintervalτ=[t1t2],themotionoftheCjcluster’scenterofthemassisdescribedby˙qCMj=pCMj(22)˙pCMj=1|Cj|Xi∈Cjσiξi(t)=σp|Cj|ξ(t)(23)wherethenumberoftherobotsintheclusterisdenotedby|Cj|.Thetermsdependentonuiarecancelledduetothesymmetryintheinteractionbetweenanytworobotsinthecluster.ThesumofwhiteGaussiannoisesisalsothewhiteGaussiannoiseandthelasttermin(23)includesthisproperlyscaledunitintensityGaussiannoise.From(23),wecanconcludethatthecenteroftheclustermassmovesrandomlythroughtheoperatingspace.Thesmallerthecluster,thefasterit”explores”theoperatingspacebeforeitmeetsotherrobotstoformalargercluster.Wequotetheword”explore”,becauseitisabehaviorthatemergesfromtheinclusionoftherandomprocessintherobotcontroller.Obviously,whileexploring,theclustermaydecomposeintotwosmallerclusters.Theprobabilityofclusterdecom-positioncanbemadeverysmallbyappropriatelychoosingtheintensityσtobesmallincomparisontotheintensityofui.Regardlessofthis,theclusterdecompositiondoesnotinﬂuenceourfurtheranalysis,becauseweanalyzethesteady-statepropertyoftherobotformationindependentlyofhowitisreached.Inthelimitwhen|Cj|islarge,thecenteroftheclustermasswillmovewithqCMj≈0.VI.SIMULATIONRESULTSBeforewepresentthesimulationresults,letusexplainsocialandhierarchicalsocialentropymetricsweusetoquantifythedegreeoftheself-organizationofagentpop-ulations.Socialentropy,inspiredbyShannon’sinformationentropy[17],hasbeenusedinmulti-agentsystems[1]asametricfordiversityinbehaviororpropertiesofagents,includingdiversityintheirspatiallocations.Thismetriccapturesanimportantfeatureofdiversity,whichisthenumberofdifferentiablegroups,i.e.,clusters,inasystemandthesizeoftheseclusters.Wedeﬁnetheclus
terasagroupofagentsinwhichthedistancebetweentheagentanditsclosestneighborfromthesameclusterissmallerthansomethresholdvalueh,whichisthecommunicationrangedistance.Ifanagentdoesnothavetheneighborwithinadistancesmallerthath,thenthisagentisaloneinthecluster.Ifagroupofnagents,representedbythesetR,canbedividedintomclustersC1,C2...Cm,suchthatthereisnoagentwhichissimultaneouslyintwoclusters,thenthesocialentropyHofthesystemRisgivenbyH(R)=−mXi=1|Ci||R|log|Ci||R|(24)with|.|representingthenumberofagentsintheclusterandPi|Ci|=|R|,where|R|=nisthenumberofrobotsinthegroup.ThesocialentropyHobviouslydependsonthethresholdh.Inordertodevelopameasurethatcanaccuratelyrepresentthediversityinapopulationofagentsregardlessofthescaleorthevalueh,hierarchicalsocialentropyhasbeenusedinliterature[1],whichisgivenby:E(R)=Z∞h=0H(R,h)dh(25)whereH(R,h)isthesocialentropygivenbyequation(24).Toillustrateﬂockingintheabsenceandthepresenceoftherandomforce,wesimulateagroupof20agentsduringa500sec.timespan.Inoursimulations,theagentshavealimitedcommunicationrange,whichismodeledwiththeartiﬁcialpotentialfunctionparameterd1=20m(seeequation(8)).Theotherparameterofthisfunctionisd0=6m,andtheintensityoftherandomprocessσ=0.25.Inthesimulationscenario,theagentsareallowedtomoveonlyinthe100×100rectangulararea(seeFig.2).Todeﬁnethisareainsidethesimulation,weintroduceanadditionalartiﬁcialpotentialfunctionresultinginshort-rangerepulsiveforcesmakingagentsmoveawayfromtheareaborder.Figure2showstheagentconﬁgurationsatdifferentpointsoftimeofthesimulationwhenrandomforceinthecontrollerisnotapplied.Theagentsstartfromaninitialrandomlychosenposition.BytheterminaltimeofthesimulationT=500sec,theagentsformthreeseparateclusters,whicharetheconsequenceofthelimitedcommunicationrange.Inthenextﬁgure,Fig.3,weillustratethesimulationinwhichtheagentsstartfromthesameconﬁgurationasin1
2 6 thepreviousexample,butnowtherandomforceinsidethecontrollerisapplied.Wenotethatbytheterminalsimulationtimetheagentsformasinglecluster.050100020406080100X Axis (m)Y Axis (m)050100020406080100X Axis (m)Y Axis (m)050100020406080100X Axis (m)Y Axis (m)050100020406080100X Axis (m)Y Axis (m)Fig.2.ConﬁgurationsofagentsattimesT=0(topleft),T=167sec(topright),T=333sec(bottomleft)andT=500sec(bottomright)whennorandommotionisapplied050100020406080100X Axis (m)Y Axis (m)050100020406080100X Axis (m)Y Axis (m)050100020406080100X Axis (m)Y Axis (m)050100020406080100X Axis (m)Y Axis (m)Fig.3.ConﬁgurationsofagentsattimesT=0(topleft),T=167sec(topright),T=333sec(bottomleft)andT=500sec(bottomright)whenrandommotionisappliedThehierarchicalsocialentropy,givenbyequation(25),forthesetwoexamplesisplottedagainstthesimulationtimeinFig.4.Fromthisﬁgure,itisevidentthattheapplicationofrandomforceresultsinlowerentropyvalues,meaningthatagentsaggregateclosertooneanotherformingbiggerclusters.Inordertoverifythattheintroductionofrandommotiondoesleadtobetterﬂockingbehaviorofagents,wecarriedouttwosetsof100simulationruns.Eachrunwascarriedoutunderthesameconditionsasinthetwopresentedexamples.Inonesimulationset,therandomforcetermofthecontrollerisappliedandintheother,itisnot.ForeachsimulationrunwecomputedthesocialentropyattheterminaltimeT=500sec.Thedistributionofthecomputedsocialentropyvalues,foreachsetofsimulations,ispresentedintheformofthecumulativedistributionfunctionshowninFig.5.Itcanbeseenthattheincreaseinthenumberof010020030040050001020304050Time (sec)Entropy010020030040050001020304050Time
 (sec)EntropyFig.4.Hierarchicalsocialentropyofthepopulationofagentsforthecasewhennorandommotionisapplied(left),andthecasewhenrandommotionisapplied(right)simulationoccurrencesforthecasewhenrandommotionisnotappliedisslowerthanthatofthecasewhenrandommotionisapplied.Thisisbecauseofthefactthatmostofthesimulationswithrandommotionhaveanentropywhichissmallerascomparedtosimulationswithnorandommotion.Asmallerentropyindicatesasmaller(spatial)diversity,i.e.,betterﬂockingbehavior.Theultimateevidencethattheapplicationoftherandomforcetermleadstoabetterﬂockingbehavioristhehistogramofnumberofsimulationswithspeciﬁcnumberofclustersintheﬁnalconﬁgurationforthetwocases,asshowninFig.6.Ascanbeseenfromtheﬁgure,alessernumberofclustersareformedforthatcasewhenrandommotionisappliedascomparedtothecasewhenrandommotionisnotapplied.Theaveragenumberofclustersforthecasewhennorandommotionisappliedis3.02,whiletheaveragenumberofclusterswhenrandommotionisappliedis1.51.Thisshowsamarkedimprovementinﬂockingbehaviorandtheformationofonegiantclusterascomparedtothecasewhennorandommotionisapplied.Itmaybenotedthattheseparateclustersformedinthesecondcasemoverandomlyinaconﬁnedspace.Thisrandommotionwithinaconﬁnedspaceresultsintotheprobabilityofclustersﬁndingeachotherapproachingunitywhenthetimeapproachesinﬁnity.00.511.52020406080100120Social entropy (H)Num.
of occurences with entropy < H No Random Motion AppliedRandom Motion AppliedFig.5.Simplesocialentropyatﬁnalconﬁgurationofagentsfor100simulationruns1 2 7 1234567010203040506070Number of clustersNumber of occurences1234567010203040506070Number of
clustersFig.6.Numberofclustersofagentsatﬁnalconﬁgurationfor100simulationrunswhennorandommotionisapplied(left)andwhenrandommotionisapplied(right)VII.CONCLUSIONInspiredbytheroleofrandomforcesinrobustnessofself-organizationofcellsandotherbiologicalsystems,weinvestigatedthepossibilityofdesigningadecentralizedcontrollerforaswarmofagentsinwhichastochasticprocessisincluded.WeconsideredtheﬂockingbehaviorofaswarmanddescribedthepreviouslyconsidereddeterministicLyapunovfunctioncontrollerdesignbasedontheartiﬁcialpotentialofinteractionsamongagents.Thedeterministicdesignmaynotleadtodesiredswarmbehaviorsbecauseoftheexistenceofagentconﬁgurationsinwhichthetotalpotentialofswarminteractionshaslocalminima.Thedesignweconsideredisanextensionofthedetermin-isticone.WeusedthesameLyapunovfunction.However,duetotheintroducedstochasticprocess,theLyapunovfunc-tionisalsoastochasticprocess.Consequently,thiscontrollerprovidesameansofescapingfromthelocalminimaofthetotalpotentialoftheswarminteractions.Theintensityoftheincludedstochasticprocessisofgreatsigniﬁcance.Ifitsintensityissmall,thentheswarmbehavioristhesameasifcontrolledbythedeterministiccontroller.Iftheintensityishigh,thentheﬂockingbehaviorcannotbeestablishedbecausetheswarmcanevenescapefromtheregionaroundtheglobalminimum.Onlyiftheintensityofthestochasticprocesslieswithinsomeintermediaterange,doestheswarmescapethelocalminimaandrandomlyexploresconﬁgurations,whichcanultimatelyleadtheswarmtowardsconﬁgurationsclosetotheglobalminimumofthetotalpotentialofinteractions.Thedrawbackofthisdesignisthatevenintheglobalminimumconﬁguration,thestochasticprocessforcestheswarmtosearchforabetterconﬁguration.Tuningtheintensityofthestochasticprocessofthecontrollerisidenticaltoconsiderationthatappearsinthedesignoftherealisticmodelofcellbehavior,ortotheproblemofprovidingphysiologicalconditionsforself-organizationinbiologicalcells.Theperformanceoftheproposedcontrollerwasillus-tratedbythederivationoftheemergentswarmbehaviorandanextensivesimulationstudy.Theperformanceofthecontrollerwiththerandomprocesstermwascompare
dtotheperformanceofthedeterministiccontroller.Thesimulationresultssuggestthattheinclusionoftherandomprocessinthecontrollercancertainlyimprovetheperformanceoftheroboticswarminachievingtheﬂockingbehavior.REFERENCES[1]T.Balch.Hierarchicsocialentropy:Aninformationtheoreticmeasureofrobotgroupdiversity.AutonomousRobots,8(3):209–237,2000.[2]T.BalchandR.Arkin.Behavior-basedformationcontrolformul-tirobotsystems.IEEETransactionsonRoboticsandAutomation,14(6):926–939,1998.[3]G.Bateson.MindandNature.NewYork:E.P.Dutton,1979.[4]J.P.Desai,J.P.Ostrowski,andV.Kumar.Modelingandcontrolofformationsofnonholonomicmobilerobots.IEEETransactionsonRoboticsandAutomation,17(6):905–908,2001.[5]D.Goldberg.GeneticAlgorithmsinSearch,Optimization&MachineLearning.AddisonWesleyLongman,Inc.,1989.[6]E.W.JusthandP.S.Krishnaprasad.Steeringlawsandcontinuummodelsforplanarformations.InIEEEInternationalConferenceonDecisionandControl,pages3609–3614,2003.[7]H.J.Kushner.StochasticStabilityandControl.AcademicPress,NewYork,1967.[8]N.LeonardandE.Fiorelli.Virtualleaders,artiﬁcialpotentialsandcoordinatedcontrolofgroups.InIEEEInternationalConferenceonDecisionandControl,pages2968–2973,2001.[9]K.Lorenz.BehindtheMirror:ASearchforaNaturalHistoryofHumanKnowledge.NewYork:HarcourtBraceJovanovich,1973.[10]D.Milutinovi´candP.Lima.CellsandRobots:ModelingandControlofLarge-SizeAgentPopulations.Springer,2007.[11]D.Milutinovi´c,P.Lima,andM.Athans.Biologicallyinspiredstochastichybridcontrolofmulti-robotsystems.InProceedingsofthe11thInternationalConferenceonAdvancedRobotics,2003.[12]G.NicolisandI.Prigogine.Self-organizationinnon-equilibriumsystems.Wiley,1977.[13]P.Ogren,M.Egerstedt,andX.Hu.AcontrolLyapunovfunctionapproachtomulti-agentcoordination.IEEETransactionsonRoboticsandAutomation,18(5):847–851,2002.[14]R.Olfati-Saber.Flockingformulti-agentdynamicsystems:algorithmsandtheory.IEEETransactionsonAutomaticControl,51(3):401–420,Mar.2006.[15]C.W.Reynolds.Flocks,herds,andschools:Adistributedbehavioralmodel.ComputerGraphics(SIGGRAPH’87
ConferenceProceed-ings),21(4):25–34,1987.[16]T.A.Schwickert,R.L.Lindquist,G.Shakhar,G.Livshits,D.Skokos,M.H.Kosco-Vilbois,M.L.Dustin,andM.C.Nussenzweig.Invivoimagingofgerminalcentresreavealsadynamicopenstructure.Nature,446:83–87,2007.[17]C.E.ShannonandW.Weaver.AMathematicalTheoryofCommuni-cation.UniversityofIllinoisPress,Urbana,IL,1949.[18]R.S.SuttonandA.G.Barto.ReinforcementLearning:AnIntroduc-tion,MITPress.MITPress,1998.[19]H.Tanner,A.Jadbabaie,andG.J.Pappas.Stableﬂockingofmobileagents,parti:Fixedtopology.InIEEEInternationalConferenceonDecisionandControl,pages2010–2015,2003.
[20]H.G.Tanner,A.Jadbabaie,andG.J.Pappas.Flockinginﬁxedandswitchingnetworks.IEEETransactionsonAutomaticControl,52(5):863–868,May2007.1 2 8