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Distributed Control of Multi-agent Systems

Permanent Link: http://ufdc.ufl.edu/UFE0044859/00001

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Title: Distributed Control of Multi-agent Systems Performance Scaling with Network Size
Physical Description: 1 online resource (132 p.)
Language: english
Creator: Hao, He
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: asymmetricdesign -- convergencerate -- distributedcontrol -- multiagentsystems -- performancescaling -- robustness
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The goal of distributed control of multi-agent systems (MASs) is to achieve a global control objective while using only locally available information. Each agent computes its own control action by using only information that can be obtained by either communication with its nearby neighbors or by on-board sensors. Recent years have witnessed a burgeoning interest in MASs due to their wide range of applications, such as automated highway system, surveillance and rescue by coordination of aerial and ground vehicles, spacecraft formation control for science missions. Most of these applications involve a large number of agents that are distributed over a broad geographical domain, in which a centralized control solution that requires all-to-all or all-to-one communication is impractical due to overwhelming communication demands. This motivates study of distributed control architectures, in which each agent makes control decisions based on only locally available information. Although it is more appealing than centralized control in this regard, distributed control suffers from a few limitations. In particular, its performance usually degrades as the number of agents in the collection increases. In this work, we examine two classes of distributed control problems: vehicular formation control and distributed consensus. Despite difference in their agent dynamics, the two problems are similar. In the vehicular formation control problem, each agent is modeled as a double-integrator. In contrast, the dynamics of each agent in distributed consensus is usually given by a single-integrator or its discrete counterpart. The goal of formation control is to make the vehicle team track a desired trajectory while keeping a rigid formation geometry, while the control objective of distributed consensus is to make all the agents' states converge to a common value. We study the scaling laws of certain performance metrics as a function of the number of agents in the system. We show that the performances for both vehicular formation and distributed consensus degrade when the number of agents in the system increases for symmetric control. Here symmetric control refers to, between each pair of neighboring agents (i,j), the weight agent i put on the information received from j is the same as the weight agent j put on the information received from i. Besides analysis, we also study how to design distributed control algorithms to improve performance scaling.   For the vehicular formation control problem, we describe a novel methodology for modelling, analysis and distributed control design. The method relies on a partial differential equation (PDE) approximation that describes the spatio-temporal evolution of each vehicle's position tracking error. The analysis and control design is based on this PDE model.  We deduce scaling laws of the closed-loop stability margin (absolute value of the real part of the least stable eigenvalue) and robustness to external disturbances (certain H-infinity norm of the system) of the controlled formation as a function of the number of vehicles in the formation. We show that the exponents in the scaling laws for both the stability margin and robustness to external disturbances are influenced by the dimension and the structure of the information graph, which describes the information exchange among neighboring vehicles. Moreover, the scaling laws can be improved by employing a higher dimensional information graph and/or using a beneficial aspect ratio for the information graph.   Apart from analysis, the PDE model is used for an asymmetric design of control gains to improve the stability margin and robustness to external disturbances. Asymmetric design means the information received from different neighbors are weighted prejudicially, instead of equally in symmetric design. We show that with asymmetric design, the system has a significantly better stability margin and robustness even with a small amount of asymmetry in the control gains. The results of the analysis with the PDE model are corroborated with numerical computation with the state-space model of the formation. Besides distributed control of vehicular formations, the progressive loss of performance has also been observed in distributed consensus, which has a wide range of applications such as distributed computing, sensor fusion and vehicle rendezvous. In distributed consensus, each agent in a network updates its state by using a weighted summation of its own state and the states of its neighbors. Prior works showed that with symmetric weights, the consensus rate became progressively smaller when the number of agents in the network increased, even when the weights were chosen to maximize the consensus rate. We show that with proper choice of asymmetric weights which are motivated by asymmetric control design for vehicular formations, the consensus rate can be improved significantly over symmetric design. In particular, we prove that the consensus rate in a lattice graph can be made independent of the size of the graph with asymmetric weights.  We also propose a weight design method for more general graphs than lattices. Numerical computations show that the resulting consensus rate with asymmetric weight design is improved considerably over that with symmetric optimal weights.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by He Hao.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Barooah, Prabir.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044859:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044859/00001

Material Information

Title: Distributed Control of Multi-agent Systems Performance Scaling with Network Size
Physical Description: 1 online resource (132 p.)
Language: english
Creator: Hao, He
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: asymmetricdesign -- convergencerate -- distributedcontrol -- multiagentsystems -- performancescaling -- robustness
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The goal of distributed control of multi-agent systems (MASs) is to achieve a global control objective while using only locally available information. Each agent computes its own control action by using only information that can be obtained by either communication with its nearby neighbors or by on-board sensors. Recent years have witnessed a burgeoning interest in MASs due to their wide range of applications, such as automated highway system, surveillance and rescue by coordination of aerial and ground vehicles, spacecraft formation control for science missions. Most of these applications involve a large number of agents that are distributed over a broad geographical domain, in which a centralized control solution that requires all-to-all or all-to-one communication is impractical due to overwhelming communication demands. This motivates study of distributed control architectures, in which each agent makes control decisions based on only locally available information. Although it is more appealing than centralized control in this regard, distributed control suffers from a few limitations. In particular, its performance usually degrades as the number of agents in the collection increases. In this work, we examine two classes of distributed control problems: vehicular formation control and distributed consensus. Despite difference in their agent dynamics, the two problems are similar. In the vehicular formation control problem, each agent is modeled as a double-integrator. In contrast, the dynamics of each agent in distributed consensus is usually given by a single-integrator or its discrete counterpart. The goal of formation control is to make the vehicle team track a desired trajectory while keeping a rigid formation geometry, while the control objective of distributed consensus is to make all the agents' states converge to a common value. We study the scaling laws of certain performance metrics as a function of the number of agents in the system. We show that the performances for both vehicular formation and distributed consensus degrade when the number of agents in the system increases for symmetric control. Here symmetric control refers to, between each pair of neighboring agents (i,j), the weight agent i put on the information received from j is the same as the weight agent j put on the information received from i. Besides analysis, we also study how to design distributed control algorithms to improve performance scaling.   For the vehicular formation control problem, we describe a novel methodology for modelling, analysis and distributed control design. The method relies on a partial differential equation (PDE) approximation that describes the spatio-temporal evolution of each vehicle's position tracking error. The analysis and control design is based on this PDE model.  We deduce scaling laws of the closed-loop stability margin (absolute value of the real part of the least stable eigenvalue) and robustness to external disturbances (certain H-infinity norm of the system) of the controlled formation as a function of the number of vehicles in the formation. We show that the exponents in the scaling laws for both the stability margin and robustness to external disturbances are influenced by the dimension and the structure of the information graph, which describes the information exchange among neighboring vehicles. Moreover, the scaling laws can be improved by employing a higher dimensional information graph and/or using a beneficial aspect ratio for the information graph.   Apart from analysis, the PDE model is used for an asymmetric design of control gains to improve the stability margin and robustness to external disturbances. Asymmetric design means the information received from different neighbors are weighted prejudicially, instead of equally in symmetric design. We show that with asymmetric design, the system has a significantly better stability margin and robustness even with a small amount of asymmetry in the control gains. The results of the analysis with the PDE model are corroborated with numerical computation with the state-space model of the formation. Besides distributed control of vehicular formations, the progressive loss of performance has also been observed in distributed consensus, which has a wide range of applications such as distributed computing, sensor fusion and vehicle rendezvous. In distributed consensus, each agent in a network updates its state by using a weighted summation of its own state and the states of its neighbors. Prior works showed that with symmetric weights, the consensus rate became progressively smaller when the number of agents in the network increased, even when the weights were chosen to maximize the consensus rate. We show that with proper choice of asymmetric weights which are motivated by asymmetric control design for vehicular formations, the consensus rate can be improved significantly over symmetric design. In particular, we prove that the consensus rate in a lattice graph can be made independent of the size of the graph with asymmetric weights.  We also propose a weight design method for more general graphs than lattices. Numerical computations show that the resulting consensus rate with asymmetric weight design is improved considerably over that with symmetric optimal weights.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by He Hao.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Barooah, Prabir.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044859:00001


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DISTRIBUTEDCONTROLOFMULTI-AGENTSYSTEMS:PERFORMANCE SCALINGWITHNETWORKSIZE By HEHAO ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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c r 2012HeHao 2

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Tomymotherandmywife 3

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ACKNOWLEDGMENTS IwouldliketoexpressmysinceregratitudetomyadvisorDr. PrabirBarooahfor leadingmethroughthiseort.Withouthisguidance,encour agementandsupport,this dissertationwouldhavenotbeenpossible.Iamverygratefu lforhissupervision,advice, andguidancefromtheinitialstageofthisresearchtothen alcompletionofthiswork. Fromhim,notonlydidIlearnhowtobearigorousandself-mot ivatedresearcher,but alsohowtodevelopcollaborativerelationshipswithother scientists.Heprovidedme unrinchingencouragementandsupportinvariouswaysandIa mindebtedtohimmore thanheknows.Ifeelveryfortunatetohavehadtheopportuni tytoworkwithhimandI wouldliketothankhimforalltheknowledgehehasimpartedt ome. IalsowanttoextendmyspecialgratitudetoDr.PrashantG.M ehta,whoisalways supportiveandhelpfultome.Iamgratefulforhisconstruct iveadviceandinspiring discussionsaswellashiscrucialcontributionstomyresea rch.Iamindebtedtohim forprovidingmetheopportunitytoworkwithhimfortwosumm ersatUIUC.His insightfullnessandspiritofadventureinresearchhavetr iggeredandnourishedmy intellectualdevelopment. ItisapleasuretothankProfessorsPramodKhargonekar,War renDixonandRichard Lindforbeinginmycommitteeandusingtheirprecioustimes toreadthisdissertation andgaveconstructivecommentstoimprovethequalityandpr esentationofthiswork. Ialsooermyregardsandgratitudetoallofthosewhosuppor tedmeinanyrespect duringthecompletionofthework. Lastbutnottheleast,Iwouldliketothanktwoofthemostimp ortantwomenin mylife:mymotherandmywife.Iamheartilythankfulforthei rlove,care,supportand encouragement.Withoutthem,mylifewouldhavebeenmuchle ssmeaningful. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 LISTOFFIGURES ....................................7 ABSTRACT ........................................8 CHAPTER 1INTRODUCTION ..................................11 1.1MotivationandProblemStatement ......................11 1.2RelatedLiterature ...............................14 1.3Contributions ..................................18 2STABILITYMARGINOFVEHICULARPLATOON ...............24 2.1ProblemFormulationandMainResults ....................27 2.1.1ProblemFormulation ..........................27 2.1.2MainResults ...............................29 2.2PDEModeloftheClosed-LoopDynamics ..................34 2.3RoleofHeterogeneityonStabilityMargin ..................36 2.4RoleofAsymmetryonStabilityMargin ....................41 2.4.1AsymmetricVelocityFeedback .....................41 2.4.2AsymmetricPositionandVelocityFeedbackwithEqual Asymmetry 44 2.4.3NumericalComparisonofStabilityMargin ..............47 2.5ScalingofStabilityMarginwithbothAsymmetryandHete rogeneity ...49 2.6Summary ....................................51 2.7TechnicalProofs ................................51 2.7.1ProofofTheorem2.2 ..........................51 2.7.2ProofofProposition2.1 .........................53 3ROBUSTNESSTOEXTERNALDISTURBANCEOFVEHICULARPLATOON 55 3.1ProblemFormulation ..............................57 3.2PDEModelsofthePlatoonwithSymmetricBidirectionalA rchitecture ..62 3.2.1PDEModelfortheCaseofLeader-to-TrailerAmplicat ion .....62 3.2.2PDEModelfortheCaseofAll-to-AllAmplication .........63 3.3RobustnesstoExternalDisturbances .....................64 3.3.1Leader-to-TrailerAmplicationwithSymmetricBid. Architecture .64 3.3.2All-to-allAmplicationwithSymmetricBidirection alArchitecture .66 3.3.3DisturbanceAmplicationwithPredecessor-Followi ngArchitecture 69 3.3.4DisturbanceAmplicationwithAsymmetricBid.Archi tecture ...70 3.3.5DesignGuidelines ............................71 3.3.6NumericalVerication .........................73 3.4Summary ....................................73 5

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4STABILITYANDROBUSTNESSOFHIGH-DIMENSIONALVEHICLETEA M 75 4.1ProblemFormulationandMainResults ....................76 4.1.1ProblemFormulation ..........................76 4.1.2MainResult1:ScalingLawsforStabilityMargin ..........81 4.1.3MainResult2:ScalingLawsforDisturbanceAmplicat ion .....84 4.2Closed-LoopDynamics:State-SpaceandPDEModels ............86 4.2.1State-SpaceModeloftheControlledVehicleFormatio n .......86 4.2.2PDEModeloftheControlledVehicleFormation ...........87 4.3AnalysisofStabilityMarginandDisturbanceAmplicat ion ........91 4.4Summary ....................................95 5FASTDISTRIBUTEDCONSENSUSTHROUGHASYMMETRICWEIGHTS .97 5.1ProblemFormulation ..............................99 5.2FastConsensusonD-dimensionalLattices ..................103 5.2.1AsymmetricWeightsinLattices ....................103 5.2.2NumericalComparison .........................107 5.3FastConsensusinMoreGeneralGraphs ...................107 5.3.1ContinuumApproximation .......................109 5.3.2WeightDesignforGeneralGraphs ...................113 5.3.3NumericalComparison .........................114 5.4Summary ....................................115 5.5TechnicalProofs ................................117 5.5.1ProofofLemma5.1 ...........................117 5.5.2ProofofLemma5.2 ...........................118 5.5.3ProofofLemma5.3 ...........................119 6CONCLUSIONSANDFUTUREWORK ......................121 6.1Conclusions ...................................121 6.2FutureWork ...................................123 REFERENCES .......................................124 BIOGRAPHICALSKETCH ................................132 6

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LISTOFFIGURES Figure page 2-1Desiredgeometryofaplatoonwith N vehiclesand1referencevehicle. .....27 2-2Numericalcomparisonofeigenvaluesbetweenstatespac eandPDEmodels. ...40 2-3Stabilitymarginoftheheterogeneousplatoonasafunct ionofnumberofvehicles. 41 2-4Stabilitymarginimprovementbyasymmetriccontrol. ..............48 2-5Therealpartofthemostunstableeigenvalueswithpoora symmetry. ......49 3-1Numericcomparisonofdisturbanceamplicationbetwee ndierentarchitectures. 72 4-1Examplesof1D,2Dand3Dlattices. ........................79 4-2Informationgraphfortwodistinctspatialformations. ...............80 4-3Informationgraphswithdierentaspectratios. ..................84 4-4Numericalvericationofstabilitymargin ......................85 4-5Apictorialrepresentationofthe i -thvehicleanditsfournearbyneighbors. ...87 4-6Originallattice,itsredrawnlatticeandacontinuousa pproximation. .......89 5-1Informationgraphfora1-Dlatticeof N agents. .................103 5-2Apictorialrepresentationofa2-dimensionallatticei nformationgraph .....105 5-3Comparisonofconvergenceratebetweenasymmetricands ymmetricdesign ...108 5-4Continuumapproximationofgeneralgraphs. ....................109 5-5Weightdesignforgeneralgraphs. ..........................113 5-6Examplesof2-DL-Zgeometric,Delaunayandrandomgeome tricgraphs. ....114 5-7Comparisonofconvergencerateswithdierentmethods .............116 7

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DISTRIBUTEDCONTROLOFMULTI-AGENTSYSTEMS:PERFORMANCE SCALINGWITHNETWORKSIZE By HeHao December2012 Chair:PrabirBarooahMajor:MechanicalEngineering Thegoalofdistributedcontrolofmulti-agentsystems(MAS s)istoachieveaglobal controlobjectivewhileusingonlylocallyavailableinfor mation.Eachagentcomputesits owncontrolactionbyusingonlyinformationthatcanbeobta inedbyeithercommunicationwithitsnearbyneighborsorbyon-boardsensors.Rec entyearshavewitnesseda burgeoninginterestinMASsduetotheirwiderangeofapplic ations,suchasautomated highwaysystem,surveillanceandrescuebycoordinationof aerialandgroundvehicles, spacecraftformationcontrolforsciencemissions.Mostof theseapplicationsinvolvealarge numberofagentsthataredistributedoverabroadgeographi caldomain,inwhichacentralizedcontrolsolutionthatrequiresall-to-allorallto-onecommunicationisimpractical duetooverwhelmingcommunicationdemands.Thismotivates studyofdistributedcontrol architectures,inwhicheachagentmakescontroldecisions basedononlylocallyavailableinformation.Althoughitismoreappealingthancentra lizedcontrolinthisregard, distributedcontrolsuersfromafewlimitations.Inparti cular,itsperformanceusually degradesasthenumberofagentsinthecollectionincreases Inthiswork,weexaminetwoclassesofdistributedcontrolp roblems:vehicular formationcontrolanddistributedconsensus.Despitedie renceintheiragentdynamics, thetwoproblemsaresimilar.Inthevehicularformationcon trolproblem,eachagentis modeledasadouble-integrator.Incontrast,thedynamicso feachagentindistributed consensusisusuallygivenbyasingle-integratororitsdis cretecounterpart.Thegoalof 8

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formationcontrolistomakethevehicleteamtrackadesired trajectorywhilekeeping arigidformationgeometry,whilethecontrolobjectiveofd istributedconsensusisto makealltheagents'statesconvergetoacommonvalue.Westu dythescalinglawsof certainperformancemetricsasafunctionofthenumberofag entsinthesystem.We showthattheperformancesforbothvehicularformationand distributedconsensus degradewhenthenumberofagentsinthesystemincreasesfor symmetriccontrol.Here symmetriccontrolrefersto,betweeneachpairofneighbori ngagents( i;j ),theweight agent i putontheinformationreceivedfrom j isthesameastheweightagent j puton theinformationreceivedfrom i .Besidesanalysis,wealsostudyhowtodesigndistributed controlalgorithmstoimproveperformancescaling. Forthevehicularformationcontrolproblem,wedescribean ovelmethodology formodelling,analysisanddistributedcontroldesign.Th emethodreliesonapartial dierentialequation(PDE)approximationthatdescribest hespatio-temporalevolution ofeachvehicle'spositiontrackingerror.Theanalysisand controldesignisbasedonthis PDEmodel.Wededucescalinglawsoftheclosed-loopstabili tymargin(absolutevalue oftherealpartoftheleaststableeigenvalue)androbustne sstoexternaldisturbances (certain H 1 normofthesystem)ofthecontrolledformationasafunction ofthenumber ofvehiclesintheformation.Weshowthattheexponentsinth escalinglawsforboththe stabilitymarginandrobustnesstoexternaldisturbancesa reinruencedbythedimension andthestructureoftheinformationgraph,whichdescribes theinformationexchange amongneighboringvehicles.Moreover,thescalinglawscan beimprovedbyemploying ahigherdimensionalinformationgraphand/orusingabene cialaspectratioforthe informationgraph. Apartfromanalysis,thePDEmodelisusedforan asymmetric designofcontrolgains toimprovethestabilitymarginandrobustnesstoexternald isturbances.Asymmetric designmeanstheinformationreceivedfromdierentneighb orsareweightedprejudicially, insteadofequallyinsymmetricdesign.Weshowthatwithasy mmetricdesign,thesystem 9

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hasasignicantlybetterstabilitymarginandrobustnesse venwithasmallamountof asymmetryinthecontrolgains.Theresultsoftheanalysisw iththePDEmodelare corroboratedwithnumericalcomputationwiththestate-sp acemodeloftheformation. Besidesdistributedcontrolofvehicularformations,thep rogressivelossofperformance hasalsobeenobservedindistributedconsensus,whichhasa widerangeofapplications suchasdistributedcomputing,sensorfusionandvehiclere ndezvous.Indistributed consensus,eachagentinanetworkupdatesitsstatebyusing aweightedsummationof itsownstateandthestatesofitsneighbors.Priorworkssho wedthatwithsymmetric weights,theconsensusratebecameprogressivelysmallerw henthenumberofagentsin thenetworkincreased,evenwhentheweightswerechosentom aximizetheconsensus rate.Weshowthatwithproperchoiceof asymmetric weightswhicharemotivatedby asymmetriccontroldesignforvehicularformations,theco nsensusratecanbeimproved signicantlyoversymmetricdesign.Inparticular,weprov ethattheconsensusrateina latticegraphcanbemadeindependentofthesizeofthegraph withasymmetricweights. Wealsoproposeaweightdesignmethodformoregeneralgraph sthanlattices.Numerical computationsshowthattheresultingconsensusratewithas ymmetricweightdesignis improvedconsiderablyoverthatwithsymmetricoptimalwei ghts. 10

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CHAPTER1 INTRODUCTION 1.1MotivationandProblemStatement Distributedcontrolhasspurredagreatinterestinthecont rolcommunityduetoits broadapplicationssuchascooperativecontrolofvehicula rformations[ 1 { 5 ],synchronizationofpowernetworksandcoupledoscillators[ 6 { 8 ],distributedconsensusofnetworked systems[ 9 { 11 ],studyofcollectivebehaviorofbirdrocksandanimalswar ms[ 12 { 14 ],and formationryingofunmannedaerialandgroundvehiclesfors urveillance,reconnaissance andrescue[ 15 { 19 ].Mostoftheseapplicationsarelarge-scalenetworkedmul ti-agentsystemsthataredistributedoverlargegeographicaldomains. Acentralizedcontrolsolution thatrequiresall-to-allorall-to-onecommunicationisim practicalduetooverwhelming communicationdemands.Thismotivatesinvestigationofdi stributedcontrolarchitectureswhereanindividualagentexchangesinformationonly withasmallsetofagents (neighbors)tomakecontroldecisions.Thegoalofdistribu tedcontrolistoachieveaglobal objectivebyusingonlylocallyavailableinformation. Inamulti-agentsystem,theinteractionbetweenneighbori ngagentsisoftendescribed byaninformationgraph.ItiswellknownthatthegraphLapla ciananditsspectral propertiesplayanimportantroleinstudyingtheperforman ceofthesystem[ 3 10 20 { 22 ].Therefore,toachievegoodclosed-loopperformance,the keyistodesignthe controlgainstooptimizecertaineigenvaluesofthegraphL aplacian.Theoptimizationof grapheigenvalueshasalwaysbeenatopicofinterestinengi neeringandscience[ 23 { 27 ]. However,mostworksassumethattheinformationgraphisund irected,whichmeansthe informationexchangebetweenneighboringagentsaresymme tric,i.e.betweentwoagents i and j thatexchangeinformation,theweightplacedby i ontheinformationreceivedfrom j isthesameastheweightplacedby j onthatreceivedfrom i .Thesymmetryassumption facilitatesanalysisanddesign.Inparticular,itmakesth eproblemofoptimizationof graphLaplacianeigenvaluesconvex.Severaldistributedc ontroldesignmethodhavebeen 11

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proposedbytakingadvantageoftheconvexityproperty[ 23 28 { 30 ].However,atypical issueindistributedcontroloflarge-scaleMASsisthatthe performanceoftheclosed-loop withsymmetricinformationgraphdegradesasthenumberofa gentsincreases.Several recentpapershavestudiedthescalingofperformanceasafu nctionofthenumberof agents[ 3 31 { 37 ]. Inthiswork,webreakthesymmetryandstudyhowtodesignadi stributedcontroller toachievereliableandscalablestabilityandrobustnessb yusingof asymmetric informationgraph.Directoptimizationisingeneralnotfeasiblei nthiscasesincetheproblem isnotconvex.Sowestartfromsymmetricdesignandexaminet heeectofintroducing smallasymmetryinthecontrolgains.Weshowthattheresult ingdesignyieldssignicant improvementofperformancemetrics(suchasconvergencera teandrobustnesstoexternal disturbances)oversymmetricdesign. Inthisdissertation,werstconsidertheproblemofcontro llingalargegroupof vehiclessothattheymaintainarigidformationgeometrywh ilefollowingadesired trajectory.Thedesiredformationgeometryisspeciedbyc onstantinter-vehiclespacings. Thedesiredtrajectoryoftheformationisgivenintermsofa ctitiousreferencevehicle, whosetrajectorycanbeaccessedbyonlyasmallsubsetofthe vehicles.Onetypical applicationofthisproblemisdistributedcontrolofvehic ularplatoons,whichaimsto maximizetracthroughputandincreasedrivingsafety.Thi stopichasgainedmuch attentioninthispastfewdecades[ 38 38 39 39 { 48 ].Intheplatoonproblem,eachvehicle intheformationmakesitsowncontroldecisionbasedonther elativeinformationsensed fromitsimmediatefrontandbackneighbors.Althoughthedy namicsofindividualvehicle isindependentoftheothers,thewholeclosed-loopbecomes acoupledsystem. Eachvehicleintheformationismodeledasadoubleintegrat or.Thedoubleintegratorisacommonlyusedmodelforvehiclesdynamics,whi chresultsfromfeedback linearizationofnon-linearvehiclemodels[ 39 49 { 51 ].Infact,itwaspointedoutin[ 52 53 ] thatintheformationcontrolproblem,foranyplantmodel P ( s )andlocalcontrollaw 12

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K ( s ),thekeyistohavetwointegratorsintheloopgain P ( s ) K ( s ).Thesingleintegratordynamicswillyieldsteadystatetrackingerrorwhilewi ththreeormoreintegrators theclosed-loopbecomesunstableforsucientlargenumber ofvehicles.Inadditionto vehicledynamics,thedoubleintegratoralsohasotherappl icationssuchasspacecraft attitudecontrol[ 54 ]andstudyingthemotionofafree-roatingparticle[ 55 ].Controlof double-integratoragentshasalsobeenextensivelystudie dforresearchandeducational purposes[ 56 { 59 ]. Westudyhowthestabilitymarginandrobustnesstoexternal disturbancesscalewith thenumberofvehicles,structureofinformationgraph,and thechoiceofthecontrolgains. Thestabilitymarginisdenedastheabsolutevalueofthere alpartoftheleaststable eigenvalueoftheclosed-loop.Itquantiesthesystem'sde cayrateofinitialerrors.The robustnesstoexternaldisturbanceismeasuredbycertain H 1 normofthesystem,which quantiesthesystem'sdisturbancerejectionability.Int hiswork,werestrictourselves toinformationgraphsthatbelongtotheclassof D -dimensional(nite)lattices.Lattices arisenaturallyasinformationgraphswhenthevehiclesint hegrouparearrangedina regularpatterninspaceandtheexchangeofinformationocc ursbetweenpairsofvehicles thatarephysicallyclose.Inaddition,latticesalsoallow forarexibilitytomodelmuch moregeneralinformationexchangearchitectures. Besidesvehicularformationcontrol,wealsostudytheprob lemofdistributedconsensusonalargenetwork,inwhicheachagentismodeledasas ingleintegratororits discretecounterpart.Indistributedconsensus,eachagen tupdatesitsstatebyusinga weightedsummationofitsownstateandthosefromitsneighb orsinthenetwork.The goalistomakealltheagents'statesasymptoticallyagreeo nacommonvalue.Distributed consensushasbeenwidelystudiedinthepastdecadeduetoit swiderangeofapplications suchasmulti-agentrendezvous,informationfusioninsens ornetwork,coordinatedcontrol ofmulti-agentsystem,randomwalkongraphs[ 9 { 11 ].Theconvergencerateofdistributed consensusisveryimportant,sinceitdeterminespractical applicabilityoftheprotocol.If 13

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theconvergencerateissmall,itwilltakemanyiterationsb eforethestatesofallagentsare sucientlyclose.Similartotheformationcontrolproblem ,distributedconsensusalsohas alimitation.Itsconvergencerateonsymmetricgraphsdegr adesasthenumberofagents inthenetworkincreases[ 36 ].Theconvergencerateischaracterizedbycertaineigenva lue ofitsgraphLaplacian.Weexaminehowdoestheconvergencer atescalewiththenumber ofagentsinthenetworkandhowtodesignthegraphweightsto improvetheconvergence rateofdistributedconsensus. 1.2RelatedLiterature Analysisofthestabilitymarginandrobustnesstoexternal disturbanceisimportanttounderstandthescalabilityofcontrolsolutionsast henumberofvehiclesinthe formation, N ,increases.Intheformationcontrolliterature,thescala bilityquestionhas beeninvestigatedprimarilyforaone-dimensionalvehicle formation,whichisusually referredtoasa platoon .It'saspecialcaseofvehicularformationwhoseinformati on graphisa1-Dlattice.Anextensiveliteratureexistsonthe platooncontrolproblem; see[ 38 43 51 60 61 ]andreferencestherein.Themostwidelystudiedinformati on exchangearchitecturesfordistributedcontrolofplatoon sare predecessorfollowing architecture, predecessor-leaderfollowing architectureand bidirectional architecture.Inthe predecessorfollowingarchitecture,everyvehicleonlyus esinformationfromitspredecessor,i.e.thevehicleimmediatelyahead.Inthepredecessor -leaderfollowingarchitecture, besidestheinformationfromitsimmediatepredecessor,th einformationoftheleaderis alsousedtocomputethecontrolaction.Inthebidirectiona larchitecture,eachvehicleuses therelativeinformationfromitsimmediatepredecessoran dfollower.Scenariosinwhich informationexchangeoccurswithvehiclesbeyondthosephy sicallyclosest,arestudied in[ 53 62 ].Withinthebidirectionalarchitecture,thefocusofmuch oftheresearchin thisareahasbeenontheso-called symmetric bidirectionalarchitecture,inwhichevery vehicleputequalweightontheinformationreceivedfromit spredecessorandfollower. Thesymmetryassumptionisusedtosimplifyanalysisanddes ign. 14

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Intheplatoonproblem,ithasbeenknownforquitesometime( see[ 31 45 46 ]and referencestherein)thatthepredecessor-followingarchi tecturesuersfromextremelypoor robustnesstoexternaldisturbances.Thisistypicallyref erredtoasstringinstabilityor slinky-typeeect[ 39 44 ].Seiler et.al. showedthatstringinstabilitywiththepredecessorfollowingarchitectureisindependentofthedesignofthec ontrolleroneachvehicle,but afundamentalartifactofthearchitecture[ 31 ].Stringinstabilitycanbeamelioratedby non-identicalcontrollersatthevehiclesbutattheexpens eofthecontrolgainsgrowing withoutboundasthenumberofthevehiclesincreases[ 39 63 ].Inaddition,itwasshown in[ 31 39 ]thatifthepredecessor-leaderfollowingarchitectureis used,theplatoonis stringstable.However,therequirementtotransmitthelea der'sinformationtoallthe othervehiclesmakesthisarchitectureunattractive.Inad dition,evenasmalltimedelay, whichisinevitableintransmittingtheleader'sinformati ontothefollowingvehicles,is enoughtocausestringinstabilityforlargeplatoons[ 64 65 ].Itshouldbementionedthat althoughstringstabilitycanalsobeachievedbyconstanth eadwaycontrolstrategy[ 39 ], theconstantheadwaypolicybyitselfisnotenough.Thehead wayhastobe largeenough toavoidtheproblemsassociatedwithconstantspacingpoli cy[ 66 ].Sinceoneofthemain motivationsforautomatedplatooningistoachievehigherh ighwaycapacitybymaking carsmovewithasmallinter-vehicleseparation,thereisan eedtostudytheconstant spacingpolicy. Thepoorrobustnesstodisturbanceofpredecessor-followi ngarchitectureledto theexaminationofthesymmetricbidirectionalarchitectu reforitsperceivedadvantage inrejectingdisturbances,especiallywithabsoluteveloc ityfeedback[ 46 ].However, thedistributedcontrolarchitectureswithsymmetriccont rolarelattershowntoscale poorlyintermsofclosed-loopstabilitymargin.Recallthe stabilitymarginisdened astheabsolutevalueoftherealpartoftheleaststableeige nvalue.Inasymmetric bidirectionalarchitecture,thestabilitymarginapproac heszeroas N increases[ 48 ].Small stabilitymarginwillcausethesystemtotakealongtimetos moothouttheinitial 15

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errors.Althoughitissuperioroverpredecessor-followin garchitectureinrobustnessto externaldisturbances(quantiedbycertain H 1 norm),itwasshownthattherobustness performanceofsymmetricbidirectionalarchitecturecann otbeuniformlyboundedwith thesizeoftheplatooneither[ 31 52 ].Indeed,thepoorrobustnesstodisturbancespersists evenformoregeneralarchitectures,wheneveryvehicleuse sinformationfrommorethan twoneighbors[ 62 ]. Asmentionedbefore,mostoftheworkonformationcontrolan ddistributedconsensusassumetheinformationgraphissymmetric.Thissymmetr yassumptioniscrucialto maketheanalysisandcontroldesigntractable.Itwasalsos hownabovethat,theformationcontrolproblemwithsymmetricinformationgraphsue rsfromfundamentallimitationinthescalabilityofclosed-loopperformance.Inaddi tion,itwasshownin[ 3 62 67 ] thatwithsymmetricinformationgraph,allowingheterogen eityinvehiclemassesandon theweightsoftheinformationgraphdoesnotsignicantlya lterthesystem'srobustness toexternaldisturbances.However,whentheinformationis asymmetric,thesituation becomestotallydierent,aswewillshowinthiswork.Witha symmetricinformation graph,theanalysisbecomesextremelydicult,asthereare fewsupportingtechniquesfor asymmetricdesign.Twonotableworkswithasymmetricdesig ninclude[ 48 68 ].In[ 48 ], Barooah et.al. proposedamistuning(asymmetric)designmethodtoimprove theclosedloopstabilitymarginofvehicularplatoonwithrelativepo sitionandabsolutevelocity feedback.Mistuningdesignreferstoallowingsmallpertur bationaroundthenominalcontrolgains.Itwasshownthattheresultingstabilitymargin withmistuningdesignyields aorderofmagnitudeimprovementoversymmetricdesign.In[ 68 ],TangermanandVeermanconsideredthecaseofrelativepositionandrelativeve locityfeedback,andtheyput equalasymmetryonthepositionandvelocitygains.Itwasco ncludedthattheconsidered asymmetriccontrolmadethesystem'srobustnesstoexterna ldisturbancemuchworsethan symmetriccontrol.Morespecically,itwasshownin[ 68 ]thatadisturbanceamplication 16

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metricgrowslinearlyin N forthesymmetricbidirectionalcasebutgrowsexponential lyin N withtheasymmetriccontrol.Thestabilitymarginwasnotex aminedintheirworks. Inadditiontothescalingofperformanceforthe1-Dvehicul arplatoons,therearealso afewothernotableworksonthevehicularformationinhighe r-dimensionalspace.Bamieh et.al. studiedcontrolledvehicleformationswitha D -dimensionaltorusastheinformation graph[ 32 ].Scalinglawswithsymmetriccontrolareobtainedforcert ainperformance measuresthatquantifytherobustnessoftheclosed-loopto stochasticnoises.Itwasshown in[ 32 ]thatthescalingoftheseperformancemeasureswith N wasstronglydependent onthedimension D oftheinformationgraph.DarbhaandYadlapalli et.al. examined thelimitationofemployingsymmetricinformationgraphfo rarbitraryformationfrom theperspectivesystem'srobustnesstosinusoidaldisturb ances[ 3 53 ].Theyconcluded thatwithsymmetricinformationgraph,the H 1 normofthesystemcannotbeuniformly boundedwiththesizeoftheformation.In[ 69 ],Pant et.al. introducedthenotionof mesh-stabilityfortwo-dimensionalformationswitha\loo k-ahead"informationexchange structure,whichreferstoaparticularkindofdirectedinf ormationrow. Thedegenerationofclosed-loopperformancewithsymmetry doesnotonlyexist intheformationcontrolliterature,itwasalsopointedout in[ 36 ]thattheconvergence rateofdistributedconsensusonlatticesandgeometricgra phswithsymmetricweights decayedtozeroasthenumberofagentsinthesystemincrease d,evenwithoptimal symmetricweightsobtainedfromconvexoptimization.Inth eformationcontrolliterature, thedynamicsofeachagentareusuallydescribedbyadoublei ntegrator,whileinthe consensusresearch,thedynamicsareingeneralgivenbyasi ngleintegratororitsdiscrete counterpart.Althoughdierentinthedynamicsmodels,the yhavethesamelimitation, i.e.theperformanceoftheclosed-loopdegradesasthenumb erofagentsinthesystem increases.Thelossofperformancecanbeattributedtothed egenerationofcertain eigenvaluesofthesymmetricgraphLaplacianwhenthesizeo fthegraphincreases. 17

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Theliteratureonconvergencerateofdistributedconsensu sisnotrich.Afewworks canbefoundin[ 70 { 72 ].TherelatedproblemofmixingtimeofMarkovchainsisstud ied in[ 73 ].In[ 36 ],convergencerateforaspecicclassofgraphs,thatwecal lL-Zgeometric graphs,wasestablishedasafunctionofthenumberofagents .Ingeneral,theconvergence ratesofdistributedconsensusalgorithmstendtobeslow,a nddecreaseasthenumberof agentsincreases.Itwasshownin[ 74 ]thattheconvergenceratecouldbearbitrarilyfastin small-worldnetworks.However,networksinwhichcommunic ationisonlypossiblebetween agentsthatarecloseenougharenotlikelytobesmall-world Oneoftheseminalworksonimprovingconvergenceratesofdi stributedconsensus protocolsisconvexoptimizationofweightsonedgesoftheg raphtomaximizethe consensusrate[ 27 29 ].Convexoptimizationimposestheconstraintthattheweig hts ofthegraphmustbesymmetric,whichmeansanytwoneighbori ngagentsputequal weightontheinformationreceivedfromeachother.However ,theconvergenceratesof distributedconsensusprotocolsongraphswithsymmetricw eightsdegradeconsiderably asthenumberofagentsinthenetworkincreases.InaD-dimen sionallattice,forinstance, theconvergencerateis O (1 =N 2 =D )iftheweightsaresymmetric,where N isthenumber ofagents.Thisresultfollowsasaspecialcaseoftheresult sin[ 36 ].Thus,theconvergence ratebecomesarbitrarilysmallifthesizeofthenetworkgro wswithoutbound. In[ 75 76 ],nite-timedistributedconsensusprotocolswerepropos edtoimprovethe performanceoverasymptoticconsensus.However,ingenera l,thenitetimeneededto achieveconsensusdependsonthenumberofagentsinthenetw ork.Thus,forlargesize ofnetworks,althoughconsensuscanbereachedinnitetime ,thetimeneededisvery large[ 75 76 ]. 1.3Contributions Inthisdissertation,westudytheperformancescalingofdi stributedcontroloflargescalemulti-agentsystemswithrespecttoitsnetworksize. Weinvestigatetwoclassesof distributedcontrolproblems:vehicularformationcontro landdistributedconsensus. 18

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Fortheformationcontrolproblem,wedescribeamethodolog yformodeling,analysis, anddistributedcontroldesignforlarge-scalevehiculart eamswhoseinformationgraphs belongtotheclassof D -dimensionallattices.The1-Dvehicularplatoonisaspeci al case,itsinformationgraphisa1-Dlattice.Theapproachis touseapartialdierential equation(PDE)basedcontinuousapproximationofthe(spat ially)discreteplatoon dynamics.OurPDEmodelyieldstheoriginalsetofordinaryd ierentialequationsupon discretization.Thisapproachismotivatedbyearlierwork onPDEmodelingofonedimensionalplatoons[ 48 ].ThePDEmodelisusedforanalysisofstabilitymarginand robustnesstodisturbancesaswellasforasymmetricdesign ofdistributedcontrollaws. Forthedistributedconsensusproblem,weproposeanasymme tricweightdesign methodtoimproveitsconvergencerate.Theasymmetricweig htdesignideaismotivated byasymmetricdesignofdistributedcontrollawsforvehicu larformations.Besides networkswith D -dimensionallatticegraphs,wealsodevelopaweightdesig nalgorithm formoregeneralgraphsthanlattices.Theweightdesignmet hodisbasedonacontinuous approximation,inwhichthegraphLaplacianofthenetworki sapproximatedbyaSturmLiouvilleoperator[ 77 ].Weshowthatwiththedevelopeddesignmethod,theconverg ence rateofdistributedconsensuswithasymmetricweightsisim provedsignicantlyoverthat withsymmetricweights. Therearevecontributionsofthisworkthataresummarized below. First,forformationwithsymmetricinformationgraph,weo btainexactquantitative scalinglawsoftheclosed-loopstabilitymarginandrobust nesstoexternaldisturbancesof thevehicularformationwithrespecttothenumberofvehicl esinthesystem.Weassume thatonlythevehiclesononeboundaryofthelatticehaveacc esstothedesiredtrajectory ofthereferencevehicle.Weshowthatthestabilitymargina ndrobustnesstoexternal disturbanceonlydependon N 1 ,where N 1 isthenumberofvehiclesalongtheaxisthatis perpendiculartotheboundarywherethereferencevehicles arelocated.Bychoosingthe structureoftheinformationgraphinsuchawaythat N 1 increasesslowlyinrelationto 19

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N ,thereductionofthestabilitymarginanddisturbanceampl icationasafunctionof N canbesloweddown.Infact,byholding N 1 tobeaconstantindependentofthenumber ofvehicles N ,thestabilitymarginanddisturbanceamplicationcanbeb oundedaway fromzeroevenasthenumberofvehiclesincreasewithoutbou nd.Itturnsout,however, thatkeeping N 1 xedwhile N increasescauseslongrangecommunicationand/orthe numberofvehiclesthathaveaccesstothedesiredtrajector yofthereferencevehicleto increase.Inaddition,whentheinformationgraphissquare ,whichmeansthereareequal numberofvehiclesineachaxisoftheinformationgraph,wes howthattheexponentsof thescalinglawsofthestabilitymarginanddisturbanceamp licationdependon D ,the dimensionoftheinformationgraph.Thestabilitymarginan ddisturbanceamplication canbeimprovedconsiderablybyapplyingahigher-dimensio nalinformationgraph. Thesecondcontributionofthisworkisaproceduretodesign asymmetric control gainssothatthestabilitymarginanddisturbanceamplica tionscalinglawsaresignificantlyimprovedoverthosewithsymmetriccontrol.Forthe 1-Dvehicularplatoon,we showthatwith asymmetricvelocityfeedback ,whichallowsanarbitrarilysmallasymmetry inthevelocitygainsfromtheirnominalsymmetricvalues,r esultsinstabilitymargin scalingas O ( 1 N ),where N isthenumberofvehiclesintheplatoon.Incontrasttothe O ( 1 N 2 )scalingseeninthesymmetriccase,thisisanorderofmagni tudeimprovement. Inaddition,whenthereisequalamountofasymmetryinbotht hepositionandvelocity feedback,thestabilitymargincanbeimprovedevenbettert o O (1),whichisindependent ofthesizeofthenetwork.Thisasymmetricdesignthuselimi natestheproblemofdecayto stabilitymarginwithincreasing N ,asseenwithsymmetricdesign.Intermsofdisturbance amplication,itwasshownbyVeermanthatasymmetricdesig nwithequalasymmetryin thepositionandvelocityfeedbackhadworserobustnesstoe xternaldisturbancescompared tosymmetriccase[ 33 ].However,ifasymmetryisonlyintroducedintotherelativ evelocity feedback( asymmetricvelocityfeedback ),numericalsimulationsshowthatthedisturbance amplicationcanbeimprovedsignicantlyoversymmetricd esign.Therefore,toachieve 20

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betterstabilitymarginandrobustnesstoexternaldisturb ancesimultaneously, asymmetric velocityfeedback isthebestdesignchoice.Theasymmetricdesignmethodcana lsobe extendedtovehicularformationswithhigher-dimensional latticeinformationgraphs. Thethirdcontributionoftheworkisweshowthatheterogene ityinvehiclemassand controlgainshaslittleeectonthestabilitymarginofave hicularplatoon.Inparticular, weshowthattheallowingheterogeneityonlychangesthecoe cientofthescalinglaw ofthestabilitymarginbutnotitsasymptotictrendwith N ,where N isthenumberof vehiclesintheplatoon.Aslongasthecontrolissymmetric, thescalinglawofthestability marginwithandwithoutheterogeneityareboth O (1 =N 2 ).Inconnectiontooptimizing theeigenvaluesofgraphLaplacian,ourresultsshowthatfo rsymmetricgraphs,evenby convexoptimization,whichallowsheterogeneityonthewei ghtsofthegraphtooptimize itseigenvalues,thedegenerationofcertaineigenvaluesi sinevitablewhenthesizeofthe graphincreases.Similarresultswereobtainedindependen tlyin[ 36 ]. Thefourthcontributionoftheworkistheapproachusedinde rivingtheresults mentionedabove.Wederiveapartialdierentialequation( PDE)basedcontinuousapproximationofthe(spatially)discreteformationdynamic s.Partialdierentialequations havebeengainingattentioninstudyinglarge-scaledistri butedsystemssuchaspowernetworks,coupled-oscillatorsandextremelylargetelescope s[ 6 78 { 81 ].APDEapproximation isalsofrequentlyusedintheanalysisofmany-particlesys temsinstatisticalphysicsand trac-dynamics;see[ 82 ]andthereferencestherein.Duetothelargescalefeatureo fthe studiedsystem,theclassicalcoupled-ODE(ordinarydier entialequation)modelseems unaptandinecient,anditprovidesnoinsightonanalysisa nddesign.ThePDEmodel providesasinglecompactmodelforthewholesystem,regard lessofhowmanyagentsare inthesystem.TheadvantageofusingaPDE-basedanalysisis thatthePDEreveals, betterthanthestate-spacemodeldoes,themechanismoflos sofstabilityandsuggests theasymmetricdesignapproachtoameliorateit.Inadditio n,thePDEmodelgivesmore insightonthesystem'sfrequencyresponse,whichaidstode rivethescalinglawofthe 21

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robustnesstoexternaldisturbance(quantiedbycertain H 1 norm).Numericalcomputationsofthestabilitymarginand H 1 normofthestate-spacemodeloftheformation areusedtoconrmthePDEpredictions.AlthoughthePDEmode lapproximatesthe (spatially)discreteformationdynamicsinthelimit N !1 ,numericalcalculationsshow thattheconclusionsdrawnfromthePDE-basedanalysishold sevenforsmallnumberof vehicles.Almostofallthescalinglawsderivedintheworkc anbeestablishedbyanalyzing thestate-spacemodelwiththecontrolgainssuggestedbyth ePDEmodel.Infact,the publicationsresultingfromthisworkcontainssuchanalys is.Wedon'tpresenttheanalysis inthisworktoavoidrepetition. Thelastbutnottheleastcontributionisamethodtoimprove theconvergencerate ofdistributedconsensusprotocolsthrough asymmetric weights.Werstconsiderlattice graphs,andshowthatwithproperchoiceofasymmetricweigh ts,theconvergencerateof distributedconsensuscanbeboundedawayfromzerouniform lyin N .Thus,theproposed asymmetricdesignmakesdistributedconsensushighlyscal able.Wenextproposeaweight designalgorithmfor2-dimensionalgeometricgraphs,i.e. ,graphsconsistingofnodesin R 2 Numericalsimulationsshowthattheconvergenceratewitha symmetricdesignedweights inlargegraphsisanorderofmagnitudehigherthanthatwith (i)optimalsymmetric weights,whichareobtainedbyconvexoptimization,and(ii )asymmetricweightsobtained byMetropolis-Hastingsmethod,whichassignsweightsunif ormlytoeachedgeconnecting itselftoitsneighbor.Theproposedweightdesignmethodis decentralized;everynode canobtainitsownweightbasedontheangularpositionmeasu rementswithitsneighbors. Inaddition,itiscomputationallymuchcheaperthanobtain ingtheoptimalsymmetric weightsusingconvexoptimizationmethod.Theproposedwei ghtdesignmethodcanbe extendedtogeometricgraphsin R D ,butinthisworkwelimitourselvesto R 2 Theremainderofthisdissertationisorganizedasfollows. Foreaseofdescription,we rstpresenttheproblemandresultson1-Dvehicularplatoo n.Chapter 2 presentsscaling lawsofstabilitymarginofthe1-Dvehicularplatoonwithsy mmetriccontrolaswellas 22

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theeectofasymmetricdesignontheclosed-loopstability margin.Chapter 3 describes thescalinglawsofrobustnesstoexternaldisturbancesoft he1-Dvehicularplatoon andasymmetricdesigntoimprovethedisturbanceamplicat ion.Distributedcontrolof vehicularformationinhigher-dimensionalspaceandthee ectofnetworkstructureonthe scalinglawsofstabilitymarginandrobustnessarepresent edinChapter 4 .Themethodof improvingconvergencerateofdistributedconsensusthrou ghasymmetricweightsdesign isdescribedinChapter 5 .Thedissertationendswithconclusionsandfutureworksin Chapter 6 23

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CHAPTER2 STABILITYMARGINOF1-DVEHICULARPLATOON Inthischapterweexaminetheclosed-loopstabilitymargin ofavehicularplatoon consistingof N vehicles,inwhicheachvehicleismodeledasadouble-integ ratorand interactswithitstwonearestneighbors(oneoneitherside )throughitslocalcontrol action.Thisisaproblemthatisofprimaryinteresttoautom atedplatooninsmart highwaysystems.Inthevehicularplatoonproblem,theform ationaimstotrackadesired trajectorywhilemaintainingarigidformationgeometry.T hedesiredtrajectoryofthe entirevehicularplatoonisgivenintermsoftrajectoryofa ctitiousreferencevehicle,and thedesiredformationgeometryisspeciedintermsofconst antinter-vehiclespacings. Althoughsignicantamountofresearchhasbeenconductedo nrobustness-todisturbanceandstabilityissuesofdoubleintegratornetw orkswithdecentralizedcontrol, mostinvestigationsconsiderthehomogeneouscaseinwhich eachvehiclehasthesame massandemploysthesamecontroller(exceptionsinclude[ 15 62 63 ]).Inaddition,only symmetriccontrollawsareconsideredinwhichtheinformat ionfromboththeneighboring vehiclesareweightedequally,with[ 33 48 ]beingexceptions.Khatir et.al. proposed heterogeneouscontrolgainstoimprovestringstability(s ensitivitytodisturbance)atthe expenseofcontrolgainsincreasingwithoutboundas N increases[ 63 ].Middleton et.al. consideredbothunidirectionalandbidirectionalcontrol ,andconcludedheterogeneityhad littleeectonthestringstabilityundercertainconditio nsonthehighfrequencybehavior andintegralabsoluteerror[ 62 ].Ontheotherhand,[ 33 ]examinedtheeectofequal asymmetryinpositionandvelocitygains(butnotheterogen eity)ontheresponseofthe platoonasaresultofsinusoidaldisturbanceintheleadveh icle,andconcludedthatthis asymmetrymadesensitivitytosuchdisturbancesworse. Inthischapterweanalyzethecasewhenthevehiclesare heterogeneous intheir massesandcontrollawsused,andalsoallowasymmetryinthe useoffrontandback 24

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information.Adecentralized bidirectional controllawnotnecessarilysymmetricisconsideredthatusesonlyrelativepositionandrelativeveloc ityinformationfromthenearest neighbors.Weexaminetheeectofheterogeneityandasymme tryonthestabilitymargin oftheclosedloop,whichismeasuredbytheabsolutevalueof therealpartoftheleast stablepole.Thestabilitymargindeterminesthedecayrate ofinitialformationkeeping errors.Sucherrorsarisefrompoorinitialarrangementoft hevehicles.Themainresult ofthechapteristhatinadecentralizedbidirectionalcont rolstrategy,heterogeneityhas littleeectonthestabilitymarginoftheoverallclosedlo op,whileevensmallasymmetry canhaveasignicantimpact.Inparticular,weshowthatint hesymmetriccase,the stabilitymargindecaysto0as O (1 =N 2 ),where N isthenumberofvehicles.Wealsoshow thattheasymptoticscalingtrendofstabilitymarginisnot changedbyvehicle-to-vehicle heterogeneity.Ontheotherhand,arbitrarysmallamountof asymmetryinthewaythe localcontrollersusefrontandbackinformationcanimprov ethestabilitymarginbya considerableamount.Wheneachvehicleweighstherelative velocityinformationfromits frontneighbormoreheavilythantheonebehindit,thestabi litymarginscalingtrendcan beimprovedfrom O (1 =N 2 )to O (1 =N ).Incontrast,ifmoreweightisgiventotherelative velocityinformationwiththeneighborbehindit,theclose dloopbecomesunstableif N is sucientlylarge.Inaddition,whenthereisequalamountof asymmetryinpositionand velocityfeedbackgains,theclosed-loopisexponentially stableforarbitrarynite N ,and thestabilitymargincanbeuniformlyboundedwiththesizeo fthenetwork.Thisresult makesitpossibletodesignthecontrolgainssothatthestab ilitymarginofthesystem satisesapre-speciedvalueirrespectiveofhowmanyvehi clesareintheformation. TheresultsareestablishedbyusingaPDEmodel.ThePDEmode lapproximates thecoupledsystemofODEsthatgoverntheclosedloopdynami csofthenetwork.This isinspiredbythework[ 48 ]thatexaminedstabilitymarginof1-Dvehicularplatoonsi na similarframework.Comparedto[ 48 ],thisworkmakestwonovelcontributions.First,we considerheterogeneousvehicles(themassandcontrolgain svaryfromvehicletovehicle), 25

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whereas[ 48 ]consideronlyhomogeneousvehicles.Secondly,[ 48 ]consideredthescenario inwhicheveryvehicleknewthedesiredvelocityoftheplato on.Incontrast,thecontrol lawweconsiderrequiresvehiclestoknowonlythedesiredin ter-vehicleseparation;the overalltrajectoryinformationismadeavailableonlytove hicle1.Thismakesthemodel moreapplicabletopracticalformationcontrolapplicatio ns.Itwasshownin[ 48 ]forthe homogeneousformationthatasymmetryinthepositionfeedb ackcanimprovethestability marginfrom O (1 =N 2 )to O (1 =N )whiletheabsolutevelocityfeedbackgaindidnotaect theasymptotictrend.Incontrast,weshowinthischapterth atwithrelativeposition andrelativevelocityfeedback,asymmetryinthevelocityf eedbackgainaloneandinboth positionandvelocityfeedbackgainsarebothimportant.Th estabilitymargincanbe improvedconsiderablybyajudiciouschoiceofasymmetry. ThePDEmodelprovidesinsightsintolossofstabilitymargi nwithsymmetriccontrol andsuggestsanasymmetricdesignmethodtoimprovethestab ilitymargin.Although thePDEapproximationisvalidonlyinthelimit N !1 ,numericalcomparisonswith theoriginalstate-spacemodelshowsthatthePDEmodelprov idesaccurateresultseven forsmall N (5to10).ThePDEapproximationisoftenusedinstudyingman y-particle systemsandinanalyzingmulti-vehiclecoordinationprobl ems[ 48 79 80 82 ].Asimilar butdistinctframeworkbasedonpartial dierence equationshasbeendevelopedby Ferrari-Trecate et.al. [ 83 ]. Therestofthischapterisorganizedasfollows.Section 2.1 presentstheproblem statementandthemainresults.Section 2.2 describesthePDEmodelofclosed-loop dynamics.Analysisandcontroldesignresultstogetherwit htheirnumericalcorroboration appearinSection 2.3 -Section 2.5 ,respectively.Thissectionendswithsummaryin Section 2.6 26

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... O X 0 ; 1 N 1 ;N 0 1 N 1 N (a)Apictorialrepresentationofaplatoon. ... 0 1 x 1 =N 1 =N Dirichlet Neumann (b)ARedrawngraphofthesameplatoon. Figure2-1.Desiredgeometryofaplatoonwith N vehiclesand1referencevehicle. 2.1ProblemFormulationandMainResults 2.1.1ProblemFormulation Weconsidertheformationcontrolof N heterogeneousvehicleswhicharemovingin1DEuclideanspace,asshowninFigure 2-1 (a).Thepositionandmassofeachvehicleare denotedby p i and m i respectively.Themassofeachvehicleisbounded, j m i m 0 j =m 0 forall i ,where m 0 > 0and 2 [0 ; 1)areconstants.Thedynamicsofeachvehicleare modeledasadoubleintegrator: m i p i = u i ; (2{1) where u i isthecontrolinput(accelerationordecelerationcommand ).Thisisacommonly usedmodelforvehicledynamicsinstudyingvehicularforma tions,whichresultsfrom feedbacklinearizationofnon-linearvehicledynamics[ 39 49 ]. Thedesiredtrajectoryoftheformationisgivenintermsofa ctitious reference vehiclewithindex0whosetrajectoryisdenotedby p 0 ( t ).Sinceweareinterestedin translationalmaneuversoftheformation,weassumethedes iredtrajectoryisaconstantvelocitytype,i.e. p 0 ( t )= v 0 t + c 0 forsomeconstants v 0 and c 0 .Theinformationonthe desiredtrajectoryofthenetworkisprovidedonlytovehicl e1.Thedesiredgeometryof theformationisspeciedbythe desiredgaps i 1 ;i for i =1 ;:::;N ,where i 1 ;i isthe desiredvalueof p i 1 ( t ) p i ( t ).Thecontrolobjectiveistomaintainarigidformation,i. e., tomakeneighboringvehiclesmaintaintheirpre-speciedd esiredgapsandtomakevehicle 1followitsdesiredtrajectory p 0 ( t ) 0 ; 1 .Sinceweareonlyinterestedinmaintaining rigidformationsthatdonotchangeshapeovertime, i 1 ;i 'sarepositiveconstants. 27

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Inthischapter,weconsiderthefollowing decentralized controllaw,wherebythe controlactionatthe i -thvehicledependsoni)the relativepositionmeasurements ii)the relativevelocitymeasurements withitsimmediateneighborsintheformation: u i = k f i ( p i p i 1 + i 1 ;i ) k b i ( p i p i +1 i;i +1 ) b fi (_ p i p i 1 ) b bi (_ p i p i +1 ) ; (2{2) where i = f 1 ;:::;N 1 g k f i ;k b i arethefrontandbackpositiongainsand b fi ;b bi arethe frontandbackvelocitygainsofthe i -thvehiclerespectively.Forthevehiclewithindex N whichdoesnothaveavehiclebehindit,thecontrollawissli ghtlydierent: u N = k f N ( p N p N 1 + N 1 ;N ) b fN (_ p N p N 1 ) : (2{3) Eachvehicle i knowsthedesiredgaps i 1 ;i and i;i +1 ,whileonlyvehicle1knowsthe desiredtrajectory p 0 ( t )ofthectitiousreferencevehicle. Combiningtheopenloopdynamics( 2{1 )withthecontrollaw( 2{2 ),weget m i p i = k f i ( p i p i 1 + i 1 ;i ) k b i ( p i p i +1 i;i +1 ) b fi (_ p i p i 1 ) b bi (_ p i p i +1 ) ; (2{4) where i 2f 1 ;:::;N 1 g : Thedynamicsofthe N -thvehicleareobtainedbycombining( 2{1 )and( 2{3 ),whichareslightlydierentfrom( 2{4 ).Thedesiredtrajectoryofthe i -thvehicleis p i ( t ):= p 0 ( t ) 0 ;i = p 0 ( t ) P ij =1 j 1 ;j .Tofacilitateanalysis,wedene thefollowingtrackingerror: ~ p i := p i p i ) ~ p i =_ p i p i : (2{5) Substituting( 2{5 )into( 2{4 ),andusing p i 1 ( t ) p i ( t )= i 1 ;i ,weget m i ~ p i = k f i (~ p i ~ p i 1 ) k b i (~ p i ~ p i +1 ) b fi ( ~ p i ~ p i 1 ) b bi ( ~ p i ~ p i +1 ) : (2{6) 28

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Bydeningthestate X :=[~ p 1 ; ~ p 1 ; ~ p 2 ; ~ p 2 ; ; ~ p N ; ~ p N ] T ,theclosedloopdynamicsofthe networkcannowbewrittencompactlyfrom( 2{6 )as: X = AX (2{7) where A istheclosed-loopstatematrixandwehaveusedthefactthat ~ p 0 ( t )= ~ p 0 ( t ) 0 sincethetrajectoryofthereferencevehicleisequaltoits desiredtrajectory. 2.1.2MainResults Themainresultsofthischapterrelyontheanalysisofthefo llowingPDE(partial dierentialequation)modelofthenetwork,whichisseenas acontinuumapproximationof theclosed-loopdynamics( 2{6 ).ThedetailsofderivationofthePDEmodelaregivenin Section 2.2 .ThePDEisgivenby m ( x ) @ 2 ~ p ( x;t ) @t 2 = k f b ( x ) N @ @x + k f + b ( x ) 2 N 2 @ 2 @x 2 + b f b ( x ) N @ 2 @x@t + b f + b ( x ) 2 N 2 @ 3 @x 2 @t ~ p ( x;t ) ; (2{8) withboundaryconditions: ~ p (1 ;t )=0 ; @ ~ p @x (0 ;t )=0 ; (2{9) where k f b ( x ) ;k f + b ( x ) ;b f b ( x )and b f + b ( x )aredenedasfollows: k f + b ( x ):= k f ( x )+ k b ( x ) ;k f b ( x ):= k f ( x ) k b ( x ) ; b f + b ( x ):= b f ( x )+ b b ( x ) ;b f b ( x ):= b f ( x ) b b ( x ) ; and m ( x ) ;k f ( x ) ;k b ( x ) ;b f ( x ) ;b b ( x )arerespectivelythecontinuumapproximationsof m i ;k f i ;k b i ;b fi ;b bi ofeachvehiclewiththefollowingstipulation: k f or b i = k f or b ( x ) j x = N i N ;b f or b i = b f or b ( x ) j x = N i N ;m i = m ( x ) j x = N i N : (2{10) 29

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Weformallydenesymmetriccontrol,homogeneityandstabi litymarginbefore statingtherstmainresult,i.e.theroleofheterogeneity onthestabilitymarginofthe network.Denition2.1. Thecontrollaw ( 2{2 ) is symmetric ifeachvehicleusesthesamefront andbackcontrolgains: k f i = k b i b fi = b bi ,forall i 2f 1 ; 2 ; ;N 1 g ,andiscalled homogeneous if k f i = k f j k b i = k b j and b fi = b fj b bi = b bj foreachpairofneighboringvehicles ( i;j ) Denition2.2. Thestabilitymarginofaclosed-loopsystem,whichisdenot edby S ,isthe absolutevalueoftherealpartoftheleaststablepoleofthe closed-loopdynamics. Theorem2.1. ConsiderthePDEmodel ( 2{8 ) ofthenetworkwithboundarycondition ( 2{9 ) ,wherethemassandthecontrolgainprolessatisfy j m ( x ) m 0 j =m 0 j k ( ) ( x ) k 0 j =k 0 and j b ( ) ( x ) b 0 j =b 0 forall x 2 [0 ; 1] where m 0 ;k 0 and b 0 are positiveconstants,and 2 [0 ; 1) denotesthepercentofheterogeneity.Withsymmetric control,thestabilitymargin S ofthenetworksatisesthefollowing: (1 2 ) 2 b 0 8 m 0 1 N 2 S (1+2 ) 2 b 0 8 m 0 1 N 2 ; (2{11) when 1 Theresultaboveisalsoprovableforanarbitrary < 1(notnecessarilysmall)when thepositiongainisproportionaltothevelocitygainusing standardresultsofSturmLiouvilletheory[ 77 ,Chapter5].Forthatcase,theresultisgiveninthefollowi nglemma anditsproofisgivenintheendofSection 2.7 Theorem2.2. ConsiderthePDEmodel ( 2{8 ) ofthenetworkwithboundarycondition ( 2{9 ) .Letthemassandthecontrolgainssatisfy 0
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networksatisesthefollowing: 2 b min 8 m max 1 N 2 S 2 b max 8 m min 1 N 2 : Themainimplicationoftheresultaboveisthat heterogeneityofmassesandcontrol gainsplaysnoroleintheasymptotictrendofthestabilitym arginwith N aslongas thecontrolgainsaresymmetric .Notethatthe O (1 =N 2 )decayofthestabilitymargin describedabovehasbeenshownforhomogeneousplatoons(al lvehicleshavethesame massandusethesamecontrolgains)independentlyin[ 35 ],althoughthedynamicsofthe lastvehicleareslightlydierentfromours.Asimilarresu ltforhomogeneousplatoons withrelativepositionandabsolutevelocityfeedbackwasa lsoestablishedin[ 48 ]. Thesecondmainresultofthisworkisthatthestabilitymarg incanbegreatly improvedbyintroducingfront-backasymmetryinthe velocity -feedbackgains.Wecall theresultingdesign mistuning -baseddesignbecauseitreliesonsmallchangesfromthe nominalsymmetricgain b 0 .Inaddition,apoorchoiceofsuchasymmetrycanalsomake theclosedloopunstable.Ingeneral,heterogeneityinmass haslittleeectonthescaling trendsofeigenvaluesofPDE[ 77 ,Chapter5].Foreaseofanalysis,welet m i = m 0 inthe sequel.Theorem2.3. Foran N -vehiclenetworkwithPDEmodel ( 2{8 ) andboundarycondition ( 2{9 ) .Let m ( x )= m 0 forall x 2 [0 ; 1] ,considertheproblemofmaximizingthe stabilitymarginbychoosingthecontrolgainswiththecons traint j b ( : ) ( x ) b 0 j =b 0 ,where isapositiveconstant,and k ( f ) ( x )= k ( b ) ( x )= k 0 .If 1 ,theoptimalvelocitygainsare b f ( x )=(1+ ) b 0 ;b b ( x )=(1 ) b 0 ; (2{12) whichresultinthestabilitymargin S = "b 0 m 0 1 N + O ( 1 N 2 )= O ( 1 N ) : (2{13) 31

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Theformulaisasymptoticinthesensethatitholdsforlarge N andsmall .Incontrast, forthefollowingchoiceofasymmetry b f ( x )=(1 ) b 0 b b ( x )=(1+ ) b 0 ; (2{14) where 0 <" 1 isasmallpositiveconstant,theclosedloopbecomesunstab lefor sucientlylarge N Thetheoremsaysthatwitharbitrarilysmallchangeinthefr ont-backasymmetry, sothatvelocityinformationfromthefrontisweightedmore heavilythantheonefrom theback,thestabilitymargincanbeimprovedsignicantly oversymmetriccontrol.On theotherhand,ifvelocityinformationfromthebackisweig htedmoreheavilythanthat fromthefront,theclosedloopwillbecomeunstableifthene tworkislargeenough.It isinterestingtonotethattheoptimalgainsturnouttobeho mogeneous,whichagain indicatesthatheterogeneityhaslittleeectonthestabil itymargin. Theastutereadermayinquireatthispointwhataretheeect sofintroducing asymmetryintheposition-feedbackgainswhilekeepingvel ocitygainssymmetric,or introducingasymmetryinbothpositionandvelocityfeedba ckgains.Itturnsoutwhen equalasymmetryinbothpositionandvelocityfeedbackgain sareintroduced,theclosed loopisexponentiallystableforarbitrary N .Moreover,thestabilitymarginscalingtrend canbeuniformlyboundedbelowin N whenmoreweightsaregiventotheinformation fromitsfrontneighbor.Westatetheresultinthenexttheor em. Theorem2.4. Foran N -vehiclenetworkwithPDEmodel ( 2{8 ) andboundarycondition ( 2{9 ) .Let m ( x )= m 0 forall x 2 [0 ; 1] .Withthefollowingasymmetryincontrol k f ( x )=(1+ ) k 0 k b ( x )=(1 ) k 0 b f ( x )=(1+ ) b 0 b b ( x )=(1 ) b 0 ,where is theamountofasymmetrysatisfying 2 (0 ; 1) ,thestabilitymarginofthenetworkcanbe uniformlyboundedbelowasfollows: S min n b 0 2 2 m 0 ; k 0 b 0 o = O (1) : 32

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Thisasymmetricdesignthereforemakestheresultingcontr ollawhighlyscalable;it eliminatesthedegradationofclosed-loopstabilitymargi nwithincreasing N .Itisnow possibletodesignthecontrolgainssothatthestabilityma rginofthesystemsatises apre-speciedvalueirrespectiveofhowmanyvehiclesarei ntheformation.Theresult aboveisforequalamountofasymmetryinthepositionfeedba ckandvelocityfeedback gains.Thisconstraintofequalasymmetryinpositionandve locityfeedbackisimposedin ordertomaketheanalysistractable. Asweseefromthepreviousresults,heterogeneityhaslittl eeectonthescalinglaw ofstabilitymargin,whileasymmetryhasahugeeect.Onema ywonderhowdoesthe stabilitymarginscalewhenthereisbothheterogeneityand asymmetryinthesystem? Thefollowingtheoremanswersthequestionforthisscenari o.Inparticular,weconsider twocases.Onecaseisasymmetricvelocityfeedbackwithsma llheterogeneity,theother caseiswhenthereisequalasymmetryinbothpositionandvel ocityfeedbacksaswellas smallheterogeneity.Theorem2.5. Consideran N -vehiclenetworkwithPDEmodel ( 2{8 ) andboundary condition ( 2{9 ) 1)Whenthereissmallasymmetryonlyinthevelocityfeedbac kandsmallheterogeneityinthecontrolgainfunctions,i.e. m ( x )= m 0 k ( f ) ( x )= k ( b ) ( x ) j k ( ) ( x ) k 0 j =k 0 b ( f ) ( x ) b ( b ) ( x )=2 "b 0 j b ( ) ( x ) b 0 j =b 0 ,where isasmallpositiveconstant.If 1 thestabilitymarginofthenetworksatises S = O ( 1 N ) : 2)Whereisequalamountofasymmetryinbothpositionandvel ocityfeedbackaswell assmallheterogeneityinthecontrolgains,i.e. m ( x )= m 0 k ( f ) ( x ) k ( b ) ( x )=2 "k 0 j k ( ) ( x ) k 0 j =k 0 b ( f ) ( x ) b ( b ) ( x )=2 "b 0 j b ( ) ( x ) b 0 j =b 0 .If 1 ,thestability marginofthenetworksatises S = O (1) : 33

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ComparingtheabovetheoremtoTheorem 2.1 andTheorem 2.4 ,weshowthatno matterthecontrolissymmetricorasymmetric,introducing heterogeneityincontrolgains doesnotchangethescalinglawofstabilitymarginwithresp ecttothenumberofvehicles intheplatoon.Thescalinglawisonlydeterminedbyasymmet ry(anditstype). 2.2PDEModeloftheClosed-LoopDynamics Inthischapter,alltheanalysisanddesignisperformedusi ngaPDEmodel,whose resultsarevalidatedbynumericalcomputationsusingthes tate-spacemodel( 2{7 ).We nowderiveacontinuumapproximationofthecoupled-ODEs( 2{6 )inthelimitoflarge N byfollowingthestepsinvolvedinanite-dierencediscre tizationinreverse.Wedene k f + b i := k f i + k b i ;k f b i := k f i k b i ; b f + b i := b fi + b fi ;b f b i := b fi b bi : Substitutingtheseinto( 2{6 ),wehave m i ~ p i = k f + b i + k f b i 2 (~ p i ~ p i 1 ) k f + b i k f b i 2 (~ p i ~ p i +1 ) b f + b i + b f b i 2 ( ~ p i ~ p i 1 ) b f + b i b f b i 2 ( ~ p i ~ p i +1 ) : (2{15) Tofacilitateanalysis,weredrawthegraphofthe1Dnetwork ,sothateachvehicleinthe newgraphisdrawnintheinterval[0 ; 1],irrespectiveofthenumberofvehicles.The i -th vehicleinthe\original"graph,isnowdrawnatposition( N i ) =N inthenewgraph. Figure 2-1 showsanexample. ThestartingpointforthePDEderivationistoconsiderafun ction~ p ( x;t ):[0 ; 1] [0 ; 1 ) R thatsatises: ~ p i ( t )=~ p ( x;t ) j x =( N i ) =N ; (2{16) suchthatfunctionsthataredenedatdiscretepoints i willbeapproximatedbyfunctions thataredenedeverywherein[0 ; 1].Theoriginalfunctionsarethoughtofassamplesof theircontinuousapproximations.Weformallyintroduceth efollowingscalarfunctions 34

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k f ( x ) ;k b ( x ) ;b f ( x ) ;b b ( x )and m ( x ):[0 ; 1] R denedaccordingtothestipulation: k f or b i = k f or b ( x ) j x = N i N ;b f or b i = b f or b ( x ) j x = N i N ;m i = m ( x ) j x = N i N : (2{17) Inaddition,wedenefunctions k f + b ( x ), k f b ( x ), b f + b ( x ), b f b ( x ):[0 ; 1] D R as k f + b ( x ):= k f ( x )+ k b ( x ) ;k f b ( x ):= k f ( x ) k b ( x ) ; b f + b ( x ):= b f ( x )+ b b ( x ) ;b f b ( x ):= b f ( x ) b b ( x ) : Dueto( 2{17 ),thesesatisfy k f + b i = k f + b ( x ) j x =( N i ) =N ;k f b i = k f b ( x ) j x =( N i ) =N b f + b i = b f + b ( x ) j x =( N i ) =N ;b f b i = b f b ( x ) j x =( N i ) =N : ToobtainaPDEmodelfrom( 2{15 ),werstrewriteitas m i ~ p i = k f b i N (~ p i 1 ~ p i +1 ) 2(1 =N ) + k f + b i 2 N 2 (~ p i 1 2~ p i +~ p i +1 ) 1 =N 2 + b f b i N ( ~ p i 1 ~ p i +1 ) 2(1 =N ) + b f + b i 2 N 2 ( ~ p i 1 2 ~ p i + ~ p i +1 ) 1 =N 2 : (2{18) Usingthefollowingnitedierenceapproximations: h ~ p i 1 ~ p i +1 2(1 =N ) i = h @ ~ p ( x;t ) @x i x =( N i ) =N ; h ~ p i 1 2~ p i +~ p i +1 1 =N 2 i = h @ 2 ~ p ( x;t ) @x 2 i x =( N i ) =N ; h ~ p i 1 ~ p i +1 2(1 =N ) i = h @ 2 ~ p ( x;t ) @x@t i x =( N i ) =N ; h ~ p i 1 2 ~ p i + ~ p i +1 1 =N 2 i = h @ 3 ~ p ( x;t ) @x 2 @t i x =( N i ) =N : Forlarge N ,Eq.( 2{18 )canbeseenasanitedierencediscretizationofthefollo wing PDE: m ( x ) @ 2 ~ p ( x;t ) @t 2 = k f b ( x ) N @ @x + k f + b ( x ) 2 N 2 @ 2 @x 2 + b f b ( x ) N @ 2 @x@t + b f + b ( x ) 2 N 2 @ 3 @x 2 @t ~ p ( x;t ) : TheboundaryconditionsoftheabovePDEdependonthearrang ementofreference vehicleintheredrawngraphofthenetwork.Forourcase,the boundaryconditionisof 35

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Dirichlettypeat x =1wherethereferencevehicleis,andofNeumanntypeat x =0: ~ p (1 ;t )=0 ; @ ~ p @x (0 ;t )=0 : 2.3RoleofHeterogeneityonStabilityMargin Thestartingpointofouranalysisistheinvestigationofth ehomogeneousand symmetriccase: m i = m 0 ;k ( ) i = k 0 ;b ( ) i = b 0 forsomepositiveconstants m 0 ;k 0 ;b 0 ,where i 2f 1 ;:::;N g .TheanalysisleadingtotheproofofTheorem 2.1 iscarriedoutusingthe PDEmodelderivedintheprevioussection.Inthehomogeneou sandsymmetriccontrol case,usingthenotationintroducedearlier,weget m ( x )= m 0 ;k f + b ( x )=2 k 0 ;k f b ( x )=0 ;b f + b ( x )=2 b 0 ;b f b ( x )=0 : ThePDE( 2{8 )simpliesto: m 0 @ 2 ~ p ( x;t ) @t 2 = k 0 N 2 @ 2 ~ p ( x;t ) @x 2 + b 0 N 2 @ 3 ~ p ( x;t ) @x 2 @t : (2{19) ThisisawaveequationwithKelvin-Voigtdamping.Duetothe linearityandhomogeneity oftheabovePDEandboundaryconditions,weareabletoapply themethodofseparation ofvariables.Weassumeasolutionoftheform~ p ( x;t )= P 1` =1 ` ( x ) h ` ( t ).Substitutingit intoPDE( 2{19 ),weobtainthefollowingtime-domainODE m 0 d 2 h ` ( t ) dt 2 + b 0 ` N 2 dh ` ( t ) dt + k 0 ` N 2 h ` ( t )=0 ; (2{20) where ` solvestheboundaryvalueproblem d 2 ` ( x ) dx 2 + ` ` ( x )=0 ; (2{21) withthefollowingboundaryconditions,whichcomefrom( 2{9 ): d ` dx (0)=0 ; ` (1)=0 : (2{22) 36

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Followingstraightforwardalgebra,theeigenvaluesandei genfunctionoftheaboveboundaryvalueproblemisgivenby(see[ 77 ]foraBVPexample) ` = 2 (2 ` 1) 2 4 ; ` ( x )= cos ( 2 ` 1 2 x ) ;` =1 ; 2 ; : (2{23) TakeLaplacetransformtobothsidesofthe( 2{20 )withrespecttothetimevariable t ,we obtainthecharacteristicequationofthePDE( 2{19 ): m 0 s 2 + b 0 ` N 2 s + k 0 ` N 2 =0 : TheeigenvaluesofthePDE( 2{19 )arenowgivenby s ` = ` b 0 2 m 0 N 2 1 2 m 0 N r 2` b 20 N 2 4 ` m 0 k 0 (2{24) Forsmall ` andlarge N sothat N> (2 ` 1) b 0 = (4 p m 0 k 0 ),thediscriminantisnegative,makingtherealpartoftheeigenvaluesequalto ` b 0 = (2 m 0 N 2 ).Theleaststable eigenvalue,theoneclosesttotheimaginaryaxis,isobtain edwith ` =1: s 1 = 2 b 0 8 m 0 1 N 2 + =) S := j Real ( s 1 ) j = 2 b 0 8 m 0 N 2 ; (2{25) where = isanimaginarynumber. WearenowreadytopresenttheproofofTheorem 2.1 ProofofTheorem 2.1 Recallthatincaseofsymmetriccontrolwehave k f i = k b i ;b fi = b bi ; 8 i 2f 1 ; ;N g : Inthiscase,usingthenotationintroducedearlier,wehave k f b ( x )=0 ;b f b ( x )=0 ; ThePDE( 2{8 )issimpliedto: m ( x ) @ 2 ~ p ( x;t ) @t 2 = k f + b ( x ) 2 N 2 @ 2 ~ p ( x;t ) @x 2 + b f + b ( x ) 2 N 2 @ 3 ~ p ( x;t ) @x 2 @t ; (2{26) 37

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Theproofproceedsbyaperturbationmethod.Tobeconsisten twiththeboundsofthe massandcontrolgainsofeachvehicle,let m ( x )= m 0 + ~ m ( x ) ; ~ m ( x ) 2 [ m 0 ;m 0 ] k f + b ( x )=2 k 0 + ~ k ( x ) ; ~ k ( x ) 2 [ 2 k 0 ; 2 k 0 ] b f + b ( x )=2 b 0 + ~ b ( x ) ; ~ b ( x ) 2 [ 2 b 0 ; 2 b 0 ] : where isasmallpositivenumber,denotingtheamountofheterogen eityand~ m ( x ) ; ~ k ( x ) ; ~ b ( x ) aretheperturbationproles.TakeLaplacetransformtobot hsidesofPDE( 2{26 )with respectto t ,wehave m ( x ) s 2 = k f + b ( x ) 2 N 2 @ 2 @x 2 + b f + b ( x ) 2 N 2 s @ 2 @x 2 ; (2{27) Lettheperturbedeigenvaluebe s = s ` = s (0)` + s ( ) ` ; theLaplacetransformof~ p ( x;t )be = (0) + ( ) ,where s (0)` and (0) correspondtotheunperturbedPDE( 2{19 ),i.e. m 0 ( s (0) ) 2 (0) = k 0 N 2 @ 2 (0) @x 2 + b 0 N 2 s (0) @ 2 (0) @x 2 : (2{28) Eq.( 2{24 )providestheformulafor s (0)` (actually, s ` ),and (0) isthesolutiontoabove equation,whichisgivenby (0) = P 1` =1 (0) ` = P 1` =1 ` ( x ) H ` ( s ),where H ` ( s )istheLaplace transformof h ( t )givenin( 2{20 ).Pluggingtheexpressionsfor s ` and into( 2{27 ),and doingan O (1)balanceleadstotheeigenvalueequationfortheunpertu rbedPDE,whichis exactlyEq.( 2{28 ): P (0) =0 ; where P := m 0 ( s (0)` ) 2 b 0 s (0)` + k 0 N 2 @ 2 @x 2 (2{29) Nextwedoan O ( )balance,whichleadsto: P ( ) = 2 m 0 s (0)` s ( ) ` (0) ~ m ( x )( s (0)` ) 2 (0) + ~ k ( x ) 2 N 2 @ 2 (0) @x 2 + s (0)` ~ b ( x ) 2 N 2 @ 2 (0) @x 2 + s ( ) ` b 0 N 2 @ 2 (0) @x 2 =: R 38

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Forasolution ( ) toexist, R mustlieintherangespaceoftheoperator P .Since P is self-adjoint,itsrangespaceisorthogonaltoitsnullspac e.Thus,wehave, =0(2{30) where ` isalsothe ` th basisvectorofthenullspaceofoperator P .Wenowhavethe followingequation: Z 1 0 2 m 0 s (0)` s ( ) ` (0) ~ m ( x )( s (0)` ) 2 (0) + ~ k ( x ) 2 N 2 @ 2 (0) @x 2 + s (0)` ~ b ( x ) 2 N 2 @ 2 (0) @x 2 + s ( ) ` b 0 N 2 @ 2 (0) @x 2 (0) ` dx =0 : Followingstraightforwardmanipulations,wegot: s ( ) ` = b 0 ` m 20 N 2 Z 1 0 ~ m ( x )( ` ( x )) 2 dx ` 2 m 0 N 2 Z 1 0 ~ b ( x )( ` ( x )) 2 dx + = ; (2{31) where = isanimaginarynumberwhen N islarge( N> (2 ` 1) b 0 = (4 p m 0 k 0 )).Usingthis, andsubstitutingtheequationaboveinto s ` = s (0)` + s ( ) ` + O ( 2 ),andsetting ` =1,we obtainthestabilitymarginoftheheterogeneousnetwork: S = b 0 2 8 m 0 N 2 b 0 2 4 m 20 N 2 Z 1 0 ~ m ( x )cos 2 2 x dx + 2 8 m 0 N 2 Z 1 0 ~ b ( x )cos 2 2 x dx + O ( 2 ) : Pluggingthebounds j ~ m ( x ) j m 0 and j ~ b ( x ) j 2 b 0 ,weobtainthedesiredresult. Wenowpresentnumericalcomputationsthatcorroboratesth ePDE-basedanalysis. Weconsiderthefollowingmassandcontrolgainprole: k f i = k b i =1+0 : 2sin(2 ( N i ) =N ) ; b fi = b bi =0 : 5+0 : 1sin(2 ( N i ) =N ) ; m i =1+0 : 2sin(2 ( N i ) =N ) : (2{32) IntheassociatedPDEmodel( 2{26 ),thiscorrespondsto k f ( x )= k b ( x )=1+0 : 2sin(2 x ), b f ( x )= b b ( x )=0 : 5+0 : 1sin(2 x ), m ( x )=1+0 : 2sin(2 x ).TheeigenvaluesofthePDE, 39

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-0.25 -0.2 -0.15 -0.1 -0.05 0 -1 -0.5 0 0.5 1 RealImaginarySSM PDE Figure2-2.Numericalcomparisonofeigenvaluesbetweenst atespaceandPDEmodels. thatarecomputednumericallyusingaGalerkinmethodwithF ourierbasis,arecompared withthatofthestatespacemodeltocheckhowwellthePDEmod elcapturestheclosed loopdynamics.Figure 2-2 depictsthecomparisonofeigenvaluesofthestate-spacemo del (SSM)( 2{7 )andthePDEmodel( 2{26 )withsymmetriccontrol.Eigenvaluesshownare foraplatoonof50vehicles,andthemassandcontrolgainspr olearegivenin( 2{32 ). Onlysomeeigenvaluesclosetotheimaginaryaxisarecompar edinthegure.Itshows theeigenvaluesofthestate-spacemodelisaccuratelyappr oximatedbythePDEmodel, especiallytheonesclosetotheimaginaryaxis.WeseefromF igure 2-3 thattheclosedloopstabilitymarginofthecontrolledformationiswellca pturedbythePDEmodel.In addition,theplotcorroboratesthepredictedbound( 2{11 ).ThelegendsofSSM,PDE andlowerbound,upperboundstandforthestabilitymarginc omputedfromthestate spacemodel,fromthePDEmodel,andtheasymptoticlowerand upperbounds( 2{11 )in Theorem 2.1 .Themassandcontrolgainsprolearegivenin( 2{32 ). 40

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5 10 20 50 100 10 -4 10 -3 10 -2 NSSSM PDE Lowerboundin( 2{11 ) Upperboundin( 2{11 ) Figure2-3.Stabilitymarginoftheheterogeneousplatoona safunctionofnumberof vehicles. 2.4RoleofAsymmetryonStabilityMargin Inthissection,weconsidertwoscenariosofasymmetriccon trol,werstpresentthe resultswhenthereisasymmetryinthevelocityfeedbackalo ne(Theorem 2.3 ).Theresults whenthereisequalasymmetryinbothpositionandvelocityf eedbacks(Theorem 2.4 ). 2.4.1AsymmetricVelocityFeedback Withsymmetriccontrol,oneobtainsan O ( 1 N 2 )scalinglawforthestabilitymargin becausethecoecientofthe @ 3 @x 2 @t terminthePDE( 2{26 )is O ( 1 N 2 )andthecoecient ofthe @ 2 @x@t termis0.Anyasymmetrybetweentheforwardandthebackward velocity gainswillleadtonon-zero b f b ( x )andapresenceof O ( 1 N )termascoecientof @ 2 @x@t .By ajudiciouschoiceofasymmetry,thereisthusapotentialto improvethestabilitymargin from O ( 1 N 2 )to O ( 1 N ).Apoorchoiceofcontrolasymmetrymayleadtoinstability ,aswe'll showinthesequel. Webeginbyconsideringtheforwardandbackwardfeedbackga inproles k f ( x )= k b ( x )= k 0 ;b f ( x )= b 0 + ~ b f ( x ) ;b b ( x )= b 0 + ~ b b ( x ) ; (2{33) 41

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where "> 0isasmallparametersignifyingthepercentofasymmetryan d ~ b f ( x ), ~ b b ( x )are functionsdenedover[0 ; 1]thatcapturevelocitygainperturbationfromthenominal value b 0 .Dene ~ b s ( x ):= ~ b f ( x )+ ~ b b ( x ) ; ~ b m ( x ):= ~ b f ( x ) ~ b b ( x ) : (2{34) Duetothedenitionof k f + b ;k f b ;b f + b and b f b ,wehave k f + b ( x )=2 k 0 ;k f b ( x )=0 ; b f + b ( x )=2 b 0 + ~ b s ( x ) ;b f b ( x )= ~ b m ( x ) : ThePDE( 2{8 )withhomogeneousmass m 0 nowbecomes m 0 @ 2 ~ p ( x;t ) @t 2 = k 0 N 2 @ 2 @x 2 + b 0 N 2 @ 3 @x 2 @t ~ p ( x;t )+ ~ b s ( x ) 2 N 2 @ 3 @x 2 @t + ~ b m ( x ) N @ 2 @x@t ~ p ( x;t ) : (2{35) Wenowstudytheproblemofhowdoesthechoiceoftheperturba tions ~ b s ( x )and ~ b m ( x )(withinlimitssothatthegains b f ( x )and b b ( x )arewithinpre-speciedbounds) aectthestabilitymargin.Ananswertothisquestionalsoh elpsindesigningbenecialperturbationstoimprovethestabilitymargin.Thefol lowingresultisusedinthe subsequentanalysis.Proposition2.1. ConsidertheeigenvalueproblemofthePDE ( 2{35 ) withmixed DirichletandNeumannboundarycondition ( 2{9 ) .Theleaststableeigenvalueisgivenby thefollowingformulathatisvalidfor 1 andlarge N : s 1 = s (0)1 4 m 0 N Z 1 0 ~ b m ( x )sin x dx 2 8 m 0 N 2 Z 1 0 ~ b s ( x )cos 2 2 x dx + O ( 2 )+ = (2{36) where s (0)1 istheleaststableeigenvalueoftheunperturbedPDE ( 2{19 ) withthesame boundaryconditionsand = isanimaginarynumberwhen N islarge( N>b 0 = (4 p m 0 k 0 ) ). 42

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TheproofofProposition 2.1 issimilartotheproofofTheorem 2.1 .Itisgiveninthe Appendix.NowwearereadytoproveTheorem 2.3 ProofofTheorem 2.3 ItfollowsfromProposition 2.1 thattominimizetheleaststable eigenvalue,oneneedstochooseonly ~ b m ( x )carefully.Thereasonisthesecondterm involving ~ b s ( x )hasthe O (1 =N 2 )trend.Therefore,wechoose ~ b s ( x ) 0 : Thismeansthattheperturbationstothe\front"and\back"v elocitygainssatisfy: ~ b f ( x )= ~ b b ( x ) ~ b m ( x )=2 ~ b f ( x ) : ThemostbenecialgainscannowbereadilyobtainedfromPro position 2.1 .Tominimize theleaststableeigenvaluewith ~ b s ( x ) 0,weshouldchoose ~ b m ( x )tomaketheintegral R 1 0 ~ b m ( x )sin( x ) dx aslargeaspossible,whichisachievedbysetting ~ b m ( x )tobethelargest possiblevalueeverywhereintheinterval[0 ; 1].Theconstraint j b ( ) i b 0 j =b 0 translates to b 0 (1 ) b ( ) ( x ) b 0 (1+ ),whichmeans k ~ b f k 1 b 0 and k ~ b b k 1 b 0 .Withthe choiceof ~ b s madeabove,wethereforehavetheconstraint k ~ b m k 2 b 0 .Thesolutiontothe optimizationproblemisthereforeobtainedbychoosing ~ b m ( x )=2 b 0 8 x 2 [0 ; 1].Thisgives ustheoptimalgains ~ b f ( x )= b 0 ; ~ b b ( x )= b 0 ; ) b f ( x )= b 0 (1+ ) ;b b ( x )= b 0 (1 ) : TheleaststableeigenvalueisobtainedfromProposition( 2.1 ): s +1 = s (0) "b 0 m 0 N 0+ O ( 2 )+ = : Since s (0) istheleaststableeigenvalueforthesymmetricPDE,weknow fromTheorem 2.1 that s (0) = O (1 =N 2 ).Therefore,itfollowsfromtheequationabovethatthesta bility marginis S = Re ( s +1 )= "b 0 m 0 N + O ( 1 N 2 ).Thisprovestherststatementofthetheorem. 43

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Toprovethesecondstatement,thecontrolgaindesign b fi =(1 ) b 0 and b bi =(1+ ) b 0 becomes b f ( x )=(1 ) b 0 and b b ( x )=(1+ ) b 0 .Withthischoice,itfollowsfrom Proposition( 2.1 )that s +1 = s (0) + "b 0 m 0 N 0+ O ( 2 )+ = : Since s (0) = O (1 =N 2 ),thesecondterm,whichis O (1 =N ),willdominateforlarge N .Since thistermispositive,thesecondstatementisproved. 2.4.2AsymmetricPositionandVelocityFeedbackwithEqual Asymmetry Whenthereisequalasymmetryinthepositionandvelocityfe edback,weconsiderthe followinghomogeneousandasymmetriccontrolgains: k f ( x )=(1+ ) k 0 ;k b ( x )=(1 ) k 0 ; b f ( x )=(1+ ) b 0 ;b b ( x )=(1 ) b 0 ; (2{37) where istheamountofasymmetrysatisfying 2 (0 ; 1). ProofofTheorem 2.4 ThePDEmodelwiththecontrolgainsspeciedin( 2{37 )becomes m 0 @ 2 ~ p ( x;t ) @t 2 = 2 "k 0 N @ ~ p ( x;t ) @x + k 0 N 2 @ 2 ~ p ( x;t ) @x 2 + 2 "b 0 N @ 2 ~ p ( x;t ) @x@t + b 0 N 2 @ 3 ~ p ( x;t ) @x 2 @t ; (2{38) Bythemethodofseparationofvariables,weassumeasolutio noftheform~ p ( x;t )= P 1` =1 ` ( x ) h ` ( t ).SubstitutingitintoPDE( 2{38 ),weobtainthefollowingtime-domain ODE m 0 d 2 h ` ( t ) dt 2 + b 0 ` dh ` ( t ) dt + k 0 ` h ` ( t )=0 ; (2{39) where ` solvesthefollowingboundaryvalueproblem L ` ( x )=0 ; L := d 2 dx 2 +2 "N d dx + ` N 2 ; (2{40) 44

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withthefollowingboundarycondition,whichcomesfrom( 2{9 ): d ` dx (0)=0 ; ` (1)=0 : (2{41) TakingLaplacetransformofbothsidesof( 2{39 )withrespecttothetimevariable t wehavethefollowingcharacteristicequationforthePDEmo del m 0 s 2 + b 0 ` s + k 0 ` =0 : (2{42) Wenowsolvetheboundaryvalueproblem( 2{40 )-( 2{41 ).Wemultiplybothsides of( 2{40 )by e 2 "Nx N 2 toobtainthestandardSturm-Liouvilleeigenvalueproblem d dx e 2 "Nx d ` ( x ) dx + ( ) ` N 2 e 2 "Nx ` ( x )=0 : (2{43) AccordingtoSturm-LiouvilleTheory,alltheeigenvaluesa rerealandhavethefollowing ordering 1 < 2 < ,see[ 77 ].Tosolvetheboundaryvalueproblem( 2{40 )-( 2{41 ),we assumesolutionoftheform, ` ( x )= e rx ,thenweobtainthefollowingequation r 2 +2 "Nr + ` N 2 =0 ; ) r = "N N p 2 ` : (2{44) Dependingonthediscriminantintheaboveequation,therea rethreecasestoanalyze: ` <" 2 ,theeigenfunctionhasthefollowingform ` ( x )= c 1 e ( "N + N p 2 ` ) x + c 2 e ( "N N p 2 ` ) x : where c 1 ;c 2 aresomeconstants.Applyingtheboundarycondition( 2{41 ),it's straightforwardtoseethat,fornon-trivialeigenfunctio ns ` ( x )toexit,thefollowing equationmustbesatised( "N N p 2 ` ) = ( "N + N p 2 ` )= e 2 N p 2 ` .For positive ,thisleadstoacontradiction,sothereisnoeigenvaluefor thiscase. ` = 2 ,theeigenfunction ` ( x )hasthefollowingform ` ( x )= c 1 e "Nx + c 2 xe "Nx : 45

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Again,applyingtheboundarycondition( 2{41 ),fornon-trivialeigenfunctions ` ( x ) toexit,wehavethefollowing "N = 1,whichimpliesthereisnoeigenvalueforthis caseeither. ` >" 2 ,theeigenfunctionhasthefollowingform ` ( x )= e "Nx ( c 1 cos( N p ` 2 x )+ c 2 sin( N p ` 2 x )) : Applyingtheboundarycondition( 2{41 ),fornon-trivialeigenfunctions ` ( x )toexit, theeigenvalues ` mustsatisfy ` = 2 + a 2` N 2 where a ` solvesthetranscendental equation a ` = ( "N )=tan( a ` ).Agraphicalrepresentationofthefunctionstan x and x="N withrespectto x showsthat a ` 2 ( (2 ` 1) 2 ;` ). Fromthelastcase,weseethat a 1 2 ( = 2 ; ),and 1 2 fromaboveas N !1 ,i.e. inf N 1 = 2 .Foreach ` 2f 1 ; 2 ; g ,thetworootsofthecharacteristicequations( 2{42 ) aregivenby s ` = b 0 ` p b 20 2` 4 m 0 k 0 ` 2 m 0 : (2{45) Dependingonthediscriminantin( 2{45 ),therearetwocasestoanalyze: If 1 4 m 0 k 0 =b 20 ,thenthediscriminantin( 2{45 )foreach ` isnon-negative,the less stable eigenvaluecanbewrittenas s +` = ` b 0 p ( ` b 0 ) 2 4 ` m 0 k 0 2 m 0 = 2 k 0 b 0 + p b 20 4 m 0 k 0 = ` : Theleaststableeigenvalueisachievedbysetting ` = 1 .Since ` !1 as ` !1 wehavethestabilitymargin S = j Re ( s +1 ) j 2 k 0 b 0 + p b 20 0 = k 0 b 0 : Otherwise,thediscriminantin( 2{45 )isindeterministic,i.e.it'snegativeforsmall ` andpositiveforlarge ` isnon-positive.Forthose ` 'swhichmakethediscriminant 46

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negative,theleaststableeigenvalueamongthemisgivenby s 1 = 1 b 0 2 m 0 + = : where = isanimaginarynumber.Forthose ` 'swhichmakethediscriminantnonpositive,wehavefromtherstcasethattheleaststableeig envalueamongthemis givenby s +1 = 2 k 0 b 0 + p b 20 4 m 0 k 0 = 1 Thestabilitymarginisgivenbytakingtheminimumofabsolu tevalueofthereal partoftheabovetwoeigenvalues, S min n b 0 1 2 m 0 ; k 0 b 0 o : Combiningtheabovetwocases,andusingthefactthat 1 2 ,weobtainthatthe stabilitymargincanbeboundedbelowasfollows S min n b 0 2 2 m 0 ; k 0 b 0 o : Thisconcludestheproof. 2.4.3NumericalComparisonofStabilityMargin Figure 2-4 depictsthenumericallyobtainedstabilitymarginsforbot hthePDE andstate-spacemodels(SSM)withsymmetricandasymmetric controlgains.Themass ofeachvehicleusedis m 0 =1.Thenominalcontrolgainsare k 0 =1, b 0 =0 : 5.The asymmetriccontrolgainsusedaretheonesgiveninTheorem 2.3 andTheorem 2.4 respectively,andtheamountofasymmetryis =0 : 1.Thelegends\SSM"and\PDE" standforthestabilitymargincomputedfromthestate-spac emodelandthePDEmodel, respectively.Thegureshowsthat1)thestabilitymargino fthePDEmodelmatches thatofthestate-spacemodelaccurately,evenforsmallval uesof N ;2)thestability 47

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5 15 40 100 300 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 NSSymmetric(SSM)Symmetric(PDE)Asymmetricvelocity(SSM)Asymmetricvelocity(PDE)Asymmetricpositionandvelocity(SSM)Asymmetricpositionandvelocity(PDE)Theorem 2.3 Theorem 2.4 Figure2-4.Stabilitymarginimprovementbyasymmetriccon trol. marginwithasymmetricvelocityfeedbackshowslargeimpro vementoverthesymmetric caseeventhoughthevelocitygainsdierfromtheirnominal valuesonlyby 10%.The improvementisparticularlynoticeableforlargevaluesof N ;3)Withequalamount ofasymmetryinboththepositionandvelocityfeedback,the stabilitymargincanbe uniformlyboundedawayfrom0,whicheliminatesthedegrada tionofstabilitymarginwith increasing N ;4)theasymptoticformulaegiveninTheorem 2.3 andTheorem 2.4 arequire accurate. Numericalvalidationthatpoorchoiceofasymmetryincontr olgainscanleadto instabilityisshowninFigure 2-5 .Themassofeachvehicleis m 0 =1.Thenominal controlgainsare k 0 =1, b 0 =0 : 5,andthecontrolgainsusedaretheonesgivenby( 2{14 ) 48

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25 50 100 200 10 -3 NRe ( s +1 )Poorasymmetricvelocity(SSM) Poorasymmetricvelocity(PDE) Theorem 2.3 Figure2-5.Therealpartofthemostunstableeigenvalueswi thpoorasymmetry. inTheorem 2.3 with =0 : 1.Notethattherealpartoftheseeigenvaluesarepositivea nd Eq.( 2{14 )alsomakesanaccurateprediction. 2.5ScalingofStabilityMarginwithbothAsymmetryandHete rogeneity Inthissection,westudythestabilitymarginofthesystemw ithbothheterogeneity andasymmetry.ThemainjobofthissectionistoproveTheore m 2.5 ProofofTheorem 2.5 Theproofalsoreliesonperturbationtechnique.Basedonth e boundsofthecontrolgainsandthedenitionof k f + b ;k f b ;b f + b and b f b ,wehave k f b ( x )=0 ;k f + b ( x )=2 k 0 + ~ k ( x ) ; ~ k ( x ) 2 [ 2 k 0 ; 2 k 0 ] b f + b ( x )=2 "b 0 ;b f + b ( x )=2 b 0 + ~ b ( x ) ; ~ b ( x ) 2 [ 2 b 0 ; 2 b 0 ] : ThePDE( 2{8 )withhomogeneousmass m 0 nowbecomes m 0 @ 2 ~ p ( x;t ) @t 2 = k 0 N 2 @ 2 @x 2 + b 0 N 2 @ 3 @x 2 @t ~ p ( x;t ) + ~ k ( x ) 2 N 2 @ 2 @x 2 + ~ b ( x ) 2 N 2 @ 3 @x 2 @t + 2 b 0 N @ 2 @x@t ~ p ( x;t ) : (2{46) 49

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LettheeigenvaluesandLaplacetransformationof~ p ( x;t )fortheaboveperturbed PDEbe s ` = s (0)` + "s ( ) ` ; = (0) + ( ) respectively,where s (0)` and (0) arecorresponding totheunperturbedPDE( 2{19 ).TakingaLaplacetransformofPDE( 2{46 ),pluggingin theexpressionsfor s ` and ,anddoingan O ( )balance,whichleadsto: P ( ) = ~ k ( x ) 2 N 2 d 2 (0) dx 2 + s (0)` 2 b 0 N d (0) dx + s (0)` ~ b ( x ) 2 N 2 d 2 (0) dx 2 2 m 0 s (0)` s ( ) ` (0) + s ( ) ` b 0 N 2 d 2 (0) dx 2 =: R; where P isdenedin( 2{29 ).Forasolution ( ) toexist, R mustlieintherangespaceof theself-adjointoperator P .Thus,wehave, =0 Wenowhavethefollowingequation: Z 1 0 ~ k ( x ) 2 N 2 d 2 (0) dx 2 + s (0)` 2 b 0 N d (0) dx + s (0)` ~ b ( x ) 2 N 2 d 2 (0) dx 2 2 m 0 s (0)` s ( ) ` (0) + s ( ) ` b 0 N 2 d 2 (0) dx 2 (0) ` dx =0 Straightforwardmanipulationsshowthat: m 0 ( s (0)` + b 0 ` 2 m 0 N 2 ) s ( ) ` = s (0)` (2 ` 1) 2 N Z 1 0 b 0 sin (2 ` 1) x dx s (0)` (2 ` 1) 2 2 8 N 2 Z 1 0 ~ b ( x )cos 2 (2 ` 1) 2 x dx (2 ` 1) 2 2 8 N 2 Z 1 0 ~ k ( x )cos 2 (2 ` 1) 2 x dx: NoticethattheexistenceofthelasttwotermsintheRHSofth eaboveequationis duetoheterogeneityinthecontrolgains,andtheircoecie ntsareordersof1 =N 2 .In addition,thersttermwhichresultsfromasymmetryhascoe cientoforder1 =N ,which dominatesthetermswithorder1 =N 2 forlarge N .Henceheterogeneityincontrolgains doesnotchangethescalingtrendofstabilitymargin,buton lyintroducingasymmetry does.TherestoftheprooffortherstpartofTheorem 2.5 followsbysubstitutingthe equationaboveinto s ` = s (0)` + "s ( ) ` ,andsetting ` =1. 50

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TheproofforthesecondpartofTheorem 2.5 issimilartotheargumentshownabove, wethereforeignoretheproof. 2.6Summary Westudiedtheroleofheterogeneityandcontrolasymmetryo nthestabilitymargin ofalarge1-Dnetworkofdouble-integratorvehicles.Theco ntrolisinadistributedsense thatthecontrolsignalateveryvehicledependsontherelat ivepositionandvelocity measurementsfromitstwonearestneighbors(oneoneeither side).Itwasshownthat heterogeneityhadlittleeectonhowthestabilitymargins caledwith N ,thenumber ofvehicles,whereasasymmetryplayedasignicantrole.If front-backasymmetryis introducedinthecontrolgains,evenbyanarbitrarilysmal lamount,thestabilitymargin canbeimprovedto O (1 =N )withasymmetricvelocityfeedback.Thestabilitymargin canbeevenimprovedto O (1)ifthereisequalamountofasymmetryinthepositionand velocityfeedback.Additionally,weshowedthatnomattert hecontrolwassymmetric ornot,vehicle-to-vehicleheterogeneitydidnotchangeth escalingofstabilitymargin. Therefore,intermsofstabilitymargin,theasymmetriccon trolwithequalasymmetry schemeprovidesabestwaytoachievethegoaloflargerstabi litymargin.Thescenarios withunequalasymmetryinpositionandvelocityfeedbackan dasymmetricposition feedbacksareopenproblems. 2.7TechnicalProofs 2.7.1ProofofTheorem 2.2 WiththeprolesandcontrolgainsgiveninTheorem 2.2 ,thePDE( 2{8 )simpliesto: m ( x ) @ 2 ~ p ( x;t ) @t 2 = b ( x ) N 2 @ 2 ~ p ( x;t ) @x 2 + b ( x ) N 2 @ 3 ~ p ( x;t ) @x 2 @t ; (2{47) where m min m ( x ) m max ;b min b ( x ) b max .Duetothelinearityandhomogeneityof theabovePDEandboundaryconditions,weareabletoapplyth emethodofseparation ofvariables.Weassumesolutionoftheform~ p ( x;t )= P 1` =1 ` ( x ) h ` ( t ).Substitutingthe 51

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solutioninto( 2{47 )anddividingbothsidesby ` ( x ) h ` ( s ),weobtain: d 2 h ` ( t ) dt 2 N 2 h ` ( t )+ 1 N 2 h ( t ) = d 2 ` ( x ) dx 2 m ( x ) ` ( x ) =b ( x ) (2{48) Sinceeachsideoftheaboveequationisindependentfromthe other,soit'snecessaryfor bothsidesequaltothesameconstant ` .Thenwehavetwoseparateequations: d 2 h ` ( t ) dt 2 + ` N 2 dh ` ( t ) dt + ` N 2 h ` ( t )=0 ; (2{49) d 2 ` ( x ) dx 2 + ` m ( x ) b ( x ) ( x )=0 : (2{50) ThespatialpartsolvesthefollowingregularSturm-Liouvi lleeigenvalueproblem d 2 ` ( x ) dx 2 + ` m ( x ) b ( x ) ( x )=0 ; d (0) dx = (1)=0 : (2{51) TheRayleighquotientisgivenby ` = R 1 0 ( d ( x ) =dx ) 2 dx R 1 0 2 ( x ) m ( x ) =b ( x ) dx : (2{52) Since m min m ( x ) m max ;b min b ( x ) b max ,wehavethat m min b max m ( x ) =b ( x ) m max b min Pluggingthelowerandupperboundsfor m ( x ) =b ( s ),wehavethefollowingrelation: b min m max R 1 0 ( d ( x ) =dx ) 2 dx R 1 0 2 ( x ) dx ` b max m min R 1 0 ( d ( x ) =dx ) 2 dx R 1 0 2 ( x ) dx Sinceweknowtheeigenvalue ` correspondingtoRayleighquotient R 1 0 ( d ( x ) =dx ) 2 dx R 1 0 2 ( x ) dx isthe eigenvalueobtainedfrom( 2{51 )with m ( x ) =b ( x )=1.And ` isgivenby ` = (2 ` 1) 2 2 4 (2{53) where ` isthewavenumber, ` =1 ; 2 ; 52

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Itisstraightforwardtoseethattheleasteigenvalue ` isobtainbysetting ` =1,i.e. 1 = 2 = 4.Sowehavethefollowingboundsfortheleasteigenvalueof ` b min 2 4 m max 1 b max 2 4 m min (2{54) TakeLaplacetransformtobothsidesof( 2{50 ),weobtainthefollowingcharacteristic equationforthePDEmodel( 2{47 ). s 2 + ` N 2 s + ` N 2 =0 : Itseigenvaluesturnouttobetherootsoftheaboveequation s ` := ` =N 2 p 2` =N 4 4 ` =N 2 2 : (2{55) Wecall s ` the ` -thpairofeigenvalues.ThediscriminantDin( 2{55 )isgivenby: D := 2` =N 4 4 ` =N 2 : Forlarge N andsmall ` D isnegative.Soboththeeigenvaluesin( 2{55 )arecomplex, thenthestabilitymarginisonlydeterminedbytherealpart sof s ` .Itfollowsfrom( 2{55 ) thattheleaststableeigenvalue(theonesclosesttotheima ginaryaxis)amongthemisthe onethatisobtainedbyminimizing ` over ` .Then,thisminimumisachievedat ` =1, andtherealpartisobtained Real ( s 1 )= 1 2 N 2 : Followingthedenitionofstabilitymargin S := j Real ( s 1 ) j aswellastheboundsfor 1 givenby( 2{54 ),wecompletetheproof. 2.7.2ProofofProposition 2.1 Theproofproceedsbyaperturbationmethod.Lettheeigenva luesandLaplace transformationof~ p ( x;t )fortheperturbedPDE( 2{35 )be s ` = s (0)` + "s ( ) ` ; = (0) + ( ) respectively,where s (0)` and (0) arecorrespondingtotheunperturbedPDE( 2{19 ).Taking 53

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aLaplacetransformofPDE( 2{35 ),pluggingintheexpressionsfor s ` and ,anddoingan O ( )balance,whichleadsto: P ( ) = s (0)` ~ b m ( x ) N d (0) dx + s (0)` ~ b s ( x ) 2 N 2 d 2 (0) dx 2 2 m 0 s (0)` s ( ) ` (0) + s ( ) ` b 0 N 2 d 2 (0) dx 2 =: R; where P isdenedin( 2{29 ).Forasolution ( ) toexist, R mustlieintherangespaceof theself-adjointoperator P .Thus,wehave, =0 Wenowhavethefollowingequation: Z 1 0 s (0)` ~ b m ( x ) N d (0) dx + s (0)` ~ b s ( x ) 2 N 2 d 2 (0) dx 2 2 m 0 s (0)` s ( ) ` (0) + s ( ) ` b 0 N 2 d 2 (0) dx 2 (0) ` dx =0 Straightforwardmanipulationsshowthat: m 0 ( s (0)` + b 0 ` 2 m 0 N 2 ) s ( ) ` = s (0)` (2 ` 1) 4 N Z 1 0 ~ b m ( x )sin (2 ` 1) x dx s (0)` (2 ` 1) 2 2 8 N 2 Z 1 0 ~ b s ( x )cos 2 (2 ` 1) 2 x dx: (2{56) Substitutingtheequationaboveinto s ` = s (0)` + "s ( ) ` ,andsetting ` =1,wecompletethe proof. 54

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CHAPTER3 ROBUSTNESSTOEXTERNALDISTURBANCESOF1-DVEHICULARPLATO ON Inthischapterwestudytherobustnesstoexternaldisturba ncesofalarge1-D platoonofvehicleswithdistributedcontrol.Weconsidert herobustnesstoexternal disturbancesfortwodecentralizedcontrolarchitectures : predecessor-following and bidirectional .Ithasbeenknownforquitesometimethatthepredecessor-f ollowing architecturehasextremelypoorrobustnesstoexternaldis turbances[ 45 46 ].Inwasshown thatstringinstabilitywiththepredecessor-followingar chitectureisindependentofthe designofthecontrolleroneachvehicle,butafundamentala rtifactofthearchitecture[ 31 ]. Thehighsensitivitytodisturbanceofpredecessor-follow ingarchitectureledtothe examinationofthebidirectionalarchitecture.Mostworks focusonsymmetricbidirectional architecture.Thesymmetryassumptionsignicantlysimpl iedanalysis.Itwasshown thatsymmetricbidirectionalarchitecturesalsosuersfr ompoorrobustnesstoexternal disturbances[ 31 52 67 ]. Althougharichliteratureexistsonsensitivitytodisturb anceswithpredecessorfollowingandsymmetricbidirectionalarchitectures,tot hebestofourknowledge,aprecise comparisonoftheperformanceofthesetwoarchitectures-i ntermsofquantitative measuresofrobustnessislacking.Thischapteraddressese xactlythisproblem.In particular,weestablishhowcertain H 1 norms,thatquantiesthesystem'srobustness, scalewiththesizeoftheplatoonforeachofthesetwoarchit ectures.Westudytwo scenariostoquantifyrobustness.First,westudytheeect ofdisturbanceactingonthe leaderonthetrackingerrorofthelastvehicle.Second,wes tudytheeectofdisturbances actingonallthevehiclesintheplatoon(excepttheleader) ontheirtrackingerrors. Correspondingly,twokindsofperformancemetricsareused toquantifytherobustness: i)the leader-to-traileramplication ,whichisdenedasthe H 1 normofthetransfer functionfromthedisturbanceontheleadertothepositiont rackingerrorofthelast 55

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vehicle;ii)the all-to-allamplication ,whichisdenedasthe H 1 normofthetransfer functionfromthedisturbancesonallthefollowerstotheir positiontrackingerrors. Forthepredecessor-followingarchitecture,itiswellkno wnthattheleader-to-trailer amplicationgrowsgeometricallyandtheall-to-allampli cationcannotbebounded aboveuniformlyin N ,thenumberofvehiclesintheplatoon[ 31 53 ].Inthischapter, weshowthattheyareboth O ( N )forsome > 1.Thus,asthesizeoftheplatoon increases,theamplicationofdisturbanceincreasesgeom etrically.Wethenshowthatwith symmetricbidirectionalarchitecture,theleader-to-tra ileramplicationis O ( N ),whereas theall-to-allamplicationis O ( N 3 ).Inaddition,theresonancefrequenciesinbothcases are O (1 =N )[ 53 ].Thus,amongthetwocontrolarchitectures,thesymmetric bidirectional architectureperformsfarbetterthanthepredecessor-fol lowingarchitectureintermsof sensitivitytodisturbance,especiallyastheplatoonsize becomeslarge. Theanalysisforthesymmetricbidirectionalarchitecture iscarriedoutwithaPDE approximationoftheclosed-loopdynamics,whichisderive dinthepreviouschapter.The asymptoticformulaeforthetwoamplicationfactorsmenti onedaboveandtheresonance frequenciesareobtainedusingaPDE-basedanalysis.Numer icalcomputationsofthe coupled-ODEmodelareprovidedtoverifytheanalysisofthe correspondingPDEmodel. AlthoughthePDEisderivedundertheassumptionthat N islarge,numericalresults showthatitmakesanaccurateapproximationevenwhen N issmall(e.g. N =10). Weassumeeachvehiclehasadouble-integratordynamicsand theplatoonishomogeneous:eachvehicleintheplatoonhasthesameopen-loo pdynamicsandusesthe samecontrollaw.Theassumptionofdouble-integratordyna micscomesfromthefactthat single-integratormodelsfailtoreproducetheslinky-typ eeectsorstringinstability[ 3 ] andhigherorderdynamicswillresultininstabilityforsu cientlarge N [ 52 53 ].In addition,heterogeneityinvehiclemassandcontrolgainsh aslittleeectonthestability marginandsensitivitytodisturbanceoftheplatoon[ 62 67 84 ].However,weshowby numericalsimulationthatasymmetryhasasubstantialeec tontherobustnessofthe1-D 56

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platoon.Judiciousasymmetryinthecontrolgainscanimpro vetherobustnessofthe1-D platoonconsiderablyoversymmetriccontrol. Therestofthischapterisorganizedasfollows.Section 3.1 presentstheproblem statement.Section 3.2 describesthePDEmodelofthe1-Dplatoonofdouble-integra tor vehicleswithsymmetricbidirectionalarchitecture.Anal ysisofthe H 1 normsofthe systemforbothsymmetricbidirectionalandpredecessor-f ollowingarchitecturesaswellas theconjectureforasymmetricbidirectionalarchitecture andtheirnumericalverications appearinSection 3.3 .ThechapterendswithasummaryinSection 3.4 3.1ProblemFormulation Weconsidertheformationcontrolof N +1homogeneousvehicles(1leaderand N followers)whicharemovingin1-DEuclideanspace,asshown inFigure 2-1 (a).The positionofthe i -thvehicleisdenotedby p i 2 R .Thedynamicsofeachvehicleare modeledasadoubleintegrator: m i p i = u i + w i ;i 2f 1 ; 2 ; ;N g ; (3{1) where m i isthemass, u i isthecontrolinputand w i istheexternaldisturbanceonthe i -th vehicle.Thedisturbanceoneachvehicleisassumedtobe w i = a i sin( !t + i ).Thisisa commonlyusedmodelforvehicledynamicsinstudyingvehicu larformations,andresults fromfeedbacklinearizationofnon-linearvehicledynamic s[ 3 39 49 ]. Thecontrolobjectiveisthatvehiclesmaintainarigidform ationgeometrywhile followingaconstant-velocitytypedesiredtrajectory.Th edesiredgeometryoftheformationisspeciedbyconstantdesiredinter-vehiclespaci ng ( i 1 ;i ) for i 2f 1 ; ;N g where ( i 1 ;i ) isthedesiredvalueof p i 1 ( t ) p i ( t ).Eachvehicle i knowsthedesiredgaps ( i 1 ;i ) ( i;i +1) .Thedesiredtrajectoryoftheplatoonisspeciedintermso faleader whosedynamicsareindependentoftheothervehicles.Thele aderisindexedby0,andits trajectoryisdenotedby p 0 ( t )= vt + (0 ;N ) ,where v isapositiveconstant,whichisthe cruisevelocityoftheplatoon.Thedesiredtrajectoryofth e i -thvehicle, p i ( t ),isgivenby 57

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p i ( t )= p 0 ( t ) (0 ;i ) = p 0 ( t ) P ij =1 ( j 1 ;j ) .Tofacilitateanalysis,wedenethetracking error: ~ p i := p i p i ) ~ p i =_ p i p i : (3{2) Weconsiderthefollowingdecentralizedcontrollaw,where thecontrolonthe i -th vehicledependsontherelativepositionandvelocitymeasu rementsfromitsimmediate predecessorandpossiblyitsimmediatefollower: u i = k f i (~ p i ~ p i 1 ) k b i (~ p i ~ p i +1 ) b fi ( ~ p i ~ p i 1 ) b bi ( ~ p i ~ p i +1 ) u N = k f i (~ p N ~ p N 1 ) b fi ( ~ p N ~ p N 1 ) ; (3{3) where i 2f 1 ; ;N 1 g and k f i ;k b i (respectively, b fi ;b bi )arethefrontandbackposition (respectively,velocity)gainsofthe i -thvehicle.Notethattheinformationneededto computethecontrolactioncanbeeasilyaccessedbyon-boar dsensors,sinceonlyrelative informationisused. Resultsin[ 62 67 84 ]showthatheterogeneityinvehiclemassandcontrolgainsh as littleeectonthesensitivitytodisturbanceandstabilit ymarginoftheplatoon.Therefore wefocuson homogeneous platoons,inwhicheveryvehiclehasthesamedynamicsand employsthesamecontrollaw.Inparticular, k f i =(1+ k ) k 0 ;k b i =(1 k ) k 0 ; b fi =(1+ b ) b 0 ;b bi =(1 b ) b 0 ; (3{4) m i =1 ;i 2f 1 ; 2 ; ;N g ; where k 2 [0 ; 1]and b 2 [0 ; 1]aretheamountsofasymmetryinthepositionandvelocity gainsrespectively.Denition3.1. Wecallthearchitecturecorrespondingto k = b =0 the symmetric bidirectional ,sincethecontrolactiononeachvehicledependsequallyon theinformation fromitsimmediatepredecessorandfollower;andthearchit ecturecorrespondingto 58

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" k = b =1 arecalledthe predecessor-following ,sincethecontrolactiononeach vehicleonlydependsontheinformationfromitsimmediatep redecessor.Thearchitecture correspondingtoothercasesiscalled asymmetricbidirectional Westudyhowthesensitivitiestoexternaldisturbancessca lewithrespecttothe numberofvehicles N intheplatoon.Wedenethefollowingtwometrics. Denition3.2. The leader-to-traileramplication H LTT isdenedasthe H 1 norm ofthetransferfunctionfromthedisturbanceontheleadert othelastvehicle'sposition trackingerror.The all-to-allamplication H ATA isdenedasthe H 1 normofthetransfer functionfromthedisturbancesactingonallthefollowerst otheirpositiontrackingerrors. Inthecaseofleader-to-traileramplication,weassumeth ereisasinusoidaldisturbanceonlyontheleader,whereastheothervehiclesareu ndisturbed,i.e. w i = 0 ;i 2f 1 ; ;N g .Weexaminetheeectofthedisturbanceontheleader W = w 0 = a 0 sin( !t + 0 ) 2 R tothepositiontrackingerrorofthelastvehicle E =~ p N 2 R .Without lossofgenerality,let a 0 =1and 0 =0forthiscase.Withthissinusoidaldisturbance, thedesiredtrajectoryoftheleaderisnowgivenby p 0 ( t )= vt + (0 ;N ) +sin( !t ).Inthe predecessor-followingarchitecture,theclosed-loopdyn amicscannowbeexpressedasthe followingcoupled-ODEmodel ~ p i = 2 k 0 (~ p i ~ p i 1 ) 2 b 0 ( ~ p i ~ p i 1 )+ 2 sin( !t ) ; (3{5) where i 2f 1 ; ;N g .Forthebidirectionalarchitecture,theclosed-loopdyna micscanbe expressedas ~ p i = k f i (~ p i ~ p i 1 ) k b i (~ p i ~ p i +1 ) b fi ( ~ p i ~ p i 1 ) b bi ( ~ p i ~ p i +1 )+ 2 sin( !t ) ; (3{6) ~ p N = k f i (~ p N ~ p N 1 ) b fi ( ~ p N ~ p N 1 )+ 2 sin( !t ) ; where i 2f 1 ; ;N 1 g 59

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Inthecaseofall-to-allamplication,weassumethereared isturbancesactingonall thefollowersbutnottheleader,andstudythe H 1 normofthetransferfunctionfromthe disturbancesonallthefollowers W =[ w 1 ;w 2 ; ;w N ] 2 R N totheirpositiontracking errors E =[~ p 1 ; ~ p 2 ; ; ~ p N ] 2 R N ,where~ p i isdenedin( 3{2 ).Sincethereisnodisturbance ontheleader,itsdesiredtrajectoryisthengivenby p 0 ( t )= vt + (0 ;N ) .Usingtheposition trackingerrorsdenedin( 3{2 ),forthepredecessor-followingarchitecture,theclosed -loop dynamicscanbeexpressedas ~ p i = k f i (~ p i ~ p i 1 ) b fi ( ~ p i ~ p i 1 )+ w i ; (3{7) where i 2f 1 ; ;N g .Forthebidirectionalarchitecture,theclosed-loopdyna micscanbe writtenas ~ p i = k f i (~ p i ~ p i 1 ) k b i (~ p i ~ p i +1 ) b fi ( ~ p i ~ p i 1 ) b bi ( ~ p i ~ p i +1 )+ w i ; (3{8) ~ p N = k f i (~ p N ~ p N 1 ) b fi ( ~ p N ~ p N 1 )+ w N ; where i 2f 1 ; ;N 1 g Forboththedisturbanceamplicationsconsideredabove,t hecoupled-ODEmodels withthepredecessor-followingandbidirectionalarchite cturescanberepresentedinthe followingstate-spaceform: X = AX + BW;E = CX; (3{9) where X isthestatevector,whichisdenedas X :=[~ p 1 ; ~ p 1 ; ; ~ p N ; ~ p N ] 2 R 2 N W is inputvector(externaldisturbances)and E istheoutputvector(positiontrackingerrors). Forexample,thestatematrixforthepredecessor-followin gandsymmetricbidirectional architecturearegivenby A p or b = I N n M 1 + L p or b n M 2 ,where I N isthe N N identity 60

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matrixand n denotestheKroneckerproduct.Theauxiliarymatrices M 1 ;M 2 aregivenby: M 1 = 264 0100 375 ;M 2 = 264 00 k 0 b 0 375 : Thematrix L ( : ) forthepredecessor-followingandsymmetricbidirectiona larchitecturesare respectivelygivenby L p = 266666664 1 11 . 11 377777775 ;L b = 266666666664 2 1 12 1 . . 12 1 11 377777777775 : Forthecaseoftheleader-to-traileramplication,theinp utmatrix B andoutputmatrix C aregivenby B = 2 [0 ; 1 ; ; 0 ; 1] T 2 R 2 N C =[0 ; 0 ; ; 0 ; 1 ; 0] 2 R 2 N respectively.The correspondingmatricesforthecaseofall-to-allamplica tionaregivenby B = I N n [0 ; 1] T C = I N n [1 ; 0]respectively.Thecasewithasymmetriccontrolcanbecon structedsimilarly, butthestatematrix A ingeneraldoesnothavesuch\nice"formasshownabove. Recallthatthe H 1 normofatransferfunction G ( s )= C ( sI A ) 1 B from W to E is denedas: jj G ( j! ) jj H 1 =sup 2 R + max [ G ( jw )]=sup W jj E jj L 2 jj W jj L 2 ; (3{10) where max denotesthemaximumsingularvalue. 1 Forthepredecessor-followingarchitecture,thedynamicsofeachvehicleonlydependontheinfo rmationfromitspredecessor. Duetothisspecialcoupledstructure,aclosed-formtransf erfunctioncanbederived. 1 Inthischapter,the L 2 normiswell-denedintheextendedspace L 2e = f u j u 2 L 2 ; 8 2 [0 ; 1 ) g ,where u ( t )=( i ) u ( t ) ; if0 t ;( ii )0 ; if t>: See[ 85 ,Chapter 5].Withalittleabuseofnotation,wesuppressthesubscrip tandwrite L 2 = L 2e 61

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Thereforewecanderiveestimatesfortheleader-to-traile randall-to-allamplications byusingstandardmatrixtheory.However,forbidirectiona larchitecture,itisingeneral diculttondaclosed-formformulafortheleader-to-trai lerandall-to-allamplications fromthestate-spacedomain.Thereareseveralreasons.Fir stofall,whenthenumberofvehiclesintheplatoonislarge,it'sveryinvolvedto computematrixinverseand multiplications,whichmakesitdiculttondaclosed-for mtransferfunctionforthis architecture.Second,thecoupled-ODEmodelprovidesnoin formationaboutatwhich frequency thesystem'sresonanceoccursandwhichinputcausesthewor stdisturbance amplication.Third,thecalculationofsingularvaluefor alargematrixisnotaeasytask. Duetothesereasons,wetakeanalternaterouteandproposea PDEmodel,whichisseen asacontinuumapproximationofthecoupled-ODEmodels( 3{6 )and( 3{8 ),toanalyze andstudythe H 1 normsofthe1-Dplatoonofdouble-integratorvehicles.Thi sPDE modelprovidesaconvenientframeworktoanalysis.Baseont hePDEmodel,closed-form formulaeofthe H 1 normsandresonancefrequencyareobtained. 3.2PDEModelsofthePlatoonwithSymmetricBidirectionalA rchitecture Theanalysisinthesymmetricbidirectionalarchitecturer eliesonPDEmodels,which areseenasacontinuumapproximationoftheclosedloopdyna mics( 3{6 )and( 3{8 )in thelimitoflarge N ,byfollowingthestepsinvolvedinanite-dierencediscr etization inreverse.ThederivationofthePDEmodelissimilartothep roceduresintheprevious chapter.3.2.1PDEModelfortheCaseofLeader-to-TrailerAmplicat ion WerstderiveaPDEmodelforthecaseofleader-to-trailera mplication,where thereisdisturbanceonlyontheleader,i.e. w i =0,for i 2f 1 ; 2 ; ;N g .Withsymmetric controlgains k f i = k b i = k 0 ;b fi = b bi = b 0 ,theclosed-loopdynamics( 3{6 )canbewrittenas ~ p i = k 0 N 2 (~ p i 1 2~ p i +~ p i +1 ) 1 =N 2 + b 0 N 2 ( ~ p i 1 2 ~ p i + ~ p i +1 ) 1 =N 2 + 2 sin( !t ) : (3{11) 62

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FollowingthesameproceduresinChapter 2 ,weconsiderafunction~ p ( x;t ):[0 ; 1] [0 ; 1 ) R thatsatises: ~ p i ( t )=~ p ( x;t ) j x =( N i ) =N ; (3{12) suchthatfunctionsthataredenedatdiscretepoints i willbeapproximatedbyfunctions thataredenedeverywherein[0 ; 1].Theoriginalfunctionsarethoughtofassamplesof theircontinuousapproximations.Usethefollowingnited ierenceapproximations: h ~ p i 1 2~ p i +~ p i +1 1 =N 2 i = h @ 2 ~ p ( x;t ) @x 2 i x =( N i ) =N ; h ~ p i 1 2 ~ p i + ~ p i +1 1 =N 2 i = h @ 3 ~ p ( x;t ) @x 2 @t i x =( N i ) =N : Undertheassumptionthat N islargebutnite,Eq.( 3{11 )canbeseenasnitedierence discretizationofthefollowingPDE: @ 2 ~ p ( x;t ) @t 2 = k 0 N 2 @ 2 ~ p ( x;t ) @x 2 + b 0 N 2 @ 3 ~ p ( x;t ) @x 2 @t + 2 sin( !t ) : (3{13) TheboundaryconditionsofPDE( 3{13 )dependonthearrangementofleaderinthe graph.Forourcase,theboundaryconditionsareoftheDiric hlettypeat x =1wherethe leaderis,andNeumannat x =0: @ ~ p @x (0 ;t )=0 ; ~ p (1 ;t )=0 : (3{14) 3.2.2PDEModelfortheCaseofAll-to-AllAmplication Forthiscase,therearedisturbancesonallthefollowersbu tnodisturbanceon theleader.Withsymmetriccontrol,theclosed-loopdynami csareslightlydierent from( 3{11 ),whicharegivenby ~ p i = k 0 N 2 (~ p i 1 2~ p i +~ p i +1 ) 1 =N 2 + b 0 N 2 ( ~ p i 1 2 ~ p i + ~ p i +1 ) 1 =N 2 + a i sin( !t + i ) : (3{15) 63

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Followingthesameprocedureasin 3.2.1 ,wederivethefollowingPDEmodel @ 2 ~ p ( x;t ) @t 2 = k 0 N 2 @ 2 ~ p ( x;t ) @x 2 + b 0 N 2 @ 3 ~ p ( x;t ) @x 2 @t + a ( x )sin( !t + ( x )) ; (3{16) where a ( x ) ; ( x ):[0 ; 1] R denedaccordingtothefollowingstipulations: a i = a ( x ) j x = N i N ; i = ( x ) j x = N i N : (3{17) TheboundaryconditionsoftheabovePDE( 3{16 )arethesameasbefore,whichisgiven in( 3{14 ). ThePDEmodels( 3{13 )and( 3{16 )areforcedwaveequationswithKelvin-Voigt damping.Theyareapproximationsofthecoupled-ODEmodels inthesensethata nitedierencediscretizationofthePDEsyield( 3{6 )and( 3{8 )respectively.Thenite dierencemethodtonumericallysolvepartialdierential equation,itsapproximation errorsandstabilityanalysisarewellstudiedin[ 77 86 ].Interestedreaderisreferred to[ 77 86 ]foracomprehensivestudy. 3.3RobustnesstoExternalDisturbances 3.3.1Leader-to-traileramplicationwithsymmetricbidi rectionalarchitecture Forasingle-input-single-outputsystem,the H 1 normoftheplatooniseectively themaximummagnitudeofthefrequencyresponse.Foranysin usoidaldisturbance w 0 = sin( !t )ontheleader,weneedtondthesinusoidaloutput~ p (0 ;t )withthemaximum amplitudeoverallfrequencies Werstpresenttherstmainresultofthischapterconcerni ngtheleader-to-trailer amplicationforsymmetricbidirectionalarchitecture.Theorem3.1. ConsiderthePDEmodel ( 3{13 ) ( 3{14 ) ofthe1-Dplatoonwithsymmetric bidirectionalarchitecture,theleader-to-trailerampli cation H sb LTT andresonancefrequency sb r havetheasymptoticformula H sb LTT 8 p k 0 N b 0 2 ;! sb r p k 0 2 N : (3{18) 64

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Theseformulaeholdforlarge N ProofofTheorem 3.1 Considerthecaseofleader-to-traileramplication,whos edynamics arecharacterizedbyPDE( 3{13 )withboundarycondition( 3{14 ).Itisanonhomogeneous PDEwithhomogeneousboundaryconditions.Thesolutionof~ p (0 ;t )canbesolvedby eigenfunctionexpansion,see[ 77 ,Chapter8].Toproceed,werstconsiderthefollowing homogeneousPDEwithhomogeneousboundaries( 3{14 ) @ 2 ~ p ( x;t ) @t 2 = k 0 N 2 @ 2 ~ p ( x;t ) @x 2 + b 0 N 2 @ 3 ~ p ( x;t ) @x 2 @t : (3{19) TheabovePDEcanbesolvedbythemethodofseparationofvari ables,weassume solutionoftheform~ p ( x;t )= P 1` =1 ` ( x ) h ` ( t ).Substitutingthesolutionintotheabove PDE( 3{19 ),wegetthefollowingspace-dependentODE 1 N 2 d 2 ` ( x ) dx 2 + ` ` ( x )=0 ; (3{20) where ` =(2 ` 1) 2 2 = (4 N 2 )and ` ( x )=cos((2 ` 1) x= 2)aretheeigenvalueand itscorrespondingeigenfunctionoftheSturm-Liouvilleei genvalueproblem( 3{20 )with followingboundaryconditions,whichcomefrom( 3{14 ), d ` dx (0)=0 ; ` (1)=0 : (3{21) Noticethattheeigenvalue 1 isthesmallesteigenvalue,whichiscalledtheprincipalmo de ofthedampedwaveequation( 3{19 ).Sincetheeigenfunctionsarecomplete(becauseof Sturm-LiouvilleTheory),anypiecewisesmoothfunctionsc anbeexpandedinaseries oftheseeigenfunctions,see[ 77 ].Therefore,weexpandtheexternalforcingtermsin PDE( 3{13 )as 2 sin( !t )= 1 X ` =1 c ` ` ( x ) 2 sin( !t ) ; (3{22) 65

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where c ` isgivenby c ` =2 R 1 0 ` ( x ) dx =( 1) ` +1 4 = ((2 ` 1) ).Substituting( 3{22 )into PDE( 3{13 ),andusing~ p ( x;t )= P 1` =1 ` ( x ) h ` ( t ),wegetthefollowingODEs d 2 h ` ( t ) dt 2 + b 0 ` dh ` ( t ) dt + k 0 ` h ` ( t )= c ` 2 sin( !t ) ; (3{23) where ` 2f 1 ; 2 ; g .Thesearesecondordersystemswithsinusoidalinputwhose amplitudedependsontheirfrequency Foreachmode ` ,thesteady-stateresponse h ` ( t )isgivenby h ` ( t )= c ` 2 p 4 +( b 20 2` 2 k 0 ` ) 2 + k 2 0 2` sin( !t + ` ) = A ` sin( !t + ` ) (3{24) forsomeconstant ` .Themaximumamplitude A ` anditsresonancefrequencyforeach modecanbedeterminedbyastraightforwardmanner,whichar e: A ` = 8 N (2 ` 1) 2 2 1 p b 20 =k 0 (2 ` 1) 2 b 40 2 = (16 k 2 0 N 2 ) ; (3{25) ` = p k 0 p 4 N 2 b 20 2 = (2 k 2 0 ) : (3{26) Thepositiontrackingerrorofthelastvehicleisnowgivenb y~ p (0 ;t )= P 1` =1 ` (0) h ` ( t )= P 1` =1 A ` sin( !t ).Togetthemaximumamplitude,thefrequency mustbeoneoftheresonancefrequency ` ofthedampedwaveequation( 3{13 ),see[ 77 ].Forlarge N ,it'snot diculttoseefrom( 3{25 )that,themaximumisachieveat sb r = 1 .Moreover,since A 1 dominatestheother A ` ( ` =2 ; 3 ; ),the H 1 normofthesystemisapproximately A 1 Usingtheassumptionthat N islargein( 3{25 )and( 3{26 ),wecompetetheproof. 3.3.2All-to-allAmplicationwithSymmetricBidirection alArchitecture Wenowpresenttheresultonall-to-allamplicationforthe 1-Dplatoonofdoubleintegratorvehicleswithsymmetricbidirectionalarchite cture. Theorem3.2. ConsiderthePDEmodel ( 3{14 ) ( 3{16 ) ofthe1-Dplatoonwithsymmetric bidirectionalarchitecture,theall-to-allamplication H sb ATA andresonancefrequency sb r 66

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havetheasymptoticformula H sb ATA 8 N 3 p k 0 b 0 3 ;! sb r p k 0 2 N : (3{27) Theseformulaeholdforlarge N ProofofTheorem 3.2 Foramulti-input-multi-outputsystem,the H 1 normisdenedas thesupremumofthemaximumsingularvalueofthetransferfu nctionmatrix G ( j! )over allfrequency 2 R + .Equivalently,itcanbeinterpretedinasinusoidal,stead y-statesense asfollows(see[ 87 ]).Foranyfrequency ,anyvectorofamplitudes a =[ a 1 ; ;a N ]with k a k 2 1,andanyvectorofphases =[ 1 ; ; N ],theinputvector W =[ w 1 ; ;w N ] =[ a 1 sin( !t + 1 ) ; ;a N sin( !t + N )](3{28) yieldsthesteady-stateresponseof E oftheform E =[~ p 1 ; ; ~ p N ]=[ b 1 sin( !t + 1 ) ; ;b N sin( !t + N )] : (3{29) The H 1 normof G ( j! )canbedenedas k G ( j! ) k H 1 =sup k b k 2 =sup 2 R + ;a; 2 R N k E k L 2 k W k L 2 ; (3{30) where b =[ b 1 ; ;b N ].Therefore,inthePDEcounterpart,the H 1 normisdeterminedby H 1 =sup 2 R + ;a ( x ) ; ( x ) jj ~ p ( x;t ) jj L 2 k a ( x )sin( !t + ( x )) k L 2 ; (3{31) where a ( x )and ( x )arepiecewisesmoothfunctionsdenedin[0 ; 1]. PDE( 3{16 )isanonhomogeneousPDEwithhomogeneousboundaryconditi ons, thereforewecanuseeigenfunctionexpansiontoexpandthen onhomogeneousterms. Beforeweproceed,noticethat a ( x )sin( !t + ( x ))= a 1 ( x )sin( !t )+ a 2 ( x )cos( !t ) ; 67

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where a 1 ( x )= a ( x )cos( ( x ))and a 2 ( x )= a ( x )sin( ( x )).Fromthesuperposition propertyoflinearsystem,theoutputisthesumoftheoutput scorrespondingtoinputs a 1 ( x )sin( !t )and a 2 ( x )cos( !t )respectively. WerstconsidertheresponseofthePDEwithinput a 1 ( x )sin( !t ).ThePDEisnow givenby @ 2 ~ p ( x;t ) @t 2 = k 0 N 2 @ 2 ~ p ( x;t ) @x 2 + b 0 N 2 @ 3 ~ p ( x;t ) @x 2 @t + a 1 ( x )sin( !t ) : (3{32) Asbefore,usingeigenfunctionexpansion, a 1 ( x )canbeexpandedasaseriesintermsof ` ( x ),i.e. a 1 ( x )= P 1` =1 d ` ` ( x ).SubstitutingtheseriesintotheabovePDEandusing ~ p ( x;t )= P 1` =1 ` ( x ) h ` ( t ),wehavethefollowingtime-dependentODEs: d 2 h ` ( t ) dt 2 + b 0 ` dh ` ( t ) dt + k 0 ` h ` ( t )= d ` sin( !t ) ; (3{33) where ` 2f 1 ; 2 ; g and d ` isgivenby d ` =2 Z 1 0 a 1 ( x ) ` ( x ) dx: (3{34) Again,foreachmode ` ,thesteady-stateresponse h ` ( t )isgivenby h ` ( t )= d ` p 4 +( b 20 2` 2 k 0 ` ) 2 + k 2 0 2` sin( !t + ` ) = A ` d ` sin( !t + ` ) ; (3{35) forsomeconstant ` .Followingstraightforwardalgebra,themaximumamplitud e A ` and itsresonancefrequencyforeachmodeis A ` = 8>><>>: 8 N 3 (2 ` 1) 3 b 0 3 1 p k 0 (2 ` 1) 2 b 20 2 = (16 N 2 ) ; if ` ` 0 1 ` k 0 ; otherwise. (3{36) ` = 8>><>>: (2 ` 1) 2 N p k 0 (2 ` 1) 2 b 20 2 = (8 N 2 ) ; if ` ` 0 0 ; otherwise. (3{37) 68

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where ` 0 = 2 p 2 k 0 N + 2 Again,when N islarge,it'snotdiculttoseefrom( 3{36 )that,themaximum of A ` isachieveat = 1 .Therefore,foranite L 2 normof a 1 ( x ),toachievethe largest L 2 normof~ p ( x;t ), a 1 ( x )shouldbeequaltotheeigenfunctionoftherstmode a 1 ( x )= 1 ( x ),i.e.theprojectionof a 1 ( x )ontoothereigenfunctionsiszero d ` =0( ` = 2 ; 3 ; ).Similarly,thefollowingrelationship a 2 ( x )= 1 ( x )shouldholdforinput a 2 ( x )cos( !t ),whichimplies ( x )= 0 isconstant,since a 1 ( x )= a ( x )cos( ( x ))and a 2 ( x )= a ( x )sin( ( x )). Consequently,theoutputwiththemaximum L 2 normisgivenby ~ p ( x;t )= A 1 1 ( x )sin( !t + 1 ) : (3{38) Therefore,the H 1 normofthesystemisobtained H 1 = A 1 k 1 ( x )sin( !t + 1 ) k L 2 k 1 ( x )sin( !t + 0 ) k L 2 = A 1 : (3{39) Usingtheassumptionthat N islargein( 3{36 )and( 3{37 ),wecompetetheproof. 3.3.3DisturbanceAmplicationwithPredecessor-Followi ngArchitecture Similarresultsasleader-to-traileramplicationwithpr edecessor-followingarchitectureexistintheliterature[ 31 45 ].Inthissection,wepresenttheseresultsforthesakeof completion.Inaddition,wehavealsoconsiderthecaseofal l-to-allamplication. Theorem3.3. Consideran N -vehicleplatoonwithpredecessor-followingarchitectur e,the leader-to-traileramplication H p LTT andall-to-allamplication H p ATA areasymptotically H p LTT N ; (3{40) H p ATA r 2 N 1 2 1 ; (3{41) 69

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wheretheaboveformulaeholdforlarge N .Inparticular, = j T ( j! p r ) j > 1 = j S ( j! p r ) j where T ( s )= 2 b 0 s +2 k 0 s 2 +2 b 0 s +2 k 0 ;S ( s )= 1 s 2 +2 b 0 s +2 k 0 ; and r istheresonancefrequencyforbothcases,whichisgivenby p r q p k 4 0 +4 k 3 0 b 20 k 2 0 b 0 : Theprooffollowsasimilarlineofattackastheworkin[ 31 ].Interestedreadersare referredtoCorollary1of[ 88 ]foranexplicitproof. 3.3.4DisturbanceAmplicationwithAsymmetricBid.Archi tecture Fortheasymmetricbidirectionalarchitecture,weconside rthefollowingcontrolgains, whichstabilizetheplatoon,seeChapter 2 : 1)Equalamountofasymmetry,i.e.0 <" k = b < 1.Inthiscase,itwasshown inTheorem3 : 5of[ 68 ]thatcertainamplicationfactor(whichisdierentfrom H LTT and H ATA denedinthischapter)growsexponentiallyin N .Weshowbynumerical simulationsthattheleader-to-trailer H as LTT andall-to-allamplications H as ATA withequal asymmetryareapproximately O ( e N ),seeSection 3.3.6 .Theasymmetricbidirectional architecturewithequalasymmetryinthepositionandveloc ityfeedbackthussuersfrom highsensitivitytodisturbances,asthepredecessor-foll owingarchitecture.However,it doesn'timplyasymmetricbidirectionalarchitecturesisn otpreferable,asshownbelow. 2)Asymmetricvelocityfeedback,i.e. k =0 ; 0 <" b < 1.ItwasshowninChapter 2 thatthestabilitymargin,whichisdenedastheabsoluteva lueoftherealpartofthe leaststableeigenvalueofthestatematrix A ,canbeimprovedconsiderablybyusingthe asymmetricvelocityfeedbackoversymmetriccontrol.Thea nalysiswasalsocarriedout basedonthePDEmodelwederivedbefore.Weconjecturethatt herobustnesscanalso beamelioratedsignicantlywithasymmetricvelocityfeed back,whichiswitnessedby extensivenumericalsimulations. 70

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Conjecture3.1. Consideran N -vehicleplatoonwithasymmetricbidirectionalarchitecture.Whenthereissmallasymmetryinthevelocityfeedback ,i.e. k =0 ; 0 <" b 1 ,the leader-to-traileramplication H av LTT andall-to-allamplication H av ATA asymptoticallysatisfy H av LTT O (1) ;H av ATA O ( N 2 ) : 3.3.5DesignGuidelines ComparingtheaboveconjecturewiththoseresultsinTheore m 3.1 ,Theorem 3.2 and Theorem 3.3 aswellasTheorem3 : 5of[ 68 ](equalasymmetry),weseethatasymmetric velocityfeedbackyieldsthebestrobustnessperformancec omparedtootherarchitectures. Thenextpreferablechoiceisthesymmetricbidirectionala rchitecture.Thepredecessorfollowingandasymmetricbidirectionalwithequalamounto fasymmetryaretheworst choicesforcontroldesignintermsofrobustness,theirlea der-to-trailerandall-to-all amplicationsgrowextremelyfastwith N Besidestherobustnessperformancemetricsanalyzedinthi schapter,itwasalsostudiedinthepreviouschapterthathowthestabilitymarginsca leswiththesizeofplatoon.It wasshowninthepreviouschapterthatwithsymmetricbidire ctionalarchitecture,thestabilitymargindecaystozeroas O (1 =N 2 ).Itcanbeimprovedto O (1 =N )withasymmetric velocityfeedback.Inaddition,itwasshownin[ 88 ]and[ 89 ]thatwithpredecessorfollowingarchitectureandasymmetricbidirectionalarch itecturewithequalasymmetry, thestabilitymarginare O (1).However,thetransienterrorsinthesearchitecturesg row considerablybeforetheydieout. Inconclusion,togetabetterstabilitymarginandrobustne ssperformance,the asymmetricvelocityfeedbackisthebestchoiceforcontrol design. 71

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10 20 50 100 250 10 0 10 2 10 4 10 6 10 8 NH LTT Asymmetricbidi.(Asymmetricvelocity) Symmetricbidi. Symmetricbidi. (Prediction( 3{18 )) (Equalasymmetry) Asymmetricbidi. Predecessorfoll. Predecessorfoll. (Prediction( 3{40 )) Conjecture 3.1 (a)Leader-to-traileramplication H LTT 10 20 50 100 250 10 0 10 5 10 10 10 15 N H ATA Asymmetricbidi.(Asymmetricvelocity) Symmetricbidi. Symmetricbidi. (Equalasymmetry) Asymmetricbidi. Predecessorfoll. Predecessorfoll. (Prediction( 3{41 )) Conjecture 3.1 Conjecture 3.1 (b)All-to-allamplication H ATA Figure3-1.Numericcomparisonofdisturbanceamplicatio nbetweendierent architectures. 72

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3.3.6NumericalVerication Inthissection,wecomparetherobustnessoftheplatoonwit hdierentcontrol architectures.Inaddition,weverifytheanalyticpredict ionsinTheorem 3.1 ,Theorem 3.2 andTheorem 3.3 withtheirnumericallycomputedvalues.Allnumericalcalc ulations areperformedinMatlab c r .Figure 3-1 showsthecomparisonbetweenthepredecessorfollowingandbidirectionalarchitecturesforboththelea der-to-traileramplicationand all-to-allamplication.Wecanseethatforbothamplicat ions,theygrowgeometricallyin thepredecessor-followingarchitectureandasymmetricbi directionalarchitecturewithequal asymmetry.Incontrast,inthesymmetricbidirectionalarc hitecture,theseamplications growmuchslowerthanthetwoarchitecturesaforementioned .Inaddition,theasymmetric velocityfeedbackarchitecturegivesthebestrobustnessp erformance.Besides,wesee thatthenumericalresultsofthetwoamplicationsintheas ymmetricvelocityfeedback architecturecoincidewithourconjecture.Moreover,thea nalyticpredictionsmatchthe numericalresultsverywell,whichveriedouranalysisinT heorem 3.1 ,Theorem 3.2 and Theorem 3.3 .Inallcases,thecontrolgainsusedare k 0 =1and b 0 =0 : 5.Theamountsof asymmetryinthecasesofequalasymmetryandasymmetricvel ocityfeedbackaregivenby k = b =0 : 2and k =0 ;" b =0 : 2respectively. 3.4Summary Westudiedtherobustnesstoexternaldisturbancesoflarge platoonofvehicles withtwodecentralizedcontrolarchitectures:predecesso r-followingandbidirectional.In particular,weexaminedhowtheleader-to-traileramplic ationandall-to-allamplication scalewith N ,thenumberofvehiclesintheplatoon.Forbothmetrics,weo btainedtheir explicitscalinglawswithrespecttothenumberofvehicles intheplatoonforsymmetric control.Inaddition,wealsoconsidertheeectofasymmetr iccontrolonthedisturbance amplication.Numericalsimulationsshowthattheasymmet ricvelocityfeedbackinthe bidirectionalarchitecturehasmuchlowersensitivitytoe xternaldisturbancethantheother 73

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architectures.Theanalysisofasymmetriccontrolonthero bustnesstodisturbanceisan ongoingwork. 74

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CHAPTER4 STABILITYMARGINANDROBUSTNESSOFVEHICLETEAMSWITH D-DIMENSIONALINFORMATIONGRAPH Weconsidertheproblemofformationcontrolofvehiclesinh igher-dimensionalspace sothatneighboringvehiclesmaintainaconstantpre-speci edspacingwhileinmotion. Thisproblemisrelevanttoanumberofapplicationssuchasf ormationryingofaerial, ground,andautonomousvehiclesforsurveillance,reconna issance,mine-sweeping.The interactionbetweenvehiclesisdescribedbyaninformatio ngraph.Inthischapter,we limitourattentiontoaspecicclassofinformationgraphs ,namely, D -dimensional (nite)lattices.Thesearenaturalchoicesforinformatio ngraphsin2Dor3Dformation problemsinwhichvehiclesarearrangedinregularpatterna ndrelativemeasurementsare possibleamongphysicallyclosestvehicles.Theplatoonpr oblemisaspecialcase,whose informationgraphisa1-Dlattice.Afewleadvehiclesarepr ovidedinformationontheir desiredtrajectoriesthattheyuseincomputingtheircontr olactions;whiletherestofthe vehiclesareallowedtouseonlylocallyavailableinformat ion. Theone-dimensionalversionofthisproblem,inwhichastri ngofvehiclesmoving inastraightlinehavetobecontrolledtomaintainaconstan tinter-vehicleseparation, hasbeenextensivelystudied[ 38 48 51 ].Thegeneraltrendoftheresultsisthatthe problemscalespoorlywiththenumberofvehicles:asthenum berofvehiclesincrease thesensitivitytodisturbancesincreases[ 31 52 53 ]andthestabilitymargindecays[ 47 48 ].Theinformationgraphsconsideredintheliteratureareu suallylimitedtoatmost twoneighbors,withnotableexceptionssuchas[ 53 62 90 ]thatconsidermoregeneral informationexchangearchitectures. Ourgoalistoexaminehowthestabilitymarginandrobustnes stoexternaldisturbancesscalewiththesizeoftheformationandthestructure oftheinformationgraphthat speciesallowableinformationexchangebetweenpairsofv ehicles.Eachvehicleismodeled asadoubleintegrator,andweassumethatthevehicleisfull yactuated,whichmeanseach coordinateofthepositionofthevehiclecanbeindependent lycontrolled.Adistributed 75

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controlalgorithmisstudiedinwhicheveryvehicle(except forafewleadvehicles)use onlyrelativepositionandrelativevelocitywithrespectt oitsneighborsintheinformation graph. Weshowthatwhenthenetworkishomogeneousandsymmetric(a llvehiclesusethe samecontrolgainsandinformationfromeachneighborisgiv enequalweight),thestability margindecaysto0as O (1 =N 2 =D )whenthegraphis\square".Therefore,increasing thedimension(whichmayneednodesphysicallyaparttoexch angeinformation)ofthe informationgraphcanimprovethestabilitymarginbyacons iderableamount.Fornonsquareinformationgraph,thestabilitymargincanbemadei ndependentofthenumberof agentsbychoosingthe\aspectratio"appropriately.Thatm ayentailanincreaseinthe numberofleadvehiclesthathaveaccesstotheformation'sd esiredtrajectory. Therestofthischapterisorganizedasfollows.Section 4.1 presentsthedistributed formationcontrolproblemandthemainresults.Thestate-s paceandPDEmodelofthe controlledformationisdescribedinSection 4.2 .Section 4.3 analyzesthescalinglawsof thestabilitymarginanddisturbanceamplicationwithD-d imensionalinformationgraph. ThechapterendswithasummarygiveninSection 4.4 4.1ProblemFormulationandMainResults 4.1.1ProblemFormulation Weconsidertheformationcontrolof N identicalvehicles.Thepositionofeachvehicle isa D s -dimensionalvector(with D s =1 ; 2or3); D s isreferredtoasthe spatialdimension oftheformation.Let p (d)i 2 R bethed-thcoordinateofthe i -thvehicle'sposition,whose dynamicsaremodeledbyadoubleintegrator: p (d)i = u (d)i + w (d) i ; d=1 ;:::;D s ; (4{1) where u (d)i 2 R isthecontrolinputand w (d) i = a i sin( !t + i ) 2 R istheexternal disturbances.Theunderlyingassumptionisthateachofthe D s coordinatesofavehicle's positioncanbeindependentlyactuated.Wesaythatthevehi clesare fullyactuated .The 76

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spatialdimension D s is1foraplatoonofvehiclesmovinginastraightline, D s =2 foraformationofgroundvehiclesand D s =3foraformationofrightvehicles.Under theaboveassumption,theeachcoordinatesofavehicle'spo sitioncanbeindependently studied;see[ 3 91 ]forexamples. Thecontrolobjectiveistomakethegroupofvehiclestracka pre-specieddesired trajectorywhilemaintainingadesiredformationgeometry .Thedesiredformation geometryisspeciedbyadesiredrelativepositionvector i;j := p i ( t ) p j ( t )for every pair ofvehicles( i;j ),where p i ( t )isthedesiredtrajectoryofthevehicle i .Thedesiredintervehicularspacingshavetobespeciedinamutuallyconsist entfashion.Desiredtrajectory oftheformationisspeciedintheformofafewctitious\re ferencevehicles",eachof whichperfectlytracksitsowndesiredtrajectory.Therefe rencevehiclesaregeneralization ofthectitiousleaderandfollowervehiclesinone-dimens ionalplatoons[ 43 47 48 ]. Asubsetofvehiclescanmeasuretheirrelativepositionswi threspecttothereference vehicles,andthesemeasurementsareusedincomputingthei rcontrolactions.Inthisway, desiredtrajectoryinformationoftheformationisspecie donlytoasubsetofthevehicles inthegroup.Inthischapterweconsiderthedesiredtraject oryoftheformationtobeofa constant-velocitytype,sothat i;j 'sdon'tchangewithtime. Nextwedenean informationgraph thatmakesitconvenienttodescribedistributed controlarchitectures.Denition4.1. An informationgraph isanundirectedgraph G =( V ; E ) ,wheretheset of nodes V = f 1 ; 2 ;:::;N;N +1 ;:::;N + N r g consistsof N realvehiclesand N r reference vehicles.Thesetofedges E V V specifywhichpairsofnodes(vehicles)areallowedto exchangeinformationtocomputetheirlocalcontrolaction s.Twonodes i and j arecalled neighbors if ( i;j ) 2 E ,andthesetofneighborsof i aredenotedby N i Notethatinformationexchangemayormaynotinvolveanexpl icitcommunication network.Forexample,ifvehicle i measurestherelativepositionofvehicle j withrespect toitselfbyusingaradarandusesthatinformationtocomput eitscontrolaction,we 77

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consideritas\informationexchange"between i and j .Ifavehicle i hasaccesstodesired trajectoryinformationthenthereisanedgebetween i andareferencevehicle. Asinthepreviouschapters,weconsiderthefollowing distributed controllaw,whereby thecontrolactionatavehicledependsoni)the relativepositionmeasurements ii)the relativevelocitymeasurements withitsneighborsintheinformationgraph: u (d)i = X j 2N i k (d) ( i;j ) ( p (d)i p (d)j (d)i;j ) b (d)( i;j ) ( v (d) i v (d) j ) ;i =1 ;:::;N; (4{2) where k (d) ( ) areproportionalgainsand b (d)( ) arederivativegains.Notethatallthevariables in( 4{2 )arescalars.Itisassumedthatvehicle i knowsitsownneighbors(theset N i ),and thedesiredspacing (d)i;j Example4.1. ConsiderthetwoformationsshowninFigure 4-2 (a)and(b).Their spatialdimensionsare D s =1 and D s =2 ,respectively.Theinformationgraph,however, isthesameinbothcases: V = f 1 ; 2 ;:::; 9 g ; E = f (1 ; 2) ; (1 ; 4) ; (1 ; 7) ; (2 ; 3) ; (2 ; 5) ; (2 ; 8) ; (3 ; 6) ; (3 ; 9) ; (4 ; 5) ; (5 ; 6) ; (7 ; 8) ; (8 ; 9) g : AdrawingoftheinformationgraphappearsinFigure 4-2 (c).Althoughtheinformation graphisthesame,thedesiredspacings i;j 'saredierentinthetwoformations.For example, (1)2 ; 5 6 =0 intheone-dimensionalformationshowninFigure 4-2 (a)whereas (1)2 ; 5 =0 inthetwo-dimensionalformationshowninFigure 4-2 (b). Inthischapterwerestrictourselvestoaspecicclassofin formationgraph,namelya niterectangularlattice:Denition4.2 ( D -dimensionallattice:) A D -dimensionallattice,specicallya n 1 n 2 n D lattice,isagraphwith n 1 n 2 :::n D nodes.Inthe D -dimensionalspace R D ,thecoordinateof i -thnodeis ~ i :=[ i 1 ;:::;i D ] T ,where i 1 2f 0 ; 1 ;:::; ( n 1 1) g i 2 2f 0 ; 1 ;:::; ( n 2 1) g ::: and i D 2f 0 ; 1 ;:::; ( n d 1) g .Anedgeexistsbetweentwo 78

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O x 1 (a)A1D4lattice. O x 1 x 2 (b)A2D4 4lattice. O x 1 x 2 x 3 (c)A3D2 3 3lattice. Figure4-1.Examplesof1D,2Dand3Dlattices.nodes ~ i and ~ j ifandonlyif k ~ i ~ j k =1 ,where kk istheEuclideannormin R D .A n 1 n 2 n D latticeisdenotedby Z n 1 n 2 n D .Withaslightabuseofnotation,\the i -thnode"isusedtodenotethenodeonthelatticewithcoordi nate ~ i Figure 4-1 depictsthreeexamplesoflattices.A D -dimensionallatticeisdrawnin R D withaCartesianreferenceframewhoseaxesaredenotedby x 1 ;x 2 ;:::;x D .Notethat thesecoordinateaxesmaynotberelatedtothecoordinateax esinthephysicalspace R D s .Wealsodene N d ( d =1 ;:::;D )asthenumberofrealvehiclesinthe x d direction. Thenwehavetherelation N 1 N 2 :::N D = N and n 1 n 2 :::n D = N + N r .Inthischapter aninformationgraph G isalwaysalattice Z n 1 n 2 n D .Foragiven N ,thechoiceof N r ;D;N 1 ;N 2 ;:::;N D servestodeterminethespecicchoiceoftheinformationgr aph withintheclass. Fortheeaseofexpositionandnotationalsimplicity,wemak ethefollowingtwo assumptionsregardingthereferencevehiclesandthedistr ibutedcontrolarchitecture( 4{2 ): Assumption4.1. Foreach ( i;j ) 2 E ,thegain k ( d ) ( i;j ) ;b ( d ) ( i;j ) doesnotdependond. Assumption 4.1 meansthatthelocalcontrolgainsdonotexplicitlydependu pon thecoordinate d .Suchanassumptionisnotrestrictivebecauseofthefullya ctuated assumption.Ifthelocalcontrolgainsareallowedtodepend upon d thenonecouldrepeat 79

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O X 1 1 v (1) t (1)5 ; 2 (1)2 ; 7 (1)5 ; 7 (a)Desiredformationgeometryofa1-Dspatialplatoonwith6vehiclesand3referencevehicles. O X 1 X 2 v (1) tv (2) t (2)2 ; 5 (2)7 ; 2 (2)7 ; 5 (1)6 ; 5 (b)Desiredformationgeometryofa2-Dspatialvehicleformationwith6vehiclesand3referencevehicles. x 1 x 2 O 111 122 (c)Theinformationgraphforboththe1-Dplatoonandthe2-D formationshownin(a)and(b). Figure4-2.Exampleoftwodistinctspatialformationsthat havethesameassociated informationgraph. theanalysisofthischapterseparatelyforeachvalueof d .Notethattheassumptiondoes notmeanthatthecontrolgainsarespatiallyhomogeneous.Assumption4.2. Thereferencevehiclesarearrangedsothatanode i intheinformation graphcorrespondstoareferencevehicleifandonlyif i 1 = n 1 1 Assumption 4.2 meansthatallreferencevehiclesareassumedtobearranged ona single\face"ofthelattice,andeveryvehicleonthisfacei sareferencevehicle.Assumption 4.2 impliesthat N 1 = n 1 1 ;n 2 = N 2 ; ;N D = n D and N = N 1 N 2 :::N D and N r = N 2 :::N D .Thisarrangementofreferencevehiclessimpliesthepres entationofthe 80

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results.Arrangementsofreferencevehiclesonotherbound ariesofthelatticecanalsobe considered,whichdoesnotsignicantlychangetheresults .Wehavecarriedsuchanalysis in[ 37 92 ],wedon'tpresentthemhereintheinterestofbrevity. Aninformationgraphissaidtobe square if N 1 = N 2 = ::: = N D = N 1 =D AsaresultoftheAssumption 4.1 ,wecanrewrite( 4{2 )as u i = X j 2N i k ( i;j ) ( p i p j i;j ) b ( i;j ) ( v i v j ) ; (4{3) wherethesuperscript(d)hasbeensuppressed.Remark4.1. Thedimension D oftheinformationgraphisdistinctfromthespatial dimension D s .Figure 4-2 showsanexampleoftwoformationsinspace,onewith D s =1 andtheotherwith D s =2 .Red(lled)circlesrepresentreferencevehiclesandblac k(unlled)circlesrepresentactualvehicles.Dashedlines(in (a),(b))representdesiredrelative positions,whilesolidlinesrepresentedgesintheinforma tiongraph.Theinformation graphforboththeformationsisthesame 3 3 two-dimensionallattice,i.e., D =2 .On accountofthefullyactuateddynamicsandAssumption 4.1 ,thespatialdimension D s plays noroleintheresultsofthischapter.Thedimensionofthein formationgraph D ,onthe otherhand,willbeshowntoplayacrucialrole.4.1.2MainResult1:ScalingLawsforStabilityMargin Therstmainresultgivesanasymptoticformulaforcontrol ledformationwith symmetriccontrol:Theorem4.1. Consideran N -vehicleformationwithvehicledynamics ( 4{1 ) andcontrol law ( 4{2 ) ,underAssumptions 4.1 and 4.2 .Withsymmetriccontrol,thestabilitymarginof theclosed-loopisgivenbytheformula S = 2 b 0 8 1 N 2 1 : (4{4) 81

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Squareinformationgraph .Forasquareinformationgraph, N = N 1 N 2 :::N D = N D 1 ,andwehavethefollowingcorollary: Corollary4.1. Consideran N -vehicleformationwithvehicledynamics ( 4{1 ) andcontrol law ( 4{2 ) ,underAssumptions 4.1 and 4.2 .Whentheinformationgraphisasquare D dimensionallattice,theclosed-loopstabilitymarginwit hsymmetriccontrolisgivenbythe asymptoticformula S = 2 b 0 8 1 N 2 =D : (4{5) TheresultfromCorollary 4.1 showsthatforaconstantchoiceofsymmetriccontrol gains k 0 and b 0 ,thestabilitymarginapproaches0as N !1 .Thedimension D of theinformationgraphdeterminesthescaling.Specically ,thestabilitymarginscales as O (1 =N 2 )for1Dinformationgraph,as O (1 =N )for2Dinformationgraph,andas O (1 =N 2 = 3 )for3Dinformationgraph.Thus, forthesamecontrolgains,increasingthe dimensionoftheinformationgraphimprovesthestabilitym arginsignicantly .Inpractice, thismayrequireacommunicationnetworkwithlongrangecon nectionsinthephysical space.Notethataninformationgraphisonlyadrawingofthe connectivity.Aneighborin theinformationgraphneednotbephysicallyclose.Remark4.2. Itwasshownin[ 47 ]thattheclosed-loopstabilitymarginforacircular platoonapproacheszeroas O (1 =N 2 ) evenwiththecentralizedLQRcontroller.Itis interestingtonotethatdistributedcontrol(withaninfor mationgraphofdimension D> 1 ) yieldsabetterscalinglawforthestabilitymarginthancen tralizedLQRcontrol. Non-squareinformationgraph .ItfollowsfromTheorem 4.1 thatbychoosing thestructureoftheinformationgraphinsuchawaythat n 1 increasesslowlyinrelation to N ,thelossofthestabilitymarginasafunctionof N canbesloweddown.Infact, when n 1 isheldataconstantvalueindependentof N ,itfollowsfromTheorem 4.1 that thestabilitymarginisaconstantindependentofthetotaln umberofvehicles.More 82

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generally,consideraninformationgraphwith n 1 = O ( N c ),where c 2 [0 ; 1]isaxed constant.UsingTheorem 4.1 ,itfollowsthat S = O (1 =N 2 c )as N !1 .If c< 1 D theresultingreductionof S with N isslowerthanthatobtainedforasquarelattice; cf.Corollary 4.1 .Thisshowsthatwithintheclassof D dimensionallattices(foraxed D ),certaininformationgraphsprovidebetterscalingofthe stabilitymarginthanothers. Thepriceonepaysforimprovingstabilitymarginbyreducin g n 1 isanincreaseinthe numberofreferencevehicles.Thisisbecausethenumberofr eferencevehicles N r isrelated to n 1 by N r = N=N 1 (seeAssumption 4.2 ). Itisimportanttostressthatnotallnon-squaregraphsarea dvantageous.For example,if N 1 = O ( N )and N 2 through N D are O (1),itfollowsfromTheorem 4.1 thatthe stabilitymarginis S = O (1 =N 2 ).Thisisthesametrendasina1-Dinformationgraph. Inthiscase,wecansaythatthe D dimensionalinformationgrapheectivelybehavesasa onedimensionalgraph. Figure 4-3 showsafewexamplesofinformationgraphthatarerelevantt othe discussionabove.Figure 4-3 (a)showsa2-dimensionalinformationgraphinwhichthe rstdimensionisheldconstant,i.e. N 1 = O (1)and N 2 = O ( N ).Figure 4-3 (b)shows a2-dimensionalinformationgraphthatis"asymptotically "1-D(as N !1 )sincethe sizeoftherstdimensionincreaseslinearlywith N ,i.e. N 1 = O ( N )and N 2 = O (1). Figure 4-3 (c)showsa2-dimensionalinformationgraphinwhichbothsi desareoflength O ( p N ). Figure 4-4 providesnumericalcorroborationofstabilitymarginpred ictedbyTheorem 4.1 foravehicleformationwithinformationgraphsofvarious\ shapes"asshown inFigure 4-3 .Thelegend"SSM"meanscomputedfromthe"statespacemodel "( 4{10 ), whichispresentedinSection 4.2 .Fortherstcase, N 1 =5and N 2 = N= 5.Theorem 4.1 predictsthatinthiscase S = O (1)evenas N !1 ,whichresultsinastabilitymargin thatisindependentof N .Inthesecondcase, N 2 =5and N 1 = N= 5,whichleadsto S = O (1 =N 2 ),whichisthesameasthatwithan1-Dinformationgraph.The thirdcase 83

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x 1 x 2 n 1 = O (1)n 2 = O ( N )O (a)Non-squareinformationgraph x 1 x 2 O n 1 = O ( N )n 2 = O (1)(b)Non-squareinformationgraph x 1 x 2 O n 1 = O ( p N )n 2 = O ( p N )(c)Squareinformationgraph Figure4-3.Informationgraphswithdierentaspectratios isthatofasquareinformationgraph, N 1 = N 2 = p N ,whichleadsto S = O (1 =N ). Theorem 4.1 andcorollary 4.1 predictsthestabilitymarginquiteaccuratelyineachof thecases.Thecontrolgainsusedinallthecalculationsare k 0 =0 : 1and b 0 =0 : 5.The stabilitymarginasafunctionof N forthreedistinct2Dinformationgraphs(thatare describedinFigure 4-3 )areshowninthisgure.Thestabilitymarginiscomputedby computingtheeigenvaluesoftheclosed-loopstatematrix; thestatespacemodelisdescribedin( 4{10 )inSection 4.2 .Theplotsshowthattheformulae( 4{6 )inTheorem 4.1 andCorollary 4.1 makeexcellentpredictionsofthetrendofstabilitymargin 4.1.3MainResult2:ScalingLawsforDisturbanceAmplicat ion Inthischapter,weonlyconsidertheall-to-allamplicati on,whichisdenedas the H 1 normofthetransferfunctionfromthedisturbancesonallth evehicles(except leaders)totheirpositiontrackingerrors.Theconceptofl eader-to-trailerhasnodirect physicalmeaningintheformationwith D -dimensionalinformationgraph,soweignore thatcase.Theorem4.2. Consideran N -vehicleformationwithvehicledynamics ( 4{1 ) and controllaw ( 4{2 ) ,underAssumptions 4.1 and 4.2 .Withsymmetriccontrol,theall-to-all 84

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25 50 100 200 400 700 10 -4 10 -3 10 -2 NSN 1 =5(SSM) N 1 =5(Theorem 4.1 ) N 1 = N= 5(SSM) N 1 = N= 5(Theorem 4.1 ) N 1 = p N (SSM) N 1 = p N (Corollary 4.1 ) Figure4-4.Numericalvericationofstabilitymarginamplicationanditspeakfrequencyoftheclosed-loopareg ivenby H ATA 8 p k 0 b 0 3 N 3 1 ;! r p k 0 2 1 N 1 : (4{6) Again,weseethattheall-to-allamplicationonlydepends on N 1 ,thenumberofreal vehiclesonthe x 1 axisoftheinformationgraph.Thus,followingthesameargu mentfor stabilitymargin,weareabletodesignanon-squareinforma tiongraphwithproperaspect ratiosuchthatthescalinglawsofthedisturbanceamplica tiongrowsmuchslowerthan N orisindependentof N ,thenumberofvehiclesintheformation. 85

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4.2Closed-LoopDynamics:State-SpaceandPDEModels 4.2.1State-SpaceModeloftheControlledVehicleFormatio n Thedynamicsofthe i -thvehicleisobtainedbycombiningtheopenloopdynamics( 4{1 )withthecontrollaw( 4{3 ),whichyields p i = X j 2N i k ( i;j ) ( p i p j i;j ) b ( i;j ) ( v i v j )+ w i ;i =1 ;:::;N: (4{7) Let p i ( t )denotethedesiredtrajectoryofthe i -thvehicle.Thetrajectoryisuniquely determinedfromthetrajectoriesofthereferencevehicles andthedesiredformation geometry.Forexample,supposethetrajectoryofareferenc evehicle r is v t .Ifthe d -th coordinateofthedesiredgapbetweenavehicle i andthereferencevehicle r is ( d ) ir ,then the d -thcoordinateofthedesiredtrajectoryof i is p ( d ) ( t )= v ( d ) t + ( d ) ir Tofacilitateanalysis,wedenethefollowingcoordinatet ransformation: ~ p i := p i p i ) ~ p i =_ p i v = v i v : (4{8) Substituting( 4{8 )into( 4{7 ),wehave ~ p i = X j 2N i k ( i;j ) (~ p i ~ p j ) b ( i;j ) ( ~ p i ~ p j )+ w i : (4{9) Sincethetrajectoryofareferencevehicleisassumedtobee qualtoitsdesiredtrajectory, ~ p i =0if i isareferencevehicle.Using( 4{9 ),thestate-spacemodelofthevehicle formationcannowbewrittencompactlyas: X = AX + BW;E = CX; (4{10) where X isthestatevector,whichisdenedas X :=[~ p 1 ; ~ p 1 ; ; ~ p N ; ~ p N ] 2 R 2 N W is inputvector(externaldisturbances)and E istheoutputvector(positiontrackingerrors). Ourgoalistoanalyzetheclosed-loopstabilitymarginandd isturbanceamplication withincreasingnumberofvehicles N .Weapproximatethedynamicsofthespatially 86

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O x 1 x 2 i i 1+ i 2+ i 1 i 2 Figure4-5.Apictorialrepresentationofthe i -thvehicleanditsfournearbyneighbors. discreteformationbyapartialdierentialequation(PDE) modelthatisvalidforlarge valuesof N .ThisPDEmodelisusedforanalysisandcontroldesign. 4.2.2PDEModeloftheControlledVehicleFormation Foragivenchoiceoftheinformationgraph,the i -thvehiclehasthecoordinate ~ i =[ i 1 ;i 2 ;:::;i D ] T in R D .Weinterpret~ p i asafunctionofthecoordinate ~ i .Inthe following,weconsideracontinuousapproximationofthisf unctiontowriteaPDEmodel. Forthe i -thnodewithcoordinate ~ i =[ i 1 ;:::;i D ] T ,weuse i d + and i d todenote thenodeswithcoordinates[ i 1 ;:::;i d 1 ;i d +1 ;i d +1 ;:::;i D ] T and[ i 1 ;:::;i d 1 ;i d 1 ;i d +1 ;:::;i D ] T ,respectively. For D =2,anode i intheinteriorofthegraphanditsfourneighbors,i.e., i 1+ i 1 i 2+ ,and i 2 .Figure 4-5 showsapictorialrepresentationofthe i -thvehicleandits fournearbyneighborsina2Dinformationgraph. i 1+ standsfortheneighborofthe i -th vehicleinthe x 1 positivedirectionrelativetovehicle i ,and i 1 standsfortheneighborof the i -thvehicleinthe x 1 negativedirectionrelativetovehicle i .And i 2+ and i 2 canbe interpretedinthesameway..Thedynamics( 4{9 )cannowbeexpressedas: ~ p i = D X d =1 k ( i;i d + ) (~ p i ~ p i d + ) D X d =1 k ( i;i d ) (~ p i ~ p i d ) D X d =1 b ( i;i d + ) ( ~ p i ~ p i d + ) D X d =1 b ( i;i d ) ( ~ p i ~ p i d )+ w i : (4{11) 87

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Wedene, k d;f + b i := k ( i;i d + ) + k ( i;i d ) ;k d;f b i := k ( i;i d + ) k ( i;i d ) ; b d;f + b i := b ( i;i d + ) + b ( i;i d ) ;b d;f b i := b ( i;i d + ) b ( i;i d ) ; (4{12) where d 2f 1 ;:::;D g ;thesuperscripts f and b denote front and back ,respectively. Substituting( 4{12 )into( 4{11 ),wehave ~ p i = D X d =1 k d;f + b i + k d;f b i 2 (~ p i ~ p i d + ) D X d =1 k d;f + b i k d;f b i 2 (~ p i ~ p i d ) D X d =1 b d;f + b i + b d;f b i 2 ( ~ p i ~ p i d + ) D X d =1 b d;f + b i b d;f b i 2 ( ~ p i ~ p i d )+ w i : (4{13) Toproceedfurther,werstredrawtheinformationgraphins uchawaysothatit alwaysliesintheunit D -cell[0 ; 1] D ,irrespectiveofthenumberofvehicles.Notethat ingraph-theoreticterms,agraphisdenedonlyintermsofi tsnodeandedgesets.A drawingofagraphinanEuclideanspace,alsocalledanembed ding[ 93 ],ismerelya convenientvisualizationtool.Fortherestofthissection ,wewillconsiderthefollowing drawing(embedding)ofthelattice Z n 1 :::n D intheEuclideanspace R D .TheEuclidean coordinateofthe i -thnode,whose\original"Euclideanpositionwas[ i 1 ;:::;i D ] T ,isnow drawnatposition[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T ,where c d := 1 n d 1 ;d =1 ;:::;D: (4{14) Figure 4-6 showsanexample,wheretheoriginallattice,showninFigur e 4-6 (a),isredrawn totinto[0 ; 1] 2 ,whichisshowninFigure 4-6 (b). ThestartingpointforthePDEderivationistoconsiderafun ction~ p ( x;t ):[0 ; 1] D [0 ; 1 ) R denedovertheunitD-cellin R D thatsatises: ~ p i ( t )=~ p ( x;t ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T (4{15) 88

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O x 1 x 2 1 11 1 1 (a)Originallattice O x 1 x 2 c 1 c 1 c 1c 2 c 2 11 (b)Redrawnlattice x 1 x 2 11 (c)functionapproximation Figure4-6.Originallattice,itsredrawnlatticeandacont inuousapproximation. Figure 4-6 pictoriallydepictstheapproach:functionsthataredene datdiscretepoints (theverticesofthelatticedrawnin[0 ; 1] D )willbeapproximatedbyfunctionsthatare denedeverywherein[0 ; 1] D .Theoriginalfunctionsarethoughtofassamplesoftheir continuousapproximations.Ingure 4-6 ,(a)isa2Dinformationgraphforaformation with3 3vehiclesand3referencevehicles.(b)showsaredrawninfo rmationgraph of(a),sothatitliesintheunit2-cell[0 ; 1] 2 .(c)givesapictorialrepresentationof continuousapproximationofadiscretefunctionwhosevalu esaredenedonthenodesin theredrawnlatticeasshownin(b).Weformallyintroduceth efollowingscalarfunctions k f d ;k b d ;b fd ;b bd :[0 ; 1] D R (for d 2f 1 ;:::;D g )denedaccordingtothestipulation: k ( i;i d + ) = k f d ( x ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T ;k ( i;i d ) = k b d ( x ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T b ( i;i d + ) = b fd ( x ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T ;b ( i;i d ) = b bd ( x ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T a ( i;i d + ) = a ( x ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T ; ( i;i d ) = ( x ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T : (4{16) Inaddition,wedenefunctions k f + b d ;k f b d ;b f + b d ;b f b d :[0 ; 1] D R as k f + b d ( x ):= k f d ( x )+ k b d ( x ) ;k f b d ( x ):= k f d ( x ) k b d ( x ) ; b f + b d ( x ):= b fd ( x )+ b bd ( x ) ;b f b d ( x ):= b fd ( x ) b bd ( x ) : (4{17) 89

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Dueto( 4{16 ),thesesatisfy k d;f + b i = k f + b d ( x ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T ;k d;f b i = k f b d ( x ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T ; b d;f + b i = b f + b d ( x ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T ;b d;f b i = b f b d ( x ) j x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T : ToobtainaPDEmodelfrom( 4{13 ),werstrewriteitas ~ p i = D X d =1 k d;f b i c d (~ p i d + ~ p i d ) 2 c d + D X d =1 k d;f + b i 2 c 2d (~ p i d + 2~ p i +~ p i d ) c 2d + D X d =1 b d;f b i c d ( ~ p i d + ~ p i d ) 2 c d + D X d =1 b d;f + b i 2 c 2d ( ~ p i d + 2 ~ p i + ~ p i d ) c 2d + a i sin( !t + i ) : (4{18) andthenusethefollowingnitedierenceapproximationsf orevery d 2f 1 ;:::;D g : h ~ p i d + ~ p i d 2 c d i = h @ ~ p ( x;t ) @x d i x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T ; h ~ p i d + 2~ p i +~ p i d c 2d i = h @ 2 ~ p ( x;t ) @x d 2 i x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T ; h ~ p i d + ~ p i d 2 c d i = h @ 2 ~ p ( x;t ) @x d @t i x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T ; h ~ p i d + 2 ~ p i + ~ p i d c 2d i = h @ 3 ~ p ( x;t ) @x d 2 @t i x =[ i 1 c 1 ;i 2 c 2 ;:::;i D c D ] T : Weemphasizethat x 1 ;:::;x D abovearethecoordinatedirectionsintheEuclideanspace inwhichtheinformationgraphisdrawn,whichareunrelated tothecoordinateaxesofthe Euclideanspacethatthevehiclesphysicallyoccupy.Subst itutingtheexpression( 4{14 )for c d ,( 4{18 )isseenasanitedierenceapproximationofthefollowing PDE: @ 2 ~ p ( x;t ) @t 2 = D X d =1 k f b d ( x ) n d 1 @ @x d + k f + b d ( x ) 2( n d 1) 2 @ 2 @x d 2 + b f b d ( x ) n d 1 @ 2 @x d @t + b f + b d ( x ) 2( n d 1) 2 @ 3 @x d 2 @t ~ p ( x;t )+ a ( x )sin( !t + ( x )) : (4{19) TheboundaryconditionsofPDE( 4{19 )dependonthearrangementofreferencevehicles intheinformationgraph.Iftherearereferencevehicleson theboundary,theboundary 90

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conditionisofDirichlettype.Iftherearenoreferenceveh icles,theboundaryconditionis oftheNeumanntype. UnderAssumption 4.2 ,theboundaryconditionsareoftheDirichlettypeonthatfa ce oftheunitcellwherethereferencevehiclesare,andNeuman nonallotherfaces: ~ p (1 ;x 2 ;:::;x D ;t )=0 ; @ ~ p @x 1 (0 ;x 2 ;:::;x D ;t )=0 ; @ ~ p @x d ( x;t )=0 ;x =[ x 1 ;:::;x d 1 ; 0or1 ;x d +1 ;:::;x D ] T ; ( d> 1) : (4{20) Ifotherarrangementsofreferencevehiclesareused,thebo undaryconditionsmaybe dierent.Itcanbeveriedinastraightforwardmannerthat thePDE( 4{19 )yieldsthe originalsetofcoupledODEs( 4{11 )uponnitedierencediscretization,see[ 77 86 ]. 4.3AnalysisofStabilityMarginandDisturbanceAmplicat ion Inthissection,weconsiderthefollowinghomogeneousands ymmetriccontrolgains k ( i;j ) = k 0 ;b ( i;j ) = b 0 ; 8 ( i;j ) 2 E ; where k 0 and b 0 arepositivescalars.Inthiscase,usingthenotationin( 4{12 )and( 4{16 ), wehave k f + b d ( x )=2 k 0 ;k f b d ( x )=0 ;b f + b d ( x )=2 b 0 ;b f b d ( x )=0 ;d =1 ;:::;D: ThePDEgivenin( 4{19 )withoutforcingsimpliesto: @ 2 ~ p ( x;t ) @t 2 = D X d =1 k 0 ( n d 1) 2 @ 2 @x d 2 + b 0 ( n d 1) 2 @ 3 @x d 2 @t ~ p ( x;t ) : (4{21) Theclosed-loopeigenvaluesofthePDEmodelrequireconsid erationoftheeigenvalue problems L ( x )= ( x ) ; (4{22) 91

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wherethelinearoperator L isdenedas: L = D X d =1 1 ( n d 1) 2 @ 2 @x d 2 ; (4{23) and isaneigenfunctionthatsatisestheboundarycondition( 4{20 )underAssumption 4.2 .Forthisboundarycondition,theeigenvalues(notethatth eyaredierentfrom theeigenvaluesofthePDEmodel)andeigenfunctionsareobt ainedbythemethodof separationofvariables([ 77 86 ]) ` = (2 ` 1 1) 2( n 1 1) 2 + ( ` 2 ) 2 ( n 2 1) 2 + + ( ` D ) 2 ( n D 1) 2 = 2 (2 ` 1 1) 2 4( n 1 1) 2 + ` 22 ( n 2 1) 2 + + ` 2D ( n d 1) 2 ; ` ( x )=cos (2 ` 1 1) x 1 2 cos( ` 2 x 2 ) cos( ` D x D ) ; (4{24) whereweusethenotation ` =( ` 1 ; ;` D )todenotethewavevectorinwhich ` 1 2 f 1 ; 2 ; g and ` 2 ; ;` D 2f 0 ; 1 ; 2 ; g .AftertakingaLaplacetransformofbothsides ofthePDE( 4{21 )withrespectto t ,andusingthemethodofseparationofvariables,the eigenvaluesofthePDEturnouttobetherootsofthecharacte risticequation: s 2 + b 0 ` s + k 0 ` =0 ; (4{25) where s istheLaplacevariableand ` istheeigenvaluein( 4{24 ). Thetworootsof( 4{25 )are s ` := b 0 ` p b 20 2` 4 k 0 ` 2 : (4{26) Wecall s ` the ` -thpairofeigenvalues. Providedeachofthe n d 'sarelargesothatthePDE( 4{19 )withtheboundary condition( 4{20 )isanaccurateapproximationofthe(spatially)discretef ormation dynamics( 4{10 )underAssumption 4.2 ,theleaststableeigenvalueofthePDE( 4{21 ) 92

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providesinformationonthestabilitymarginoftheclosedloopformationdynamics.We arenowreadytoproveTheorem 4.1 thatwasstatedinSection 4.1 ProofofTheorem 4.1 ConsidertheeigenvalueproblemforPDE( 4{21 )withmixed DirichletandNeumannboundaryconditions( 4{20 ).Let'srstexaminethediscriminant in( 4{26 ), D := b 20 2` 4 k 0 ` = 4 b 20 (2 ` 1 1) 2 4( n 1 1) 2 + ` 22 ( n 2 1) 2 + + ` 2D ( n d 1) 2 2 4 2 k 0 (2 ` 1 1) 2 4( n 1 1) 2 + ` 22 ( n 2 1) 2 + + ` 2D ( N d 1) 2 ; Undertheassumption n d ( d =1 ;:::;D )areverylarge,forsmall ` d D isnegative.So boththeeigenvaluesin( 4{26 )arecomplex,thenthestabilitymarginisonlydetermined bytherealpartsof s ` .Forlarge ` d D ispositive,soboththeeigenvaluesin( 4{26 )are real.Itiseasytoverifythattherealpartinthiscasearemu chlargerthanthatwith negativediscriminant D .Therefore,weonlyconsiderthecasewhentheeigenvaluesa re complex. Itfollowsfrom( 4{26 )thattheleaststableeigenvalues s min (theonesclosestto theimaginaryaxis)amongthemistheonethatisobtainedbym inimizing ` overthe D -tuples( ` 1 ;:::;` D ).Using( 4{24 ),thisminimumisachievedat ` 1 =1 ;` 2 = = ` D =0, s min = s (1 ; 0 ;:::; 0) ; andtherealpartisobtained Re ( s min )= b 0 ` 2 = 2 b 0 8( n 1 1) 2 : Followingthedenitionofstabilitymargin, S := j Re ( s min ) j = 2 b 0 8( n 1 1) 2 = 2 b 0 8 N 2 1 ; (4{27) wherethelastequalityfollowingfrom N 1 = n 1 1. 93

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WenowproveTheorem 4.2 thatwasstatedinSection 4.1 ProofofTheorem 4.2 Werstobservethatthesmallesteigenvalueoftheoperator L givenin( 4{23 )isobtainedbyminimizing ` overthe D -tuples( ` 1 ;:::;` D ).Using( 4{24 ), thisminimumisachievedat ` 1 =1 ;` 2 = = ` D =0, min = (1 ; 0 ;:::; 0) = 2 4( n 1 1) 2 = 2 4 N 2 1 ; wherethelastequalityfollowingfrom N 1 = n 1 1. WenowwritethePDEmodelwithexternaldisturbancesas @ 2 ~ p ( x;t ) @t 2 = D X d =1 k 0 ( n d 1) 2 @ 2 @x d 2 + b 0 ( n d 1) 2 @ 3 @x d 2 @t ~ p ( x;t )+ u ( x;t ) ; where u ( x;t )= a ( x )sin( !t + ( x ))istheexternalsinusoidaldisturbance.TakeLaplace transformtobothsidesoftheabovePDEwithrespecttotheti mevariable t ,weget s 2 P ( x;s )= D X d =1 k 0 ( n d 1) 2 @ 2 P ( x;s ) @x d 2 + b 0 s ( n d 1) 2 @ 2 P ( x;s ) @x d 2 + U ( x;s ) ; (4{28) where s istheLaplacevariableand P ( x;s ) ;U ( x;s )aretheLaplacetransformsof~ p ( x;t ) and u ( x;t )respectively.Usingthemethodofseparationofvariables ,weassumeasolution oftheform P ( x;s )= ( x ) h ( s ),where ( x )istheeigenfunctionofthelinearoperator L Substituting P ( x;s )= ( x ) h ( s )into( 4{28 ),weget s 2 ( x ) h ( s )= D X d =1 k 0 ( n d 1) 2 @ 2 ( x ) @x d 2 + b 0 s ( n d 1) 2 @ 2 ( x ) @x d 2 h ( s )+ U ( x;s ) ; Now,substituting L ( x )= ` ( x )intotheaboveequation,wehave s 2 ( x ) h ( s )=( k 0 ` b 0 ` s ) ( x ) h ( s )+ U ( x;s ) ; whichimplies ( s 2 + k 0 ` + b 0 ` s ) P ( x;s )= U ( x;s ) ; 94

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Wethusobtainthefollowingtransferfunctionfrom U ( x;s )to P ( x;s )(see[ 94 ]) G ( s )= P ( x;s ) U ( x;s ) = 1 s 2 + b 0 ` s + k 0 ` ; (4{29) where ` isthe ` -theigenvalueofthelinearoperator L ,itisgivenin( 4{24 ).Similarto nite-dimensionalsystem,the H 1 normofatransferfunctionisgivenbythesupremumof thesquarerootofthelargesteigenvalueof G ( j! ) G ( j! ),wehave k G ( j! ) k H 1 = s sup sup ` 1 2 b 0 ` j! + k 0 ` 1 2 + b 0 ` j! + k 0 ` =sup sup ` 1 p ( k 0 ` 2 ) 2 +( b 0 ` ) 2 =sup ` A ` : (4{30) where A ` = 8>><>>: 2 3 = 2 ` b 0 p 4 k 0 ` b 20 ; if ` 2 k 0 =b 20 ; 1 ` k 0 ; otherwise. (4{31) ` = 8>><>>: p 4 ` k 0 2 2` b 20 2 ; if ` 2 k 0 =b 20 ; 0 ; otherwise. (4{32) Foranyxed k 0 ;b 0 ,when n d islarge,wehave ` 2 k 0 =b 20 .The H 1 normandthepeak frequencyofthetransferfunction G ( s )aregivenby k G ( j! ) k H 1 = A (1 ; 0 ; ) = 2 3 = 2 min b 0 p 4 k 0 min b 20 ; (4{33) r = p 4 min k 0 2 2min b 20 2 : (4{34) Recallthat min = 2 4( n 1 1) 2 = 2 4 N 2 1 ,usetheassumptionthat n d islarge,wenishtheproof. Similarproofbasedonthestate-spacemodel( 4{10 )canbefoundin[ 88 95 ]. 4.4Summary Westudiedtheproblemofdistributedcontrolofalargeform ationofvehicleteams with D -dimensionalinformationgraph.Weshowedthatthestabili tymarginscalesas 95

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O (1 =N 2 =D )andtheall-to-allamplicationscalesas O ( N 3 =D )fora D -dimensionalsquare informationgraph.Therefore,increasingthedimensionof theinformationgraphcan improvethestabilitymarginandrobustnesstoexternaldis turbancesbyaconsiderable amount.Fornon-squareinformationgraph,thestabilityma rginandall-to-allamplicationcanbemadeindependentofthenumberofagentsbychoo singthe\aspectratio" appropriately.However,itshouldbetakenintoaccounttha tincreasingthedimensionof theinformationgraphorchoosingabenecialaspectratiom ayrequirelongrangecommunicationorentailanincreaseinthenumberofleadvehicl es.Theseresultsaretherefore usefultothedesignerinmakingtrade-osbetweenperforma nceandcostindesigning informationexchangearchitecturesfordecentralizedcon trol. Ourresultsforsquare D -latticesarecomplementarytothoseof[ 90 ],inwhichthe eectofgraphdimensionontheresponseoftheclosedloopto stochasticdisturbancesis quantiedintermsof\microscopic"and\macroscopic"meas ures.Itwasshownin[ 90 ] thatfor D> 3,theseperformancemeasuresbecomeindependentof N ,whileforsmaller D ,theperformancebecomesworsewithoutboundasthenumbero fvehiclesincrease. Incontrast,weshowedthatthestabilitymargindecaysto0a ndall-to-allamplication increaseto 1 as N increasesinevery D .Thoughthedecayisslowerforlarger D ,itis neverindependentof N .Toachieveasize-independentstabilitymarginandall-to -all amplication,thegraphneedstobenon-square.Sincethean alysisof[ 90 ]isdonein thespatialFourierdomain,itisnotclearifnon-squarelat ticeswithboundariescanbe handledinthatframework. 96

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CHAPTER5 IMPROVINGCONVERGENCERATEOFDISTRIBUTEDCONSENSUSTHROU GH ASYMMETRICWEIGHTS Studyofconsensushasalonghistoryinsystemsandcontrolt heoryaswellas computerscience.Earlyworkscanbedatedbacktothe1960s( see[ 96 ]andthereferences therein).Distributedconsensushasbeenwidelystudiedin thepastfewdecadesdueto itsbroadapplicationsindistributedcomputing,multi-ve hiclerendezvous,datafusionin largesensornetwork,coordinatedcontrolofmulti-agents ystemandformationrightof unmannedvehiclesandclusteredsatellites,etc.(see[ 1 5 9 { 11 97 98 ]).Indistributed consensus,eachagentinanetworkupdatesitsstatebyusing aweightedsummationofits ownstateandthestatesofitsneighborssothatalltheagent s'stateswillreachacommon value. Thetopicofthischapteristheconvergencerateofdistribu tedlinearconsensus protocolongraphswithxed(timeinvariant)topology.Wes tudyhowtodesignthe graphweightstoimprovetheconvergencerateofdistribute dconsensusprotocol.The convergencerateisextremelyimportant,sinceitdetermin espracticalapplicabilityof theprotocol.Iftheconvergencerateistoosmall,itwillta keextremelylargenumberof iterationstodrivethestatesofallagentssucientlyclos e.Thisisunfavorableforagents suchaswirelesssensorswhohavelimitedbatterylifetimes Comparedtothevastliteratureondesignofconsensusproto cols,however,the literatureonconvergencerateanalysisismeager.Afewwor kscanbefoundin[ 70 { 72 99 100 ].TherelatedproblemofmixingtimeofMarkovchainsisstud iedin[ 73 ]. In[ 36 ],convergenceratesforaspecicclassofgraphs,thatweca llL-Zgeometricgraphs, areestablishedasafunctionofthenumberofagents.Genera llyspeaking,theconvergence ratesofdistributedconsensusalgorithmstendtobeslow,a nddecreaseasthenumberof agentsincreases.Itwasshownin[ 74 ]thattheconvergenceratecanbearbitrarilyfastin small-worldnetworks.However,networksinwhichcommunic ationisonlypossiblebetween agentsthatarecloseenougharenotlikelytobesmall-world 97

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Oneoftheseminalworksonthissubjectisconvexoptimizati onofweightsonedges ofthegraphtomaximizetheconsensusconvergencerate[ 27 29 ].Convexoptimization imposestheconstraintthattheweightsofthegraphmustbes ymmetric,whichmeans anytwoneighboringagentsputequalweightontheinformati onreceivedfromeachother. Theconvergencerateofconsensusprotocolsongraphswiths ymmetricweightsdegrades considerablyasthenumberofagentsinthenetworkincrease s.InaD-dimensionallattice, forinstance,theconvergencerateis O (1 =N 2 =D )iftheweightsaresymmetric,where N is thenumberofagents.Thisresultfollowsasaspecialcaseof theresultsin[ 36 ].Thus,the convergenceratebecomesarbitrarilysmallifthesizeofth enetworkgrowswithoutbound. In[ 75 76 ],nite-timedistributedconsensusprotocolsarepropose dtoimprovethe performanceoverasymptoticconsensus.However,ingenera l,thenitetimeneededto achieveconsensusdependsthenumberofagentsinthenetwor k.Thus,forlargesizeof networks,althoughconsensuscanbeachievedinnitetime, thetimeneededtoreach consensusbecomeslarge. Inthischapter,westudytheproblemofhowtoincreasetheco nvergencerateof consensusprotocolsbydesigning asymmetric weightsonedges.Werstconsiderlattice graphsandderivepreciseformulaeforconvergencerateint hesegraphs.Inparticular,we showthatinlatticegraphs,withproperchoiceofasymmetri cweights,theconvergence rateofdistributedconsensuscanbeboundedawayfromzerou niformlyin N .Thus,the proposedasymmetricdesignmakesdistributedconsensushi ghlyscalable;thetimetoreach consensusisnowindependentofthenumberofagentsinthene twork.Bytimetoreach consensuswemeanthetimeneededforthestatesofallnodest oreachan neighborhood oftheasymptoticconsensusvalue.Weprovidetheformulaef orasymptoticsteady-state consensusvalue.Withasymmetricweights,theconsensusva lueingeneralisnotthe averageoftheinitialconditions. 98

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Wenextproposeaweightdesignschemeforarbitrary2-dimen sionalgeometric graphs,i.e.,graphsconsistingofnodesin R 2 .Hereweusetheideaofcontinuumapproximationtoextendtheasymmetricdesignfromlatticestogeo metricgraphs.Weshow howaSturm-Liouvilleoperatorcanbeusedtoapproximateth egraphLaplacianinthe caseoflattices.ThespectrumoftheLaplacianandtheconve rgencerateofconsensus protocolsareintimatelyrelated.Thediscreteweightsinl atticescanbeseenassamplesof acontinuousweightfunctionthatappearsintheS-Loperato r.Basedonthisanalogy,a weightdesignalgorithmisproposedinwhichanode i choosestheweightontheedgetoa neighbor j dependingontherelativeanglebetween i and j .Numericalsimulationsshow thattheconvergenceratewithasymmetricdesignedweights inlargegraphsisanorderof magnitudehigherthanthatwith(i)optimalsymmetricweigh ts,whichareobtainedby convexoptimization[ 27 29 ],and(ii)asymmetricweightsobtainedbyMetropolis-Hast ings method,whichassignsweightsuniformlytoeachedgeconnec tingitselftoitsneighbor. Theproposedweightdesignmethodisdecentralized,everyn odecanobtainitsownweight basedontheangularpositionmeasurementswithitsneighbo rs.Inaddition,itiscomputationallymuchcheaperthanobtainingtheoptimalsymme tricweightsusingconvex optimizationmethod.Theproposedweightdesignmethodcan beextendedtogeometric graphsin R D ,butinthischapterwelimitourselvesto R 2 Therestofthischapterisorganizedasfollows.Section 5.1 presentstheproblem statement.Resultsonsize-independentconvergencerateo nlatticegraphswithasymmetricweightarestatedinSection 5.2 .Asymmetricweightdesignmethodformoregeneral graphsappearinSection 5.3 .ThechapterendswithasummaryinSection 5.4 5.1ProblemFormulation Tostudytheproblemofdistributedlinearconsensusinnetw orks,werstintroduce someterminologies.Thenetworkof N agentsismodeledbyagraph G =( V ; E )with vertexset V = f 1 ;:::;N g andedgeset E V V .Weuse( i;j )torepresentadirected edgefrom i to j .Anode i canreceiveinformationfrom j ifandonlyif( i;j ) 2 E .In 99

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thischapter,weassumethatcommunicationisbidirectiona l,i.e.( i;j ) 2 E ifandonlyif ( j;i ) 2 E .Foreachedge( i;j ) 2 E inthegraph,weassociateaweight W i;j > 0toit.The setofneighborsof i isdenedas N i := f j 2 V :( i;j ) 2 E g .TheLaplacianmatrix L ofan arbitrarygraph G withedgeweights W i;j isdenedas L i;j = 8>>>>>><>>>>>>: W i;j i 6 = j; ( i;j ) 2 E ; P Nk =1 W i;k i = j; ( i;k ) 2 E ; 0otherwise. (5{1) Alinearconsensusprotocolisaniterativeupdatelaw: x i ( k +1)= W i;i x i ( k )+ X j 2 N i W i;j x j ( k ) ;i 2 V ; (5{2) withinitialconditions x i (0) 2 R ,where k = f 0 ; 1 ; 2 ; g isthediscretetimeindex. Followingstandardpracticeweassumetheweightmatrix W isastochasticmatrix,i.e. W i;j 0and W 1 = 1 ,where 1 isavectorwithallentriesof1.Thedistributedconsensus protocol( 5{2 )canbewritteninthefollowingcompactform: x ( k +1)= Wx ( k ) ; (5{3) where x ( k )=[ x 1 ( k ) ;x 2 ( k ) ; ;x N ( k )] T isthestatesofthe N agentsattime k .It's straightforwardtoobtainthefollowingrelation L = I W ,where I isthe N N identity matrixand L istheLaplacianmatrixassociatedwiththegraphwith W i;j asitsweights onthedirectededge( i;j ).Inaddition,theirspectraarerelatedby ( L )=1 ( W ), i.e. ` ( L )=1 ` ( W ),where ` 2f 1 ; 2 ; ;N g and ` ; ` aretheeigenvaluesof L and W respectively.Thelineardistributedconsensusprotocol( 5{3 )implies x ( k )= W k x (0). Weassume W isstronglyconnected(irreducible)andprimitive.Inthat casethespectral radiusof W is1andthereisexactlyoneeigenvalueontheunitdisk.Let 2 R 1 N be theleftPerronvectorof W correspondingtotheeigenvalueof1,i.e. W = i > 0and 100

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P Ni =1 i =1,wehave lim k !1 W k = 1 ; (5{4) Therefore,allthestatesofthe N agentsasymptoticallyconvergetoasteadystatevalue x as k !1 lim k !1 x ( k )= 1 x (0)= 1 x; (5{5) where x = P Ni =1 i x i (0). Itiswellknownthatforaprimitivestochasticmatrix,ther ateofconvergence R can bemeasuredbythespectralgap R =1 ( W ),where ( W )istheessentialspectralradius of W ,whichisdenedas ( W ):=max fj j : 2 ( W ) nf 1 gg : Iftheeigenvaluesof W arerealandtheyareorderedinanon-increasingfashionsuc hthat 1= 1 2 N ,thentheconvergencerateof W isgivenby R =1 ( W )=min f 1 2 ; 1+ N g : (5{6) Inaddition,fromGerschgorincircletheorem,wehavethat N 1+2max i W ii .If max i W ii 6 =0,then1+ N isaconstantboundedawayfrom0.Therefore,thekeyto ndalowerboundfortheconvergencerateof W istondanupperboundonthesecond largesteigenvalue 2 of W .Equivalently,wecanndalowerboundofthesecondsmalles t eigenvalue 2 oftheassociatedLaplacianmatrix L ,since 2 =1 2 Denition5.1. Wesayagraph G has symmetric weightsif W i;j = W j;i foreachpairof neighboringagents ( i;j ) 2 E .Otherwise,theweightsarecalled asymmetric Iftheweightsaresymmetric,thematrix W isdoublystochastic,meaningthateach rowandcolumnsumis1. 101

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Thefollowingtheoremsummariestheresultsin[ 36 ]ontheconvergencerateof consensuswithsymmetricweightsinabroadclassofgraphst hatincludelattices.A D -dimensionallattice,specicallya N 1 N 2 N D lattice,isagraphwith N = N 1 N 2 N D nodes,inwhichthenodesareplacedattheintegerunitcoord inate pointsofthe D -dimensionalEuclideanspaceandeachnodeconnectstoothe rnodes thatareexactlyoneunitawayfromit.A D -dimensionallatticeisdrawnin R D witha Cartesianreferenceframewhoseaxesaredenotedby x 1 ;x 2 ; ;x D .Wecallagraphisa L-Zgeometricgraph ifitcanbeseenasaperturbationofregularlatticein D -dimensional space;eachnodeconnectsothernodeswithinacertainrange .Theformaldenitionis givenin[ 36 ]. Theorem5.1 ([ 36 ]) Let G bea D -dimensionalconnectedL-Zgeometricgraphorlattice andlet W beanydoublystochasticmatrixcompatiblewith G .Then c 1 N 2 =D R c 2 N 2 =D ; (5{7) where N isthenumberofnodesinthegraph G and c 1 ;c 2 aresomeconstantsindependent of N TheabovetheoremstatesthatforanyconnectedL-Zgeometri c/latticegraph G theconvergencerateofconsensuswithsymmetricweightsca nnotbeboundedaway from0uniformlywiththesize N ofthegraph.Theconvergencerateofthenetwork becomesarbitrarilyslowas N increaseswithoutbound.Thelossofconvergencerate withsymmetricinformationgraphhasalsobeenobservedinv ehicularformations;as discussedinChapter 2 andChapter 4 .Infact,anotherimportantconclusionoftheresult aboveisthatheterogeneityinweightsamongnodes,aslonga s W issymmetric,does notchangetheasymptoticscalingoftheconvergencerate.A tbestitcanchangethe constantinfrontofthescalingformula(see[ 73 ]also).Therefore,evencentralizedweight optimizationschemeproposedin[ 27 29 ]-thatconstraintheweightstobesymmetricin ordertomaketheoptimizationproblemconvex-willsuerfr omthesameissueasthatof 102

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... 3 12 N 1 N W 1 ; 2 W 2 ; 1 W 2 ; 3 W 3 ; 2 W N 1 ;N W N;N 1 x 1 o Figure5-1.Informationgraphfora1-Dlatticeof N agents. un-optimizedweightsontheedges.Namely,theconvergence ratewilldecayas O (1 =N 2 =D ) ina D -dimensionallattice/L-Zgeometricgraphevenwiththeopt imizedweights.Inthe restofthechapter,westudytheproblemofspeedinguptheco nvergenceratebydesigning asymmetric weights. 5.2FastConsensusonD-dimensionalLattices FirstweestablishtechnicalresultsonthespectrumandPer ronvectorsof D dimensionallatticeswithpossiblyasymmetricweightsont heedges.Wethensummarize theirdesignimplicationsattheendofsection 5.2.1 5.2.1AsymmetricWeightsinLattices Werstconsiderdistributedconsensusona1-dimensionall attice.Thiswillbeuseful ingeneralizingto D -dimensionallattices.Eachagentinteractswithitsneare stneighbors inthelattice(oneoneachside).Itsinformationgraphisde pictedinFigure 5-1 .The updatinglawofagent i isgivenby x i ( k +1)= W i;i x i ( k )+ W i;i 1 x i 1 ( k )+ W i;i +1 x i +1 ( k ) : where i 2f 2 ; 3 ; ;N 1 g .Theupdatinglawsofthe1-stand N -thagentsareslightly dierentfromtheaboveequation,sincetheyonlyhaveonene ighboreach. 103

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Theweightmatrix W (1) forthe1-dimensionallatticeistridiagonal: W (1) = 266666666664 W 1 ; 1 W 1 ; 2 W 2 ; 1 W 2 ; 2 W 2 ; 3 . W N 1 ;N 2 W N 1 ;N 1 W N 1 ;N W N;N 1 W N;N 377777777775 : Thefollowinglemmagivesthespectrumandtheleft-handPer ronvectorfortheweight matrix W (1) .TheproofofthelemmaisgiveninSection 5.5 .. Lemma5.1. Let W (1) betheweightmatrixassociatedwiththe 1 -dimensionallattice withtheweightsgivenby W i;i +1 = c;W i +1 ;i = a ,where a 6 = c arepositiveconstantsand a + c 1 .Thentheeigenvaluesof W (1) are 1 =1 ; ` =1 a c +2 p ac cos ( ` 1) N ; where ` 2f 2 ; ;N g ,anditsleftPerronvectoris = 1 c=a 1 ( c=a ) N [1 ;c=a; ( c=a ) 2 ; ; ( c=a ) N 1 ] : Wenextconsiderconsensusona D -dimensionallatticewiththefollowingweights W i;i d + = c d ;W i;i d = a d ;d 2f 1 ; ;D g ; (5{8) where a d 6 = c d arepositiveconstantsand P Dd =1 a d + c d 1.Thenotation i d + denotesthe neighboronthepositive x d axisofnode i and i d denotestheneighboronthenegative x d axisofnode i .Forexample,2 1+ and2 1 inFigure 5-2 denotenode3andnode1, respectively,and2 2+ isnode5. 104

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x 1 x 2 o 1 2 3 4 5 6 a 1 c 1 c 1 a 1 a 2 c 2 Figure5-2.Apictorialrepresentationofa2-dimensionall atticeinformationgraphwith theweights W (2) i;i d + = c d ;W (2) i;i d = a d ,where d =1 ; 2. Lemma5.2. Let W ( D ) betheweightmatrixassociatedwiththe D -dimensionallatticewith theweightsgivenin ( 5{8 ) .Thenitseigenvaluesaregivenby ~` ( W ( D ) )=1 D X d =1 (1 ` d ( W (1) d )) ; where ~ ` =( ` 1 ;` 2 ; ;` D ) ,inwhich ` d 2f 1 ; 2 ; ;N d g and W (1) d isthe N d N d weight matrixassociatedwitha 1 -dimensionallatticewiththeweightsgivenby W (1) d ( i;i +1)= c d ;W (1) d ( i +1 ;i )= a d and i 2f 1 ; ;N d 1 g .ItsleftPerronvectoris = (1) D n (1) D 1 nn (1) 1 ,where (1) d istheleftPerronvectorof W (1) d ,and n denotestheKronecker product. TheproofofLemma 5.2 isgiveninSection 5.5 .Thenexttheoremshowstheimplicationsoftheprecedingtechnicalresultsontheconver genceratein D -dimensional lattices.Theorem5.2. Let G bea D -dimensionallatticegraphandlet W ( D ) beanasymmetric stochasticmatrixcompatiblewith G withtheweightsgivenin ( 5{8 ) .Thentheconvergence ratesatises R c 0 ; (5{9) where c 0 2 (0 ; 1) isaconstantindependentof N 105

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ProofofTheorem 5.2 AccordingtoLemma 5.1 ,theeigenvaluesof W (1) d (denedin Lemma 5.2 )aregivenby: 1 ( W (1) d )=1 ; ` ( W (1) d )=1 a d c d +2 p a d c d cos ( ` d 1) N d : FromLemma 5.2 ,thesecondlargesteigenvalue 2 ( W ( D ) )andthesmallesteigenvalue N ( W ( D ) )of W ( D ) aregivenby 2 ( W ( D ) )=1 max d 2f 1 ; ;D g (1 2 ( W (1) d )) 1 max d 2f 1 ; ;D g ( a d + c d 2 p a d c d ) ; (5{10) N ( W ( D ) )=1 D X d =1 (1 N d ( W (1) d )) =1 D X d =1 ( a d + c d 2 p a d c d cos ( N d 1) N d ) 1 D X d =1 ( a d + c d 2 p a d c d ) : (5{11) Recallthat R =min f 1 2 ; 1+ N g .Inaddition, a d ;c d arexedconstantsandsatisfy a d 6 = c d P Dd =1 a d + c d 1,thereforethelowerboundsof1 2 ( W ( D ) )and1+ N ( W ( D ) ) arexedpositiveconstants.Wethenhavethattheconvergen cerateof W ( D ) satisfy R =1 ( W ( D ) ) c 0 ,where c 0 isaconstantindependentof N Remark5.1. RecallfromTheorem 5.1 ,foranyL-Zgeometricorlatticegraphs,aslong astheweightmatrix W issymmetric,nomatterhowdowedesigntheweights W i;j ,the convergenceratebecomesprogressivelysmallerasthenumb erofagents N increases,and itcannotbeuniformlyboundedawayfrom 0 .Incontrast,Theorem 5.2 showsthatfor latticegraphs,asymmetryintheweightsmakestheconverge ncerateuniformlybounded awayfrom 0 .Infact,anyamountofasymmetryalongthecoordinateaxeso fthelattice ( a d 6 = c d ),willmakethishappen.Asymmetricweightsthusmaketheli neardistributed 106

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consensuslawhighlyscalable.Iteliminatestheproblemof degenerationofconvergencerate withincreasing N Thesecondquestioniswheredothenodestatesconvergetowi thasymmetricweights? Recallthattheasymptoticsteadystatevalueofallagentsi s x = P Ni =1 i x i (0) .Foralattice graph,itsPerronvector isgiveninLemma 5.1 andLemma 5.2 .Thuswecandetermine thesteadystatevalue x iftheinitialvalue x (0) isgiven.Thisinformationisparticularly usefultondtherendezvouspositioninmulti-vehiclerend ezvousproblem.Ontheother hand,weseefromLemma 5.1 andLemma 5.2 thatif a d 6 = c d ,then i 6 = 1 N ,whichimplies thesteady-statevalueisnottheaverageoftheinitialvalu es.Theasymmetricweight designisnotapplicabletodistributedaveragingproblem. 5.2.2NumericalComparison Inthissection,wepresentthenumericalcomparisonofthec onvergenceratesof thedistributedprotocol( 5{3 )betweenasymmetricdesignedweights(Theorem 5.2 )and symmetricoptimalweightsobtainedfromconvexoptimizati on[ 27 29 ].Forsimplicity, wetakethe1-Dlatticeasanexample.Theasymmetricweights usedare W i;i +1 = c = 0 : 3 ;W i +1 ;i = a =0 : 2.WeseefromFigure 5-3 thattheconvergenceratewithasymmetric designedweightsismuchlargerthanthatwithsymmetricopt imalweights.Inaddition, giventheasymmetricweightvalues c =0 : 3 ;a =0 : 2,weobtainfrom( 5{10 )and( 5{11 ) that 2 0 : 5+2 p 0 : 06 ; N 0 : 5+2 p 0 : 06,whichimplies R =min f 1 2 ; 1+ N g 0 : 5 2 p 0 : 06 0 : 01 : (5{12) WeseefromFigure 5-3 thattheconvergencerate R isindeeduniformlyboundedbelow by( 5{12 ). 5.3FastConsensusinMoreGeneralGraphs Inthissection,westudyhowtodesigntheweightmatrix W toincreasetheconvergencerateofconsensusingraphsthataremoregeneralthanl attices.Weusetheidea ofcontinuumapproximation.Undersome\niceness"propert ies,agraphcanbethought 107

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20 40 80 150 10 -4 10 -3 10 -2 RN Symmetricoptimal Asymmetricdesign Lowerbound( 5{12 ) Figure5-3.Comparisonofconvergencerateof1-Dlatticebe tweenasymmetricdesignand convexoptimization(symmetricoptimal). ofasapproximationofa D -dimensionallattice,andbyextension,oftheEuclideansp ace correspondingto R D [ 101 ].Thesepropertieshavetodowiththegraphnothavingarbitrarilylargeholesetc.Preciseconditionsunderwhichagr aphcanbeapproximatedby the D -dimensionallatticeareexploredin[ 102 ](forinnitegraphs)andin[ 36 ](fornite graphs).Thedimension D ofthecorrespondinglattice/Euclideanspaceisalsodeter mined bytheseproperties. Thekeyistoembedthediscretegraphproblemintoacontinuu m-domainproblem.WeuseaSturm-LiouvilleoperatortoapproximatetheLa placianmatrixofa D -dimensionalgeometricgraph.AD-dimensionalgeometricg raphissimplyagraph withamappingofnodestopointsin R D .Basedonthisapproximation,were-derivethe asymmetricweightsforlatticesdescribedinthepreviouss ectionasvaluesofcontinuous functionsdenedover R D alongtheprincipalaxesin R D .Inalattice,theneighborsofa nodeliealongtheprincipalcanonicalaxesof R D .Foranarbitrarygraph,theweightsare nowchosenassamplesofthesamefunctions,alongdirection sinwhichtheneighborslie. 108

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x 1 x 1 x 1 x 2 x 2 x 2 o o o 1 1 1 1 1 1 L Figure5-4.Continuumapproximationofgeneralgraphs. Themethodisapplicabletoarbitrarydimension,butweonly considerthe2-Dcasein thischapter.Graphswith2-Ddrawingsareoneofthemostrel evantclassesofgraphsfor sensornetworkswhereconsensusislikelytondapplicatio n. 5.3.1ContinuumApproximation Recallthattheconvergencerateisintimatelyconnectedto theLaplacianmatrix. WewillshowthattheLaplacianmatrixassociatedwithalarg e2-Dlatticewithcertain weightscanbeapproximatedbyaSturm-Liouvilleoperatord enedona2-Dplane.Thus it'sreasonabletosupposethattheSturm-Liouvilleoperat orisalsoagood(continuum) approximationoftheLaplacianmatrixoflargegraphswith2 -Ddrawing.Westartfrom 2-DlatticegraphandderiveaSturm-Liouvilleoperator.We thenusethisoperator toapproximatethegraphLaplacianofmoregeneralgraphs.T heideaisillustratedin Figure 5-4 Foreaseofdescription,werstconsidera1-Dlattice,with thefollowingasymmetric weights,whichareinspiredbytheasymmetriccontrolgains forvehicularplatoonsthat wasdiscussedinChapter 2 W i;i +1 = c = 1+ 2 ;W i +1 ;i = a = 1 2 ; (5{13) 109

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where i 2f 1 ; 2 ; ;N 1 g and 2 (0 ; 1)isaconstant.ThegraphLaplaciancorresponding totheweightsgivenin( 5{13 )isgivenby L (1) = 266666666664 1+ 2 1 2 1+ 2 1 1 2 . . 1+ 2 1 1 2 1+ 2 1 2 377777777775 : (5{14) Recallthattondalowerboundoftheconvergencerateofthe weightmatrix W (1) ,it's sucienttondalowerboundofthesecondsmallesteigenval ueoftheassociateLaplacian matrix L (1) WenowuseaSturm-LiouvilleoperatortoapproximatetheLap lacianmatrix L (1) Werstconsiderthenite-dimensionaleigenvalueproblem L (1) = .Expandingthe equation,wehavethefollowingcoupleddierenceequation s 1+ 2 i 1 + i + 1 2 i +1 = i ; where i 2f 1 ; 2 ; ;N g and 0 = 1 N +1 = N .Theaboveequationcanberewrittenas 1 2 N 2 i 1 2 i + i +1 1 =N 2 N i +1 i 1 2 =N = i : Thestartingpointforthecontinuumapproximationistocon siderafunction ( x ): [0 ; 1] R thatsatises: i = ( x ) j x = i= ( N +1) ; (5{15) suchthatafunctionthatisdenedatdiscretepoints i willbeapproximatedbyafunction thatisdenedeverywherein[0 ; 1].Theoriginalfunctionisthoughtofassamplesofits continuousapproximation.Undertheassumptionthat N islarge,usingthefollowing 110

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nitedierenceapproximation: h i 1 2 i + i +1 1 =N 2 i = h @ 2 ( x;t ) @x 2 i x = i= ( N +1) ; h i +1 i 1 2 =N i = h @ ( x;t ) @x i x = i= ( N +1) ; thenite-dimensionaleigenvalueproblemcanbeapproxima tedbythefollowingSturmLiouvilleeigenvalueproblem L (1) ( x )= ( x ) ; where L (1) := 1 2 N 2 d 2 dx 2 N d dx ; (5{16) withNeumannboundaryconditions: d (0) dx = d (1) dx =0 : (5{17) Lemma5.3. TheeigenvaluesoftheSturm-Liouvilleoperator L (1) ( 5{16 ) withboundary condition ( 5{17 ) for 0 <"< 1 arerealandthersttwosmallesteigenvaluessatisfy 1 ( L (1) )=0 ; 2 ( L (1) ) 2 = 2 : WeseefromLemma 5.3 thatthesecondsmallesteigenvalueoftheSturm-Liouville operator L (1) isuniformlyboundedawayfromzero.Thisresultisnotsurpr ising,sinceit's acontinuumcounterpartofLemma 5.1 ,whichshowsthatthesecondsmallesteigenvalue correspondingtothe1-Dlatticewithdesignedasymmetricw eightsisuniformlybounded below.TheproofofLemma 5.3 isgiveninSection 5.5 Wenowconsiderthefollowingweightsfortheconsensusprob lemwithD-dimensional latticegraph W ( D ) i;i d + = c d = 1+ 2 D ;W ( D ) i;i d = a d = 1 2 D ; (5{18) where 2 (0 ; 1)isaconstant. 111

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TheLaplacianmatrixofaD-dimensionalsquarelatticewith theweightsgiven in( 5{18 )isgivenby L ( D ) = I W ( D ) .Followingsimilarprocedureofeigenvalueapproximationforthe1-dimensionallattice,thesecondsmal lesteigenvalueoftheLaplacian matrix L ( D ) canbeapproximatedbythatofthefollowingSturm-Liouvill eoperator L ( D ) = D X ` =1 ( 1 2 DN 2 d d 2 dx 2d + DN d d dx d ) ; (5{19) withthefollowingNeumannboundaryconditions @ ( ~x ) @x d x d =0or1 =0 ; (5{20) where d =1 ; 2 ; ;D and ~x =[ x 1 ;x 2 ; ;x D ] T Continuumapproximationhasbeenusedtostudythestabilit ymarginoflarge vehicularplatoonsinChapter 2 ,inwhichthecontinuummodelgivesmoreinsightinto theeectofasymmetryonthestabilitymarginofthesystems .Inthischapter,weusethe secondsmallesteigenvalueoftheSturm-Liouvilleoperato r L ( D ) toapproximatethatofthe Laplacianmatrix L ( D ) Theorem5.3. Thesecondsmallesteigenvalues 2 ( L ( D ) ) oftheSturm-Liouvilleoperator L ( D ) ( 5{19 ) withboundarycondition ( 5{20 ) for 0 <"< 1 isrealandsatises 2 ( L ( D ) ) 2 2 D ; (5{21) whichisapositiveconstantindependentof N ProofofTheorem 5.3 Bythemethodofseparationofvariables[ 77 86 ],theeigenvaluesof theSturm-Liouvilleoperator L ( D ) isgivenby ( L ( D ) )= D X d =1 ( L (1)d ) ; (5{22) where L (1)d isthe1-dimensionalSturm-Liouvilleoperatorgivenby L (1)d = 1 2 DN 2 d d 2 dx 2d DN d d dx d ; 112

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withNeumannboundaryconditions.FollowingLemma 5.3 ,wehavethatthesmallest eigenvalueof L (1)d is0andthesecondsmallesteigenvalueof L (1)d isboundedbelowby L (1)d 2 = 2 D .Therefore,wehavefrom( 5{22 )thatthesecondsmallesteigenvalueis 2 ( L ( D ) )=min d f 2 ( L ( d ) ) g 2 2 D : 5.3.2WeightDesignforGeneralGraphs x 1 x 2 o 12 13 1 1 1 2 3 (a)Relativeangle 0 2 3 2 2 1 4 1+ 4 g (b)Weightfunction Figure5-5.Weightdesignforgeneralgraphs. Theinspirationoftheproposedmethodcomesfromthedesign forlattices.The4 weightsforeachnode i ina2-Dlatticecanbere-expressedassamplesofacontinuou s function g :[0 ; 2 ) [ 1 4 ; 1+ 4 ]: W i;i 1+ = g ( i;i 1+ ) ;W i;i 2+ = g ( i;i 2+ ) ; W i;i 1 = g ( i;i 1 ) ;W i;i 2 = g ( i;i 2 ) where i;j istherelativeangularpositionof j withrespectto i .Giventheangular positionsof i 'sneighborsandthevaluesoftheweights,weknowthatthefu nction g must satisfy: g ([0 ; 2 ;; 3 2 ])=[ 1+ 4 ; 1+ 4 ; 1 4 ; 1 4 ] : (5{23) 113

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Thus,wechoosethefunction g asshowninFigure 5-5 (b). Foranarbitrarygraph,wenowchoosetheweightsbysampling thefunctionaccording totheangleassociatedwitheachedge( i;j ): W i;k = g ( i;k ) P j 2 N i g ( i;j ) ; (5{24) where g ( )isthefunctiondescribedinFigure 5-5 (b).Theaboveweightfunction( 5{24 ) canbeseenasalinearinterpolationof( 5{23 ).Weseefrom( 5{24 )thattheweighton eachedgeiscomputableinadistributedmanner;anodeonlyn eedstoknowtheangular positionofitsneighbors.Thisdesignmethoddoesnotrequi reanyknowledgeofthe networktopologyorcentralizedcomputation.5.3.3NumericalComparison Inthissection,wepresentthenumericalcomparisonofconv ergenceratesamong asymmetricdesign,symmetricoptimalweightsandweightsc hosenbytheMetropolisHastingsmethod.Thesymmetricoptimalweightsareobtaine dbyusingconvexoptimizationmethod[ 29 73 ].TheMetropolis-Hastingsweightsarepickedbythefollow ingrule: W i;j =1 = j N i j ,where N i denotesthenumberofneighborsofnode i .Theweightsgenerated bythismethodareingeneralasymmetric.Weplottheconverg encerate R asafunction of N ,where N isthenumberofagentsinthenetwork.Theamountofasymmetr yusedis =0 : 5. 0 0.5 1 0 0.5 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 (a)L-Zgeometric 0 0.5 1 0 0.5 1 (b)Delaunay 0 0.5 1 0 0.5 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 (c)Randomgeometric Figure5-6.Examplesof2-DL-Zgeometric,Delaunayandrand omgeometricgraphs. 114

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WerstconsideraL-Zgeometricgraph[ 36 ],whichisgeneratedbyperturbingthe nodepositionsinasquare2-Dlattice( N 1 = N 2 = p N )withGaussianrandomnoise (zeromeanand1 = (4 p N )standarddeviation)andconnectingeachnodewithotherno des thatarewithina2 = p N radius.Second,weconsideraDelaunaygraph[ 5 ],whichis generatedbyplacing N nodesona2-Dunitsquareuniformlyatrandomandconnecting anytwonodesiftheircorrespondingVoronoicellsintersec t,aslongastheirEuclidean distanceissmallerthan1 = 3.Finally,weconsiderarandomgeometricgraphs[ 103 ],which isgeneratedbyplacing N nodesona2-Dunitsquareuniformlyatrandomandconnecting pairsofnodesthatarewithinadistance3 = p N ofeachother.Figure 5-6 givesexamplesof L-Zgeometricgraphs,Delaunaygraphsandrandomgeometric graphs. Figure 5-7 showsthecomparisonofconvergenceratesamongasymmetric design, symmetricoptimalandMetropolis-Hastingsweights.Forea ch N ,theconvergencerateof 10samplesofthegraphsareplotted.WeseefromFigure 5-7 thatforalmosteverysample ineachofthethreeclasses,theconvergenceratewiththeas ymmetricdesignisanorderof magnitudelargerthantheothers,especiallywhen N islarge. 5.4Summary Westudiedtheproblemofhowtodesignweightstoincreaseth econvergencerate ofdistributedconsensusinnetworkswithstatictopology. Weprovedthatonlattice graphs,withproperchoiceofasymmetricweights,theconve rgenceratecanbeuniformly boundedawayfromzero.Inaddition,weproposedadistribut edweightdesignalgorithm for2-dimensionalgeometricgraphstoimprovetheconverge ncerate,byusingacontinuum approximation.Numericalcalculationsshowthattheresul tingconvergencerateis substantiallylargerthanthatoptimalsymmetricweightsa ndMetropolisHastingsweights. Animportantopenquestionisaprecisecharacterizationof graphsforwhichtheoreticalguaranteesonsize-independentconvergenceratecanb eprovidedwiththeproposed design.Inaddition,characterizingtheasymptoticsteady statevalueformoregeneral graphsthanlatticesisalsoon-goingwork. 115

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100 200 500 1,000 10 -2 10 -1 RN Symmetricoptimal AsymmetricDesign Metropolis-Hastings (a)L-Zgeometricgraphs 100 200 500 1000 10 -2 10 -1 RN Symmetricoptimal AsymmetricDesign Metropolis-Hastings (b)Delaunaygraphs 100 200 500 1,000 10 -2 10 -1 RN Symmetricoptimal AsymmetricDesign Metropolis-Hastings (c)Randomgeometricgraphs Figure5-7.Comparisonofconvergencerateswithproposeda symmetricweights, Metropolis-Hastingsweights,andsymmetricoptimal.Fore ach N ,resultsfrom 5samplegraphsareplotted. 116

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5.5TechnicalProofs 5.5.1ProofofLemma 5.1 Thestochasticmatrix W (1) hasasimpleeigenvalue 1 =1.FollowingTheorem3.1 of[ 104 ],theothereigenvaluesof W (1) aregivenby ` =1 a c +2 p ac cos ` ;` 2f 2 ; ;N g ; where ` ( 6 = m;m 2 Z Z beingthesetofintegers)istherootofthefollowingequati on 2sin( N ) cos ( )=( a + c ) r 1 ac sin N; whichimplies sin( N )=0 ; orcos = ( a + c ) 2 r 1 ac : Since a> 0 ;c> 0and a 6 = c ,wehave ( a + c ) 2 q 1 ac > 1,thuscos 6 = ( a + c ) 2 q 1 ac .Inaddition,we havethat 6 = m ,whichyields ` = ( ` 1) N ;` = f 2 ; ;N g : (5{25) Wenowobtaintheeigenvaluesof W (1) ,whichisgivenby ` =1 a c +2 p ac cos ( ` 1) N ;` = f 2 ; ;N g : Let =[ 1 ; 2 ; ; N ]betheleftPerronvectorof W (1) .Fromthedenitionof Perronvector,wehave W (1) = .Thankstothespecialstructureofthetridiagonalform of W (1) ,wecansolvefor explicitly,whichyields i =( c=a ) i 1 1 ; (5{26) where i 2f 2 ; 3 ; ;N g .Inaddition,wehave i > 0and P Ni =1 i =1.Therefore, 1= N X i =1 i = N X i =1 ( c=a ) i 1 1 ) 1 = 1 c=a 1 ( c=a ) N : 117

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Substitutingtheaboveequationinto( 5{26 ),wecompletetheproof. 5.5.2ProofofLemma 5.2 Withtheweightsgivenin( 5{8 ),itisstraightforward-throughabittedious-toshow thatthegraphLaplacian L ( D ) associatedwiththe D -dimensionallatticehasthefollowing form: L ( d ) = I N d n L ( d 1) + L (1)d n I N 1 N 2 N d 1 ; 2 d D; where L (1) = L (1)1 and L (1)d =1 W (1) d istheLaplacianmatrixofdimension N d N d ,which isgivenby L (1)d = 266666666664 c d c d a d a d + c d c d . . a d a d + c d c d a d a d 377777777775 : (5{27) Sincea D -dimensionallatticeistheCartesianproductgraphof D 1-dimensional lattices,theeigenvaluesofthegraphLaplacianmatrix L ( D ) aresumoftheeigenvaluesof the D 1-dimensionalLaplacianmatrix L (1)d .Thus,wehave ` 1 ;:::;` D ( L ( D ) )= D X d =1 ` d ( L (1)d ) : Inaddition,wehavethat W ( D ) = I N L ( D ) and W (1) d = I N d L (1)d ,thustheeigenvalues ~` of W ( D ) aregivenby ~` ( W ( D ) )=1 ~` ( L ( D ) )=1 D X d =1 ` d ( L (1)d ) =1 D X d =1 (1 ` d ( W (1) d )) : 118

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Tosee = (1) D n (1) D 1 nn (1) 1 istheleftPerronvectorof W ( D ) ,werstnotice that (1) d W (1) d = (1) d ; (1) d L (1)d =0 ; where d 2f 1 ; ;D g .Therestoftheprooffollowsbystraightforwardinduction method, weomittheproofduetospacelimit. 5.5.3ProofofLemma 5.3 Multiplybothsidesof( 5{16 )by2 N 2 e 2 "Nx ,weobtainthestandardSturm-Liouville eigenvalueproblem d dx e 2 "Nx d ( x ) dx +2 N 2 e 2 "Nx ( x )=0 : (5{28) AccordingtoSturm-LiouvilleTheory,alltheeigenvaluesa rereal,see[ 77 86 ].Tosolve theSturm-Liouvilleeigenvalueproblem( 5{16 )-( 5{17 ),weassumesolutionoftheform, ( x )= e rx ,thenweobtainthefollowingequation r 2 +2 "Nr +2 N 2 =0 ; ) r = N ( p 2 2 ) : (5{29) Dependingonthediscriminantintheaboveequation,therea rethreecasestoanalyze: 1. <" 2 = 2,thentheeigenfunction ( x )hasthefollowingform ( x )= c 1 e N ( + p 2 2 ) x + c 2 e N ( p 2 2 ) x ,where c 1 ;c 2 aresomeconstants.Applyingtheboundarycondition( 5{17 ),it'sstraightforwardtoseethat,fornon-trivialeigenf unctions ( x )to exit,thefollowingequationmustbesatised + p 2 2 + p 2 2 = e 2 N p 2 2 + p 2 2 + p 2 2 : Thus,wehave =0. 2. = 2 = 2,thentheeigenfunction ( x )hasthefollowingform ( x )= c 1 e "Nx + c 2 xe "Nx : 119

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Applyingtheboundarycondition( 5{17 )again,it'sstraightforwardtoseethatthere isnoeigenvalueforthiscase. 3. >" 2 = 2,thentheeigenfunctionhasthefollowingform ( x )= e "Nx ( c 1 cos( N p 2 2 x )+ c 2 sin( N p 2 2 x ).Applyingtheboundarycondition( 5{17 ),fornon-trivialeigenfunctionstoexit,theeigenvalues mustsatisfy = 2 2 + ` 2 2 2 N 2 ,where ` =1 ; 2 ; Combiningtheabovethreecases,theeigenvaluesoftheStur m-Liouvilleoperatorare 2f 0 ; 2 2 + ` 2 2 2 N 2 g ,where ` 2f 1 ; 2 ; g .Thesecondsmallesteigenvalue 2 ( L )ofthe Strum-Liouvilleoperator L isthengivenby 2 ( L )= 2 2 + 2 2 N 2 2 2 ; whichisaconstantthatisboundedawayfrom0. ontinuumapproximationhasbeenusedtostudythestability marginoflargevehicularplatoons[ 91 105 ],inwhichthecontinuummodelgivesmoreinsightontheeec t ofasymmetryonthestabilitymarginofthesystems.Inthisc hapter,weusethesecondsmallesteigenvalueoftheSturm-Liouvilleoperator L ( D ) toapproximatethatofthe Laplacianmatrix L ( D ) 120

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CHAPTER6 CONCLUSIONSANDFUTUREWORK Thischaptersummarizesthecontributionsofthisdisserta tionanddiscussespossible directionsforfutureresearch. 6.1Conclusions Thisdissertationstudiedperformancescalingofdistribu tedcontrolofmulti-agent systemswithrespecttonetworksize.Weinvestigatedtwocl assesofdistributedcontrol problemsthatarerelevanttovehicularformationcontrola nddistributedconsensus.In thevehicularformationcontrolproblem,eachvehicleismo deledbyadoubleintegrator, whilethedynamicsofeachagentindistributedconsensusar egivenasingleintegrator. Despitedierenceinagentdynamics,thetwoproblemssuer fromsimilarperformance limitations.Inparticular,theirperformancesdegradewh enthenumberofagentsinthe systemincreaseswithsymmetriccontrol,wheresymmetricc ontrolrefersto,betweeneach pairofneighboringagents,theinformationreceivedfrome achotherisgiventhesame weight.Oneofthemaincontributionsofthisworkisthatwep roposedanasymmetric controldesignmethodtoamelioratetheperformancescalin glawsforbothvehicular formationcontrolanddistributedconsensus.Asymmetricd esignmeansbetweeneachpair ofneighboringagents,theinformationreceivedfromeacho therisweighteddierently, insteadofequallyinsymmetricdesign.Weshowedtheresult ingperformancescalinglaws wereimprovedconsiderablyoverthosewithsymmetriccontr ol. Forthevehicularformationcontrolproblem,wedescribeda novelframeworkfor modeling,analysisanddistributedcontroldesign.Thekey componentofthisframework isaPDE-based(partialdierentialequation)continuousa pproximationofthe(spatially) discreteclosed-loopdynamicsofthecontrolledformation .BasedonthisPDEmodel,we derivedexactquantitativescalinglawsofthestabilityma rginandrobustnesstoexternal disturbances,withrespecttothenumberofvehiclesinthef ormation.Theresultsshowed thatwithsymmetriccontrol,thestabilitymarginandrobus tnessperformancesdegraded 121

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progressivelywhenthenumberofvehiclesintheteamincrea sed.Thescalinglawsof stabilitymarginandrobustnessperformancesdevelopedin thisdissertationarehelpfulto understandthelimitationsofdistributedcontrolarchite cture. Besidesanalysisofperformancescalings,thePDEmodelisa lsoconvenientfor distributedcontroldesign.Bytakingadvantagesofthewel ldevelopedPDEandoperator (suchasSturm-Liouville)theoryaswellasperturbationte chnique,weproposedan asymmetric designmethod,whichimprovedthestabilitymarginandrobu stnessto disturbancesconsiderablyoversymmetriccontrol.Numeri calexperimentsshowedthat thePDEmodelmadeanaccurateapproximationofthestate-sp acemodelevenfora smallvalueof N ,where N isthenumberofvehiclesintheformation.Moreover,the resultingasymmetriccontrolissimpletoimplementandthe reforeattractiveforpractical applications. Wenextappliedtheasymmetricdesignmethodtoanotherclas sofdistributedcontrol problem:distributedconsensus.Indistributedconsensus ,eachagentinanetworkupdates itsstatebyusingaweightedsummationofitownstateandthe statesofitsneighbors. Thegoalistomakealltheagents'statesreachacommonvalue .Itwasshownthatwith symmetricweight,theconsensusratebecameprogressively smallerwhenthenumberof agentsinthenetworkincreased,evenwhentheweightswerec hoseninanoptimalmanner. Weproposedamethodtodesign asymmetric weightstospeeduptheconvergencerate ofdistributedconsensusinnetworkswithstatictopology. Weprovedthatonlattice graphs,withproperchoiceofasymmetricweights,theconve rgenceratecouldbeuniformly boundedawayfromzerowithrespecttothenumberofagentsin thenetwork.Inaddition, wedevelopedadistributedweightdesignalgorithmformore generalgraphsthanlattices toimprovetheirconvergencerates.Numericalcalculation sshowedthattheresulting convergenceratewassubstantiallylargerthanthatwithop timalsymmetricweightsor MetropolisHastingsweights. 122

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6.2FutureWork Thereareseveralpossibletopicsoffutureinvestigations thataresummarizebelow. TheinformationgraphsstudiedinChapter 2 4 arelimitedtoD-dimensionallattices. Morecomplexgraphstructuresshouldbeexploredinfuturew ork.Webelievethatthe PDEapproximationwillbebenecialhere,byallowingustos amplefromthecontinuous gainfunctionsdenedoveracontinuousdomaintoassigngai nstospatiallydiscrete agents. InChapter 3 ,numericalsimulationsshowthatwith asymmetricvelocity feedback,the system'srobustnesstoexternaldisturbancecanbeimprove dsignicantlyoversymmetric controlandthecasewithequalasymmetryinthepositionand velocityfeedback.These resultsweresummarizedasaconjecture.Futureresearchwi llfocusonthetheoretical analysistoverifysuchanimprovement. Additionally,regardingthedistributedconsensusproble minChapter 5 ,animportant openquestionisaprecisecharacterizationofgraphsforwh ichtheoreticalguaranteeson size-independentconvergenceratecanbeprovidedwiththe proposeddesign.Characterizingtheasymptoticsteadystatevalueformoregeneralgraph sthanlatticesisvaluableas well. Lastbutnottheleast,webelievetheasymmetricdesignwill haveapotential importantimpactonotherapplicationsofdistributedcont roloflargenetworkedsystems. Besidesvehicularformationsanddistributedconsensus,w ebelievetheasymmetricdesign methodcanalsobeappliedtoimprovemixingtimeofrandomwa lksandperformanceof distributedKalmanlter.Futureworkwilllookattheseapp lications.Inaddition,the asymmetricdesignmayalsohelpanswerthequestionofhowto avoidactuatorsaturation inlarge-scalemulti-agentsystemwhichresultsfromlarge transienterrorsand/orhighgain controller,asevidencedin[ 95 106 107 ]. 123

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BIOGRAPHICALSKETCH HeHaowasborninMarch,1984inHaicheng,China.Hereceived hisBachelorof Sciencedegreeinmechanicalengineeringandautomationin 2006fromNortheastern University,Shenyang,China,andamaster'sdegreeinmecha nicalengineeringin2008 fromZhejiangUniversity,Hangzhou,China.Hethenjoinedt heDistributedControl SystemsLaboratoryattheUniversityofFloridatopursuehi sdoctoraldegreeunderthe advisementofDr.PrabirBarooah. 132