ConnectedVehicularSystems
Communication,Control,andOptimization
GeGuo
TheStateKeyLaboratoryofSyntheticalAutomation forProcessIndustries,NortheasternUniversity
TheSchoolofControlEngineering,Northeastern UniversityatQinhuangdao,China
ShixiWen
SchoolofInformationandEngineering
TheKeyLaboratoryofAdvancedDesignandIntelligentComputing, MinistryofEducation,DalianUniversity,China
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LibraryofCongressCataloging-in-PublicationData
Names:Guo,Ge(OfDongbeidaxue(1993)),author.|Wen,Shixi,author.
Title:Connectedvehicularsystems:communication,control,and optimization/GeGuo,ShixiWen.
Description:Hoboken,NewJersey:Wiley,[2024]|Includesindex.
Identifiers:LCCN2023022911(print)|LCCN2023022912(ebook)|ISBN 9781394205462(cloth)|ISBN9781394205479(adobepdf)|ISBN 9781394205486(epub)
Subjects:LCSH:Automatedvehicles.|Intelligenttransportationsystems.
Classification:LCCTL152.8.G862024(print)|LCCTL152.8(ebook)|DDC 629.04/6–dc23/eng/20230601
LCrecordavailableathttps://lccn.loc.gov/2023022911
LCebookrecordavailableathttps://lccn.loc.gov/2023022912
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Setin9.5/12.5ptSTIXTwoTextbyStraive,Pondicherry,India
Contents
Preface ix Acknowledgments xiii
PartIVehicularPlatoonCommunicationandControl 1
1ControlwithVaryingCommunicationRange 3
1.1Introduction 3
1.2ProblemFormulation 5
1.3SwitchingControlofConnectedVehicles 9
1.4SimulationsandExperiments 16
1.5ConclusionsandFutureWork 23 References 24
2ControlSubjecttoCommunicationInterruptions 26
2.1Introduction 26
2.2ProblemFormulation 27
2.3MixedCACC-ACCControl 28
2.4Finite-TimeSliding-ModeControl 32
2.5NumericalSimulations 34
2.6ConclusionsandFutureWork 39 References 41
3ControlandCommunicationTopologyAssignment 42
3.1Introduction 42
3.2ProblemStatement 44
3.3CommunicationTopologyandControlCo-Design 48
3.4SimulationStudies 57
3.5ConclusionsandFutureWork 70 References 70
4ControlwithCommunicationDelayandSwitchingTopologies 72
4.1Introduction 72
4.2ProblemFormulation 73
4.3StabilityAnalysis 77
4.4ControllerSynthesis 82
4.5SimulationStudies 86
4.6ConclusionsandFutureWork 95
References 96
5ControlwithEvent-TriggeredCommunication 97
5.1Introduction 97
5.2ProblemFormulation 99
5.3Event-TriggeredCommunicationandPlatoonControl 104
5.4SimulationStudy 107
5.5ConclusionsandFutureWork 119
References 120
PartIIPerformanceGuaranteeUnderActuatorLimitation 121
6AdaptiveFault-TolerantControlwithActuatorSaturation 123
6.1Introduction 123
6.2SystemModelingandProblemFormulation 124
6.3QuadraticSpacingPolicyandAdaptivePID-TypeSlidingSurface 127
6.4ControllerDesignandStabilityandAnalysis 128
6.5SimulationResults 135
6.6ConclusionsandFutureWork 139
References 142
7Fault-TolerantControlwithInputQuantizationandDeadZone 143
7.1Introduction 143
7.2SystemModelingandProblemFormulation 144
7.3ImprovedQuadraticSpacingPolicyandAdaptivePID-TypeSlidingSurface 148
7.4ControllerDesignandStabilityAnalysis 149
7.5SimulationResults 155
7.6ConclusionsandFutureWork 157
References 163
8PrescribedPerformanceConcurrentControl 165
8.1Introduction 165
8.2ProblemFormulation 166
8.3ControllerDesignGuaranteedPrescribedPerformance 168
8.4SimulationStudies 175
8.5ConclusionsandFutureWork 179
References 179
9AdaptiveSlidingModeControlwithPrescribedPerformance 181
9.1Introduction 181
9.2ProblemFormulation 181
9.3ModelTransformation 184
9.4VehiclesTrackingControllerDesign 185
9.5SimulationStudies 190
9.6ConclusionsandFutureWork 197 References 198
PartIIISpeedTrajectoryPlanningandControl 199
10SpeedPlanningandTrackingControlofVehicles 201
10.1Introduction 201
10.2ProblemFormulations 202
10.3SpeedPlanning 205
10.4SpeedTrackingControllerDesign 207
10.5SimulationandExperiments 213
10.6ConclusionsandFutureWork 221 References 224
11AnalyticalSolutionforSpeedPlanningandTrackingControl 225
11.1Introduction 225
11.2SystemModelingandProblemFormulation 226
11.3SpeedOptimizationBasedonPMP 228
11.4SpeedTrackingControlandStringStability 232
11.5SimulationStudies 237
11.6ConclusionsandFutureWork 240 References 241
12SpeedPlanningandSliding-ModeControltoReduceIntervehicleSpacing 242
12.1Introduction 242
12.2ProblemStatement 243
12.3IntervehicleSpacingOptimization 246
12.4Sliding-ModeControllerDesign 250
12.5SimulationStudies 253
12.6ConclusionsandFutureWork 265 References 266
13TrajectoryPlanningandPID-TypeSliding-ModeControltoReduceIntervehicle Spacing 268
13.1Introduction 268
13.2ProblemDescription 269
13.3DistributedTrajectoryOptimization 271
13.4PID-TypeSliding-ModeControllerDesign 275
13.5SimulationResults 278
13.6ConclusionsandFutureWork 288 References 288
14TrajectoryPlanningandFixed-TimeTerminalSliding-ModeControl 290
14.1Introduction 290
14.2ProblemFormulation 291
14.3VehiclesTrajectoryOptimization 293
14.4Fixed-TimeTrackingControlDesign 297
14.5NumericalSimulations 301
14.6ConclusionsandFutureWork 307
References 307
Index 309
Preface
Withtheubiquitousapplicationofvehicle-to-vehicle(V2V)andvehicle-to-infrastructure(V2I) communicationtechnologies,connectedandautomatedvehicles(CAVs)arecapableofgathering andsharingroadandtrafficinformationandevenvehiclestateswithneighboringvehicles.Inparticular,enabledbytheinformationshared,CAVsallowautomatedvehiclemotion,carfollowing, cooperateddrivingandplatooning,andvehicle-trafficsignalcooperativecontrol.Therefore,CAVs arebelievedtobeapromisingtechnologytodelivergreatersafetyandmobilitybenefitstothenew generationofintelligenttransportationsystems(ITSs)withincreaseddrivingsafety,ridecomfort, trafficefficiency,andthroughput,alongwithreducedcongestion,accidents,emissions,andair pollution.However,theoperationofCAVsandtheassociatedITSsdependsheavilyontimely andreliableinformationgatheringandsharing,properdecision-making,andeffectiveactuation ofthedrivingdecision.However,criticalchallengingissuesarefacingCAVsfromallaspects includingsensing,communication,controldesign,andcommandactuating,which,ifnot properlyaddressed,canresultinsafetyrisksandlosses.
ThisbookcontainsourresearchadvancesinthepastdecadeintheanalysisandsynthesisofCAV systemsfromallaspectsoftrajectoryplanning,cooperativecontrol,andcommunication.Thefocus ofthisbookisonthedevelopmentofmathematicalmodelsandmethodologiesfortrajectoryoptimizationandtrackingcontrol,communicationsconflictresolution,cooperativecontrolsubjectto communicationconstraints,andsensor/actuatorfailures/faultsforCAVsfromdifferentperspectives.Thisbookiscomposedof14chapters.Thecontentsaredividedintothreeparts,with Chapter1 – Chapter5asPartI,Chapter6 – Chapter9asPartII,andChapter10 – Chapter14 asPartIII,respectively,concernedwithcooperativevehicularcommunicationandcontrol,performanceguaranteeunderactuatorlimitations,andspeedtrajectoryplanningandtrackingcontrol ofCAVs.
Chapter1 studiestheplatooncontrolproblemsubjecttovaryingcommunicationrangewitha constant-spacingpolicy.Accordingtotheconnectivitystatusbetweentheleaderandeachfollower, connectedvehiclescontrolismodeledasaswitchedplatooncontrolsystemwithaconnectivitystatus-matrix-dependentcontroller.Byusingswitchedsystemtheory,aseriesofsufficientconditions areobtainedasacriterionforthestabilityanalysisandcontrolsynthesisoftheleaderfollowing platoon.Basedontheobtainedconditions,ausefulcontrolalgorithmisproposedforconnected vehicles.Foreachobtainedconnectivity-status-matrix-dependentcontroller,thestringstability andazerosteady-statespacingerrorcanbeguaranteedbyadditionalconditions.
Chapter2 investigatesplatooningofconnectedvehiclesconsideringcommunicationinterruptionandlatency.Foraheterogeneousplatoonofvehicles,ahybridreferencemodelwithcooperativeadaptivecruisecontrol(CACC)andadaptivecruisecontrol(ACC)isestablished.Thenanovel CACC-ACCswitchingcontrolmethodissuggested,whichactivateseitheraCACCschemeoran
x Preface
augmentedACCstrategydependingonthestatusofcommunications.Byintroducingaplatoon statetrackingerrorsystem,acontrolalgorithmisderivedusingfinite-timesliding-modecontrol theory,whichcanrobustlyguaranteestringstabilityandzerosteady-statespacingerroroftheconnectedvehicles.
Chapter3 studiestheco-designproblemofplatoon controllerandinter-vehiclecommunicationtopology(IVCT)inLTE-V2Vnetworks. Thecommunicationassignmentisachieved basedonthecooperativeawarenessmessagedisseminationmechanism.Asampled-datafeedbackcontrollerisproposedforconnectedvehicl estoeliminatetheeffectofstochasticpacket dropoutsandexternaldisturbance,wherethecontrollergaindependsontheIVCT.Toguaranteethestabilityrequirementofconnectedvehicleswiththeminimizedcostfunction,aunifiedcontrolframeworkisestablishedtojointlydeterminetheoptimalIVCTfromallthe availableonesandtheassociatedfeedbackcontrollergain.Thisco-designprocedureisbased ontheoptimalcontrolanddynamicprogrammingtechnique,wherebothfixedandperiodic switchingIVCTsareavailable.Ausefulalgor ithmisproposedtoimplementtheestablished co-designframework.
Chapter4 addressestheplatooncontrolprobleminasampled-datasetupwithswitchingcommunicationtopologyandtransmissiondelays.Atrackingerror-basedsampled-datacontrolmethod isproposed,wheretheneighboringvehicle’sstateinformationistransmittedviatheVANETwith communicationdelay.ByrepresentingtheswitchingcommunicationtopologybyaMarkovian chain,theconnectedvehicularcontrolsystemismodeledasaMarkovianswitchingtime-delaysystemwithdisturbance.InthecontextofMarkovianjumpingsystemtheory,acontrolmethodologyis obtainedforconnectedvehiclestoguaranteethatthetrackingerrorscanbestabilizedmean-square exponentiallywithagivendisturbanceattenuationlevel.Thecontrollersofconnectedvehicleswith bothfixedandvariablegainsaresuggested.Theresultsareextendedtocoverpartiallyunknown transitionratesoftheMarkovchains.
Chapter5 studiestheco-designproblemconsideringadynamicevent-triggeredcommunication mechanism(DECM).UndertheDECM,thetransmissionsofsampledvelocityandacceleration fromaprecedingvehicletothecontrollercanbesignificantlyreduced.Asampled-dataplatoon controllerisdesignedbasedonthetrackingerror(spacingerror,velocityerror,andacceleration error).SufficientconditionsforthestabilityoftheCACCsystemareobtainedfortheDECM-based sampled-datafeedbackcontroller.Accordingtotheobtainedconditions,parameterdesigncriteria areestablishedfortheDECMtoguaranteethestableperformanceofconnectedvehiclescontrol systems.
Chapter6 addressesafault-tolerantcontrolproblemforconnectedvehiclessubjecttoactuator faultsandsaturation.Tocompensatefortheeffectsofactuatorfaultsandsaturation,anadaptive fault-tolerantcontrolmethodisproposedbasedonnonlinearvehicledynamicsandanewquadratic spacingpolicy.Theimprovedquadraticspacingpolicyisintroducedtoremovetheassumptionof zeroinitialspacingerrors.Thenonlinearvehicledynamicsisapproximatedbyaradialbasisfunctionneuralnetwork(RBFNN).Theadaptivefault-tolerantcontrolmethodisdevelopedinthecontextofthePID-typesliding-modecontroltechniqueandprovedtobecapableofguaranteeing individualvehiclestability,stringstability,andtrafficflowstability.
Chapter7 revisitsthefault-tolerantcontrolproblemforconnectedvehiclesconsideringactuator faults,inputquantization,anddead-zonenonlinearity.Theoccurrenceofactuatorfaultsmaycause abruptvelocityandaccelerationchange,whichmayyieldaviolationofthespacingpolicy.So,an improvedquadraticspacingpolicyreflectingtheeffectofactuatorfaultisproposed,whichremoves theconditionofzeroinitialspacingerrors.Then,anadaptivefault-tolerantcontrolschemeisdevelopedbyemployingRBFNNandPID-typesliding-modecontrolmethod.
Chapter8 investigatesprescribedperformanceconcurrentcontrol(PPCC)ofconnectedvehicles withunknownparameters,disturbances,andactuatorsaturation.Acloserspacingpolicyisintroducedtoachievestringstabilitywithavirtualleader-bidirectionalinformationflow.Basedona newtransformedtrackingerrorfunctionandanauxiliarysystemintroducedtodealwithactuator saturation,adistributedadaptivetrackingPPCCcontrollerisdesignedtoachieveindividualvehicle stabilityandstringstabilityinthesensethatallthesignalsinthesystemareuniformlyultimately bounded.
Chapter9 revisitstheprescribedperformanceplatooncontrolprobleminthepresenceof actuatorsaturation,uncertainparameters,andunknowndisturbances.Twoadaptivesliding-mode controlschemesbasedonleader-predecessorandleader-bidirectionalinformationflows,respectively,arepresentedtoensurestringstabilityandstrongstringstabilitywithprescribedtracking performance.Theactuatorsaturationnonlinearityisapproximatedwithasmoothhyperbolic tangentfunction.Theeffectsofuncertainparametersandexogenousdisturbancesaredealtwith byintroducingasetofadaptationlaws.
Chapter10 studiesthespeedoptimizationandtrackingcontrolproblemforheavy-dutytruck platoons.Thespeedplanningalgorithmisderivedwithregardtoanaveragevehiclebasedona combinedfuel-timecostandrecedingdynamicprogramming.Theideaofusinganaveragevehicle insteadoftheleaderforspeedplanningmakesthespeedprofilemorefuel-efficientforplatooningof vehiclesdifferentinweightandsize.Thevehiclecontroller,adiscrete-timeback-steppingcontrol law,isdesignedonthebasisofanonlinearvehiclemodelconsideringroadslopeandheterogeneity ofvehicles.Thecontrolalgorithmisstrengthenedbyanovelstringstabilitycriterion.
Chapter11 isconcernedwithspeedoptimizationandtrackingcontrolproblemsforplatooning ofconnectedvehicles.Atwo-layeredcontrolarchitectureispresented:aset-pointoptimization layerandavehicletrackingcontrollayer.Inthefirstlayer,aspeed-planningalgorithmisderived tocalculatethespeedset-pointfortheconnectedvehiclesbyaveragingtheoptimalspeedofeach vehicle,whichisobtainedbysolvingafuel-timeoptimizationproblembasedonPontryagin’sminimumprinciple.Thesecondlayercontainsasetofdistributedsliding-modecontrollersforvehicle trackingcontrol,whichcanguaranteestringstabilityoftheconnectedvehicleswiththedesired inter-vehiclespacing.
Chapter12 investigatesdistributedtrajectoryoptimizationandadaptiveplatooncontrolofconnectedvehicles.Adistributedhierarchicalframeworkisproposedfortrajectoryoptimizationand trackingcontrol.Theupperlayerprovidesanoptimaltrajectoryforeachvehicle,whichisrealized byminimizingtheinter-vehiclespacingwithregardtothedesiredvaluesusingconvexoptimization.Thelowerlayercontainsanadaptivesliding-modecontrollertotracktheoptimaltrajectory. Tocompensateforuncertainvehicledynamics,aparameteradaptationlawbasedonthetracking errordynamicsisinvolvedinthecontroller,whichguaranteesbothindividualvehiclestabilityand stringstability.
Chapter13 studiesdistributedtrajectoryoptimizationandplatooncontrolproblemsforconnectedvehicleswithaquadraticspacingpolicy.Thequadraticspacingpolicybasedontheexpected teamspeedisintroducedtoimprovetheflexibilityofspeedplanningandregulation.Thetrajectory optimizationproblemissolvedusingdistributedconvexoptimizationbasedonspacingerrorminimization,resultinginanalgorithmtoprovidetheoptimaltrajectoryforallfollowingvehicles. Then,aPID-typesliding-modecontrollerwithadoublehigh-powerreachinglawispresented forspeed-trackingcontrolofeachfollower.Themethodologycanguaranteeindividualvehiclestability,stringstability,andtrafficflowstabilitywithignorableturbulenceofspacingandspeed.
Chapter14 extendsthedistributedtrajectoryoptimizationandplatooningmethodtoachieve fixed-timetracking.Theoptimaltrajectoryforallfollowingvehiclesisobtainedbyminimizing
thespacingerrorsviadistributedconvexoptimization.Thetrajectorytrackingcontrollerisderived basedonthetrackingerrordynamicsinthecontextoffixed-timestabilityandterminalslidingmodecontrol.Themethodcanrobustlyguaranteezerosteady-statespacingerrorsandindividual vehiclestabilityandstringstabilitysimultaneously.
Inthisbook,specialattentionisgiventoaclearpresentationoftheformulations,algorithms,and theirimplementationinnumericalsimulationsandexperimentalstudies.Relatedworksandreferencesaregivenattheendofeachchaptertoguidethereaderstowardfurtherknowledgeinthis area.Thebookcanbeusedincoursesforgraduatestudentsinmodelingandcontrolofconnected automatedvehiclesandtransportationsystems.Researchersandengineerscanalsodrawuponthe bookindevelopingmathematicalmodelsandalgorithmsfortheoreticalstudyandapplication purposes.
GeGuo,NortheasternUniversity,Shenyang,China ShixiWen,DalianUniversity,Dalian,China
2March2023
Acknowledgments
OurresearchworkhasbeenfundedbytheNationalNaturalScienceFoundationofChinaunder Grants60974013,61273107,61573077,U1808205,61803062,and62173079,and,inpart,bytheNaturalScienceFoundationofLiaoningProvinceunderGrant2022-MS-406.Wearegratefultothe fundersfortheirsupportinthepastdecade.TheauthorsacknowledgethecontributionsofDandan Li,QiongWang,PingLi,HongboLei,ZiweiZhao,DongqiYang,JianKang,Ren-Yong-Kang Zhang,andmanyothersfortheirhardworkandcommitmentthatmadethismonographcome tofruition.Withouttheireffortandtimeindevelopingqualitycontributionsinoneortwoof thechapters,thisbookwouldnothavebeenpossible.
VehicularPlatoonCommunicationandControl
Communicationissuesduetobandwidthshortageareinevitableinvehicularadhocnetworks (VANET),whichmayinducetimedelay,packetdropouts,andmediumaccessconstraintsthat canleadtoperformancedegradationoreveninstabilityinconnectedvehicularcontrolsystems. Moreover,inter-vehiclecommunicationandcontrolaretwoaspectsthatarestronglycoupled.This partconsistsoffivechapters. Chapter1 studiestheplatooncontrolproblemsubjecttovarying communicationrange.Basedontheconnectivitystatusbetweentheleaderandothervehicles, theplatooncontrolsystemismodeledasaswitchingsystem.Thenasetofsufficientconditions forstabilityanalysisandcontrolsynthesisaregiven,yieldingaplatooncontrolalgorithmthatguaranteesstringstabilityandzerosteady-statespacingerrors. Chapter2 revisitstheproblemsubjectto communicationinterruptionandlatency.Basedonahybridreferencemodelofcooperativeadaptivecruisecontrol(CACC)andadaptivecruisecontrol(ACC),aCACC-ACCswitchingcontrol methodissuggested.Toensurestringstabilityandzerosteady-statespacingerrorswithafinite time,afinite-timesliding-modecontrolalgorithmisderived. Chapter3 studiestheco-designproblemofplatooncontrollerandinter-vehiclecommunicationtopology.Aunifiedco-designframeworkispresentedtojointlydeterminetheoptimalcommunicationtopologyandthesampleddatacontrollergainsinassociationwiththetopology.Theco-designalgorithmisderivedbased onthestabilityrequirementandminimizationofthecostfunctionviadynamicprogramming. Chapter4 studiestheplatooncontrolproblemwithaswitchingcommunicationtopologyand transmissiondelays.TheswitchingcommunicationtopologyisdescribedasaMarkovchain, andtheconnectedvehicularcontrolsystemismodeledasaMarkovianjumpingsystemwithdelay. Theresultingcontrolmethodcanguaranteeexponentialconvergenceofthetrackingerrorsinthe mean-squaresense. Chapter5 revisitstheco-designproblemoftheCACCcontrollerandeventtriggeredcommunicationmechanism.Sufficientconditionsforthesampled-datacontrollerandthe event-triggeredcommunicationmechanismareobtainedtoguaranteesystemstability.
ControlwithVaryingCommunicationRange
1.1Introduction
Connectedvehiclescanshareinformation(position,speed,etc.)withneighboringvehiclesandthe infrastructureviavehicularadhocnetworks(VANETs),whichmakesplatooningorcooperative adaptivecruisecontrol(CACC)ofvehiclesanimportanttechnologytoalleviatetrafficcongestion andreducetrafficaccidents[1].Inaplatooningsystem,vehiclescantravelatasmallintervehicle spacingandhencelowerairdragtoreducefuelconsumptionandimprovethethroughputofroad traffic.Therefore,vehicularplatooninghasattractedmuchattentioninrecentyears[2,3].Asan extensionoftheACCtechnologyonboardmostcars,CACCisapromisingtechniqueinintelligent transportationsystems[4,5]enablingcooperativecontrolandplatooningofvehiclesviaintervehiclecommunicationsinadditiontoonboardsensors.WhenacarisfollowinganotherintheACC mode,amplificationsofacceleration/decelerationareinevitableduetotheunavailabilityofinformationfromtheprecedingvehicles.Thelongertheplatoon,themoresignificanttheshockwave effectintheplatoonofvehicles[6],whichisknownasstringinstability.CACCreliesoncommunicationnetworkstoexchangeinformationbetweenvehicles,allowingvehiclestoadapttothose aheadand,hence,caneffectivelymitigatetheshockwaveeffect.Inaddition,thetimeheadwayin CACCissmallerthanACC,leadingtoimprovedroadcapacityandtrafficthroughput[7],and reducedaerodynamicdragandfuelconsumption[8].
Generally,typicalspacingpoliciesusedforconnectedvehiclesaretheconstant-spacing(CS)policyandtheconstant-time-headway-spacing(CTHS)policy,dependingonwhethertherequired spacingofthevehicleisdependentonitsspeed[9,10].CSpolicyhastheadvantagetoimprove roadcapacity,whileCTHSpolicycanhelpimproveroadsafety.Fromtheviewpointofstringstability,connectedvehicleswithCSpolicyarehardertobeguaranteed.
Withtherapiddevelopmentofintervehiclecommunicationtechnologies,thevarioustypesof interactiontopologies(seeFigure1.1)[11]areavailableforplatoonsnow.Nevertheless,stringstabilityisnotalwayssatisfiedunderinteractiontopologieswhenaCSpolicyisinvolved.Forexample, [12]provedthatstringstabilitycannotbesatisfiedforhomogeneousplatoonswithpredecessorfollowing(PF)topologyandCSpolicyowningtoacomplementarysensitivityintegralconstraint.Reference[13]showedthatplatoonswithbidirectionaltopologyalsosufferedfundamentallimitations onstringstability.AstheleaderPF(LPF)topology,theleadercanperiodicallybroadcastitsinformationtoeachfollower,whichhasbeenproventoyieldbetterstringstabilityperformance[12,14] withaCSpolicy.
Todate,considerableresearchhascontributedtosolvingthestringstabilityproblemforLPFplatoonswithCSpolicy.Theimportanttechnicalissuesincludecommunicationconstraints,optimal
ConnectedVehicularSystems:Communication,Control,andOptimization,FirstEdition.GeGuoandShixiWen. ©2024JohnWiley&Sons,Inc.Published2024byJohnWiley&Sons,Inc.
Figure1.1 Communicationtopologyinvehicleplatooncontrol:(a)leaderpredecessorfollowing; (b)predecessorfollowing;(c)bidirectionaltopology.
control,nonlineardynamics,range-limitedsensors,uncertainvehiclemodels,anddisturbances.To nameafewstudies,[15]analyzedthestringstabilityinthepresenceofcertaintimedelaysinthe leaderstatereception.Reference[16]investigatedtheeffectofcommunicationdelaysonstringstability.TheeffectsofcommunicationlimitationsonLPFplatooncontrolwerestudiedin[17,18]to includecommunicationdelays,packetdropouts,andquantization.Reference[19]proposeda decentralizedoverlappingcontrollawbyusingtheinclusionprinciple,whichcanpreservethe stringstabilityandsteady-statebehaviorfortheplatooncontroller.Reference[20]establisheddistributedrecedinghorizoncontrolalgorithmsforplatoonstodealwithnonlinearvehicledynamics. Inthesealgorithms,theaccelerationinformationoftheleadingcarisnotneededinthecontroller. ThealgebraicgraphtheoryandRouth-Hurwitzstabilitycriterionwereusedin[11]toanalyzethe stabilityandscalabilityofaplatoon.Reference[21]proposedaguaranteed-costcontrollerforvehicleplatoonstodealwithactuatordelays(e.g.fuelingandbrakingdelays)andtheeffectsofsensing rangelimitations.
Inthischapter,weareinterestedinstabilityanalysisandcontrollersynthesisforLPFplatoon withaconstantspacingpolicy,wheretheconnectivitystatusbetweentheleaderandeachfollowerinVANETsvarieswithtime.Undersuchcircumstances,none,part,orallofthefollowing vehiclescanreceivethestateinformationbroadc astbytheleaderatanytime.Consequently,the platoonismodeledasaswitchedcontrolsystem[22,23],whereacompactformoftheplatoon controllerswitchesaccordingtotheconnectiv itystatusofeachfollower.Usingtheaverage dwelltimetechniqueandpiecewiseLyapunov-Krasovskiifunctional,thecriteriaforstability analysisandcontrollersynthesisarepropose dforplatoons.Accordingtotheobtainedconditions,ausefulLPFplatooncontrolalgorithmisproposed.Thecontributionsofthischapter aresummarizedasfollows:
1)Anovel-switchedcontrolmodelisestablishedtodescribetheeffectsofvaryingcommunication rangesonLPFplatoonswithaCSpolicy.
2)Anefficientplatooncontrolstructureisproposedinwhichthecontrollergainoftheplatoonis dependentontheconnectivitystatusbetweentheleaderandeachfollower.
1.2ProblemFormulation
3)Aseriesofnewconditionsareobtainedasthecriteriaforstabilityanalysisandcontrollersynthesisfortheswitchedtime-delayplatooncontrolsystem.
4)Ausefulplatooncontrolalgorithmisproposedtodeterminetheconnectivity-status-matrixdependentfeedbackcontroller,whichsimultaneouslyguaranteestherequirementofstringstabilityandazerosteady-statespaceerrorforeachobtainedcontrollergain.
Theremainderofthischapterisorganizedasfollows.Section1.2analyzestheeffectsofvarying communicationrangesandestablishesaswitchedplatooncontrolsystemmodel.Section1.3gives themainresultonthestabilityandcontrollersynthesisforplatoonsinordertosatisfythestring stabilityrequirementforeachobtainedcontrollergain.Numericalsimulationsandexperiments withlaboratory-scaleArduinocarsaregiveninSection1.4.Theconcludingremarksandfuture researchtopicsaregiveninSection1.5.
Notation Throughoutthistechnicalnote, AT denotesthetransposeofamatrix A and I isthe identitymatrixwithappropriatedimensions. A ≥ 0(A >0)meansthat A isapositivesemidefinite (positive-definite)matrix,and A ≥ B (A > B)meansthat A B ≥ 0(A B >0). Rn and Rn × m denote the n-dimensionalrealEuclideanspaceandthesetofall n × m realmatrices,respectively. representstheEuclideannormforavector.diag{ }denotesablockdiagonalmatrix.
1.2ProblemFormulation
Consideraplatoonof N +1vehiclesinafreewayscenario.Weassumethatthisplatoonisrunning withinahorizontalstraightlineundertheconstantspacingpolicy(seeFigure1.2).Denoteby zr, vr, and ar the r-th(r =0,1, …, N)vehicle’sposition,velocityandacceleration,with r =0standingfor theleaderandtheothersasthefollowers.Eachfollowerisequippedwithanonboardsensorto measurethedistancebetweenitandtheprecedingvehicle.Theleaderperiodicallybroadcasts itsstateinformation(velocityandacceleration)toallfollowersviaVANETs.Itisassumedthat thecommunicationrangeoftheleaderislongenoughtocoverallfollowersinaplatoon.Toavoid theinstabilityoftheplatooncausedbysaturationonthevelocity,themaximumvelocityofeach followerisdefinedas vmax suchthat v0 ≤ vmax.Moreover,itisassumedthattheleadermovesata constantspeed,i.e. z 0 t = v0 t and v0 t =0astime t ∞
Definethespacingerrorofthe r-thfollowingvehicleas
Platoonofvehicles.
Figure1.2
where δd isthedesiredintervehiclespacingand L isthelengthofthevehicle.Thedynamicsof vehicle r, r =1,2, …, N,aredescribedbythefollowingdifferentialequations:
where ς denotestheenginetimeconstant,and ur(t)isthecontrolinputforthe r-thvehicle. Inordertoensuresafebrakingduringdeployment,thedesiredintervehiclespacingisrequiredto nosmallerthanthebrakingdistance.AccordingtothePATHalgorithmproposedby[24],braking distance dbrk isadoptedbythefollowingkinematicapproach
where Δvr = vr vr 1 istherelativevelocitybetweenthesuccessivevehicles. κ 1 isthemaximum decelerationofvehicle i and κ 2 isthemaximumdecelerationofitsprecedingvehicle i 1. τ isthe delaytimeofthebrakingsystemand d0 istheminimumdistance. Considertheexistingactuatortimedelay τ (e.g.thefuelingdelayandbrakingdelay)whenavehicleexecutesthecontrolcommand.Thenamorerealisticdynamicmodelfor(1.4)isrepresentedas
Define xt =Col x r t N r =1 and ut =Col ur t N r =1 torespectivelydenotethestateandthecontrolvectors,where ‘Col’ representsthecolumnvector,and x r t =
r δr v0 vr a0 ar T . Accordingto(1.2),(1.3),and(1.6),thestatespaceequationoftheentirevehicleplatoonisrepresentedas
r
Inthischapter,thelong-termevolution(LTE)V2VnetworkisutilizedtosupportreliableV2V communication.Asspecifiedinthestandard,tensubframesof1mseachmakeanLTEframe.Normally,thesizeofthestateinformationoftheleadervariesbetween50bytesand500bytes.Ina commonLTEcelldeployment(i.e.bandwidthis10MHZ),onesubframeissufficientforavehicle
1.2ProblemFormulation
totransmititsstateinformationundersuitablemodulationandcodingscheme.Here,thesignal-tointerference-plus-noise(SINR)isusedtocharacterizethereliabilityofV2Vcommunication.Tobe specific,theSINRattheintendedreceiverside(the r-thfollower)inasubframeisrepresented as[25]
where pv2v isthetransmissionpowerfromtheleadertofollower i. h r isthechannelgain,which containspathloss,shadowingeffectfromtheleadertofollower i Ir istheinterferencefromcell user. H0 isthepowerofadditivenoiseineachsubframe.WhentheSINRisaboveatargetedSINR threshold γ V2 V,thedatatransmissionrateintheV2Vcommunicationisreliable.Duetohigh mobility,theconditionofthewirelesschannel(e.g. h r )variesdynamically.However,unreliable datatransmissionratesmaycausecommunicati onconstraintssuchascommunicationdelay, packetdropoutsandmediumaccessconstraints .Therefore,weassumethatthevelocityofthe platooncanguaranteethattheSINRoftheV2Vchannelsatisfies γ r > γ V2 V.Namely,thedata transmissionrateinV2Vcommunicationisreliablewhenfollower i iscommunicatingwith theleader.
ThevaryingcommunicationrangeinaVANETinfluencesontheconnectivitystatusbetweenthe leaderandeachfollower.Defineabinaryvariable γ r(t) {0,1}thatdenotestheconnectivitystatus betweenthe r-thfollowerandtheleader.Specifically,ifthedistancebetweenthe r-thfollowerand theleaderiswithinthecurrentcommunicationrange,then γ r(t)=1;otherwise γ r(t)=0.Thisisthe distancebetweenthe r-thfollowingvehicleandtheleadingonebeyondthecurrentcommunication range.Definematrix Mσ (t) =diag{Δ1(t), Δ2(t), , ΔN(t)}where Δr(t)=diag{1,1,1, γ r(t), γ r(t)}representsthefollowers’ connectivitystatuswiththeleader,and σ (t) {0,1,2, …N}denotestheswitching signal.Allpossiblevaluesoftheconnectivitystatusmatrix Mσ (t) aretakeninafiniteset{M0, M1, M2, MN},where Mσ , σ {0,1,2, N}isgivenas
M 0 =diag Π 12 , Π 22 , , Π N 2 ,
M 1 =diag Π 11 , Π 22 , , Π N 2 ,
M 2 =diag Π 11 , Π 21 , Π 32 , …, Π N 2 ,
M 3 =diag Π 11 , Π 21 , Π 31 , Π 42 , …, Π N 2 ,
M N =diag Π 11 , Π 21 , , Π N 1 where Π r1 =diag{1,1,1,1,1}and Π r2 =diag{1,1,1,0,0}.Anillustrativeexampleoftheconnectivitystatusmatrix Mσ (t) isshowninFigure1.3. Foreachfollower,the r-thfollowercanusethereceivedvalues v0 and a0 fromtheleadertocomputethevelocityerror v0r = v0 vr andaccelerationerror a0r = a0 ar when γ r(t)=1;otherwise,the velocityerrorandaccelerationerroraresetto v0r =0and a0r =0bythezerostrategy.Asaresult,the controllerofthe r-thfollowerhasthefollowingswitchedform
ur t =
1ControlwithVaryingCommunicationRange
Figure1.3 Connectivitystatusmatrixfor(a)
where k p σ , k v σ , k a σ , k ve σ ,and k ae σ aretheconnectivitystatusmatrix Mσ dependentcontrollergainstobe designed.Accordingto(1.9),thecompactformofafeedbackcontroller(asshowninFigure1.4)for aplatooncanberepresentedas
where
istheconnectivity-status-matrixdependentfeedbackcontrollergainwith K r σ =diag k p
, k ae σ .Let ασ (t)denotethetotal timewhentheconnectivitystatusmatrixofthefollowingvehiclesisequalto Mσ inthetimeinterval [0, t).Then ασ = ασ t N σ =0 ασ t iscalledthemoderate. Takingthefeedbackcontroller(1.10)into(1.7)andassumingtheinitialcondition x(t)= ϕ(t)= ϕ(t
for t [ τ ,0),theplatoonstatespaceequationcanberewrittenas
Figure1.4 Controllerstructureofvehicleplatoon.
1.3SwitchingControlofConnectedVehicles 9
Thecontrolobjectiveoftheplatooncontrolsystemistodesigntheconnectivity-status-dependent feedbackcontrollerof(1.10)foraplatoontoensurethateachfollowermovesatthesamespeed withtheleaderwhilemaintainingaconstantdesiredintervehiclespacing.Namely,theplatoon controlsystem(1.11)isneededtomeetthefollowingcriteriawithvaryingcommunication rangesandactuatordelays:
1)Individualvehiclestability:Theentireclosed-loopplatooncontrolsystem(1.11)ismean-square exponentialstable.
2)Steady-stateperformance:Thespacingerror δr(t)approachestozeroforallvehicles.
3)Stringstability:Thetransienterrorsarenotamplifiedwiththevehicleindexunderanymaneuveroftheleadingvehicleforeachobtainedfeedbackcontrollergain, Gr(s) ≤ 1,where Gr(s)= δr(s)/δr 1(s)with δr(s)istheLaplacetransformofthespacingerror δr(t).
Toproceed,weneedthefollowingdefinitionandlemma.
Definition1.1[26] System(1.11)issaidtobeexponentiallystabilizedifthereexistpositive scalar c and ρ <0suchthatforanyinitialcondition ϕ(t0) Rn,thesolution x(t)satisfies
xt ≤ ceρ t t 0 ϕ t 0 , t ≥ t
1 12 where ρ issaidtobethedecayrate.
Definition1.2[27] Foranyswitchingsignal σ (t)and t ≥ t0,let N(k)denotethenumberofswitchingtimesof σ (t)overthetimeinterval[t0, t).If N(t) ≤ N0 +(t t0)/Ta holdsfor N0 >0and Ta >0, then Ta iscalledtheaveragedwelltimeand N0 thechatterbound.Forsimplicity,butwithoutlossof generality,wechoose N0 =0.
Lemma1.1[28,29] Forascalar λ (0,1),definethepositive-definitematrix R Rn × n andtwo realvectors W1 Rn and W2 Rn.Definethefollowingfunction f(λ)asareciprocallyconvexcombinationon f λ = 1 λ W T 1 RW 1 + 1 1 λ W T 2 RW 2
Ifthereexistsamatrix S Rn × n suchthat RS SR ≥ 0,thenthefollowinginequalityholds: f λ ≥ W 1 W 2 T RS SR W 1 W 2
Proof: Accordingtothelower-boundstheoremandreciprocallyconvexinequalitygivenintheorem1of[28],itisstraightforwardtoobtainLemma1.1.
1.3SwitchingControlofConnectedVehicles
Inthissection,wefirstgiveastabilityanalysisandcontrollerdesigncriteriafortheswitchedtimedelayvehicleplatooncontrolsystem(1.11).Then,weaddtheconstraintsfortheobtainedcontroller toguaranteethestringstabilityrequirementandthezerosteady-statespacingerrorforeach obtainedconnectivity-status-matrix-dependentcontrollergain.
Fortheswitchedsystem(1.11),introduceapiecewisequadraticLyapunov-Krasovskiifunctional inthefollowingform
with λσ >0, Pσ >0, Qσ >0,and Rσ >0.ThepropertyofthedefinedLyapunov-Krasovskiifunctional Vσ (t)isshowninthefollowinglemma.Forsimplicityofpresentation,let ℓr, r =1,2,3,4beablock entrymatrixwith ℓ rg = ℓ r ℓ g.Tomakethisclear,wegiveanexample, ℓ 1 =[In,0,0,0]T and ℓ 12 =[In, In,0,0]T .
Lemma1.2 Givenscalars λσ >0,ifthereexistrealmatrices Pσ >0, Qσ >0, Rσ >0,andrealmatrix Sσ ofappropriatedimensionssuchthatthefollowinginequalitieshold
and Gσ = Aℓ 1 + B
M
(t)definedin(1.13)has thefollowingproperties
Proof: By V
Define
Notethat
ApplyingLemma1.1to ξσ (t),onecanobtain
1.3SwitchingControlofConnectedVehicles
Itisstraightforwardtoknowthat xt = Gσ ϕ t .Combining(1.15)–(1.17),onecanobtain
If ψ σ + τ 2 GT σ Ri Gσ <0,then V σ t + λσ V σ t ≤ 0canbeguaranteed.UsingtheSchurcomplementwithmatrixinequality Ωσ canyieldtheinequality V σ t + λσ V σ t ≤ 0.Hence,theproof ofthelemmaiscompleted.
Wearenowinapositiontogivethemainresultsofthischapter.
Theorem1.1 Consideraleader-followingvehicularplatooncontrolarchitecturewithavarying communicationrange.Forthescalars λσ >0, κ >0,and μ ≥ 1,iftheaveragedwelltimeofthe platoonsystems(1.11)satisfies
19 andifthereexistrealmatrices P σ >0, Qσ >0,and Rσ >0,andrealmatrix K σ ,and Sσ isofappropriatedimensionssuchthatthefollowinginequalitieshold
theneachfollowingvehiclecanbeexponentiallystabilizedwiththedecayrate ρ/2,where ρ =1 T a ln μ N
,andthestatedecayestimationisgivenby xt < b a
1 22 where
and
Moreover,theconnectivitystatus-dependentplatooncontrollercanbesolvedby
Proof: First,wewillprovethatthecontrollergainobtainedin(1.23)canguaranteeproperty(1.15)fortheLyapunov-Krasovskiifunctional Vσ (t)definedin(1.13)ifcondition(1.20)of Theorem1.1issatisfied.Let P σ = P σ ininequality(1.14)ofLemma1.1.Definevariables
K
σ = P σ K σ and S =diag I n , R 1 σ P σ .Bypre-andpost-multiplyingbothsidesof(1.14)by S,one canobtain(1.20),wheretheterm P σ R 1 σ P σ dealswithinequality P
P σ .Since theconditionsinLemma1.1aresatisfied,theinequality(1.15)holds.
Next,weprovethattheswitchedplatooncontrolsystem(1.11)isexponentiallystable.Let0< t1 < t2, ,< tk < t denotethetimeswitchinginstantoftheswitchingsignal σ (t).Accordingto(1.15), wecanobtain
Combinedwithconditions(1.21)ofTheorem1.1andtheLyapunov-Krasovskiifunctional Vσ (t)(t) definedin(1.13),onecanobtain V i t ≤ μV j t , i, j 0,1,2,
forany xi(t).
Accordingto(1.25),wehave
where t k denotesthetimeinstantisimmediatelybeforethetimeinstant tk. Itiseasytoknowthat
Applying(1.26)recursively,oneobtains
where ρ =1 T
.Thecondition(1.19)ofTheorem1.1guaranteesthat ρ ≤ 0. IfaquadraticpiecewiseLyapunov-Krasovskiifunctional Vσ (t)(t)isconsidered,thenthereexist a >0and b >0suchthat
where a x(t) 2 ≤ Vσ (
Byinequality(1.28),wefurtherobtain
xt <
whichmeansthateachvehiclecanbemean-squareexponentiallystabilizedwithadecayrate of ρ/2.
Thecontrollerobtainedintheprevioussectioncanensuretheintervehiclestabilityofeachfollowingvehicle.However,theintervehiclestabilitydoesnotguaranteethestringstabilityforthe dynamicallycoupledplatooncontrolsystem.Therefore,additionalconditionsareneededto
1.3SwitchingControlofConnectedVehicles
complementtheobtainedcontrollerinordertosatisfythestringstabilityandzerosteady-stateerror requirements.Sincestringstabilityishighlydependentontheinteractiontopologyofplatoon,how toanalyzestringstabilityforaswitchedplatooncontrolsystemstillremainsanopenandchallengingproblem[12,30].Consequently,thediscussionsonstringstabilityandtheproblemofzero steadyspacingerrorhavemainlyfocusedoneachobtainedconnectivity-status-matrix-dependent controllergain.
Accordingto(1.3),wehave
r t = ar 1 t ar t 1 30
TakingLaplacetransformsto(1.30),thetransformfunction δr(s)/δr 1(s)canbeobtained.Forthe varyingcommunicationrange,weconsiderthefollowingcaseforobtainingtransformfunction δr(s)/δr 1(s).
Case1.1
Bothfollower i anditsprecedingvehicle(follower i 1)arewithinthecommunicationrangeof theleader.Then,wehave
Case1.2
Neitherfollower i noritsprecedingvehicle(follower i 1)arewithinthecommunicationrange oftheleader.Undersuchcircumstance,
=0and
=0,thenonecanderive
Case1.3
Follower i 1arewithinthecommunicationrangeoftheleader,butfollower i isout,thenone canobtain
j t isusedin(1.31c).
Inthenext,wewanttoillustratethat(1.31c)canbeseenasaspecialcaseof(1.31b).Consideringtheterm r 1 j =1 sδj s inthedenominatoroftransformfunction G3(s),onecanobtainthat
1 j =1
G1 s 1 , where(δj(s))/( δj 1(s))= G1(s)isused.Accordingtothefinaltheorem,wecanderivethat lim s 0 sG 1 s δr 1 s =0,whichmeanslim s 0 r 1 j =1 sδj s =0.Similarly,itiseasytoknowthat lim s 0 r 1 j =1 s2 δj s =0.Therefore,transformfunction(1.31c)isequalto(1.31b).
Aspreviouslymentioned,wehavethefollowingsufficientconditionsaboutstringstability condition|δr(jw)/δr 1(jw)| ≤ 1.
Theorem1.2 Forthespacingerrortransformfunction(1.31),iftheconnectivity-statusdependentfeedbackcontrollersatisfiesthefollowingconditions:
a k p σ >0
b 1+ k a σ + d
c k a σ + d∗ k ae σ <0 d
thencondition|δr(jw)/δr 1(jw)| ≤ 1alwaysholdstrueforeachobtainedfeedbackcontrollergain, where d∗ =1if γ i(t)=1;otherwise d∗ =0.
Proof: |δr(jw)/δr 1(jw)| ≤ 1isrewrittenas:
Itcanbeclearlyseenthat a >0.Inordertoguarantee|G(jw)|holdstrue,weneedtoprovethatthe conditioninTheorem1.2canmake b ≥ 0holdtrue.Accordingtocondition(a)of(1.32)andusing cos(wτ ) ≤ 1andsin(wτ ) ≤ wτ ≤ 1,for w >0,wehave
