INTERNATIONALJOURNALOFROBUSTANDNONLINEARCONTROL Int Robust Nonlinear Control    DOI
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INTERNATIONALJOURNALOFROBUSTANDNONLINEARCONTROL Int Robust Nonlinear Control DOI

1002rnc592 Zenohybridsystems JunZhang arl1enrikJohansson JohnLygeros andShankarSastry Department of Electrical Engineering and Computer Sciences Uni ersity of California Berkeley CA 947201770 USA Department of Signals Sensors Systems Royal Institute

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INTERNATIONALJOURNALOFROBUSTANDNONLINEARCONTROL Int Robust Nonlinear Control DOI




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INTERNATIONALJOURNALOFROBUSTANDNONLINEARCONTROL Int Robust Nonlinear Control 2001; 11 :435 451(DOI:10.1002/rnc.592) Zenohybridsystems JunZhang ./arl1enrikJohansson .JohnLygeros andShankarSastry Department of Electrical Engineering and Computer Sciences, Uni ersity of California, Berkeley, CA 94720-1770, U.S.A. Department of Signals, Sensors & Systems Royal Institute of echnology 100 44 Stockholm, Sweden Department of Engineering, Uni ersity of Cambridge, Cambridge CB2 1PZ, U.K. SU33AR4 TheinteractingcontinuousanddiscretedynamicsinhybridsystemsmayleadtoZenoexecutions.whichare

solutionso7thesystemhavingin nitelymanydiscretetransitionsin nitetime.Althoughphysicalsystems donotshowZenobehaviour.modelso7realsystemsmaybeZenoduetomodellingabstraction.Itishardto analyse such models with the existing theory. Since abstraction is an important tool in the hierarchical design o7 hybrid systems. one would like to determine when it may lead to Zeno models. Zeno hybrid systems are studied in detail in the paper. Necessary and su cient conditions 7or the existence o7 Zeno executionsaregiven.The Zenosetisintroducedasthe limitseto7 aZenoexecution.:ropertieso7the

Zenosetarederived7ora7airlylargeclasso7hybridsystems.Copyright2001 John;iley&Sons.Ltd. /E4;ORDS : hybridautomata;zenoexecution;zenosets 1. INTRODUCTION 1ybrid systems have proved to be an e ective tool 7or the modelling. analysis and design o7 alargenumbero7evolvingtechnologicalsystems.inwhichdigitaldevicesinteractwithanana- log environment. Systems o7 this type are common in embedded computation. robotics. mechatronics. avionics. and process control. Owing to the rapid advances in computer tech- nology.hybridsystemsarebecomingincreasinglyrelevantandimportantandconsequentlyhave

attractedconsiderableresearchinterest.1owever.despiterecentprogress.thereareanumbero7 7undamental properties o7 hybrid systems that have not been investigated to su cient detail. Theseincludeexistenceanduniquenesso7executions.whichhaveonlyrecentlybeenaddressed [1 4A.Anothersuchissue isZenoexecutions. Correspondenceto:JunZhang.Departmento7ElectricalEngineeringandComputerSciences.Universityo7Cali7ornia. Berkeley.CA 94B20-1BB0.U.S.A. E-mail:zhangCun eecs.berkeley.edu Contract/grantsponsor:ARO/3URI;contract/grantnumber:DAA104-9D-1-0341

Contract/grantsponsor:ONR;contract/grantnumber:N00014-9B-1-094D Contract/grantsponsor:DAR:A;contract/grantnumber:F33D15-9E-C-3D14 Contract/grantsponsor:SwedishFoundation7orInternationalCooperationinResearchand1igherEducation Contract/grantsponsor:Tele7onaktiebolagetL3Ericsson sFoundation Copyright 2001John;iley&Sons.Ltd.
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Roughlyspeaking.anexecutiono7 a hybrid systemis called Zeno .i7 ittakes in nitelymany discretetransitionsina nitetimeinterval.:hysicalsystemsare.o7course.notZeno.butahybrid modelo7aphysicalsystemmaybeZeno.duetomodellingover-abstraction.Sinceabstractionis an

important tool 7or handling complex systems. understanding when it leads to Zeno hybrid systemsisessential.7orexample7orthedevelopmento7simulationtools7orhybridsystems.Zeno hybrid systems. or systems && close '' to Zeno. make computer simulations imprecise and time- consuming.3ostsimulationpackagesdeveloped7orhybridsystems.suchasDymola[5A.Omola [DA. and S1IFT [BA. get stuck when a large number o7 discrete transitions take place within ashorttimeinterval.Itisthere7oreimportanttounderstandtheZenophenomenoninorderto develope cientcomputationaltools 7orhybridsystems. Itisdi

culttodrawconclusionsaboutZenosystemsusingtheavailabletheory.Zenohybrid automatahavebeenstudiedtosomeextentinthetheoreticalcomputerscienceliterature[E 12A. Thecontinuousdynamicsinthosecases.however.arequitelimited.Zenohybridautomatawith moregeneralnon-linearvector eldshaveonlyrecentlybeeninvestigated[13 1DA.Thelacko7 theoretical resultshas o7ten lead researchersto impose non-Zeno assumptions by de7ault. For example.thisisthecaseinrecentworkonhybridcontroldesign[1B 20A.Theworkpresentedin thispaperisa rststeptowardsbuildingasuiteo7resultstocharacterizeZenohybridsystems.

Ourresultsareuse7ul.7orinstance.whendesigninghybridcontrollers.Sinceauni edtheory7or hybrid control design does not yet exist. one has to prove that the closed-loop system is well posedonacase-by-casebasis;thisincludesprovingthatthesystemisnon-Zeno(see7orexample Re7erence[21A). The main contribution o7 the paper is to present some 7undamental properties o7 Zeno executionsandZenohybridautomata.;eintroducetheZenosetasthe limitseto7a Zeno execution.Acompletecharacterizationo7theZenosetisgiven7ora7ewquitegeneralclasseso7

hybridsystems.The7eatureso7theresetsturnouttobeveryimportant.Forexample.weshow thati7theresetsareallidentitymapsorallcontractingontheguard.thecontinuousparto7the stateconverges.;ealsoinvestigatetheconditionsunderwhichtherearenoZenoexecution.In particular.itisprovedthat7orhybridautomatawithidentityresetsontheguard.i7theguards andtheinterioro7thedomainsaredisCointandi7theboundarieso7thedomains7oranycyclesare alsodisCoint.thenthehybridautomatadonotacceptZenoexecutions. Theoutlineo7thepaperisas7ollows.InSection2weintroducethenotationandthede nitions

o7hybridautomataandexecution(Section2.1).7ollowedbyanumbero7exampleso7Zenohybrid automata 7rom the areas o7 modelling. simulation. veri cation. and control (Section 2.2). The examplesareusedtomotivatetheanalysisintheresto7thepaper.Zenohybridautomataand ZenosetsareintroducedinSection3.andsomepropertieso7ZenosetsarediscussedinSection 3.2. Section 3.3 presents both necessary and su cient conditions 7or the existence o7 Zeno executions.AsummaryandconclusionsaregiveninSection4.Tomaintainthe owo7thepaper. alltechnicalproo7saregivenin theappendix. 2. BAC/FROUNDAND3OTIGATION 2.1. Hybrid automata

executions and underlying assumptions Fora nitecollection o7variables.let denotetheseto7valuationso7thesevariables.;euse lowercaseletterstodenotebothvariablesandtheirvaluations.;ere7ertovariableswhoseseto7 43D J.Z1ANF Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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Thedomainissometimescalledtheinvariantset inthehybridsystemliteratureincomputerscience. Thiscan bedonewithoutloss o7generality.sincei7 ahybrid automatonviolatestheseconditions.one can construct anotherhybridautomatonthat acceptsexactlythesameseto7executionsandsatis

estheconditions[2A. valuationsis niteorcountableas discrete andtovariableswhoseseto7valuationsisasubseto7 aEuclideanspaceas continuous .Foraseto7continuousvariables with 7or 0.we assumethat isgiventheEuclideanmetrictopology.anduse todenotetheEuclideannorm. Foraseto7discretevariables .weassumethat isgiventhediscretetopology(everysubsetis anopenset).generatedbythemetric 0i7 and 1i7 .;edenote thevaluationso7theuniono7 and by .whichisgiventheproducttopologygenerated bythemetric (( ).( )) .;eassumethatasubset o7atopological space is given the induced topology. and we use to denote its closure. its

interior. its boundary. its complement. its cardinality. and ) the set o7 all subsetso7 The7ollowingde nitionsarebasedonRe7erences[2.13.22A. De nition 1 Hybridautomaton Ahybridautomaton H is a collection H .Init. ).where isa nitecollectiono7discretevariables; isa nitecollectiono7 continuousvariableswith Init isaseto7initialstates; P isa vector eld; )ismapassigningtoeach asubseto7 calledthedomain o7 isa seto7 edges; X) is amapassigningtoeachedge a subseto7 calledtheguardo7 ;and X) isa resetmap.assigningtoeachedge andeach a subseto7 ;ere7erto( asthe state o7 .Throughoutthepaper.itisassumedthat

and that is Lipschitz continuous in its second argument. Further. we assume that 7or all and7orall ). Ahybridautomatoncanberepresentedbyadirected graph( ).withvertices andedges .Foranexample.seeFigure1.Foreachvertex. .we speci7y a vector eld. ) and a domain. ). For each edge we speci7y a guard. ). and aresetmap. )(whichissuppressedi7 ).Thediscreteparto7 theinitialstateis indicated by a double circle and the continuous part by an arrow. Since there is a unique graphical representation 7or each hybrid automaton. we will use the corresponding graph as a7ormalde nitionintheexamples. De nition

Hybridtime trajectory AhybridtimetraCectoryisa niteorin nitesequenceo7intervals .suchthat A7orall0 i7 theneither Aor ). 7orall and 7or all0 AhybridtimetraCectoryisasequenceo7intervalso7therealline.whoseendpointsoverlap.The interpretationisthattheendpointso7theintervalsarethetimesatwhichdiscretetransitionstake place.Notethat isallowed.there7oremultiplediscretetransitionsmaytakeplaceatthe sametime.Sinceallhybridautomatathatwillbediscussedaretimeinvariant.weassume.without 43B ZENO14BRIDS4STE3S Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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Figure1. ;atertanksystemandcorrespondingZenohybridautomation. loss o7 generality. that 0. 1ybrid time traCectories can extend to in nity i7 is an in nite sequenceori7itisa nitesequenceendingwithanintervalo7the7orm[ ). Fora hybridtime traCectory . let denotetheset 0.1. i7 is niteand 0.1. i7 is in nite. ;e use and to denote the time evolution o7 the discrete and continuousstate.respectively(withaslightabuseo7notation).1ere isamap7rom to and isacollectiono7 maps.Anexecutionisnowde nedasatriple )in the7ollowingway. De nition 3 Execution An execution o7 a hybrid automaton is a collection ). where

is a hybrid time traCectory. isamap.and isacollectiono7 maps .such that (0). (0)) Init. 7orall and7orall ). ))and7or all ). )). 7orall ). 1)) ).and )). ;e say a hybrid automaton accepts an execution . For an execution ). we use (0). (0)) to denote the initial state o7 . The execution time )isde ned as lim . where 1 is the number o7 intervals in the hybrid time traCectory.Anexecutioniscalled nite i7 isa nitesequenceendingwithacompactinterval.itis called in nite i7 iseitheranin nitesequenceori7 .anditiscalled Zeno i7itisin nite but .Fora Zenoexecution .wecall )the Zeno time .;euse )to

denotetheseto7allin niteexecutionso7 with initialcondition( Init.Allthehybrid automata considered in this paper are assumed to be non-blocking . in the sense that 7orall( Init.Conditions7ordeterminingwhenthisisthecasearegivenin Re7erence[2A. Astate( iscalled reachable by .i7thereexistsa niteexecution )with and ( ). )) ). ;e use Reach to denote the set o7 states reachable by a hybrid automaton . Throughout this paper. we assume that Reach Conditionsunderwhichthisisthecasecanbeestablishedusinginvariantassertions.provedby inductionargumentsoverthelengtho7theexecutions[2A. 43E J.Z1ANF Copyright

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2.2. ,oti ating examples ItshouldbenotedthattheZenophenomenonisastrictlyhybridphenomenoninthesensethat bothcontinuousdynamicsanddiscretedynamicsareneededtogenerateZenoexecutions.Inthis section.weillustratehowZenohybridautomataappearindi erentareaso7hybridsystems.In particular.wegiveexampleso7Zenohybridautomatainmodelling.simulation.veri cation.and control. 2.2.1. ,odelling and simulation. 1ybridautomataprovideanatural7ramework7ordeveloping

modelswithabstracteddynamics.Thisisause7ulapproachwhenanalysingcomplexsystems.and 7or control design it o7ten leads to an appealing hierarchical system structure. 1owever. i7 the abstraction is not done care7ully. erroneous conclusions may be derived 7rom the model. as illustratedbythe7ollowingexample. Example -ater tank system Considerthewatertanksystemo7Alurand1enzinger[9A.showninFigure1.1ere denotes thevolumeo7waterinTank .and 0denotestheconstant owo7waterouto7Tank .Let denotetheconstant owo7waterintothesystem.directedexclusivelytoeitherTank1orTank

2ateachpointintime.TheobCectiveistokeepthewatervolumesabove and .respectively (assumingthat (0) and (0) ).Thisistobeachievedbyaswitchedcontrolstrategy that switches the in ow to Tank 1 whenever and to Tank 2 whenever .Itis straight7orward to show that the unique in nite execution that the hybrid automaton accepts 7or each initial state is Zeno. i7 max . The Zeno time is (0) (0) )/( ).O7course.arealimplementationo7thewatertank systemcannotbeZeno.butinsteadoneorbotho7thetankswilldrain.TheZenomodeldoesnot capturethis.Theactualscenariodependsonthedynamicso7theswitch.whichinthemodelwas assumedto

beinstantaneous.For7urtherdiscussionsonthisexample.seeRe7erence[13A. It is di cult to run e cient computer simulations 7or systems that show a large number o7 discrete transitions during a short time interval. O7ten. either the numerical error or the simulationtime(orboth)willbeunsatis7actory[13.14A.Oneclasso7systemswherethisproblem arisesismechanicalsystemswith7riction.I7the7rictionismodelledasCoulomb7riction.thenthe 7rictional7orce isgivenby sgn .where istherelativevelocityo7thecontactsur7aces and 0 is a constant. ;e may easily model a system with Coulomb 7riction as a hybrid automaton.

where the domains and the guards depend on the sign o7 the velocity. Frictional systemssometimeshavetheso-calledstick slipmotion.whichmeansthatthemotionsisdivided into two phases both o7 non-zero duration:one when 0 and one when 0 (7or a simple example see Re7erence [23A). For the hybrid automaton. the latter corresponds to a Zeno behaviour. because it implies that the velocity switches in nitely 7ast between a positive and anegativevalue.ResolvingZenonessbyintroducinganewdiscretestate.whichhasavector eld givenbythecontinuousdynamicscorrespondingtothestickingmotion.hasbeenproposed7or

simpleexamples[24.25A.butso7arnorigorousmethodseemstohavebeendeveloped.Inmany cases.suchamethodwouldspeedupthesimulationconsiderably.Othersimulationmethodsto avoid Zeno include time-stepping methods [2DA. It should be pointed out that modelling o7 7riction and impacts 7or rigid bodies is. o7 course. by itsel7 a very active eld. where advanced mathematicaltools are used to handlecontacts in a consistent way [2BA. The example here is meantasanillustrationo7themulti-domainmodellingapproachtakenin hybridsystems. 439 ZENO14BRIDS4STE3S Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control

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2.2.2. Analysis and eri cation. 3ost o7 the veri cation methods proposed 7or hybrid systems seek to determine whether the set o7 states reachable by a hybrid automaton satis es certain properties.For example. model checking techniquesinvolvecomputeralgorithmsthat explore the set o7 reachable states automatically. This approach. developed in theoretical computer science 7or purely discrete systems. has been extended to timed automata [2EA. multi-rate automata [29A. hybrid automata with constant di erential inclusions [30A and. most recently.

classeso7hybridautomatawithlinearvector elds[31A. Deducti techniques.ontheotherhand. seektodirectlyestablishpropertieso7theexecutionso7 thehybridautomaton.byproving.7or example.invariantassertions[32A.Thoughtheanalysisisnotcompletelyautomatedinthiscase. theproo7smaybeassistedbytheoremprovers[33.34A. These veri cation techniques may lead to misleading claims when applied to Zeno hybrid automata.sincetheseto7statesreachablebyaZenomodelmaynotre ectthestatesreachableby theunderlyingphysicalsystem.Forexample.7orthewatertankhybridautomaton(whichisin

7actamulti-rateautomaton)onecanshowthat7orallthereachablestatesthewaterinthetanks willbeabovethedesiredlowwatermarks.Clearlythiscannotbethecase7orthephysicalsystem. whentherateatwhichwaterisaddedtothetanks. .islessthanthetotalrateatwhichwateris removed. .i.e..whenthehybridautomatonmodelisZeno. SimilarproblemsareencounteredwhenonetriestoextendLyapunovtypeanalysistechniques tohybridsystems.Thishasledresearchersinthisareatoexplicitlyaddassumptionsrequiringthe systemtobe non-Zeno[1E.1BA. 2.2.3. Safe and optimal control. 3ethods 7or designing controllers that ensure that a hybrid systemis sa7e.in

thesense that it does not reachan undesirable con guration.have also been developed.Thesemethods.motivatedbyearlierworkonpurelydiscreteandpurelycontinuous systems.havebeenextendedtotimedautomata[35A.hybridautomatawithconstantdi erential inclusions [3DA. and hybrid automata with non-linear vector elds [22A. All the proposed approaches su er 7rom the drawback that they allow the controller to cheat by 7orcing the systemtobeZeno.andtherebyhidingthe7actthatunsa7estatescanbereached.Thishasagain 7orced researchers to a priori introduce non-Zeno assumptions. As an example consider the

7ollowingproblem7rom7ree ightairtra cmanagement[3BA. Example Aircraft con ict resolution Consider two aircra7t moving at the same constant altitude along straight line traCectories. Introduce a set o7 co-ordinates that centres one aircra7t at the origin and let ( ) denote the relative co-ordinates o7 the other aircra7t. The dynamics o7 this system is given by cos sin . where (0. ) is the velocity o7 the rst air- cra7t. (0. ) the velocity o7 the second. and ) the constant relative orientationo7theaircra7ts.I7 istreatedasacontrolsignaland asadisturbancesignal.the

modeldescribesapursuit-evasiongamewiththe rstaircra7tbeingtheevaderandthesecondthe pursuer.Theevaderwouldliketopreventthepursuer7romgettingcloserthanacertaindistance. which would de ne an unsa7e ight con guration. Solving the game [3BA leads to a saddle solutiondescribedbythehybridautomatoninFigure2.where sgn sin sgn cos Itiseasytoveri7ythatallexecutionsacceptedbythishybridautomatonavoidtheunsa7eset.The con gurationmay.however.stillbeunsa7einpractice.Thereasonisthatthehybridautomaton acceptsZenoexecutions.7orexample.governedbytheinitialstatesdepictedinthe gure.They

correspondtoasituationwheretheevaderconstantlyswitchesitsvelocitybetween and 440 J.Z1ANF Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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Figure2. Zenohybridautomatondescribingacon ictbetweentwoaircra7ts. Figure3. ZenohybridautomatondescribingFuller sphenomenon. . This is. o7 course. not realistic. because there are some dynamics involved in the switching.I7thiscontrollerwasimplementedinpractice.thesystemwouldmostlikelyreachthe unsa7esetthrougha chatteringtraCectory. Zenotypebehaviourmayalsoarise7orcertainclasseso7 optimalcontrolproblems.

Example .uller s Phenomenon ThehybridautomatoninFigure3generatestheoptimalcontrols7ortheproblemo7minimiz- ing the per7ormance index )d 1. with respect to the dynamics (0) (0.0).andthecontrolconstraint 1.Thedomainsandtheguardsinvolvethe constant (0.1/2).ItispossibletoshowthatthishybridautomatonisZeno[3EA.In optimalcontrol.thisisre7erredto asFuller sphenomenon[39A. 441 ZENO14BRIDS4STE3S Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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Figure4. Zenohybridautomatontogetherwiththecontinuousparto7anexecution solid and dotted). 3.

ZENO14BRIDAUTO3ATAANDZENOEHECUTIONS Zenohybrid automataare de ned in this section.The notion o7 Zenoset is introduced as the limitset7orin niteexecutionswith niteexecutiontime.Anumbero7examplesarepresented toillustratethecharacteristicso7theZenoset.ForarelateddiscussionseeRe7erences[13 1DA. De nition Zeno hybridautomaton A hybrid automaton is Zeno i7 there exists an initial state ( Init. such that all executionsin )areZeno. 3anymodelso7realsystemsareZeno.7orexample.thehybridsystemsdiscussedinSection(2.2). Anexampleo7 aZenoexecutionisgivennext. Example Multi level bouncing ball The hybrid

automaton in Figure 4 (which is a variant o7 the bouncing ball automaton o7 Re7erence[13A)acceptstheexecutionillustratedbythesimulationtotheright.Thecontinuous parto7theexecutionisshown7or (2.0).Itiseasilycheckedthatthehybridautomatonis Zeno by explicitly deriving the sequence o7 time intervals . which is a converging geometricseries. It is clear that Zenoness is due to the interplay between the vector eld. the resets. and the guards. Forexample. i7 in Example4 the resets o7 is replaced by /( dx 1). where 1/ 20 (0) .thenitiseasytoveri7ythat divergesas 1/ .1ence.thehybrid

automatonwillnotacceptanyZenoexecutionsinthiscase. 3.1. limit sets and Zeno sets Toinvestigatethelimitingpropertieso7in niteexecutions.wegeneralizedtheconcepto7 limit set[40Ato hybridautomata. De nition imit set Apoint( isan limitpointo7anin niteexecution ).i7thereexistsa sequence with 7or some . such that as ) and ). )) ).The limitset o7anexecution istheseto7all limitpointso7 442 J.Z1ANF Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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TheZenosetisintroducedasthe limitseto7anin niteexecutionthathas niteexecution time. De nition Zeno set An

limitpointo7aZenoexecutioniscalledaZenopoint.TheZenoseto7aZenoexecutionis theseto7 allZenopoints. ;euse todenotetheZenoset.Inotherwords. consistso7allclusterpointso7 sequences ). )) with and such that as . ;e write .( 7or the discrete part o7 and 7orthecontinuouspart(noticethat.ingeneral. ).TheZenosetcanbe empty. nite.countable.orevenuncountable.ForalltheexamplesinSection(2.2)thecontinuous state converged to a unique value. i.e. 7or some . In Example 4. we have . (0.0)). ( . (3.0)). ( .(5.0)) . so that and 0.0).(3.0).(5.0) .;e presenta 7ewmoreexamplesto illustrateotherpossibilities. Example

ncountable Zeno set Considerahybridautomaton acceptingaZenoexecutionwith .3odi7ythis hybridautomatoninto a new hybrid automaton by adding two components( ) to the continuous state o7 . For and . let the continuous dynamics in all discrete states be 0.theresets7oralledgesbe cos sin sin cos where isarationalconstantandtheinitialconditionbe (1.0) .Theguardsandthedomainso7 are the obvious extensions o7 those o7 . Then. the Zeno set o7 is .1ence.theZenosetisanuncountableset. Example Empty Zeno set Consider a hybrid automaton that accepts a Zeno execution with . Append acomponent

tothecontinuousstatewithtrivialdynamics 0.reset andinitial condition (0) 1.Then.7orall with and suchthatas PR thesequence hasnoclusterpoint.Themodi edhybridautomatonhasnoZenopoint (itsZenoset isempty).sincetheaugmentedcontinuousstateblowsup. The discrete part o7 the Zeno set. . will be visited in nitely o7ten by a Zeno execution. Adiscretestatebeingvisitedin nitelyo7tenis.however.notnecessarilyin .asshownbythe 7ollowingexample. Example Discrete states visitedin nitely often ConsiderthehybridautomatoninFigure5.ItiseasytoseethatitacceptsaZenoexecution withZenotime 1.TheZenosetis .(0.1))

.Thediscretestate isvisitedin nitely o7tenbytheZenoexecution.butstill .Thereason7orthisisthat blowsupin Lemma2inSection3.3givesconditionsunderwhichadiscretestatethatisvisitedin nitely o7tenbelongsto Formosthybridautomata.thediscreteevolutiono7Zenoexecutionsbecomesperiodicasthe Zenotimeisapproached.1owever.inRe7erence[41Ahybridautomatathatdonotexhibitthis periodicbehaviourarepresented. 443 ZENO14BRIDS4STE3S Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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Figure5. ZenohybridautomatoninExampleB. 3.2. Properties of Zeno sets

Determiningthestructureo7theZenosetcanbeveryimportantinsomecases.Forexample.i7it turnsoutthatthecontinuousstateconverges( 1).onemayhopetode neextensionso7the Zenoexecutionbeyond usingregularizationtechniques[13A.Tostudysuchpropertieso7the Zenoset we introducethe7ollowingde nitions.The reset is calledan identity on i7 7or all and 7orall ). is called non-expanding on i7 there exists [0.1A suchthat7orall( . 7orall )and7orall is called contracting on i7 the same is true 7or some [0.1). Finally. is called non- contracting on i77orall( . 7orall ) and7orall ). In the proo7s. the 7ollowing lemma is

used. which is an immediate generalization o7 the correspondingresult7orcontinuous-timesystems[40.:roposition5.3A. 1emma 1 Considerahybridautomatonwith non-expandingon .Then.thereexists 0suchthat 7orallexecutions ).all andall 1) 1.I7instead is non-contractingon .then 1)e 1. Forcontinuous-timesystems.Lipschitzcontinuityo7 thevector eldexcludesthepossibility 7or niteescapetime.Lemma1allowsustodrawasimilarconclusion7orhybridsystemswhose reset is non-expanding on the guards. This. in particular. implies that all Zeno executions o7 ahybridsystemremainbounded.There7ore.sincethevector

eldsareassumedtobeLipschitz continuous. there exists some 0 such that 7or all and ). )) 3oreover. 7or sequences with and such that as PR . the ;eierstrasstheoremimpliesthat ). hasatleastoneclusterpoint.There7ore.Zeno executionso7hybridautomatawhoseresetsarenon-expandingontheguards.haveatleastone Zenopoint( ). Foridentityresets.thecontinuousparto7 theZenosetisasinglepoint.asisshownnext. 2heorem 1 ForallZenoexecutionso7 ahybridautomatonwith identityon Asimilarresultholdsi7 iscontractingon 444 J.Z1ANF Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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heorem ForallZenoexecutionso7 ahybridautomatonwith contractingon Noticethatthede nitiono7contractinggivenaboveimplicitlyrequiresthat .0) 7orall( suchthat0 ).Theresultcanbeextendedtocaseswheretheresetsshare any common xed point. . and are contracting a7ter a change o7 co-ordinates taking to .Thiswould.7orexample.allowustoextendTheorem2tocoverappropriateclasseso7 ne7unctions. 3.3. Existence of Zeno executions SinceZenoexecutionsdonotre ectthetruebehaviouro7aphysicalsystem.itisimportantto study under what conditions hybrid automata accept only Zeno executions. Su cient and

necessaryconditions7orthisarepresentedinthissection. First.recallthateveryhybridautomatoncanbeassociatedwithadirectedgraph( ).Itis obviousthatahybridautomatonisZenoonlyi7thatgraphhasacycle.The7ollowingobservation isalsostraight7orward. Proposition 1 I7thereexistsa nitecollectiono7states suchthat ). )7orall 1. 1;and Reach 7orsome 1. thenthehybridautomatonacceptsa Zenoexecution. ExampleBshowsthatitispossible7oradiscretestatetobevisitedin nitelyo7tenbyaZeno execution.butstillnotappearin .1owever.i7theresetisnon-expandingontheguards.thisis notthecase. 1emma 2 ForallZenoexecutions

)o7ahybridautomatonwith non-expandingon .there existssome suchthat7orall Lemma2canbeusedto establishthelocationo7Zenopointswithrespecttothedomains. 2heorem 3 Consider a hybrid automaton with non-expanding on and assume it accepts a Zeno executionwithZenoset 7orsome 0.I7 7orall( with . then )7or all 1. A consequence o7 Theorem 3 is that ). The 7ollowing non-Zeno condition 7ollowsdirectly7romthetheorem. Corollary 1 Ahybridautomatonwith identityon acceptsnoZenoexecutionsi7 7orall( 7orallcycles with and( .1 1. 445 ZENO14BRIDS4STE3S Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control

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It is interesting to notice that the standing assumption by Tavernini in Re7erence [42A is implied by the two conditions in Corollary 1. Under this assumption. it is proved that each solution has nitely many switching points in nite time. i.e.. the system is non-Zeno. ThesecondconditioninCorollary1.ondisCointboundarieso7thedomains.canbereplacedby thetwoassumptionsthattheboundariesonlymeetatasinglepointwhichisanequilibriumpoint 7orthevector eldineachdiscretestate.;ithoutlosso7generalitythispointcanbeassumedto betheorigin. heorem For a hybrid automaton with

identity on . an execution ) with 0 isnotZenoi7 7orall( 7or all cycles with and ( .1 1. and .0) 0. Note that the rst assumption o7 Theorems 3 and 4 are 7ul lled 7or systems under logic- based switching [43A. piecewise linear systems [20A. complementarity systems [44A and mixed logical dynamical systems [45A. 1ence. these two results provide assumptions to guarantee non-Zenoness 7or these large classes o7 hybrid systems. Similar result to Corollary 1 and Theorem 4 holds also 7or hybrid automata with contracting resets having the origin as a xedpoint. 4. CONCLUSIONS 3otivated by a number o7

examples appearing in di erent applications o7 hybrid systems. wehavestudiedsomepropertieso7Zenoexecutions.;esawthatitisimportanttounderstand Zeno in order to develop e cient tools 7or modelling. veri cation. simulation. and design o7 hybrid systems. The Zeno set was introduced to capture the limiting behaviour o7 Zeno executions. In general. the Zeno set can have a complex structure. It was proved. however. that 7or hybrid automata having either only identity resets or only resets con- tractingontheguard.thecontinuousparto7theZenosetisasingleton.Thesehybridsystems

include.7orexample.7eedbackcontrolsystemswithlogic-basedswitchingandcomplementarity systems. For such systems. our results guarantee that every Zeno execution converges to a single point in the continuous state space. It may then be possible to extend the execution beyond the Zeno time in a consistent way [13. 3A. Both necessary and su cient conditions 7or a hybrid automaton to accept Zeno executions were presented. In particular. it was proved that 7or hybrid automata with identity resets on the guard. i7 the guards and the

interioro7thedomainsaredisCointandi7theboundarieso7thedomainsarealsodisCoint.thenthe hybrid automaton does not accept Zeno executions. 3oreover. it was shown that the last condition can be replaced by that the boundaries o7 the domains have the single intersection pointbeingtheorigin.whichshouldalsobeanequilibriumpoint7orthevector eldinalldiscrete states. 44D J.Z1ANF Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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A::ENDIH Proofof emma . ByLipschitzcontinuity.thereexists 0suchthat7orall and ). )) 1). Since it7ollowsthat sothat 1) 1) 1)

ApplyingtheBellman FronwallLemma[40Atwice.wehave 1)e 1)e and Bytheassumptiononnon-expandingresets.wehave .which yields 1)e 1)e Byinduction. (0) 1)e there7ore. 1)e Theproo77ornon-contractingresetsissimilar. Proof of heorem 1. Consider a Zeno execution ). For all with and suchthat as PR .wehave ). ))d )( ). )). ). ))). 7orsome . 1ence7orall 0. )( ). )). ). ))) )( ). )). ). ))) )( ). )). ). ))) 44B ZENO14BRIDS4STE3S Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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whichgivesthat where 0issuchthat ). )). ). ))) 7orall and (recall that such a exists

by continuity o7 and the 7act that remains bounded). Since is a Cauchy sequence. The space is complete. so the sequencehasalimit lim ).3oreover.the7ollowingargumentshowsthatthislimitis independent o7 the choice o7 sequence . Consider two sequences and and such that and as PR . ;ithout loss o7 generality.supposethat )( ). )). ). ))) )( ). )). ). ))) )( ). )). ). ))) This gives that ). 1ence. 0as PR whichshowsthatbothsequenceshavethesamelimit.Thiscompletestheproo7. Proof of 2heorem 2. Asintheproo7o7Theorem1.wegetthat7oraZenoexecution )it holdsthat ). ))d Usingthe7actthat 7or some

[0.1).it7ollowsthat 1). ))d Byinduction. and (R There7ore. 0as PR .whichyieldsthat 0as PR .hence 44E J.Z1ANF Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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Proof of 1emma 2. Supposethat7orall 0thereexistsome .suchthat .By the assumption that is nite. the sequence has a subsequence with . By Lemma 1. the sequence is bounded.Then. there exists some suchthatlim (bypossiblypassingtoasubsequence).Bythede nitiono7the Zenoset.( .whichgivesa contradiction. Poof of 2heorem 3. For every ( . there exists a sequence with and suchthatas PR and

.Noticethat.since isgiventhediscrete topology. implies that 7or su ciently large. 3oreover. by an argument similar to the one in the proo7 o7 Theorem 1. there exists 0 such that .There7ore. 0as PR .Bythestandingassump- tions. we have that . 3oreover.7rom Lemma 2. i7 is su ciently large. 1ence.7or su cientlylarge. there exists some such that ). which gives that . Since . it 7ollows that ). 3oreover. since )is aclosedset.thelimit lim )belongsto ). Proof of 2heorem 4. Assume that ) with 0 is a Zeno execution. By Theorem 1. it holds that 7or some . Then by Theorem 3 and the second

assumption.it7ollowsthat .Let bethelargestLipschitzconstanto7 )7orall .By Lemma 2. there exists some such that 7or all . Then. 7rom Re7erence [40. :roposition5.3A.wehavethat7orall and Since ). :roceeding7urther. Since isaZenoexecution.lim 0.Thiscontradicts.however.the7act thatlim 0.1ence.thehybridautomatonacceptsnoZenoexecutions. AC/NO;LEDFE3ENTS This work was supported by ARO under the 3URI grant DAA104-9D-1-0341. ONR under grant N00014-9B-1-094D. DAR:A under contract F33D15-9E-C-3D14. Swedish Foundation 7or International CooperationinResearchand1igherEducationandTele7onaktiebolagetL3Ericsson

sFoundation.The authorswouldliketothankSlobodanSimic and3aurice1eemels7orcommentsonthemanuscript.and AntonioBicchi7orpointingouttheexamplebyFuller. REFERENCES 1. vanderScha7tAJ.SchumacherJ3.Complementaritymodelingo7hybridsystems. IEEE ransactionson Automatic Control 199E; 43 (4):4E3 490. 2. Lygeros J. Johansson /1. Sastry S. Egerstedt 3. On the existence o7 executions o7 hybrid automata. IEEE Conference on Decision andControl .:hoenix.AZ.1999. 449 ZENO14BRIDS4STE3S Copyright 2001John;iley&Sons.Ltd. Int Robust Nonlinear Control 2001; 11 :435 451
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