DEVS相关文献.pdf

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DEVSFrameworkforModelling,Simulation,Analysis,andDesignof HybridSystems BernardP.Zeigler ElectricalandComputerEngineering TheUniversityofArizona Tucson,AZ,U.S.A. HaeSangSongandTagGonKim DepartmentofElectricalEngineering KoreaAdvancedInstituteofScienceandTechnology Taejon-,Korea HerbertPraehofer SystemsTheoryandInformationEngineering JohannesKeplerUniversityofLinz A-Linz,Austria Abstract WemakethecasethatDiscreteEventSystemSpecication(DEVS)isauniversalformalismfor discreteeventdynamicalsystems(DEDS).DEVSoersanexpressiveframeworkformodelling,design, analysisandsimulationofautonomousandhybridsystems.WereviewsomeknownfeaturesofDEVS anditsextensions.WethenfocusontheuseofDEVStoformulateandsynthesizesupervisorylevel controllers. Introduction Formaltreatmentofdiscreteeventdynamicalsystemsisreceivingevermoreattention[].Work onamathematicalfoundationofdiscreteeventdynamicmodelingandsimulationbeganinthes [,,]whenDEVS(discreteeventsystemspecication)wasintroducedasanabstractformalism fordiscreteeventmodeling.Becauseofitssystemtheoreticbasis,DEVSisauniversalformalismfor discreteeventdynamicalsystems(DEDS).Indeed,DEVSisproperlyviewedashort-handtospecify systemswhoseinput,stateandoutputtrajectoriesarepiecewiseconstant[].Thestep-liketransitions inthetrajectoriesareidentiedasdiscreteevents. Recently,interestinhybridsystems(mixeddiscrete-continuous)systemshasbeengrowing.Such systemsareprominentinsuchareasasintelligentcontrol[,]andreactivesystemdesign[].This articleproposesthatDEVS-basedsystemstheoryprovidesasound,generalframeworkwithinwhichto addressmodelling,simulation,design,andanalysisissuesforhybridsystems. Fig.showsthebasicapproachtohybridsystemsresearchproposedhere.Webrieyaddressthe ideasraisedinthisgure.

MUNIC A TIO M O C T W O R K E N N L ANALY SIS MANUFA C T U RIN PERF O R M A N G S Y S T E M C + V A W A A M E D E V R G E E DEVS M E T S Y Formalism L U A T I O N A IC G O L D E V Sim++ IMPLEMENT A T I O N R S E T U MP CO S Formal i s m in C+ DISCRETE EVENT C O N T R O L Figure:DEVS-CenteredHybridSystemsResearch . Modelling Discreteeventmodellinghasitsorigininjobshopschedulingsimulationsinoperationsresearch. Morerecentlycontroltheoristsdevelopedtheirowndiscreteeventformalisms[].Theuniversalityclaims oftheDEVSjustcitedareaddressedbycharacterizingtheclassofdynamicalsystemswhichcanbe representedbyDEVSmodels.PrahoferandZeigler(inpress)showedthatanycausaldynamicalsystem whichhaspiecewiseconstantinputandoutputsegmentscanberepresentedbyDEVS.Wecallthisclass ofsystemsDEVS-Representable[].Inparticular,DierentialEquationSpeciedSystems(DESS)are usuallyusedtorepresenttheplant(systemundercontrol)inhybridsystemsandarecontrolledby high-level,symbolic,event-drivencontrolschemes.Sensingandactuationinsuchsystemsareevent- likeandtheyhavepiecewiseconstantinputandoutputtrajectorieswhenviewedwithintheframeof theirsensing/actuatorinterface.Consequently,withinsuchaninterface,theplantisrepresentableby aDEVS.Likewise,thecontrollerhasnaturalDEVSrepresentation. Closureundercoupling[, ]isadesirablepropertyforsubclassesofdynamicalsystemssinceit guaranteesthatcouplingofclassinstancesresultsinasysteminthesameclass.TheclassofDEVS- representabledynamicalsystemsisclosedundercoupling.Thisjustieshierachical,modularconstruc- tionofbothDEVSmodelsandthe(continuousordiscrete)counterpartsystemstheyrepresent. . Simulation DEV&DESSisanextensionoftheDEVSformalismforcombineddiscrete/continuousmodelingand simulation[ ,].DEV&DESSincludesboththeDEVSandDESSclassesofsystems. Inaddition itprovidesameansofspecifyingsystemswithinteractingcontinuousanddiscretetrajectories.The formalismisfullyexpressiveofhybridsystemsinthatitisshowntobeclosedundercoupling. In particular,theresultantsofcoupledmodelsthatcontaincomponentsofeitherclassareexpressibelas basicmodelsintheformalism.Sinceitisalsoimplementedinsoftware,DEV&DESSoersapowerful modelingformalismforsimulatinghybridsystems. . DesignandAnalysis RamageandWonham[]weretherstcontroltheoriststodevelopanapproachtodesigningdiscrete eventcontrollers.Theymodelledtheplantasanautomatonandusedlanguagetheorytodesigna controllerthatforcestheplanttoexhibitbehaviorconsistentwithgivenobjectives.Thegoalofthis paperistodemonstratetheapplicabilityofDEVSasunifyingframeworkforhybridcontrol.Wedoso byreformulatingtheRamage-WonhamapproachtocontrollerdesignwithintheDEVSformalism.The

resultingmethodologyisbothmoregeneralandmoreintuitivelyappealingthantheoriginal.Sinceit isuniversalforgeneralpiecewiseconstantsystems,DEVS,canrepresenttiminginformationaboutthe plantthateludesanautomatonformulation.Suchinformationcanbeemployedtoimprovecontroller designs.Moreover,DEVSdistinguishesbetweeninternalandexternalstatetransitions,afactthat enablesustoderiveacontrollerDEVSfromaplantDEVSmodelinaratherdirectmanner. SincetheappealoftheproposedmethodologyrestsupontheuniversalityofDEVSrepresentation,we rstbrieyreviewsystemsformalismsandtheirmappingintoDEVS.ThenwediscusstheDEVS-based methodologyfordiscreteeventcontrolofhybridplants.Weconcludewithopenquestionsforfuture research. ReviewofGeneralDynamicalSystemsandDEVS . GeneralDynamicalSystems Basedon[,,]wedeneageneraldynamicalsystemasfollows: DS=(T;X;Y;;Q;;) withTisthetimebase, Xisthesetofinput-values, Yisthesetofoutput-values, isthesetofadmissibleinputsegments!:!XoverTandXandisclosedunder concatenationaswellasunderleftsegmentation[], Qisthesetofstates; :Q!Qistheglobalstatetransitionfunction, :QX!Yistheoutputfunction. TheglobalstatetransitionfunctionofthegeneraldynamicalsystemDShastofulllthefollowing properties[,]: Consistency:(q;!(t;t>)=q; () Semigroupproperty:!:!X;t:(q;!)=((q;!);!);() Causality:!;!ift:!(t)=!(t) then(q;!)=(q;!):() Causality,thesemigrouppropertyandclosureofadmissiblesegmentsunderleftsegmentationjusties deningthestatetrajectoryresultingfromeveryinitialstateqQandinputsegment!:! Xinthefollowingway: STRAJq;!:!Q with tSTRAJq;!(t)=(q;!): () SimilarlywedenetheoutputtrajectoryOTRAJq;!:!Yby:foreveryinitialstateqQ, inputsegment!:!Xandt OTRAJq;!(t)=(STRAJq;!(t);!(t)): () Nowtheinput/outputbehaviorRDSofthedynamicalsystemisgivenby () RDS=f(!;OTRAJq;!):!;qQg:

. TheDiscreteEventSystemSpecication(DEVS)Formalism AnatomicDEVSisastructure[,,] DEVS=(XM;YM;S;ext;int;;ta) whereXMisthesetofinputs, YMisthesetofoutputs, Sisthesetofsequentialstates, ext:QX!Sistheexternalstatetransitionfunction, int:S!Sistheinternalstatetransitionfunction, :S!Yistheoutputfunction, ta:S!R+[fgisthetimeadvancefunction. ADEVSspeciesadynamicalsystemDSinthefollowingway: thetimebaseTistherealnumbersR; X=XM[fg,i.e.,theinputsetofthedynamicalsystemistheinputsetoftheDEVStogether withasymbol=XMspecifyingthenon-event, Y=YM[fg,i.e.,theoutputsetofthedynamicalsystemistheoutputsetoftheDEVStogether with. Q=f(s;e):sS;eta(s)gthesetofstatesofthedynamicalsystemconsistsofthe sequentialstatesoftheDEVSpairedwitharealnumberegivingtheelapsedtimesincethelast event, theadmissibleinputsegmentsisthesetofallDEVS-segmentsoverXandT(DEVS-segments []arecharacterizedbythefactthatforany!:!X,thereisonlyanitenumber ofeventtimesf;;:::;ng;iwith!(i)=.Asaspecialcase,aso-called segmenthasnoevents. foranyDEVSinputsegment!:!Xandstateq=(s;e)attimettheglobalstate transitionfunctionisdenedasfollows: (q;!)= (s;e+tt) () ife+tt) ife+ttta(s)^: t:!(t)=; ((ext(s;e+tt;!(t)););!(t;t>) ( ) if!(t)=^tt+ta(s)e^!restrictedtoisasegment.

theoutputfunctionofthedynamicalsystemisgivenby ((s;e);x)=(s): () Foragiveninputsegment,!threedierentcasescanbeidentied: First,if!isanon-eventsegmentandaninternaleventdoesnotoccurininterval,then thesequentialstateisleftunchangedbuttheelapsedtimecomponenteofthetotalstatehasto beupdated(). Second,ifatimeeventscheduledbythetimeadvancefunctiondoesoccurinthetimeinterval andthereisnoinputeventpriortotheinternaleventtime,thentheinternaltransition functiondenesanewsequentialstate,theelapsedtimecomponentissettozero,andtherestof theinputsegmentisappliedtothisnewstate(). Third,iftheinputsegment!denesaninputeventpriortotheoccurrenceofaninternalevent, thentheexternalstatetransitionfunctionisappliedwiththeextenalinputvalueandelapsed timeandtheelapsedtimeissettozero.Therestoftheinputsegmentisappliedtothisnewstate ( ). . DEVS-RepresentationofConstantInput/OutputDynamicalSystems Wedeneapiecewiseconstanttrajectory!:!Xinthefollowingway:thereisanite (possiblyempty)subsetf;;:::;ng;iforwhich!(i)=!(i)andR+is arbitrarilysmall.PiecewiseconstanttrajectoriesareequivalenttoDEVS-segmentsinthesensethat theycanbetransformedintoeachother.Thetransformationsareasfollows: Piecewiseconstanttrajectory!toDEVS-segment!:Foreveryeventtimeif;;:::;ngof !thereisanvalue!(i)=!(i)andforall=i)!()=. DEVS-segment!topiecewiseconstanttrajectory!:Letf;;:::;ng;ibethe eventtimesoftheDEVS-segment!with!(i)=.Thenforeverywedenethe respectivevalueofthepiecewiseconstanttrajectory!()bythevalue!(x)withxbeingthe largesttimeinf;;:::;ngwithx.Ifsuchanumberxdoesnotexistthen!()=. ForxXletxdenotethesegment!:!Xwithconstantvaluex,i.e.,t !(t)=x.Ifitisclearfromthecontext,wewritexforx. Wenowdeneaconstantinput/outputdynamicalsystemasadynamicalsystemwhoseinputtra- jectories!andassociatedoutputtrajectoriesOTRAJq;!arepiecewiseconstantonly.Ourmain interestistoshowhowsuchasystemcanberepresentedintheDEVSformalism. Givenaconstantinput/outputdynamicalsystemDS=(T;X;Y;;Q;;),wedenea DEVSDS=(XM;YM;S;ext;int;;ta) inthefollowingway: XM=X, YM=Y, S=QX,and

x1 x1 x s1 s2 x s1 state event external event x2 state event thr 1,b1 thr 2,b2 T T T internal event T s1' (a) continuous system internal event external event x2 s2 s2' thr 1 thr 2 T T ta ((thr1, s2'), x2) (a) discrete system Figure:IllustratingDiscreteEventTrajectories foreverytotalstate((q;x);)andinputsegment!:!Xwedene ta((q;x))=minfejOTRAJq;x(t)=OTRAJq;x(t+e); () () int((q;x))=(STRAJq;x(t+ta(s));x); ext((q;x);e;x)=(STRAJq;x(t+e);x); () () ((q;x))=(q;x): Theorem:TheDEVSDSasconstructedaboveandtheoriginalDSarebehaviorallyequivalent, i.e.,theyhavethesameinput/outputbehaviorwhenstartedinthesamestate. Proof:Themainworkistoshowthatforallinputtrajectories,!andinitialstates,qthestatein theconstructeddiscreteeventsystemDEVSDSisequaltothestateoftheoriginaldynamicalsystem DSatboththeeventtimesefe;e;:::;engof!andtheeventtimesifi;i;:::;ingofthe associatedoutputtrajectoryOTRAJq;!.Thisisdonebyexaminingthestateandoutputtrajectories ofthedynamicalsystemDS=(T;X;Y;;Q;;)speciedbyDEVSDS.Fromthedenitionofthe

outputin(),andfromthefactthatoutputsdonotchangebetweeneventtimesitfollowsthatthe outputbehaviorofthediscreteeventsystemDEVSDSisequaltothatoftheoriginaldynamicalsystem DS.TheformalproofispresentedinPrahoferandZeigler(inpress). . CoupledModels ModelsconstructedfromcomponentsareformalizedintheDEVSformalismascoupledmodels,DN DN= whereX:inputeventsset; Y:outputeventsset; M:DEVScomponentsset; EICDN:INM:IN:externalinput couplingrelation; EOCM:OUTDN:OUT:externaloutput couplingrelation; ICM:OUTM:IN:internalcouplingrelation; SELECT:M;!M:tie-breakingselector. HereDN.IN,DN.OUT,M.IN,andM.OUTrefertotheinputandoutputportsofthenewlycon- structedcoupledmodelandcomponentmodels,respectively.Thethreeelements,EIC;EOC;ICspecify theconnectionsbetweenthesetofmodelsMandinput,outputportsX;Y.SELECTfunctionacts asatie-breakingselector. ClosureundercouplingofDEVS-representablesystemsisstatedas: Theorem:AmodularcoupledsystemwhosecomponentsareDEVS-representableandwhichdoes notcontainalgebraicloopsisaDEVS-representabledynamicalsystem. ThedenitionsrequiredandformalproofarepresentedinPraehoferandZeigler(inpress). . DEVS-basedCombinedDiscrete/ContinuousModelingofHybridSystems Inthissection,wedemonstratehowtheDEVSrepresentationappliestoaspecialsubclassofdy- namicalsystems{thosewhosestatedynamicisdenedbyasetofrstorderdierentialequations andwhoseinputandoutputtrajectoriesarepiecewiseconstant.Weintroducetheoutput-partitioned DEV&DESSformalismforhybridsystemmodelingandshowhowaDEVSabstractionisconstructed. Letusconsideracontinuoussystemwithann-dimensionalcontinuousstatespaceSc=ScSc :::Scn=Rnwhosecontinuousbehaviorcanbemodeledbyasetofrstorderdierentialequations =f;;:::;ng.However,weassumethatthesystemreceivesinputstimulationswhicharepiecewise constanttimefunctions.Moreover,thesystemisoutttedwithasetofthresholdlikesensors.Such athresholdsensoronlychangesitsstateatparticularxpointsfi;bi-thresholdvalues-ofcontinuous statevariablessi.Forsimplicityletsassumethatonlybooleanvaluedsensorsthrfi;biareusedwhich deneabooleanvaluex=thrfi;bforallsifi;bi. ThenwedenetheoutputpartitionedDEV&DESSinthefollowingway: Denition:Anoutput-partitionedDEV&DESSisastructure: opDEV&DESS=(X;Y;Q;;;F) () where

Xisthesetofinputvalueswhichisanarbitraryset, S=SS:::Sn=Rn,i.e.,isthecontinuousstatespacewhichisann-dimensional vectorspace(ithasnstatevariables), =f;;:::;ngisasetofrateofchangefunctionwithi:SX!Sci, Sif;:::;ngFiwithFi=ffi;;:::;fi;migisthesetofthresholdvaluesforalldimensionsi, F= Y=BFistheoutputvaluesetwhichisthecrossproductofallbooleanvaluesetsofthethreshold sensorsthrfi;bi;fi;biF, :S!YMistheoutputfunctionwhichisdenedby (s)=(thrf;(s);thrf;(s);:::;thrfi;bi(si);:::;thrfn;bn(sn)) () i.e.,theoutputisdenedbythevaluesofthethresholdsensors. SuchaspecicationdenesadynamicalsystemDSinthefollowingway: thetimebaseTistherealnumbersR; inputsetX,outputsetY,andstatesetQ=Softhedynamicalsystemareidenticaltothoseof theDEV&DESS, theadmissibleinputsegmentsisthesetofallpiecewiseconstantsegmentsoverXandT, foranypiecewiseconstantinputsegment!:!Xandstatesattimettheglobal statetransitionfunctionisdenedbythesetofrstorderdierentialequationsasfollows: (s;!)=s+Zttf(STRAJs;!();!())d; () theoutputfunctionofthedynamicalsystemisgivenby () (s;x)=(s): BythethresholdvaluesfibiFeachdimensioniofthecontinuousstatespaceispartitionedintoa setofoutputblocks ( ) OBi=f(fi;;fi;>;;:::;;:::;;:::; ;:::;)oftheoutputpartitionischaracterizedbythefactthatallstatesinthesame outputblockhavethesamethresholdvalues,i.e., () d(s)=d(s),i fi;bi:sci^sci: Theoutputvaluesdonotchangeaslongasthecontinuousstatesstayinoneoutputblockand,there- fore,theoutputtrajectoryisapiecewiseconstanttimefunctionOTRAJs;!:!Ywith OTRAJs;!(t)=(STRAJs;!(t)). Forsimplicityreasons,wedonotdistinguishbetweenopenorclosedintervalsinthiswork.

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