线性系统理论第二章课件

LinearSystemTheoryLecture1IntroductionContentsIntroductiontocontrolDevelopmentofcontroltheoryStepstodesigncontrolMathematicalmodelofsystemHowtobuildmathematicalmodelofasystemRelationofdifferentmodelsBackgroundknowledgeMilestonesindevelopmentofControlTheoryClassicalControlTheory:Before1950’s.CharacterizedbyTransferFunctionanalysismethod.Notverygoodformulti-variableorlarge-scalesystemRootLocus,FrequencyResponsemethodsareextensivelyusedVerymuchdependentondesignexperienceMilestonesindevelopmentofControlTheoryModernControlTheory:After1960’s.Characterizedbystatevariableanalysismethod.Verygoodformulti-variableorlarge-scalesystem,andcanbeeasilyextendedtotime-variablesystemorevennon-linearsystemSystematicmethodsaredevelopedtoanalyzethecontrollability,observabilityandstabilityPerformancecanbeclearlyspecifiedandanalyticalmethodsaredevelopedtoaccomplishthedesignMilestonesindevelopmentofControlTheoryLarge-scaleSystemTheoryandIntelligentControlAfter1970’sProposedfordealingwithlarge-scale,complex,andhierarchicalcontroloflargesystem.Highlyintelligent,adaptiveandrobustControlExamplesFlyingballcontrolboilerManualcontrolfluidControlExamplesBoiler-generatorControlControlExamplesPowerGenerationControlModelingofphysicalsystemVerificationSimulation/VisualizationValidationAcontrolmodelRoughlyspeaking,controlsystemdesigndealswiththeproblemofmakingaconcretephysicalsystembehaveaccordingtodesiredspecificationsOpen-loopcontrolClosed-loopcontrolDefinitionof“system”Definition:Literally:agrouporcombinationofinterrelated,interdependent,orinteractingelementsformingacollectiveentity;amethodicalorcoordinatedassemblageofparts,facts,concepts,etc.Here:Mathematicaldescriptionofarelationshipbetweenexternallysuppliedquantities(I.e.,thosecomingfromoutsideofthesystem)andthedependentquantitiesthatresultfromtheactionoreffectonthoseexternalquantitiesSystemisdescribedby“model”,whichisagroupofdifferentialequations(partialorordinary),oralgebraicequationsconcerningrelevantvariablesWhyweneedtostudyasystem?Becausewewanttocontrolit.Whatiscontrol?Roughlyspeaking,controlistomakeasystembehavelikewedesire.EssentialelementsSometermsSISO:Single-Input-Single-OutputMIMO:Multiple-Input-Multiple-OutputSIMO:MISO:Someterms--MemoryAsystemwithmemoryisonewhoseoutputdependsonitselffromanearlierpointintime;E.g.,CapacitorC,InductorLAmemorylesssystemisonewhoseoutputdependsonlyonthecurrenttimeandcurrentinput.E.g.,ResistorRSometerms--CausalityAsystemissaidtobecausalifthevalueoftheoutputattimet0dependsonthevaluesoftheinputandoutputforallt0uptot0,i.e.,t<=t0EverypracticalsystemiscausalForeasyunderstanding,thismeansthatwecannotuseanywaytochangewhathavealreadyhappened.Someterms—statevariableStateVariable:asetofvariablesthatdefinethesystemresponse.Oncetheinitialconditionsaregiven,thedifferentialequationscompletelycharacterizeanychosenoutputfunctionfromtheinitialtimeforanyadmissibleinputfunction.Vc,IL:statevariableVR,IL:notstatevariableSometerms—LumpednessLumpedness:AsystemissaidtobelumpedifitsnumberofstatevariableisfiniteLimitedstatevariablesInfinitestatevariablesSometerms--TimeInvarianceTimeInvariance:Atime-invariantsystemisonewhoseoutputdependsonlyonthedifferencebetweentheinitialtimeandthecurrenttime,y=y(t-t0).Otherwise,thesystemistime-variantForyoureasyunderstanding,TImeanstheresponseofasystemisindependentofthetimewhenanexcitationisapplied.Timeinvarianceisamathematicalfiction.Noman-madeelectronicsystemistimeinvariantinthestrictsense.y(t)=(H(x))(t)y(t-)=(H(x))(t-)y(t-)=(H(x,))(t-)Someterms--LinearityLinearity:Asystemislinearifitsatisfiesthesuperpositionproperty,i.e.,Theoutputofasystemtotheinputf(t)

isy(t)=Prove,orgiveanexampleastowhetherthesystemis:A.LinearB.CausalC.TimeinvariantSometermsZero-stateresponseOutputexcitedexclusivelybytheinputZero-inputresponseOutputexcitedexclusivelybytheinitialstateSuperpositionpropertyoflinearsystemForalinearsystem,Response=zero-inputresponse+zero-stateresponseTherefore,theoverallresponsecanbedecomposedandstudiedseparatelyLinearizationofnon-linearsystemTaylorserieslinearizationFormultivariablecase,MathematicaldescriptionofasystemMathematicaldescriptionofasystemInput-outputDescription:ForacausalsystemMathematicaldescriptionFormultivariablecase,theGisamatrix,Where,Mathematicaldescription—transferfunctionContinuoussystemDiscretesystemZero-pole:Mathematicaldescription----StatespacemodelEverylinearlumpedsystemcanbedescribedbyasetofstateequation,ForaLinearTimeInvariant(LTI)system,thetransferfunctionsatisfiestimeshiftingproperty,i.e.,Mathematicaldescription----StatespacemodelHencethestateequationhastheform,(constantcoefficients)LaplaceTransformDefinition:AfterLaplacetransformation,asystemcanbedescribed,Expamplestextexample2.2,2.3,2.4,2.5PropertiesofpolynomialLaplacetransformtransfersatransferfunction(ofalumpedsystem)intoafractionoftwopolynomials

issaidtobeproperifissaidtobestrictlyproperifisbiproperifisimproperifZeroesandpolesArationaltransferfunctioncanbedecomposedintoaformlike,ziandzjarenotnecessarilydifferentand,thereforeziisazeroofpiandpjarenotnecessarilydifferentand,thereforepiisapoleofLaplacetransformofstatespaceequationForanLTIsystem,ItsLaplacetransformis,Solveand,Relationshipbetweeninput-outputandstatespacedescriptionHence,forarelaxedsystem(x(0)=0),Therefore,thetransferfunction,HowtobuildamodelofasystemAllmodelsarewrong,butsomemodelsareuseful----G.E.BoxCasestudy:howtomodelasystemFreebodymotionandNewton’slaw:StatevariablemodelPhysicalmodelMathematicalmodelStatevariableinputoutputTransferfunctionmodelSimulation(stepresponse)M1=1kgM2=0.5kgk=