北理自控 刘世岳课件 自动控制理论C

AUTOMATICCONTROLTHEORY:CTheseriesnumber:C016203Coursehours:48Credits:3Prerequisites:BasicTheoryofAutomaticControl:ASignal&SystemsGrading:FinalExaminationandHomeworkAssignmentsEverything'soutofcontrolorundercontrol?aObjectives:ThiscourseisacorecourseforundergraduatestudentsmajoringinAutomaticControloftheSchoolofInformationTechnologyatabeginninglevel.Itaimstoenablethestudentstohaveopportunitiesforfurtheracademictrainingandworkinginindustry.aSomeguresarefrominternetandreferencetexts.AutomaticControlC–p.1/22OUTLINEANDTOPICSChapter1:Discrete-timeControlSystems(10lecturehours)1-1Introduction1-2SamplingandDataHold1-3DiscretizationofContinuous-timeSystems1-4zTransformandInversezTransform1-5PulseTransferFunctions1-6StabilityandSteady-stateError1-7DynamicalBehaviorandDesignofDiscrete-timeControlSystems1-8MatlabToolboxAutomaticControlC–p.2/22OUTLINEANDTOPICS,CONT.Chapter2:Non-linearControlSystems(8lecturehours)2-1Introduction2-2DescribingFunctionMethod2-3Phase-planeAnalysisMethod2-4InverseSystemMethod2-5MatlabToolboxChapter3:LyapunovStabilityAnalysis3-1StabilityintheLyapunovsense3-2LinearSystems3-3Non-linearSystems3-4Applications3-5MatlabToolboxA.M.Lyapunov,1857–1918,RussianAutomaticControlC–p.3/22OUTLINEANDTOPICS,CONT.Chapter4:OptimalControlTheory(18lecturehours)4-1MathematicalDescriptionofOptimalControlProblem4-2VariationalPrinciples4-3OptimalControlforContinuous-timeSystems2-4OptimalControlforDiscrete-timeSystems2-5MatlabToolboxminJ=min12x0(N)Sx(N)+12PNk=0[x0(k)Qx(k)+u0(k)Ru(k)] ReferencetextsOgata,K.,Discrete-timeControlSystems,2nded.,,2005,andModernControlEngineering,4thEdition.(),,2002.,,,,2005.AutomaticControlC–p.4/22Ch.1:DISCRETE-TIMECONTROLSYSTEMS1.1IntroductionRoughlyspeaking,adiscrete-timecontrolsystemconsistsofadigitalcon-troller(computer),plant(orprocess),actuator,andsensor.Thedigitalcontrollercontainsdigitalcomputer,sampler,holdcircuit,andD/AandA/Ddevices:AutomaticControlC–p.5/221.1INTRODUCTIONaAdiscrete-timecontrolsystemcanalsoberepresentedinblockdiagram:1.2Impulsesamplinganddatahold(thefollowingdiagramisfrominternet)aThelecturematerialsarefromOgata'sbook“Discrete-timeControlSystems”(Chapters1–5).AutomaticControlC–p.6/221.2IMPULSESAMPLINGANDDATAHOLDAsamplerconvertsananalogsignalintoasequenceofamplitude-modulatedpulses.Theholdcircuitholdsthevalueofthesampledpulsesignaloveraperiod.AutomaticControlC–p.7/221.2IMPULSESAMPLINGANDDATAHOLDTherearetwomodesofoperationforasample-and-holdcircuit:thetrackingmodeandtheholdmode.AutomaticControlC–p.8/221.2IMPULSESAMPLINGANDDATAHOLDAnalog-to-DigitalConverters:A/DTheprocessbywhichthesampledanalogsignalisquantizedandconvertedtoabinarynumberiscalledanalog-to-digitalconversion.AnA/Dconvertertransformsananalogsignalintoadigitalsignal.TypesofSamplingOperationsAsignalwhoseindependentvariabletisdis-creteiscalledadiscrete-timesignal.Thereareseveraldifferenttypesofsam-plingoperations:1.Periodicsampling.Thesamplinginstantsareequallyspaced,ortk=kT;k=0;1;2;:::.Tisreferredtoasthesampleperiod.2.Multiple-ordersampling.Thepatternofthetk'sisrepeatedperiodically;thatis,tk+rtkisconstantforallk.3.Multiple-ratesampling.Adigitalcontrolsystemmayhavedifferentsam-plingperiodsindifferentfeedbackpaths.4.Randomsampling.Thesamplinginstantsarerandom,ortkisarandomvariable.AutomaticControlC–p.9/22MATHEMATICALMODELOFASAMPLERAsamplerisconsideredasaimpulsesamplershownbythefollowingdiagram:Thesampledsignalx(t)canberepresentedbytheinniteseries:x(t)=1Xk=0x(kT)(tkT)Here(t)istheDiracfunctionwhichisdenedas(t)=(1;ift=0;0;otherwise.AutomaticControlC–p.10/22IMPULSESAMPLERASAMODULATORThesamplermaybeconsideredamodulatorwiththeinputx(t)asthemod-ulatingsignalandthetrainofunitimpulses(t)asthecarrier,asshowninFigure3-2.Notethattheimpulsesamplerisintroducedformathematicalconvenience;itisactitioussampleranditdoesnotexistintherealworld.Weassumethattheholdingtime,i.e.,thesampleperiodTisverylargerthansamplingduration.Hencethismodelisreasonableandapplicable.AutomaticControlC–p.11/22zTRANSFORMOFx(t)ConsidertheLaplacetransformofx(t).X(s)=L(x(t))=L1Xk=0x(kT)(tkT)
=1Xk=0x(kT)L((tkT))=1Xk=0x(kT)ekTs:HereweuseL[(tkT)]=ekTs.DeneeTs=z.ThenX(s)js=lnz=T=1Xk=0x(kT)zk:Wedenetheright-handsideofthelastequationastheztransformofx(t).Thatis,Z(x(t)):=P1k=0x(kT)zkwhichisreferredtoastheone-sidedztransformandconsideredasthedis-cretetimecounterpartofLaplacetransforms.AutomaticControlC–p.12/22DATA-HOLDCIRCUITData-holdisaprocessofgeneratingacontinuous-timesignalh(t)fromadiscrete-timesequencex(t).Aholdcircuitapproximatelyreproducesthesig-nalappliedtothesamplerbyusingann-orderpolynomialextrapolatorasfol-lows:h(kT+)=ann+an1n1+


+a1+x(kT);0tT:Ifletn=0,weobtainazero-orderhold,inwhichtheamplitudeofthesampleisheldfromonesamplinginstanttothenext.AutomaticControlC–p.13/22TRANSFERFUNCTION:Gh0(s)Considerthesamplerandzero-orderholdshowninFigure3-4(a).Assumex(t)=0fort0.Thentheoutputh1(t)canberepresentedbyh1(t)=1Xk=0x(kT)[1(tkT)1(t(k+1)T)]SinceL[1(tkT)]=ekTss;t