第三讲数字介绍

Lecture3

IntroductoryDigitalcontrolLecturer:Dr.NingyunLUDepartmentofAutomaticControlNUAA,2007E-mail:C031006:ModernDigitalControlOutlineIntroductorydigitalcontrolDigitization->Emulation(3.1,3.3)Effectofsampling(3.2)MoreaboutthesamplingprocessAliasing&anti-aliasingSamplingtheorem(Chap5,11)Continuousvs.digitalcontrolBasically,wewanttosimulatethecont.filterD(s),calledEmulation?GivenacontinuouscontrollerD(s),whatisitsbestdigitalapproximation?D(s)containsdifferentialequations(timedomain)–mustbetranslatedintodifferenceequations.Analogcontrolsystemctrl.filterD(s)plantG(s)sensor1r(t)u(t)y(t)e(t)+-continuouscontrollerForexample,PIDcontrolTisthesampletime(s),Sampledsignal:x(kT)=x(k)

control:differenceequationsD/Aandholdsensor1r(t)u(kT)u(t)e(kT)+-r(kT)plantG(s)y(t)clockA/DTTy(kT)digitalcontrollervoltage→bitbit→voltageDigitalControlSystemElementsofdigitalcontrol-1Sampler:convertacontinuous-timesignalintoadiscrete-timesignal.ThevaluesofthesignalaresampledrepetitivelyatregularinstanceT(samplingperiodT,samplingrate1/T,frequency2pi/T)(timediscrete)Samplesofcontinuoussignalareconvertedinto(binary)numbersforprocessinginsideadigitalcomputer.(valuediscrete)Impulsesamplingx(t)x*(k)T2TImpulse-sampledoutputisasequenceofimpulses,withthestrengthofeachimpulseequalthemagnitudeofx(t)atthecorrespondingtimekTDefineatrainofunitimpulsesasT2TModulatorcarrierElementsofdigitalcontrol-2D/AandholdDigital-to-Analog(D/A)converter,toconvertabinarynumberintoanalogvoltagesHold,toholdvoltagesintoacontinuous-timesignal.Zero-OrderHoldx(t)x*(k)Zero-Orderholdh1(t)ApproximationApproximateadifferentialequationbyadifferenceequationusingtechniquessuchasEuler’smethodElementsofdigitalcontrol-3DifferenceEquationusingEuler’smethodUsingEuler’smethod,findthedifferenceequations.DifferentialequationUsingEuler’smethodDifferenceequationDigitalPIDusingEuler’smethodSignificanceofsamplingtimeTCompare–investigateusingMatlab1) Closedloopstepresponsewithcontinuouscontroller.2) Closedloopstepresponsewithdiscretecontroller. Samplerate=20Hz3) Closedloopstepresponsewithdiscretecontroller. Samplerate=40HzExample:controllerD(s)andplantG(s)MatlabimplementationcontinuouscontrollernumD=70*[12];denD=[110];numG=1;denG=[110];sysOL=tf(numD,denD)*tf(numG,denG);sysCL=feedback(sysOL,1);step(sysCL);discretecontrollernumD=70*[12];denD=[110];sysDd=c2d(tf(numD,denD),T);numG=1;denG=[110];sysOL=sysDd*tf(numG,denG);sysCL=feedback(sysOL,1);step(sysCL);ControllerD(s)andplantG(s)StepresponseswithdifferentsamplingrateEffectofsamplingD/AinoutputfromcontrollerThesinglemostimportantimpactofimplementingacontroldigitallyisthedelayassociatedwiththehold.AnalysisonsampletimedelayApproximately1/2sampletimedelayCanbeapprox.byPadè (andcont.analysisasusual)ctrl.filterD(s)PadéP(s)sensor1r(t)u(t)y(t)e(t)+-plantG(s)MoreaboutthesamplingprocessConsiderthecontinuous-timesignal•Thecorrespondingdiscrete-timesignaliswhereisthenormalizeddigitalangularfrequencyofAliasing–anexample•Thethreecontinuous-timesignalsoffrequencies3Hz,7Hz,and13Hz,aresampledatasamplingrateof10Hz,i.e.withT=0.1sec.generatingthethreesequences•Plotsofthesesequences(shownwithcircles)andtheirparenttimefunctionsareshownbelow:NotethateachsequencehasexactlythesamesamplevalueforanygivenkWhy?ThisfactcanalsobeverifiedbyobservingthatAsaresult,allthreesequencesareidenticalanditisdifficulttoassociateauniquecontinuous-timefunctionwitheachofthesesequencesTheabovephenomenonofacontinuoustimesignalofhigherfrequencyacquiringtheidentityofasinusoidalsequenceoflowerfrequencyaftersamplingiscalledaliasing•Sincethereareaninfinitenumberofcontinuous-timesignalsthatcanleadtothesamesequencewhensampledperiodically,additionalconditionsneedtoimposedsothatthesequencecanuniquelyrepresenttheparentcontinuoustimesignalAnti-aliasing?•RecallThusif ,thenthecorrespondingnormalizeddigitalangularfrequency ofthediscrete-timesignalobtainedbysamplingtheparentcontinuous-timesinusoidalsignalwillbeintherange

NoaliasingAnalysisAnalysis(cont.)Ontheotherhand,if ,thenormalizeddigitalangularfrequencywillfoldoverintoalowerdigitalfrequencyintherangebecauseofaliasingTopreventaliasing,thesamplingfrequency shouldbegreaterthan2timesthefrequency ofthesinusoidalsignalbeingsampledTheconditiontobesatisfiedbythesamplingfrequencytopreventaliasingiscalledthesamplingtheoremSamplingTheoremNyquistsamplingtheoremOnecanrecoverasignalfromitssamplesifthesamplingfrequencyfs=1/T(ws=2p/T)isatleasttwicethehighestfrequencyinthesignal,i.e.ws>2w0(closedloopband-width)Inpractice,weneed 20w0<ws<40w0

GraphicalExplanationofSamplingtheoremToshowthevalidityofthesamplingtheorem,wefirstshouldfindthefrequencyspectrumofthesampledsignalx*(t)x(t)x*(k)0ω1-ω1101/T01/T1010ω1-ω1101/T0ω1-ω11/TIdealLow-PassfilterX(s)X*(s)Y(s)X*(s)01/TFolding:Thephenomenonoftheoverlapinthefrequencyspectraisknownasfolding01/TSummaryDigitizationmethodsallowthedesignertoconvertacontinuouscompensationD(s)intoasetofdifferenceequationsthatcanbeprogrammeddirectlyintoacontrolcomputerEuler’smethodcanbeusedforthedigitizationWhenthesamplerateisfastenough(30*bandwidth),thedigitallycontrolledsystemwillbehaveclosetoits