離散時(shí)間控制系統(tǒng) 課件

Computer-ControlledSystems12/24/2022Computer-ControlledSystems12CourseInformationTime:13:30-15:10pm,Wednesday 10:00-11:40am,Friday(evenweeks)Venue:綜B204References:離散時(shí)間控制系統(tǒng)(英文版第2版)，KatsuhikoOgata，機(jī)械工業(yè)出版社，2004離散時(shí)間控制系統(tǒng)(中文版)，KatsuhikoOgata，陳杰，蔡濤等譯，機(jī)械工業(yè)出版社，2006GradingProcedure:in-termevaluationexperimentalresultsfinalpaperexamination.2CourseInformationTime:1Chapter1IntroductiontoDiscrete-Time3Chapter1IntroductiontoDisc4Contents1-1Introduction1-2DigitalControlSystems1-3QuantizingandQuantizationError1-4DataAcquisition,ConversionandDistributionSystems1-5ConcludingComments44Contents1-1Introduction41-1IntroductionDigitalControllerArapidincreaseintheuseofdigitalcontrollers51-1IntroductionDigitalControDigitalControllerThetrendisduetoDigitalcontrolcanachieveoptimalperfor-mance,haveDecision-makingcapabilityandflexibilityinthecontrolprogramAvailablityoflow-costdigitalcomputersAdvantagesofdigitalsignals1-1Introduction6DigitalController1-1IntroducTypesofSignalsContinuous-timesignal:AsignaldefinedoveracontinuousrangeoftimeAnalogsignal:AsignaldefinedoveracontinuousrangeoftimewhoseamplitudecanassumeacontinuousrangeofvaluesAcontinuous-timequantizedsignalDiscrete-timesignal:Asignaldefinedonlyatdiscreteinstantsoftime1-1Introduction7TypesofSignals1-1IntroductiSampled-datasignal:Adiscrete-timesignaliftheamplitudecanassumeacontinuousrangeofvaluesDigitalsignal:Adiscrete-timesignalwithquantizedamplitudeComparisonDiscrete-time,digital,sampleddatasignal(controlsystem)Continuous-time,analogsignal(controlsystem)1-1Introduction8Sampled-datasignal:Adiscret1-1Introduction91-1Introduction9SystemsDealtWithinThisBookLinearandtimeinvariantDiscrete-TimeControlSystemsoneormorevariablescanchangeonlyatdiscreteinstantsoftime.TheseinstantsmayspecifythetimesatwhichsomephysicalmeasurementisperformedorthememoryofadigitalcomputerisreadoutDescribedinlineardifferenceequationswithconstantcoefficients1-1Introduction10SystemsDealtWithinThisBoo1-2DigitalControlSystemsFigure1-2Blockdiagramofadigitalcontrolsystem111-2DigitalControlSystemsFigS/HandA/D(A/D)Sample-and-Hold(S/H)SamplingProcesses,replaceoriginalcontinuous-timesignalbyasequenceofvaluesatdiscrete-timetimepointsacircuitthatreceivesanaloginputsignalandholdsthissignalataconstantvalueforaspecifiedperiodoftime.Analog-to-DigitalConverter(A/D)Alsocalledanencoder,isadevicethatconvertsananalogsignalintoadigitalsignal,usuallyanumericallycodedsignal.AS/HcircuitisoftenanintegralpartofacommerciallyavailableA/Dconverter.1-2DigitalControlSystems12S/HandA/D(A/D)1-2DigitalTypesofSamplingOperationsPeriodicsamplingtk=kT(k=0,1,2,…)Multiple-ordersamplingtk+r-tk=constantMultiple-ratesamplingAdigitalcontrolsystemhavedifferentsam-plingperiodsindifferentfeedbackpathsRandomsamplingtkisarandomvariable1-2DigitalControlSystems13TypesofSamplingOperations1-SignalFormsinaDigitalControlSystem1-2DigitalControlSystemsFigure1-3Blockdiagramofadigitalcontrolsystemshowingsignalsinbinaryorgraphicform14SignalFormsinaDigitalContD/AandholdDigital-to-AnalogConverter(D/A)Alsocalledadecoder,isadevicethatconvertsadigitalsignalintoansampled-datasignal.Holdreconstructtheanalogsignalthathasbeentransmittedasatrainofpulsesamples,i.e.fillinthespacesbetweensamplingperiodsandthusroughlyreconstructtheoriginalanalogsignal1-2DigitalControlSystems15D/Aandhold1-2DigitalControPlantorProcessAplantisanphysicalobjecttobecontrolled.Wecallanyoperationtobecontrolledaprocess.AccuratemodelingisperhapsthemostdifficultpartinthedesignofcontrolsystemTransducerIsadevicethatconvertsaninputsignalintoanoutputsignalofanotherform,suchasadevicethatconvertsapressuresignalintoanvoltageoutput.Analogtransducer,sampled-datatransducer,digitaltransducer1-2DigitalControlSystems16PlantorProcess1-2DigitalCoThemainfunctionsinvolvedinA/Dconversionaresampling,amplitudequantizingandcodingAmplitudequantizingRepresentacontinuousoranalogsignalbyafinitenumberofdiscretestatesiscalledamplitudequantizationCodingorEncodingRepresentasamplevaluebyanumericalcode1-3QuantizingandQuantizationError17ThemainfunctionsinvolvedinQuantizingThestandardnumbersystemisthebinarynumbersystem.Thecodegroupconsistsofnpulseseachindicatingeither‘on’(1)or‘off’(0).Inthecaseofquantizing,n‘on-off’pulsescanrepresent2namplitudelevelsoroutputstates.ThequantizationlevelQ:therangebetweentwoadjacentdecisionpoints:Q=FSR/2n,FSRisthefull-scalerange.MSBisthemostsignificantbit,hasthemostweight(onehalfofthefullscale)LSBistheleastsignificantbit,hastheleastweightLSB=FSR/2n1-3QuantizingandQuantizationError18Quantizing1-3QuantizingandQQuantizationErrorSincedigitaloutputcanassumeonlyafinitenumberoflevels,ananalognumbermustberoundedofftothenearestdigitallevel.Quantizationerrorvariesbetween0and1/2Q.QuantizationerrordependsonfinenessoftheQ,andcanbemadeassmallasdesiredbymakingQsmaller.1-3QuantizingandQuantizationError19QuantizationError1-3Quantizi1-3QuantizingandQuantizationErrorTodeterminethedesiredsizeofthequantizationlevelinagivendigitalcontrolsystem,theengineermusthaveagoodunderstandingoftherelationshipbetweenthesizeofthequantizationlevelandtheresultingerror.201-3QuantizingandQuantizatioForananaloginputx(t),theoutputy(t)takesononlyafinitenumberoflevels,whichareintegralmultiplesofthequantizationlevelQ1-3QuantizingandQuantizationErrorFigure1-4(a)Blockdiagramofaquantizeranditsinput-outputcharacteristics21Forananaloginputx(t),theRound-offerrorTheerrorresultingfromneglectingtheremainingdigitsiscalledround-offerror.Quantizationerrorisaround-offerrorThefinerthequantizationlevelis,thesmallertheround-offerror.1-3QuantizingandQuantizationError22Round-offerror1-3QuantizingRound-offerror1-3QuantizingandQuantizationErrorFigure1-4(b)Analoginputx(t)anddiscreteoutputy(t)23Round-offerror1-3QuantizingQuantizationnoise:theuncertaintypresentinthequantizationprocess.ForasmallquantizationlevelQ,thequan-zationerrorissimilartothatofnoise.Soquantizationprocessactsasasourceofrandomnoise.Thevarianceofthequantizationnoiseis24Quantizationnoise:theuncert1-4DataAcquisition,ConversionandDistributionSystemsFigure1-5(a)Blockdiagramofadata-acquisitionsystem;251-4DataAcquisition,ConversiTransducerAphysicalvariablesuchasposition,velocity,acceleration,temperatureisfirstconvertedintoanelectricalsignal(avoltageorcurrent)AmplifierAmplifiesthevoltageoutputofthetransducerConvertsacurrentsignalintoavoltagesignalBuffersthesignal1-4DataAcquisition,ConversionandDistributionSystems26Transducer1-4DataAcquisitionLow-passfilterAttenuatesthehigh-frequencysignalcomponents,suchasnoise(electronicnoisesarerandominnatureandmaybereducedbylow-passfilters.However,suchcommonelectricalnoisesaspower-lineinterferencearegenerallyperiodicandmaybereducedbymeansofnotchfilters.)AnalogMultiplexingAdevicethatperformsthefunctionoftime-sharinganA/Dconverteramongmanyanalogchannels.1-4DataAcquisition,ConversionandDistributionSystems27Low-passfilter1-4DataAcquisIfmanysignalsaretobeprocessedbyasingleA/Dandadigitalcontroller,thentheseinputsignalsmustbefedtothecontrollerthroughamultiplexer.Isamultipleswitchthatsequentiallyswitchesamonginputchannelsinsomeprescribedfashion.Atagiveninstantoftime,onlyoneswitchisinthe“on”position.Whentheswitchisoninagiveninputchannel,theinputsignalisconnectedtotheoutputofthemultiplexerforaspecifiedperiodoftime.1-4DataAcquisition,ConversionandDistributionSystems28Ifmanysignalsaretobeproc1-4DataAcquisition,ConversionandDistributionSystems291-4DataAcquisition,ConversiDemultiplexerSeparatesthecompositeoutputdigitaldatafromthedigitalcontrollerintotheoriginalchannelsSample-and-HoldCircuitsSampler:covertananalogsignalintoatrainofamplitude-modulatedpulses.Holdcircuit:holdthevalueofthesampledpulsesignaloveraspecifiedperiodoftime1-4DataAcquisition,ConversionandDistributionSystems30Demultiplexer1-4DataAcquisittwooperationmodesThetrackingmode:theswitchisclosed,i.e.,theinputsignalisconnectedTheholdmode:theswitchisopen,i.e.,theinputsignalisdisconnectedWhenthesamplingdurationisnegligible,thesamplermaybeconsideredanidealsampler1-4DataAcquisition,ConversionandDistributionSystems31twooperationmodes1-4DataAc1-4DataAcquisition,ConversionandDistributionSystemsFigure1-7Sample-and-holdcircuit321-4DataAcquisition,Conversi1-4DataAcquisition,ConversionandDistributionSystemsFigure1-8Trackingmodeandholdmode331-4DataAcquisition,ConversiAnalog-to-DigitalConvertersTheprocessbywhichasampledanalogsignalisquantizedandconvertedtoabinarynumber.TypesoffrequentlyusedA/DConvertersSuccessive-approximationtypeIntegratingtypeCountertypeParalleltypeSelectioncriterionsofA/DconvertersConversionspeed,accuracy,sizeandcost1-4DataAcquisition,ConversionandDistributionSystems34Analog-to-DigitalConverters1-CountertypeA/D(thesimplestA/D)ClockpulsesareappliedtothedigitalcounterinsuchawaythattheoutputvoltageoftheD/Aconverter(thatis,partofthefeedbackloopintheA/Dconverter)issteppeduponeleastsignificantbit(LSB)atatime.Thentheoutputvoltageiscomparedwiththeanaloginputvoltageonceforeachpulse.Whentheoutputvoltagehasreachedthemagnitudeoftheinputvoltage,theclockpulsesarestopped.Thecounteroutputvoltageisthenthedigitaloutput.1-4DataAcquisition,ConversionandDistributionSystems35CountertypeA/D(thesimplestSuccessive-approximationtype(mostfrequentlyused)Theprincipleis:Thesuccessive-approximationregister(SAR)firstturnsonthemostsignificantbit(halfthemaximum)andcomparesitwiththeanaloginput.Thecomparatordecideswhethertoleavethebitonorturnitoff.Iftheanaloginputvoltageislarger,themostsignificantbitisseton.Next,turnonbit2andthencomparetheanaloginputvoltagewiththree-fourthsofthemaximum.1-4DataAcquisition,ConversionandDistributionSystems36Successive-approximationtypeAfterncomparisonsarecompleted,thedigitaloutputofthesuccessive-approximationregisterindicatesallthosebitsthatremainonandproducesthedesireddigitalcode.Thus,thistypeofA/Dconvertersets1biteachclockcycle,andsoitrequiresonlynclockcyclestogeneratenbits,wherenistheresolutionoftheconverterinbits.(Thenumbernofbitsemployeddeterminestheaccuracyofconversion.)Thetimerequiredfortheconversionisapproximately2μsecorlessfora12-bitconversion.1-4DataAcquisition,ConversionandDistributionSystems37Afterncomparisonsarecomple1-4DataAcquisition,ConversionandDistributionSystemsFigure1-9Schematicdiagramofasuccessive-approximation-typeofA/Dconverter381-4DataAcquisition,ConversiErrorsinA/DConvertersTheinput-outputcharacteristicsofA/DConverterschangewithtimeandtemperature.Actualanalog-to-digitalsignalconvertersalwayshavesomeerrors,suchasoffseterror,linearityerror,andgainerror.CommercialconvertersarespecifiedforthreebasictemperaturerangesCommercial(0oCto70oC)Industrial(-25oCto85oC)Military(-55oCto125oC)1-4DataAcquisition,ConversionandDistributionSystems39ErrorsinA/DConverters1-4DaErrorsinA/DConverters1-4DataAcquisition,ConversionandDistributionSystemsFigure1-10ErrorsinA/Dconverters(a)offseterror;(b)linearityerror;(c)gainerror40ErrorsinA/DConverters1-4DaThereverseofthedata-acquisitionAdata-distributionsystemconsistsofregisters,ademultiplexer,digital-to-analogconverters,andholdcircuits.Itconvertsthesignalindigitalform(binarynumbers)intoanalogform.Theoutputoftheholdcircuitisfedtotheanalogactuator,which,inturn,directlycontrolstheplantunderconsideration.1-4DataAcquisition,ConversionandDistributionSystemsFigure1-5(b)blockdiagramofadatadistributionsystem411-4DataAcquisition,ConversiDigital-to-AnalogConvertersForthefullrangeofthedigitalinput,thereare2ncorrespondingdifferentanalogvalues,including0.Forthedigital-to-analogconversion,thereisaone-to-onecorrespondencebetweenthedigitalinputandtheanalogoutputTwocommonD/AmethodsWeightedresistors:simpleincircuitconfiguration,butitsaccuracymaynotbeverygoodR-2Rladdernetwork:alittlemorecomplicatedincircuitconfiguration,butismoreaccurate.1-4DataAcquisition,ConversionandDistributionSystems42Digital-to-AnalogConverters1-Weightedresistors1-4DataAcquisition,ConversionandDistributionSystemsFigure1-11SchematicdiagramofaD/Aconverterusingweightedresistors43Weightedresistors1-4DataAcqWeightedresistorsTheinputresistorsoftheoperationalamplifierhavetheirresistancevaluesweightedinabinaryfashion.Whenthelogiccircuitreceivesbinary1,theswitchconnectstheresistortothereferencevoltage;whenthelogiccircuitreceivesbinary0,theswitchconnectstheresistortogroundNoticethat:asthenumberofbitsisincreased,therangeofresistorvaluesbecomeslargeandconsequentlytheaccuracybecomespoor1-4DataAcquisition,ConversionandDistributionSystems44Weightedresistors1-4DataAcqR-2Rladdernetwork1-4DataAcquisition,ConversionandDistributionSystemsFigure1-12n-bitD/AconverterusinganR-2Rladdercircuit45R-2Rladdernetwork1-4DataAcNotethat:withtheexceptionofthefeedbackresistor(whichis3R),allresistorsinvolvedareeitherRor2R.Thismeansthatahighlevelofaccuracycanbeachieved1-4DataAcquisition,ConversionandDistributionSystems46Notethat:withtheexceptionHoldCircuitsFillinthespacesbetweensamplingperiodsandthusroughlyreconstructtheoriginalanalogsignalTheholdcircuit:toextrapolatetheoutputsignalbetweensuccessivepointsaccordingtosomeprescribedmanner

Zero-order-hold:producesastaircasewaveform1-4DataAcquisition,ConversionandDistributionSystems47HoldCircuits1-4DataAcquisit1-4DataAcquisition,ConversionandDistributionSystemsFigure1-13Outputfromazero-orderhold481-4DataAcquisition,ConversiFirst-order-hold:generatesanoutputslopeequaltotheslopeofalinesegmentconnectingpreviousandpresentsamplesandprojectingitfromthevalueofthepresentsample.Moreaccuratelythanazero-orderhold.Iftheslopeoftheoriginalsignaldoesnotchangemuch,thepredictionisgood.If,however,theoriginalsignalreversesitsslope,thenthepredictioniswrong.Andtheoutputgoesinthewrongdirection,thuscausingalargeerrorforthesamplingperiodconsidered.1-4DataAcquisition,ConversionandDistributionSystems49First-order-hold:generatesan1-4DataAcquisition,ConversionandDistributionSystemsFigure1-14Outputfromafirst-orderhold501-4DataAcquisition,ConversiInterpolativefirst-orderhold:generatesastraight-lineoutputwhoseslopeisequaltothatjoiningtheprevioussamplevalueandthepresentsamplevalue,buttheprojectionismadefromthepredictionpoint.Itsaccuracyisbetterthanthatofotherholdcircuits,butthereisaone-samplingperioddelay.Fromtheviewpointofthestabilityofclosed-loopsystems,suchadelayisnotdesirable,andsotheinterpolativefirst-orderhold(polygonalhold)isnotusedincontrolsystemapplications.1-4DataAcquisition,ConversionandDistributionSystems51Interpolativefirst-orderhold1-4DataAcquisition,ConversionandDistributionSystemsFigure1-14Outputfromaninterpolativefirst-orderhold(polygonalhold)521-4DataAcquisition,ConversiDigitalControllersandAnalogControllerAnalogControllersrepresentthevariablesinanequationbycontinuousphysicalquantities.caneasilybedesignedtoservesatisfactoryasnon-decision-makingcontrollersthecostincreasesrapidlyasthecomplexityofthecomputationsincreases1-5ConcludingComments53DigitalControllersandAnalogDigitalControllersoperateonlyonnumbersdecisionmakingisoneoftheirimportantfunctionshandlenonlinearcontrolequationsinvolvingcomplicatedcomputationsorlogicoperationsuseverymuchwiderclassofcontrollawsflexibleinallowingprogrammingchangessuperiorinperformanceandlowerinpricethantheiranalogcounterpartshighlyreliable,andoftencompactandlightweightlesssensitivetonoisesignals1-5ConcludingComments54DigitalControllers1-5ConcluOutlineoftheBookTheobjectiveofthebook:Understand,analyzeanddesignCCSChapter1:introductorymaterialsChapter2:ztransformtheoryztransformsofelementaryfunctions,importantpropertiesandtheoremsoftheztransform,theinverseztransform,andthesolutionofdifferenceequationsbytheztransformmethodChapter3:zplaneanalysisofdiscrete-timecontrolsystemsImpulsesamplingandreconstructionoforiginalsignalsfromsampledsignals,pulsetransferfunctions,andrealizationofdigitalcontrollersanddigitalfilters1-5ConcludingComments55OutlineoftheBook1-5ConcluChapter4:Designofdiscrete-timecontrolsystemsbyconventionalmethodsmappingbetweenthesplaneandthezplane,stabilityanalysis,transientandsteady-stateresponseanalysis,designbytheroot-locusandfrequencyresponsemethods,andananalyticaldesignmethodChapter5:state-spaceanalysisState-spacerepresentation,thesolutionofdiscrete-timestatespaceequations,andthepulsetransferfunctionmatrix,discretizationofcontinuous-timestate-spaceequationsandLyapunovstabilityanalysisChapter6:Poleplacementandobserverdesigncontrollabilityandobservability,designtechniquesbasedonpoleplacement,fu11-orderstateobserversandminimum-orderstateobservers,thedesignofservosystem1-5ConcludingComments56Chapter4:Designofdiscrete-Computer-ControlledSystems12/24/2022Computer-ControlledSystems12CourseInformationTime:13:30-15:10pm,Wednesday 10:00-11:40am,Friday(evenweeks)Venue:綜B204References:離散時(shí)間控制系統(tǒng)(英文版第2版)，KatsuhikoOgata，機(jī)械工業(yè)出版社，2004離散時(shí)間控制系統(tǒng)(中文版)，KatsuhikoOgata，陳杰，蔡濤等譯，機(jī)械工業(yè)出版社，2006GradingProcedure:in-termevaluationexperimentalresultsfinalpaperexamination.58CourseInformationTime:1Chapter1IntroductiontoDiscrete-Time59Chapter1IntroductiontoDisc60Contents1-1Introduction1-2DigitalControlSystems1-3QuantizingandQuantizationError1-4DataAcquisition,ConversionandDistributionSystems1-5ConcludingComments604Contents1-1Introduction41-1IntroductionDigitalControllerArapidincreaseintheuseofdigitalcontrollers611-1IntroductionDigitalControDigitalControllerThetrendisduetoDigitalcontrolcanachieveoptimalperfor-mance,haveDecision-makingcapabilityandflexibilityinthecontrolprogramAvailablityoflow-costdigitalcomputersAdvantagesofdigitalsignals1-1Introduction62DigitalController1-1IntroducTypesofSignalsContinuous-timesignal:AsignaldefinedoveracontinuousrangeoftimeAnalogsignal:AsignaldefinedoveracontinuousrangeoftimewhoseamplitudecanassumeacontinuousrangeofvaluesAcontinuous-timequantizedsignalDiscrete-timesignal:Asignaldefinedonlyatdiscreteinstantsoftime1-1Introduction63TypesofSignals1-1IntroductiSampled-datasignal:Adiscrete-timesignaliftheamplitudecanassumeacontinuousrangeofvaluesDigitalsignal:Adiscrete-timesignalwithquantizedamplitudeComparisonDiscrete-time,digital,sampleddatasignal(controlsystem)Continuous-time,analogsignal(controlsystem)1-1Introduction64Sampled-datasignal:Adiscret1-1Introduction651-1Introduction9SystemsDealtWithinThisBookLinearandtimeinvariantDiscrete-TimeControlSystemsoneormorevariablescanchangeonlyatdiscreteinstantsoftime.TheseinstantsmayspecifythetimesatwhichsomephysicalmeasurementisperformedorthememoryofadigitalcomputerisreadoutDescribedinlineardifferenceequationswithconstantcoefficients1-1Introduction66SystemsDealtWithinThisBoo1-2DigitalControlSystemsFigure1-2Blockdiagramofadigitalcontrolsystem671-2DigitalControlSystemsFigS/HandA/D(A/D)Sample-and-Hold(S/H)SamplingProcesses,replaceoriginalcontinuous-timesignalbyasequenceofvaluesatdiscrete-timetimepointsacircuitthatreceivesanaloginputsignalandholdsthissignalataconstantvalueforaspecifiedperiodoftime.Analog-to-DigitalConverter(A/D)Alsocalledanencoder,isadevicethatconvertsananalogsignalintoadigitalsignal,usuallyanumericallycodedsignal.AS/HcircuitisoftenanintegralpartofacommerciallyavailableA/Dconverter.1-2DigitalControlSystems68S/HandA/D(A/D)1-2DigitalTypesofSamplingOperationsPeriodicsamplingtk=kT(k=0,1,2,…)Multiple-ordersamplingtk+r-tk=constantMultiple-ratesamplingAdigitalcontrolsystemhavedifferentsam-plingperiodsindifferentfeedbackpathsRandomsamplingtkisarandomvariable1-2DigitalControlSystems69TypesofSamplingOperations1-SignalFormsinaDigitalControlSystem1-2DigitalControlSystemsFigure1-3Blockdiagramofadigitalcontrolsystemshowingsignalsinbinaryorgraphicform70SignalFormsinaDigitalContD/AandholdDigital-to-AnalogConverter(D/A)Alsocalledadecoder,isadevicethatconvertsadigitalsignalintoansampled-datasignal.Holdreconstructtheanalogsignalthathasbeentransmittedasatrainofpulsesamples,i.e.fillinthespacesbetweensamplingperiodsandthusroughlyreconstructtheoriginalanalogsignal1-2DigitalControlSystems71D/Aandhold1-2DigitalControPlantorProcessAplantisanphysicalobjecttobecontrolled.Wecallanyoperationtobecontrolledaprocess.AccuratemodelingisperhapsthemostdifficultpartinthedesignofcontrolsystemTransducerIsadevicethatconvertsaninputsignalintoanoutputsignalofanotherform,suchasadevicethatconvertsapressuresignalintoanvoltageoutput.Analogtransducer,sampled-datatransducer,digitaltransducer1-2DigitalControlSystems72PlantorProcess1-2DigitalCoThemainfunctionsinvolvedinA/Dconversionaresampling,amplitudequantizingandcodingAmplitudequantizingRepresentacontinuousoranalogsignalbyafinitenumberofdiscretestatesiscalledamplitudequantizationCodingorEncodingRepresentasamplevaluebyanumericalcode1-3QuantizingandQuantizationError73ThemainfunctionsinvolvedinQuantizingThestandardnumbersystemisthebinarynumbersystem.Thecodegroupconsistsofnpulseseachindicatingeither‘on’(1)or‘off’(0).Inthecaseofquantizing,n‘on-off’pulsescanrepresent2namplitudelevelsoroutputstates.ThequantizationlevelQ:therangebetweentwoadjacentdecisionpoints:Q=FSR/2n,FSRisthefull-scalerange.MSBisthemostsignificantbit,hasthemostweight(onehalfofthefullscale)LSBistheleastsignificantbit,hastheleastweightLSB=FSR/2n1-3QuantizingandQuantizationError74Quantizing1-3QuantizingandQQuantizationErrorSincedigitaloutputcanassumeonlyafinitenumberoflevels,ananalognumbermustberoundedofftothenearestdigitallevel.Quantizationerrorvariesbetween0and1/2Q.QuantizationerrordependsonfinenessoftheQ,andcanbemadeassmallasdesiredbymakingQsmaller.1-3QuantizingandQuantizationError75QuantizationError1-3Quantizi1-3QuantizingandQuantizationErrorTodeterminethedesiredsizeofthequantizationlevelinagivendigitalcontrolsystem,theengineermusthaveagoodunderstandingoftherelationshipbetweenthesizeofthequantizationlevelandtheresultingerror.761-3QuantizingandQuantizatioForananaloginputx(t),theoutputy(t)takesononlyafinitenumberoflevels,whichareintegralmultiplesofthequantizationlevelQ1-3QuantizingandQuantizationErrorFigure1-4(a)Blockdiagramofaquantizeranditsinput-outputcharacteristics77Forananaloginputx(t),theRound-offerrorTheerrorresultingfromneglectingtheremainingdigitsiscalledround-offerror.Quantizationerrorisaround-offerrorThefinerthequantizationlevelis,thesmallertheround-offerror.1-3QuantizingandQuantizationError78Round-offerror1-3QuantizingRound-offerror1-3QuantizingandQuantizationErrorFigure1-4(b)Analoginputx(t)anddiscreteoutputy(t)79Round-offerror1-3QuantizingQuantizationnoise:theuncertaintypresentinthequantizationprocess.ForasmallquantizationlevelQ,thequan-zationerrorissimilartothatofnoise.Soquantizationprocessactsasasourceofrandomnoise.Thevarianceofthequantizationnoiseis80Quantizationnoise:theuncert1-4DataAcquisition,ConversionandDistributionSystemsFigure1-5(a)Blockdiagramofadata-acquisitionsystem;811-4DataAcquisition,ConversiTransducerAphysicalvariablesuchasposition,velocity,acceleration,temperatureisfirstconvertedintoanelectricalsignal(avoltageorcurrent)AmplifierAmplifiesthevoltageoutputofthetransducerConvertsacurrentsignalintoavoltagesignalBuffersthesignal1-4DataAcquisition,ConversionandDistributionSystems82Transducer1-4DataAcquisitionLow-passfilterAttenuatesthehigh-frequencysignalcomponents,suchasnoise(electronicnoisesarerandominnatureandmaybereducedbylow-passfilters.However,suchcommonelectricalnoisesaspower-