江南大学计算机控制系统巴基斯坦老师课件

ComputerControlSystemDesign-2PresentedBy:MJunaidKhan

AssociateProfessor,Dept.ofElectronicandPowerEngineeringcontactjunaidationalUniversityofScienceandTechnologyPakistan1ContentsReviewoflastlectureDesignbyEmulation-IndirectDesignMethodMethodstoDiscretizeContinuousControllerForwardRectangularRuleBackwardRectangularRuleTrapezoidalRuleBilinearORTustin’sTransformationZOHEquivalent–StepInvarianceMethodPoleZeroMapping–MatchedPoleZeroMappingBilinearTransformationwithFrequencyPre-warpingAnalyzingPerformanceofDiscreteSystemNumericalIntegrationMethods2Review-

TwoWaystoDesignaDigitalControllerIndirectDesign:

Firstdesignacontinuous

time

controllerandthendiscretizeit usingsomediscretization

techniquetoobtainanequivalent digitalcontroller.DirectDesign:

Discretizetheplantfirsttoobtainadiscrete-timesystemand thenapplydigital

controlsystemdesigntechniquesIndirectDesignDirectDesign3Review-

StrategyofIndirectDesignHavingacontinuoustransferfunctionD(s),findthebestdiscreteequivalentD(z)usinganysuitablemethodofconversion.Judgetheeffectivenessofthedigitaldesignbycomparingit’sfrequencyresponsewiththatofD(s)SelectedsamplingfrequencyiskepthigherthanfrequencyoftheinputsignalsForinputsignalswhichareathighfrequencyi.e.approachingtheNyquistrate(fs/2)orfoldingfrequency,thefidelityofD(z)comparedwithD(s)willdeteriorate.Thismeansthat:

ifthesamplingfrequency‘fs’islessthandoubleofsignalfrequency,theperformanceofD(z)willbebad.4Review-

DiscreteApproximationsForwardRectangularRule:

Itissimpletoapply,butastablesystemcanbecomeunstable,soitisimpracticaltousethisapproximation.BackwardRectangularRule

:Astablesystemwillresultinastablesystem,buttherearelargedistortionsindynamicresponseandfrequencyresponsepropertiesTrapezoidalRule

:Astablesystemwillremainstable,howeveritcancausefrequencydistortionorwarping.Frequencypre-warpingcandecreasethedistortioninfrequencyresponse.5Review-DiscreteApproximations

RemarksForwardRectangularRuleisnotusedinpracticalapplications.BackwardRectangularRulealwaysmapsastablecontinuouscontrollertoastablediscretecontroller.However,someunstablecontinuouscontrollercanalsobetransformedintostablediscretecontrollersThebilineartransformation(trapezoidalorTustin’sapproximation)mapsthelefthalfsplaneintotheunitdisc.Hence,stablecontinuouscontrollersareapproximatedbystablediscretecontrollersandunstablecontinuouscontrollersaremappedtounstablediscretecontrollersInpractice,theTustin’sapproximation(bilineartransformation)istheapproximationofchoiceforconvertingcontinuous-timecontrollerstodiscrete-timecontrollers.Infact,somecomputer-aidedprograms(e.g.MATLAB)don’tevenhavetheoptiontoapproximatewithforwardorbackwarddifferencemethods6IndirectdesignmethodStrategies:1.EmulationbyZOHEquivalent-Step-invariancemethod

Thismethodsimplyassumesthatthesignalenteringthemicroprocessorisconstantoverthesamplingtime(thefunctionoftheZOHDAContheoutputsignal)7Indirectdesignmethod

ZOHEquivalentorStep-invariancemethodConvertD(s)to(D(z)(suitableforimplementationonamicroprocessor)basedonasamplingtimeof0.1secondbyZOHmethod.Example…8Indirectdesignmethod

ZOHEquivalentorStep-invariancemethodRemarks:

1.Astablesystemwillremainstable2.Frequencyfoldingphenomenamayoccur,butthankstothelow-passcharacteristicsoftheZOH,itisalittlebetter.3.Complexcomputationforlarge-scalesystems4.Steady-statevalueisinvariant,i.e., G(s)|s=0=H(z)|z=19Indirectdesignmethod2.PoleandZeroMappingSinceeverypoleandzeroofD(s)inthes-planehasitsequivalentpositioninthez-planethroughthemapping:thenitsseemsreasonabletoformD(z)fromD(s)bymappingthepositionsofthepolesandzeroesinterms's'topositionsinthez-planeusingequationsabove.AsimpleexamplewilldemonstratetheMethod.IfThenthepositionsofthefinitepolesandzeroesofD(s)are:10Indirectdesignmethod2.PoleandZeroMappingUsingthemapping,thesemaptopositionsinthes-planegivenby:ThusD(z)isgivenby:ThevalueofK'isselectedtoensurethegainofD(s)andD(z)arethesameatsomespecificfrequency,usuallyzerofrequency(DCgain).TheDCgaininthes-planeisdeterminedwhens=0andinthez-planewhenz=111IndirectdesignmethodPoleandZeroMappingThusforequalDCgain:Andthustheequivalenttransferfunctionisgivenby:12IndirectdesignmethodThisisapopularmethodandhasavalidrational,andfortransferfunctionswithasmanyzeroesaspolesinD(s)itisareasonableapproach.Howeverinmanycontrollertransferfunctionsthisisnotthecase.Forexamplethetransferfunction:Ithastwopoless=0andbandtwozeroess=aand∞.Thedifficultyismappingthesat∞.Somedesignersplaceitatz=0andsomeatz=-1whichbecauseofthenatureofthez-plane(duetothepeculiarnatureofthemappingequations)arebothreasonabledecisions.However,thisisnotverysatisfactoryandevenwithoutthisproblemthemethoddoesnotalwaysworkPoleandZeroMapping13IndirectdesignmethodExample2…Obtainanexpressionforthecontrollerindiscreteformusingthepole/zeromappingmethod.Expressyouranswersinrecursiveformsuitableforimplementationonamicroprocessor.PoleandZeroMapping14IndirectdesignmethodTheBilinearorTustin'sTransformationInsteadofassumingtheinputsignalisheldconstantbetweensamples(theZOHmethod),thismethodassumesthattheprocessismoreaccurateifastraightlinebetweensuccessivesamplesoftheinputisconsidered(sameasinTrapezoidalMethod)andisabetterapproximationtowhatishappeningbetweensamplesasshownbelow:15IndirectdesignmethodTheBilinearorTustin'sTransformationTustinsuggestedthatforthesampledsystemtheprocessofsignalintegrationcanbeapproximatedby:Intheaboveyrepresenttheintegralofx.Takingthez-transformoftheaboveandre-arrangingintotransferfunctionformgives:16IndirectdesignmethodTheBilinearorTustin'sTransformationIntegrationincontinuoussystemsisrepresentedbytheLaplacetransferfunction1/s

,hencethemappingfromthes-domaintothez-domainisapproximatedby:yrepresenttheintegralofx.Takingthez-transformoftheaboveandRe-arrangingintotransferfunctionformgives:17IndirectdesignmethodTheBilinearorTustin'sTransformationThisEquationisTustin'smappingandtheideaisthateverywheresappearsinD(s),theequationissubstitutedforit.18Indirectdesignmethod

Designexample:19Indirectdesignmethod

Designexample:20Indirectdesignmethod

Designexample:CalculatingdesiredcontrollerparametersTheclosed-looptransferfunctionofthecruisecontrolsystemwiththePIcontrollaw,i.e.,21Indirectdesignmethod

Designexample:VerificationthroughSIMULINKTheclosed-looptransferfunctionofthecruisecontrolsystemwiththePIcontrollaw,i.e.,22Indirectdesignmethod

Designexample:Digitalcontrollerwithasamplingrate30timesthebandwidth23Indirectdesignmethod

Designexample:Digitalcontrollerwithasamplingrate6timesthebandwidth24SummaryIndirectDigitalcontrollerdesigncanbeobtainedasfollows: Approximationusingforwardrectangularrule

Approximationusingbackwardrectangularrule

DesignbyEmulationwithZOH

Designthroughpole-zeromapping

DesignUsingBilinearTransformationThemethodoftransformationplaysasignificantroleintheperformanceoftheobtaineddigitalsystemChoiceofsamplingtime/frequencyplaysamajorroleintheperformanceoftheobtaineddigitalsystem25Indirectdesignmethod

FrequencyWarpinginBilinearTransformationNotethattheentire

axismapsintoonecompleterevolutionoftheunitcircle.

(mapsaxisintoinfinitenumberofrevolutionsoftheunitcircle)Bilinearand

transformationshaveconsiderabledifferencesbetweenthemintheirtransientandfrequencyresponsecharacteristics.26Indirectdesignmethod

FrequencyWarping-DefinitionFrequencywarpingtransformationisaprocesswhereonespectralrepresentationonacertainfrequencyscale(e.g.,z,s-domain)andwithacertainfrequencyresolution(mostoftenuniform)istransformedtoanotherrepresentationonanewfrequencyscale.Thenewrepresentationhasauniformfrequencyresolutiononthenewscale-however,ithasanon-uniformresolutionwhenobservedfromtheoldscale.Thewarpingfunctiondefineshowindividualfrequencycomponentsanddifferentfrequencyrangesaremappedonthenewscale.Italsodefineshowtheresolutionofthenewrepresentationisallocated,i.e.whichrangesintheoriginalrepresentationarecompressed(shrinked,resolutionreduced)andwhichexpanded(stretched,resolutionincreased).27Indirectdesignmethod

FrequencyWarpingFrequencyWarping

Itiseasytocheckthatthebilineartransformgivesaone-to-one,order-preserving,conformalmapbetweentheanalogfrequencyaxis

andthe

digitalfrequencyaxis

,where

isthesamplinginterval.Therefore,theamplituderesponsetakesonexactlythesamevaluesoverbothaxes,withtheonlydefectbeingafrequencywarpingsuchthatequalincrementsalongtheunitcircleinthe

planecorrespondtolargerandlarger

bandwidthsalongthe

axisinthe

plane.Somekindoffrequencywarpingisobviouslyunavoidableinanyone-to-onemapbecausetheanalogfrequencyaxisisinfinitewhilethedigitalfrequencyaxisisfinite.Therelationbetweentheanaloganddigitalfrequencyaxesmaybederivedimmediately28Indirectdesignmethod

FrequencyWarpingForFrequencypre-warping,thecontinuoustimefilterisUsingbilinear/Tustintransformation,transferfunctioninz-domainisSetComparingfrequencyresponsesThisshowsthefrequencydistortionorwarping29NowifisverysmallAndifTheresponsesareequalwhenalsocalledpre-warpingequalityIndirectdesignmethod

FrequencyWarping30Indirectdesignmethod

FrequencyPre-WarpingProcedureforpre-warping1.

Warpthefrequencyscalebeforetransforming2.

TransformusingBilinear31Indirectdesignmethod

FrequencyPre-WarpingExampleAssumethattheintegrator

hastobeimplementedasadigitalfilterUsingBilinearTransformationPre-warpinggivesThefrequencyfunctionofisgivenby:32Indirectdesignmethod

FrequencyPre-WarpingExampleAssumethattheintegrator

hastobeimplementedasadigitalfilterUsingBilinearTransformationwithpre-warpingThefrequencyfunctionAt:Thedistortioninthefrequencyresponsecanbecorrectedatasinglefrequencyusingthepre-warpingequality33Indirectdesignmethod

FrequencyPre-WarpingRemarks1.Thepre-warpingequalityisgivenby3.Thechoiceofpre-warpingfrequencydependsonthemappedfilter4.Incontrolapplications,asuitablechoiceofisthe3-dBfrequencyforaPIorPDcontrollerandtheupper3-dBfrequencyforaPIDcontroller5.InMATLAB,thebilineartransformationisaccomplishedusingthefollowingcommand>>Gd=c2d(Gc,T,‘tustin’)6.Ifpre-warpingisrequestedatafrequencyw,thenthecommandis:>>Gd=c2d(Gc,T,‘prewarp’,w)2.Thedistortioninthefrequencyresponsecanbecorrectedatasinglefrequencyusingthepre-warpingequality34EquivalentDiscreteTimeFiltersforaContinuousTimeFilterMappingMethodMappingEquationEquivalentDiscreteTimeFiltersfor

ForwardRectangularRule

NotrecommendedBackwardRectangularRuleTrapezoidalRuleBilinear/Tustin35EquivalentDiscreteTimeFiltersforaContinuousTimeFilterMappingMethodMappingEquationEquivalentDiscreteTimeFiltersforBilinear/TustinBilinearwithfrequencyprewarpingZOHEquivalentorStepInvarianceMatchedPoleZeroMappingApole/zeroats=-aismappedtoAninfinitepole/zeroismappedtoz=-1

36Example1:bode(1,[11])holdondbode([11],[3-1],1)Indirectdesignmethod37Example2:sys_c=tf([119],[129]);s