机械工程师手册,仪器仪表,系统,控制系统,和MEMch15.doc

620CHAPTER15CONTROLLERDESIGNThomasPeterNealConsultantLakeView,NY1INTRODUCTION6202FUNDAMENTALSOFCLOSED-LOOPPERFORMANCE6212.1AccuracyandLoopGain6212.2DynamicResponseandStability6253FREQUENCYCOMPENSATIONTOIMPROVEOVERALLPERFORMANCE6323.1Well-DampedSystems6333.2PoorlyDampedSystems6373.3HigherOrderEffects6414INNERFEEDBACKLOOPS6454.1DerivativesoftheControlledVariable6464.2AlternativeInnerLoopVariables6494.3NonelectronicInnerLoops6505PREFILTERSANDFEEDFORWARD6515.1LagPrelters6515.2LeadPrelters6525.3Feedforward6546PIDCONTROLLERS6566.1EquivalencetoFrequencyCompensation6566.2SystemsHavingNoInherentIntegration6596.3SystemsHavinganInherentIntegration6607EFFECTSOFNONLINEARITIES6607.1SimpleNonlinearities6637.2ComplexNonlinearities6687.3ComputerSimulation6688CONTROLLERIMPLEMENTATION6698.1AnalogControllers6698.2Hard-WiredDigitalControllers6728.3Computer-BasedDigitalControllers673REFERENCES6771INTRODUCTIONThepurposeofthischapteristoprovideabasisforthespecicationandfunctionaldesignofelectronicservocontrollers.Noattemptismadetotreatthesubjectofelectroniccircuitdesign.Instead,thegoalistoaidtheengineerinselectingandapplyingasuitableoff-the-shelfcontrollerorinspecifyingthecontrollerrequirementstoacircuitdesigner.Theem-phasisisonposition,velocity,orforcecontrolofmechanicalloads,althoughmanyofthetechniquesareapplicabletocontrollerdesigningeneral.Specializedsubjectssuchasmul-tiaxiscontrolandadaptivecontrolarebeyondthescopeofthischapter.Asastartingpoint,itispresumedthataservoactuatorhasbeenselectedandmounted,togetherwithasuitablepowersupply,driveamplier,andmechanicaldrivemechanism.Inaddition,theprimaryfeedbacktransducerhasbeenchosen,andasimpleloopclosurehasbeenanalyzedtodeterminewhetherthespeciedclosed-loopperformancecanbeobtained.ReprintedfromInstrumentationandControl,Wiley,NewYork,1990,bypermissionofthepublisher.Mechanical Engineers Handbook: Instrumentation, Systems, Controls, and MEMS, Volume 2, Third Edition.Edited by Myer KutzCopyright 2006 by John Wiley & Sons, Inc.2FundamentalsofClosed-LoopPerformance621Figure1Generalizedservomechanism.TheprocessofaccomplishingthesetasksistreatedinChapter14.Ifasimpleloopclosureprovidesadequateperformance,thecontrollerdesignproblemprimarilyconsistsofmakingsomebasicdecisionsconcerningelectronicimplementation.Inmanyapplications,asimpleloopclosureisinadequate,andmoreelaboratecontrollerfunctionsarerequired.Theselattercasesaretheprimarysubjectofthischapter.Thethrustofthediscussionissynthesisofthecontrollerfunction,ratherthananalysisofanexistingdesign.Forthisreason,continuous-timefrequency-domaintechniqueswillbeusedexten-sively,allstartingfromblockdiagramsandtransferfunctionsbasedontheLaplaceopera-tor.13Thesetechniques,manyofwhicharegraphical,areparticularlyusefulintheearlydesignstages.Sincemostpeopleintuitivelyrelatetotime-domainresponses,relationshipsbetweenthefrequency-domainresultsandtimehistorieswillbediscussedasappropriate.Ifthecontrolsystemistobeimplementedindigitalhardwareorsoftware,itiscertainlypossibletohandletheentiredesigntaskusingthemathematicsofsampled-datasystems.47Thisapproachhasnotbeenusedherebecausetheso-calledclassicaltechniquesbasedontheLaplacetransformaremoreillustrative,anddesigntrade-offsareeasiertoevaluate.Forthecontrolsystemapplicationsbeingconsideredhere,itisgenerallynecessarytokeeptheresolutionhighandthesamplingintervalsmall.Inthiscase,controllercharacteristicsde-scribedastransferfunctionsinLaplaceformcanbeaccuratelytransformedintovariousmathematicalformsappropriatefordigitalimplementation,forexampletheZ-transform.2FUNDAMENTALSOFCLOSED-LOOPPERFORMANCEToproperlydesignaservocontroller,itisnecessarytomaintainaclearpictureofthedesiredendresult,namely,theachievementofsomepredeterminedperformancegoals.Theseper-formancespecicationsshouldbeestablishedearlyinthedesign.Asaminimum,theyshoulddenethedesiredstaticanddynamicaccuracy,bandwidth(responsetime),andstability.Thefollowingsectionsofferabriefreviewoftheseimportantfactorsandhowtheyrelatetobasicloopparameters.2.1AccuracyandLoopGainThemostbasicrequirementofaservomechanismisprobablystaticaccuracy;thatis,thecontrolledvariablemustaccuratelyholdthecommandsetpoint.ReferringtothegeneralizedblockdiagramofFig.1,severalsourcesofinaccuracycanbedescribed.Anexternaldis-turbancecancausetheloadtomovewithoutanychangeinthecommandsignal.Theloadwillcontinuetomoveuntiltheresultingerrorsignalcausestheactuatortobalancethe622ControllerDesignFigure2Simpliedvelocityservo.disturbance.Anomaliesintheactuatorandloadmustalsobeoffsetbyaniteerrorsignal.Examplesaretemperature-inducednullshifts,hysteresis,threshold,friction,andlostmotion.Themagnitudeoftheseerrorsignalsisminimizediftheampliergainishigh.Ideally,theampliergainwouldbesethighenoughthattheaccuracyoftheservobecomesdependentonlyupontheaccuracyofthetransduceritself.Inpractice,however,theampliergainislimitedbystabilityconsiderations.Therefore,itisdesirabletoprovidehighgainsatlowfrequenciesforaccuracyandlowgainsathighfrequenciestominimizestabilityproblems.Sincetherateofamplituderoll-offwithfrequencyisdirectlyrelatedtophaselag,excessiveroll-offcancreatemorestabilityproblemsthanitsolves.Agoodcompromiseistomaketheentireforwardpathlooklikeanintegratoroverthefrequencyrangeofinterest(atypeIsystem).Thistechniqueisverycommonlyusedtogivenearlyinnitestaticgainandalineargainroll-offwithfrequency,atthecostof90phaselag.Itisimportanttonotethatsomeservoloopscontainaninherentintegrator,whichcom-plicatestheaccuracy-versus-stabilityproblem.Forexample,manyactuatorsareinherentlyratedeviceswhenoperatedopenloop,sothatasteadyinputresultsinaproportionalvelocityoutput.Avelocityservousingsuchanactuatorwillinherentlyhaveaproportionalforwardloop,andanintegratingservoampliercanbeused(Fig.2).However,thecorrespondingpositionservowillinherentlyhaveanintegrationintheforwardloop,asshowninFig.3.Inthislattercase,theuseofanintegratingservoampliercancauseseverestabilityproblems.Fromthesegures,transferfunctionscanbewrittenfortheclosed-loopresponsestocom-mandanddisturbanceinputs.Notethatthedynamicresponsecharacteristicsofallelementshavebeenneglected,anditisassumedthattheloadincludesnospringtoground:V11(1)eKs/K1cvvvVsK(2)4FsKdvvX11(3)eKs/K1cxvxXK14(4)FKs/K1dvxvx2FundamentalsofClosed-LoopPerformance623Figure3Simpliedpositionservo.whereeccommandsignalVeeerrorsignal,Vevvelocityfeedbacksignal,Vexpositionfeedbacksignal,VFddisturbanceforceappliedtotheload,NK1integratingservoampliergain,(V/s)/VK2proportionalservoampliergain,V/VK3actuatorgain,(mm/s)/VK4actuatorvelocitydroopduetoforcedisturbance,(mm/s)/NKvvelocitytransducergain,volts/(mm/s)Kxpositiontransducergain,volts/mmKvvopen-loopgainofvelocityservoK1K3Kv,s1Kvxopen-loopgainofpositionservoK2K3Kx,s1Xloadposition,mmVX,mm/sAsshowninFig.4,thevelocityandpositionresponsestocommandsarebothchar-acterizedbyarst-orderlaghavingabreakfrequencyequaltotheopen-loopgain.However,theresponsestodisturbanceforcesarequitedifferentinthetwocases(Fig.5).Whenthedisturbanceisdownstreamoftheintegrator(velocityservo),theservoerrorisK4Fdathighfrequenciesbutrollsoffatfrequencies(inradianspersecond)belowKvvandiszerostatically.Whenthedisturbanceisupstreamoftheintegrator(positionservo),thereisastaticerrorinverselyproportionaltotheopen-loopgain,whichrollsoffatfrequenciesaboveKvx.NotethatEqs.(1)(4)remainreasonablyvalidwhenthedynamicresponsecharacteristicsofthevariousopen-loopelementsareconsidered,forthosecasesinwhichKvv(orKvx)iswellbelowthelowestbreakfrequenciesofthoseelements.Athigherloopgains,theclosed-loopdynamicscanchangeconsiderably,asshowninthefollowingdiscussion.Theconclusionsregardingtheeffectsofdisturbanceforcesonservoaccuracycanbegeneralizedtoanyforward-loopoffsetoruncertainty.ReferringagaintoFig.2,itcanbeseenthattheintegratingamplierwillcompensateforanyforward-loopoffsetdownstreamoftheintegrator,sothatthestaticerrorsarezero.FromFig.3,itisapparentthatstaticerrorsduetooffsetsupstreamoftheintegrationcanbequantiedasX1e(5)VK0vx624ControllerDesignFigure4Simpliedresponsetocommands.Figure5Simpliedresponsetodisturbance.whereXe裠outputerror裠Xc裠X,mmV0裠forward-loopoffset,convertedtoanequivalentopen-loopoffsetinvolts,mm/sXc裠positioncommand裠ec/Kx,mmEvenwhennoforward-loopoffsetsordisturbancesarepresent,servosexhibitfollowingerrors,sometimescalledtrackingerrors.Inservoshavingforward-loopintegrations,thesequasi-staticerrorsresultwheneverthecommandsignalchangesataconstantrate,asinaposition-trackingservo.ForthepositionservoofFig.3,thefollowingerroriseX1ee裠裠(6)eXKccvxForthevelocityservoofFig.2,thefollowingerroris2FundamentalsofClosed-LoopPerformance625eV1ee(7)eVKccvvwhereVeoutputerrorVcV,mm/sVcvelocitycommandec/Kv,mm/sNotethatthefollowingerrorforthepositionservoisthepositionerrorresultingfromasteadyrateofchangeofpositioncommand.Forthevelocityservo,thefollowingerroristhevelocityerrorresultingfromasteadyrateofchangeofvelocitycommand.Theservoerrorsdiscussedthusfarhavebeenthosethatcanbeminimizedbyatightservoloop(highloopgain).Tothesemustbeaddederrorsinthetransducermechanism.Evenifinniteloopgainwereachievable,theservocanbenomoreaccuratethanthetransduceritself.Themostimportanttypesoftransducerinaccuraciesarerepeatability,res-olution,andlinearity.Errorsduetotransducerlocationandmountinggeometrymustalsobetakenintoaccount.Manyoftheforegoingconceptscanbeappliedtoservosingeneral.Forexample,aforceorpressureservoworkingagainstaspringloadissimilartoapositionservointhesensethatoutputforceisproportionaltoactuatorposition.Also,temperaturecontrolservostendtobehavelikepositionservossincethecontrollingdevicetendstoprovideheatowproportionaltotemperatureerror,andthermalloadstendtoproducetemperaturerateofchangeproportionaltoheatow.2.2DynamicResponseandStabilityAsdiscussedinSection2.1,open-loopgainhasastronginuenceonservoaccuracy.Highloopgainsalsoprovidefastdynamicresponseinmostcases.However,stabilityconsidera-tionswilllimitthemaximumusefulloopgain.Thedynamicresponseandstabilityofaservoaredeterminedbythedynamiccharacteristicsofthevariousloopcomponents.Inmanysituations,theforward-loopdynamicsaredominatedbyarelativelysmallnumberoflow-frequencylagelements,andthetransducerdynamicsarenegligible.Inthesecases,itisoftenpossibletoobtainanadequateestimateofservoperformancebyapproximatingthecombinedforward-loopcharacteristicswithanintegratorplusarst-orderorsecond-orderlag.Theadequacyofthisapproximationcanbedeterminedbythematchofthefrequencyresponsegainandphaseforthefrequencyrangeinwhichthephaselagislessthan180.Usingthesetworatherbasicdynamicforms,therelationshipsamongloopgain,stability,anddynamicresponseareeasilyseen.AblockdiagramusingthebasicdynamicformsisshowninFig.6,whereUgeneralizedcontrolledvariableUcgeneralizedcommandinputDgeneralizeddisturbanceinputGdopen-loopresponseofUtoDTherst-orderlagistypicalofsimpletemperaturecontrolsystemsanddcservoshavingshortelectricaltimeconstants.Thesecond-orderlagisoftenrepresentativeofelectrohy-draulicservosanddcservoshavinglongelectricaltimeconstants.Theclosed-loopresponsestocommandinputsareU11(8)2U(/K)s(1/K)s1c1u1u1and626ControllerDesignFigure6Basicdynamiccongurations:(a)rst-orderlag;(b)second-orderlag.Figure7Rootlociforbasicdynamiccongurations:(a)rst-orderlag;(b)second-orderlag.U12岠(9)322Us/K岠岠(2岠/K岠)s岠(1/K)s岠1cu222u22u2RepresentativerootlociareshowninFig.7forbothforms.InFig.7a,thelagcombineswiththeintegratortoproducesecond-orderclosed-looppoles.Theclosed-loopnaturalfre-quencyincreaseswith1oopgain,whilethedampingratiodecreases.InthecaseofFig.7b,2FundamentalsofClosed-LoopPerformance627Figure8Closed-loopfrequencyresponsesforU1/Uc.theclosed-looptransferfunctionconsistsofarst-orderandasecond-orderlag.Thebreakfrequencyoftherst-orderlagincreaseswith1oopgain,whilethesecond-orderdampingratiorapidlydecreases.Inbothloopclosures,thereareclearlytrade-offsbetweenclosed-loopbandwidthandstability.Therearenumerousmethodsforquantifyingtherelationshipsbetweenbandwidthandstability.Closed-loopfrequencyresponsestocommandinputsareshownforbothbasicformsinFigs.8and9,whileFigs.10and11presentthecorrespondingstepresponses.Usefulnumericalmeasuresofstabilityarephasemargin,gainmargin,anddampingratiooftheclosed-loopcomplexpair.ThesearegiveninFigs.12and13.NotethatthegainmarginforFig.12isinnite.ReferringtoFig.6,closed-loopresponsestodisturbanceinputscanbewrittenasUGs(爠s爠1)1d1爠(10)2DK(爠/K)s爠(1/K)s爠1u11u1u1and22UGss/爠爠(2爠/爠)s爠12d222爠(11)322DK(s/K爠)爠(2爠/K爠)s爠(1/K)s爠1u2u222u22u2Todeterminethenaldynamicformoftheseresponses,itisnecessarytohaveatransferfunctionforGd.Thistransferfunctioncanbeobtainedfromthephysicalmodelofthesystem,byderivingtheresponseofthecontrolledvariabletothedisturbanceinputwiththecontrolleroutputequaltozero.Asanexample,consideradcmotordrivinganinertialloadandhavingashortelectricaltimeconstant.Foranintegratingvelocityloop,thedisturbancetransferfunctionhastheform628ControllerDesignFigure9Closed-loopfrequencyresponsesforU2/Uc:(a)钠2钠0.1;(b)钠2钠0.2.2FundamentalsofClosed-LoopPerformance629Figure9(Continued)(c)頠2頠0.4;(d)頠2頠0.8.630ControllerDesignFigure10Closed-loopstepresponsesforU1/Uc.Figure11Closed-loopstepresponsesforU2/Uc:(a)鬠2鬠0.1.2FundamentalsofClosed-LoopPerformance631Figure11(Continued)(b)20.2;(c)20.4.632ControllerDesignFigure11(Continued)(d)20.8.KdG(12)ds11andUKs1d(13)2DK(/K)s(1/K)s1u11u1u1ForapositionloopKdG(14)ds(s1)1andUK11d(15)2DK(/K)s(1/K)s1u11u1u1ItisinstructivetocompareEqs.(13)and(15)withEqs.(2)and(4),respectively.3FREQUENCYCOMPENSATIONTOIMPROVEOVERALLPERFORMANCEInSection2,itisclearthatopen-loopdynamiccharacteristicsimposeprofoundlimitationsuponclosed-loopperformance.However,itisoftenpossibletoextendtheselimitationsby3FrequencyCompensationtoImproveOverallPerformance633Figure12StabilityparametersforU1/Uc.modifyingtheinherentopen-loopdynamicswithfrequencycompensation(shaping).Therearemanytechniquesfordesigningcompensators.Thebesttechniquetouseisafunctionoftheparticularopen-loopdynamicsunderconsideration,aswellasclosed-loopperformancegoals.Thefollowingsectionsdescribesometechniquesthatareusefulinvariouscommonlyencounteredsituations.3.1Well-DampedSystemsAsmentionedinSection2.1,anidealformforthecombinedforward-looptransferfunctionisanintegrator.Thisensuresveryhighgainsatlowfrequencies,alineargainroll-offwithfrequency,andonly90ofphaselag.Forsystemsinwhichthedominantopen-looppolesarereasonablywelldamped,loopgainisusuallylimitedbyphaselag.Thisisclearlyillus-tratedbyFigs.12and13,inwhichphasemarginsdeterioratefasterthangainmargins(exceptwhen2islow).Anobviouswaytoimprovephasemarginsistomaketheopen-looptransferfunctionlooklikeanintegratorouttohigherfrequencies.Thiscanbeaccomplishedbyusingaleadcompensator,whosezerosareidenticaltothedominantforward-looppoles.Tomakethecompensatorphysicallyrealizable,itmusthaveatleastasmanypolesaszeros,butthesepolescanbeplacedathigherfrequencies.Theneteffectofsuchaleadcompensatoristomovethebreakfrequenciesoftheforward-looppolestohigherfrequencies.FortheexampleofFig.6a,theformofthecompensatorwouldbe634ControllerDesignFigure13StabilityparametersforU2/Uc:(a)closed-loopdampingratio;(b)gainmargin.3FrequencyCompensationtoImproveOverallPerformance635Figure13(Continued)(c)phasemargin.s1czG(16)c1s1cpwherecz1.FortheexampleofFig.6b,theformwouldbe22s/(2/)s1czczczG(17)c222s/(2/)s1cpcpcpwherecz2andcz2.Inbothcases,theclosed-loopperformanceisnowdeterminedbythecompensatorpolessincetheoriginalpolesarecanceledbythecompensatorzeros.ThisaugmentedperformancecanbequantiedbyusingFigs.813,withcp,cp,cpsub-stitutedfor1,2,2,respectively.Thereareanumberofpracticallimitationsontheuseofleadcompensation,primarilyrelatedtothelargehigh-frequencygainofthecompensatoritself.FortheexamplesofEqs.(16)and(17),thehigh-frequencygainsarecp/czand(cp/cz)2,respectively.Asamini-mum,thischaracteristicwillamplifyanyhigh-frequencyelectricalnoiseinthesystem.Withreasonablecareintheelectricaldesign,high-frequencycompensatorgainsof10ormoreareoftenpractical.Forarst-ordercompensator,thismeansthattheforward-loopbreakfre-quenciescanbeboostedbyafactorof10,whiletheboostisonlythesquarerootofthisfactorforasecond-ordercompensator.Anotherproblemassociatedwiththegainboostofaleadcompensatoristhatpoorlydampedhigh-frequencymodescanbeexcitedorevende-stabilized.ThislattereffectwillbefurtherdiscussedinSection3.3.Thepracticalityofleadcompensationinanygivenapplicationcanbebestdeterminedexperimentally(additionalhigh-frequencylagsaresometimesneeded).Thehigh-frequencynoisesituationisimprovedconsiderablyforsystemsinwhichtheintegratoriselectronic(e.g.,thevelocityservodescribedinSection2.1).Inthiscase,theintegratorcanreplaceoneofthecompensatorpolessothatEqs.(16)and(17)arereplacedby636ControllerDesignFigure14Firs