620620CHAPTER15CHAPTER15 CONTROLLERDESIGNCONTROLLERDESIGNThomasPeterNealThomasPeterNeal ConsultantConsultant LakeView,NYLakeView,NY1INTRODUCTION6201INTRODUCTION6202FUNDAMENTALSOFCLOSED-2FUNDAMENTALSOFCLOSED- LOOPPERFORMANCE621LOOPPERFORMANCE621 2.1AccuracyandLoopGain621 2.2DynamicResponseandStability6253FREQUENCYCOMPENSATION3FREQUENCYCOMPENSATION TOIMPROVEOVERALLTOIMPROVEOVERALL PERFORMANCE632PERFORMANCE632 3.1Well-DampedSystems633 3.2PoorlyDampedSystems637 3.3HigherOrderEffects6414INNERFEEDBACKLOOPS6454INNERFEEDBACKLOOPS645 4.1DerivativesoftheControlled Variable646 4.2AlternativeInnerLoopVariables649 4.3NonelectronicInnerLoops6505PREFILTERSAND5PREFILTERSAND FEEDFORWARD651FEEDFORWARD6515.1LagPrefilters6515.2LeadPrefilters6525.3Feedforward6546PIDCONTROLLERS6566PIDCONTROLLERS656 6.1EquivalencetoFrequency Compensation656 6.2SystemsHavingNoInherent Integration659 6.3SystemsHavinganInherent Integration6607EFFECTSOFNONLINEARITIES6607EFFECTSOFNONLINEARITIES660 7.1SimpleNonlinearities663 7.2ComplexNonlinearities668 7.3ComputerSimulation6688CONTROLLER8CONTROLLER IMPLEMENTATION669IMPLEMENTATION669 8.1AnalogControllers669 8.2Hard-WiredDigitalControllers672 8.3Computer-BasedDigital Controllers673REFERENCES677REFERENCES6771INTRODUCTION1INTRODUCTIONThepurposeofthischapteristoprovideabasisforthespecificationandfunctionaldesign ofelectronicservocontrollers.Noattemptismadetotreatthesubjectofelectroniccircuit design.Instead,thegoalistoaidtheengineerinselectingandapplyingasuitableoff-the- shelfcontrollerorinspecifyingthecontrollerrequirementstoacircuitdesigner.Theem- phasisisonposition,velocity,orforcecontrolofmechanicalloads,althoughmanyofthe techniquesareapplicabletocontrollerdesigningeneral.Specializedsubjectssuchasmul- tiaxiscontrolandadaptivecontrolarebeyondthescopeofthischapter. Asastartingpoint,itispresumedthataservoactuatorhasbeenselectedandmounted,togetherwithasuitablepowersupply,driveamplifier,andmechanicaldrivemechanism.In addition,theprimaryfeedbacktransducerhasbeenchosen,andasimpleloopclosurehasbeenanalyzedtodeterminewhetherthespecifiedclosed-loopperformancecanbeobtained.ReprintedfromInstrumentationandControl,Wiley,NewYork,1990,bypermissionofthepublisher.Mechanical Engineers Handbook: Instrumentation, Systems, Controls, and MEMS, Volume 2, Third Edition. Edited by Myer Kutz Copyright 2006 by John Wiley thatis,the controlledvariablemustaccuratelyholdthecommandsetpoint.Referringtothegeneralized blockdiagramofFig.1,severalsourcesofinaccuracycanbedescribed.Anexternaldis- turbancecancausetheloadtomovewithoutanychangeinthecommandsignal.Theload willcontinuetomoveuntiltheresultingerrorsignalcausestheactuatortobalancethe622622ControllerDesignFigure2Figure2Simplifiedvelocityservo.disturbance.Anomaliesintheactuatorandloadmustalsobeoffsetbyafiniteerrorsignal. Examplesaretemperature-inducednullshifts,hysteresis,threshold,friction,andlostmotion.Themagnitudeoftheseerrorsignalsisminimizediftheamplifiergainishigh.Ideally,theamplifiergainwouldbesethighenoughthattheaccuracyoftheservo becomesdependentonlyupontheaccuracyofthetransduceritself.Inpractice,however,theamplifiergainislimitedbystabilityconsiderations.Therefore,itisdesirabletoprovidehigh gainsatlowfrequenciesforaccuracyandlowgainsathighfrequenciestominimizestability problems.Sincetherateofamplituderoll-offwithfrequencyisdirectlyrelatedtophaselag, excessiveroll-offcancreatemorestabilityproblemsthanitsolves.Agoodcompromiseis tomaketheentireforwardpathlooklikeanintegratoroverthefrequencyrangeofinterest(atypeIsystem).Thistechniqueisverycommonlyusedtogivenearlyinfinitestaticgainandalineargainroll-offwithfrequency,atthecostof90phaselag. Itisimportanttonotethatsomeservoloopscontainaninherentintegrator,whichcom- plicatestheaccuracy-versus-stabilityproblem.Forexample,manyactuatorsareinherently ratedeviceswhenoperatedopenloop,sothatasteadyinputresultsinaproportionalvelocity output.Avelocityservousingsuchanactuatorwillinherentlyhaveaproportionalforwardloop,andanintegratingservoamplifiercanbeused(Fig.2).However,thecorresponding positionservowillinherentlyhaveanintegrationintheforwardloop,asshowninFig.3.Inthislattercase,theuseofanintegratingservoamplifiercancauseseverestabilityproblems.Fromthesefigures,transferfunctionscanbewrittenfortheclosed-loopresponsestocom- mandanddisturbanceinputs.Notethatthedynamicresponsecharacteristicsofallelements havebeenneglected,anditisassumedthattheloadincludesnospringtoground:V11(1) eKs/K1cvvvVsK(2)4FsKdvvX11(3)eKs/K1cxvxXK14(4)FKs/K1dvxvx2FundamentalsofClosed-LoopPerformance623623Figure3Figure3Simplifiedpositionservo.whereeccommandsignalV eeerrorsignal,V evvelocityfeedbacksignal,V expositionfeedbacksignal,V Fddisturbanceforceappliedtotheload,N K1integratingservoamplifiergain,(V/s)/V K2proportionalservoamplifiergain,V/V K3actuatorgain,(mm/s)/V K4actuatorvelocitydroopduetoforcedisturbance,(mm/s)/N Kvvelocitytransducergain,volts/(mm/s) Kxpositiontransducergain,volts/mm Kvvopen-loopgainofvelocityservoK1K3Kv,s1 Kvxopen-loopgainofpos