DC Motor Control：直流电机的控制

.

.

.

.

.

.

.

.

.

PAGE

2

PAGE

2

DCMotorControl

RogerAarenstrup

roger.aarenstrup@

Contents

Introduction

3

Thedcmotormodel

4

Speedcontrolwithpid

6

Continuouscontrol

6

DiscreteControl

7

Choosingparameters

9

HandCodetesting

10

UsingSimulinkControlDesignProducts

10

RapidPrototyping

10

FixedPoint

11

ProductionCodeGeneration

11

Positioncontrolusingstatefeedbackpoleplacement

12

Attempt1-statefeedbackandstaticgain

12

Attempt2–integralaction

12

Attempt3–Observer

13

Attempt4–Theservocase

13

Considerations

14

Bandwidth

14

Samplerate

16

References

17

Introduction

ThisexampledescribeshowtodevelopspeedandpositioncontrolsystemsforaDCmotorwithaload.Variousmethodsareusedandthefocusisonhowtomodelandimplementthevariouspartsandnotofparametertuning.Animportantthingtonoteisthatagoodcontrollerisnotjustatextbookimplementationbutrequiresanumberofadditionalpartstoworkproperly.Itisagoodideatogothroughthistexttogetherwiththemodelsandsomecontroldesignliteraturethatgivesmoredetailsaboutthetheory.Pleaseseethereferenceschapterforsomesuitablebooks.

Thistextandmodelscomewithabsolutelynoguaranteeifyoufindanythingincorrectpleaseletmeknow:

Roger.aarenstrup@

Thedcmotormodel

Themodelsusedherecanbedownloadedfrommatlabcentralwiththelink;

/matlabcentral/fileexchange/loadFile.do?objectId=11829&objectType=file

ThemodelisaquitesimplelinearDCmotormodelwithaflexibleload.InthelibrarythereareMATLABfilesdescribingastate-spacerepresentationofthemodel.Howtoderivethemisdescribedinmanycontrolsystembooksandalsointhecontrolsystemtoolboxdocumentation:

/access/helpdesk/help/toolbox/control/getstart/buildmo4.html

Themodeldescribedtheredoesn’tincludetheflexibleloadhowtoaddthatisdescribedhereandinthemodelsincluded.

IhavefoundthatalinearmodelisgoodenoughformanycontrolsystemswithDCmotors.Modelingisnotaboutmakingthemostdetailedmodelbuttomakeamodelgoodenoughforthetask.

Thereareafewthingsworthtonotice.Ihavetakenthedcmotorparametersfromthemaxonmotorsproductcatalog.InthefutureIhopethatitispossibletodownloadmodelsdirectlyfromtheirwebpage,justlikeitnowispossibletodownloadCADmodelsoverthemechanics.Inthecatalogthereisnospecificvalueforthemechanicaldamping.Usingthecommonexpressionforthemechanicaltimeconstant,Tm=bm/Jm,wherebmisthemechanicaldampingandJmistherotorinertiadoesn’tgivethecorrectno-load-current.Thismightbeduetonon-linearpartshere.Iusedsimulationstoestimateamechanicaldampingthatgivesaclosematchforthemechanicalrisetimeandno-load-current.Ifyouknowmoreaboutthis,pleaseletmeknow.

ThefigurebelowpresentstheDCmotormodelandload.Theloadisrepresentedbyadoubleintegratorwithinertia.Sinceitiscoupledwiththemotorrotorthroughasomewhatflexiblelinktherewillbeaspringactionifthepositionsoftherotorandtheloaddiffer.Thedifferenceinposition(angle)isthusfeedbackasacountertorquemultipliedbythespringconstant.Thesameistrueforthedamping.Thedifferenceinvelocitiesbetweenthetwobodieswilldampthesystem.Fordetails,pleaseseetherealmodel.

we

Figure,DCmotormodelwithflexibleload.

Speedcontrolwithpid

InthischapterwewillusemodelelaborationtogofromacrudemodeltoadiscreteimplementationofaPIDbasedcontrolsystemforaDCmotorwithload.

Continuouscontrol

Thisfirstattempttocontrolthedcmotorandloadusesacontinuoustime(LaPlacerepresentation)PIDwithapproximatederivative,seefigurebelow.Seemodela_pid_cont.mdlforthecompletemodel.Whentuningtheparametersitisimportanttoconsiderthecontrolsignal(outputfromthecontroller)tomakesureitwon’texceedthelimitsoftheamplifier.Tomeetthisfortheentirestepthecontrollergain(proportional)hastobequitelow,wewillseehowtoimprovethisfurtheron.

Figure,PIDwithapproximatederivative(froma_pid_cont.mdl).

Insomecasesitisnotdesirabletoderivethecommandedinputsignalbecauseitmightbeasteporsimilarthatwillgivebadresultswhenderived.Inthenextmodelb_pid_cont.mdlavariantofthePIDcontrollerisusedwhereonlytheoutputoftheplantisderived.

Figure,PIDwithapproximatederivativeoftheplantoutput.Notethatthederivativepartusesonlyy,theoutputfromtheplant,andnottheerror.(fromb_pid_cont.mdl)

Now,letsaddsomemoredetailstothemodel.Thisismodel-based-design,MBD,wheremoreandmoredetailsareaddeduntilthefinalimplementationisreached.AndduringdevelopmentthemodelcanbeverifiedcontinuouslybySimulationandlatercodegeneration.Let’saddasimplemodelofanamplifier.Theamplifierworksasalow-passfilterandasaturationoftheoutputsignal.Sometimesitisusefultouseanamplifierthatcanoutputavoltagelargerthanthenominalvoltageforthedcmotor,toimprovecontrolduringshortperiodsoftime.Let’salsoaddthetachometermodel.Theresultingmodelisc_pid_cont.mdl.

Itisoftenagoodideanottohaveapurestepasinputbecausethestepcancausespikesandmightexcitemodesinthesystem.Tomakethereferencesignalmoresmoothasimplelow-pasfilterisadded.However,itisusuallyagoodideatocalculateafunction,aspolynomialsforexample,fortheinput.

Themainresultfromasimulationwiththismoredetailedmodelisthattherearenowrippleontheoutputcausedbythetachometer.

DiscreteControl

TobeabletoimplementthecontrolleronamicroprocessorwefirstneedtoconvertittodiscretetimeinSimulink.Thereareanumberofdifferentwaytodothat,theonechosenhereistakenfrom[1].Seed_pid.mdlforamodelexample.Heresampletimecolorsareturnedonshowingthatthecontrollerpartisdiscretewithasampletimeof100us.Moreaboutchoosingsampletimeslater.Thesampletimeissetinamodelcallback,calledwhenthemodelisloaded.

Inthediscreteversionthederivativepartdependsonlyontheplantoutput,justlikethesecondcontinuousversion.Itisalsoadvantageous,inmanycases,tolettheproportionalgainonlytodependonapartofthecommandedsignal.Thisfactorisalsointroducedinthediscretecontrollerandiscalledb.

Figure,DiscretePIDcontroller.(Fromd_pid_disc.mdl)

Let’scontinuewithanti-windupfortheintegrator.Thereareseveralwaystoaddanti-windupforanintegrator.Theeasiestwayistojustlimittheinternalstateoftheintegratortoareasonablevalue,thiscanbedonedirectlyintheDiscrete-TimeIntegratorBlock.Itisalsopossibletojuststopupdatingtheintegratorstatewhentheactuatorissaturated.Sincewindupgenerallyisaproblembecausethereisanactuator(amplifier)thathasabounded(non-linear)output,anotherwayistotakethedifferencebetweentheinputtotheactuatorandsubtractthatfromtheoutputoftheactuatorandfeedthatsignalbacktotheintegratorpathinthecontroller.Thismeansthatiftheactuatorissaturatedthefeedbackwilldecreasetheintegralaction.Anadvantagewiththismethodisthatitcanbeappliedtoanykindofactuators,notjustactuatorswithlimitedoutput.

Thelastcaseaboveisimplementedinthenextmodel.Theimplementationismainlytodemonstratehowtoimplementanti-windup,sincethereisnotanyrealwindupprobleminthismodel.Youcanexperimentwiththeinputtoseeifyougetwindup.

Thenewcontroller,inthefigurebelowandmodelfilee_pid_cont.mdl,usesamodeloftheactuator,asaturationblock,togetthesaturatedvalue.Anotherwaywouldbetomeasuretheactualvaluebutthatwouldaddcosts.

Figure,DiscretePIDwithanti-windup.(Seee_pid_disc.mdl)

Themodelwiththiscontroller,e_pid.mdl,alsosamplesthereferencesignalfromanexternalsourceandisolatesthesampledpartsthatwilllaterbeimplementedonamicrocontroller.

Choosingparameters

ThereareanumberofparametersthathavetobechosenforthediscretePIDcontroller;K,Ti,Td,Tt,b,N,Umin,UmaxandTs.ParametertuningandselectionisnotthepurposeofthistutorialbutIincludesomeguidelines.

Forthecontrollerparameters,K,Ti,TdandTi,thereareseveralmethodstochoosefrom,see[1].

Tt,theintegratoranti-windupfeedback,isrelatedtoTiandcansometimesbeequaltoitbuttypicallyitis0.1-0.5Ti[1].

Parameterb,howmuchofthereferencesignaltobeusedfortheproportionalgain,shouldbeintheinterval0.1to1.

Nistypically3-20butcanbeashighas100.

UminandUmax,thesaturationpointsfortheactuatormodelinthecontroller,shouldbeasclosetotherealactuatoraspossible.

Ts,thesampletime,isrelatedtomanythings.ForPIDcontrollersithastobelowerthanforPIforexample.Itisalsorelatedtodisturbances.IntheexamplesaboveIuse100uswhichisprobablyfasterthannecessary.

HandCodetesting

Nowifyou’renotusinganyadvancedtoolsforcodegenerationandyouareusingafloatingpointprocessor,youcanwriteanequivalentdiscretecontrollerinc-codeandincludeit,withans-functionblock,inthesimulationtoverifytheimplementation.Inthemodeldirectorythereisac-filewiththediscretecontrollerimplemented,withoutanti-windup.Youcancompilefrommatlabwith:

mexsfun_wrp_pid.chandcode_pid.c

Thefilesfun_wrp_pid.cisans-functionwrapperforthecontrollerhandlingtheinterfacebetweenSimulinkandthecontrollercode.

Nowyoucanexperimentandverifyyourfinalimplementationsoyouaresureitworksbeforemessingwiththerealhardware,usemodelf_pid_disc.mdl.Ifyouhaveafixed-pointprocessoryoushouldfirstmodelthecontrollerusingSimulinkFixedPointthenyoucandothesamethingwithit,seelatersectionaboutfixed-point.

UsingSimulinkControlDesignProducts

ThereareseveraltoolsfromtheMathWorksthatcanbeusedforcontroldesign.ThemostwidelyusedisControlSystemToolboxthatincludesmanyusefultools.ForSimulinktherearethreeproductsinparticularthatarethemostuseful;SimulinkControlDesign,SimulinkParameterEstimationandSimulinkResponseoptimization.WiththeresponseoptimizationtoolforexampleyoucanspecifyadesiredtimedomainresponseandletSimulinkoptimizethecontrollerparameterstofitthedesiredresponse.

RapidPrototyping

Modelsdon’talwaysdescribetherealsystemperfectly.Beforeputtingeffortinthefinalimplementationitisgenerallyagoodideatotestitwiththerealplant.ThisisdonebyrapidprototypingwherecodeisautomaticallygeneratedfromSimulinkwithReal-TimeWorkshopanddownloadedtoatargetsystemlikethexPCtargetbox.Thenyoucantrytheimplementationofthecontrollerandtuneparametersagainsttherealplant.

Figure,examplemodelthatcanbecompiledforthexPCtargetforrapid-prototyping,g_pid_disc.mdl.

IfyouontheotherhandhavethecontrollerimplementedonthetargetprocessorandwanttotestityoucancompiletheplantinsteadofthecontrollerandruntheplantonthexPCtargetagainstyourprocessor.Inthiswayyoucanverifyyourimplementationbeforeyouhavetheplantandwithoutdamagingit.ThisiscalledHardware-In-The-Loop.

FixedPoint

Ifyouareusingafixed-pointprocessoritisnowtimetoconvertthefloatingpointmodeltofixedpointusingSimulinkFixed-Point.Itisoutofthescopeheretoshowthat.

ProductionCodeGeneration

Thenextstepwouldbetoproducecodethatissuitableforproduction.Itcanofcoursebehandcoded,liketheoneweusedbeforeinans-functionbutthereareanumberofbigadvantagesusingproductioncodegenerationwithReal-TimeWorkshopEmbeddedCoder.Itisnotjustatimesavingthing;youalsoavoidhandcodingerrorthatisusuallyrandomtothenature.Ifyouhavetochangeyourcontrollerlateritisaloteasiertojustregeneratethecodethanstartingaprojecttore-writesomeoldhandcode.Itisalsoalothardertomakesurethatthefinalimplementationactuallymatchesthesimulatedversion.ISimulinkyoudobit-accuratesimulations.

InthenextsectionthereisexamplecodeproducedbyReal-TimeWorkshopEmbeddedCoderintheController_ert_rtwfolder.

Positioncontrolusingstatefeedbackpoleplacement

Nowwearegoingtomakeapositioncontrollerinsteadofspeedcontroller.Inthisversionwewilluseadifferentamplifierthatcontrolsthecurrentfedtothedcmotorinsteadofthevoltage.Thecurrentismeasuredintheamplifierandwithasimplefeedbackthecurrentiscontrolled.ThisprovidesagreatadvantagesincethetorqueofthedcmotorisdirectlyproportionaltothecurrentthroughthetorqueconstantKt.Theresultisthatwecanneglecttheelectricalpartofthedcmotorandthushaveonestatelessinourcontroldesign,savingcomputationaltimeanddevelopmenteffort.

Inthefiless_dc_motor_load.mthereisareducedstatespacemodelforthisnewapplication.Thefilealsoincludesthecontroldesigncodeforthemodelsinthispart.

TheideaistofeedallstatesbackwithavectorgainL,bydoingsoitispossibletouseamethodtoplacetheclosedlooppolesondesiredlocations.Poleplacementisveryefficientingettingdesiredperformancebutmightnotgiveoptimalsolutionsforpowerconsumptionforexample.Itisalsonotsuitableforhigherordersystems.Higherordercanbe5-7andup.InthesecasesanLQGapproachmightbeabetterchoice.ButfortheDCmotorwithloadinthiscaseapoleplacementapproachworksfine.

Attempt1-statefeedbackandstaticgain

Thefirstapproachcanbeseenina_ss_controller.mdl,andinthefiless_dc_motor_load.mitiscalledattempt1.Herethefourclosedlooppolesareplacedonthepositiverealaxis(discretesystemsarestablewhenpolesareinsidetheunitcircle)on0.9875,0.9863,0.9850and0.9838.Forthepoleplacementalgorithmtoworkwellthepolescan’tbetooclosetoeachother.Whensimulatingonlythispart,withouttheKstatconstant,givesareasonableresult.Howeverthestaticgainisquitehigh,ca478.4.Usuallyitisdesirabletohaveastaticgainof1.ByjustaddingtheKstat=1/478.4thisiscorrected.TheKstatcanbeobtainedbysimulationorbycalculatingtheclosedloopstaticgain,asinthess_dc_motor_load.mfile.

Thisgivesaquitereasonableresultandthedesignisveryeasy,comparedtoPIDtuning.However,thestaticgainwillbesensitivetomodelerrorsandalsothereisnothingthatcompensatesfordisturbancesinthisfirstapproach.

Attempt2–integralaction

Asalwaystogetastaticgaintobe1andtoreducesensitivitytodisturbancesweintroduceintegralaction.Thisintegralpartshouldhaveananti-windupmechanismsimilartothePIDversionbutthatisnothandledhere,takeitasanexercisetoaddit.Byintroducinganintegratorweintroduceonemorestateinourmodel,seeattempt2inss_dc_motor_load.m.Theintegratorshouldbeconnectedtotheerroroftheoutputweneedtocontrol,inthiscasethepositionoftheload.Itis,however,morecommonthatanencoderisattachedtotherotoroftheDCmotorthantotheload.Alsosincethestaticpositionoftheloadandtherotorisequal,itispossibletousetherotorencoderinsteadofthepositionoftheloadfortheintegralaction.

Nowwehaveonemorepoletoplacebecauseoftheintegrator.Whereshouldweputit?Itistemptingtomakeitfaster(closerto0)thantheotherpolesnottointerferewiththeresponsetimeofthesystem.However,bydoingsowillincreasethesensitivitytodisturbancessignificantly.Abetterchoiceistoputitslightlyslowerthantherest.Thiswillmakethesystemresponsesomewhatsloweralsobutlesssensitivetodisturbances.Thisisacceptableforthecontrolcasebutnotfortheservocase.Intheservocasewewillnottrackthereferencesignalfastenough.Sowhatshouldwedo?IfwekeepthefeedforwardgainKffwewillintroduceazerointheclosedloopsystem.Byselectingthatzerocarefullywecanuseittocanceltheslowerpoleintroducedbytheintegrator.BydoingthesymbolicmathitturnsoutthatcancellationhappenswhenKff=Lint/(pint–1).WhereLintisthefeebackfortheintegratorandpintisthepoleplacedfortheintegrator.Seess_dc_motor_load.mfileforhowthisisdone.

IwouldalsoliketoremindyouthattheSimulinkControlDesigntoolthatcreatesalinearmodelisaveryefficienttooltoexamineanyverifydesigns.Justselectinputsandoutputsandcreatethelinearmodel.Thenyoucanwatchpolesandzeroes,stepresponse,bodeplotsetc,withtheLTIViewer.

Attempt3–Observer

Inattempt2wegotaquiteniceresult,wehavecontroloverthemovementandwearenotverysensitivetodisturbancesandmodelingerrors.However,tocontrolweuseinformationaboutallfourstatesplustheoutputwearetryingtocontrol.Thisinformationiseasytoobtaininthemodelforsimulationbutinarealsystemthatwouldrequiteustomeasureallthestateswhichgenerallyisnotpossibleorsuitable.Thesolutionistouseanobserver.Theobservertakestheinputtotheamplifierandtheoutputfromtheencoderandcalculatesanestimationofthestates.Thosestatescanthenbeusedbyourcontroller.Modelc_ss_controller.mdlshowshowthiscanbeimplemented.

Itturnsoutthatobtaininganobserverfromthestate-spacemodeloftheplantisverysimilartodesigningthefeedbackgain.Seeattempt3inss_dc_motor_load.m.Aninputisaddedforthemeasuredpositionandthenthesamepoleplacementisappliedandwegettheobserver.Pleaseseereferenceliteraturefordetails.Tomaketheobserverworkwelltogetherwiththecontrolleritspolesshouldbeplacedquitealotcloserto0oreveninthenegativerealaxis.Seec_ss_controller.mdl.

Note1!Thiscaseillustrateshowefficientsimulationtoolsarefordesigningcontrolsystems.Wecandesigntheinitialcontrollerandverifyitbysimulationthenadddetails,modelelaboration,suchastheobserverandsimulateagaintoverify.Withoutsimulationtoolsweneedtoimplementtheentirecontrollerwithobserverwithoutpossibilitytoverifytheimplementationnortheparametersforit.Ifweaddrapidprototypingtotestthealgorithmandtuneparameterswehaveprobablysaved90%developmenttime.

Note2!Thereasonwhyanobserver,withbothinputandoutputfromtherealplant,isusedinsteadofjustamodelovertheplantisthatwiththefeedbackK,wearemorerobusttomodelingerrors.

Attempt4–Theservocase

Thiscaseisjusttoshowanexampleofhowtheservercontrolcanbeimplemented.Irefertoliteratureabouthowtodesignthesecontrollers,forexample[1].Theexamplemodelisnamedd_ss_controller.mdl.

Considerations

Bandwidth

Herewewillmakethreedifferentdefinitionsofbandwidth,see[2]formoredetails.

Definition1:Bandwidthisthefrequencyareainwhichcontroliseffective[ω1ω2].

Usuallywewantthelowerfrequencytobe0,sincewedon’twantanystaticerror,wecanthencallω2=ωBthebandwidth.

Thewordeffectiveisnotclearbuthereweassumethatitmeansthatwegainsomethingbyusingthecontrollerinthisarea.

IfTistheclosedloopsystem,Sisthesensitivityfunction(S=1-T),risthereferenceinput,yistheoutputandeistheerror,e=y–r,wegetthefollowing:

y=Tr

e=y–r=(T–1)r=-Sr

FromthisweseethatTshouldbeascloseto1aspossible.–Sshouldbeaslowaspossible,sinceittellshowmuchoftheerrorthataffectstheoutput.Thisisveryinterestingbecauseitgivesustwomoreprecisedefinitionsofbandwidth.

Westartwiththeonebasedony=Tr.Ifwesaythatcontroliseffectiveaslongas|T(jω)|>-3dB,thebandwidthwillbethefrequencywhere|T(jω)|crosses–3dBfromabove.|T(jω)|isplottedinthefigurebelow.

Figure,|T(jω)|forthesysteminmodelc_ss_controller.mdl.Itispositioninputtoencoderoutputthatisplotted.

TheSandTfunctionsaredefinedinthess_dc_motor_load.mfileandcanbeexaminedwithltiview(S,T).

Fromthefigurewecanseethatthebandwidth,accordingtothedefinitionabove,issomewherebetween50and60rad/s.Thebandwidthisdirectlyrelatedtothepolesweplaced.Ifyouincreasethefrequencyofthepolesyouwillgetahigherbandwidth.Sowiththisdefinitionwecanspecifyourbandwidthwiththepole