cvt(无级变速器)英文文献

Vol.44,No.5,May2006,387–406

Controlofahydraulicallyactuatedcontinuously

variabletransmission

MICHIELPESGENS*†§,BASVROEMEN†,BARTSTOUTEN‡,FRANSVELDPAUS‡

andMAARTENSTEINBUCH‡

†DrivetrainInnovationsb.v.,Horsten1,5612AX,TheNetherlands

‡TechnischeUniversiteitEindhoven,POBox513,5600MBEindhoven,TheNetherlands

Vehiculardrivelineswithhierarchicalpowertraincontrolrequiregoodcomponentcontrollertracking,enablingthemaincontrollertoreachthedesiredgoals.Thispaperfocusesonthedevelopmentofatransmissionratiocontrollerforahydraulicallyactuatedmetalpush-beltcontinuouslyvariabletransmission(CVT),usingmodelsforthemechanicalandthehydraulicpartoftheCVT.Thecontrollerconsistsofananti-windupPIDfeedbackpartwithlinearizingweightingandasetpointfeedforward.Physicalconstraintsonthesystem,especiallywithrespecttothehydraulicpressures,areaccountedforusingafeedforwardparttoeliminatetheirundesiredeffectsontheratio.Thetotalratiocontrollerguaranteesthatoneclampingpressuresetpointisminimal,avoidingbeltslip,whiletheotherisraisedabovetheminimumleveltoenableshifting.ThisapproachhaspotentialforimprovingtheefficiencyoftheCVT,comparedtonon-modelbasedratiocontrollers.Vehicleexperimentsshowthatadequatetrackingisobtainedtogetherwithgoodrobustnessagainstactuatorsaturation.Thelargestdeviationsfromtheratiosetpointarecausedbyactuatorpressuresaturation.Itisfurtherrevealedthatallfeedforwardandcompensatortermsinthecontrollerhaveabeneficialeffectonminimizingthetrackingerror.

Keywords:Continuouslyvariabletransmission;Feedforwardcompensation;Feedbacklinearization;Hydraulicactuators;Constraints

1.Introduction

Theapplicationofacontinuouslyvariabletransmission(CVT)insteadofasteppedtransmis-sionisnotnew.Alreadyinthe50sVanDoorneintroducedarubberV-beltCVTforvehiculardrivelines.Modern,electronicallycontrolledCVTsmakeitpossibleforanyvehiclespeedtooperatethecombustionengineinawiderangeofoperatingpoints,forinstanceinthefueloptimalpoint.Forthisreason,CVTsgetincreasinglyimportantinhybridvehicles,seeforexample[1–3].AccuratecontroloftheCVTtransmissionratioisessentialtoachievetheintendedfueleconomyand,moreover,ensuregooddriveability.

Theratiosetpointisgeneratedbythehierarchical(coordinated)controlleroffigure1.ThiscontrollerusestheacceleratorpedalpositionastheinputandgeneratessetpointsforthelocalcontrollersofthethrottleandoftheCVT.

*Correspondingauthor.Email:pesgens@dtinnovations.nl

§MichielPesgenswaspreviouslyaffiliatedwithTechnischeUniversiteitEindhoven.

VehicleSystemDynamics

ISSN0042-3114print/ISSN1744-5159online©2006Taylor&Francis

http://www.tandf.co.uk/journalsDOI:10.1080/00423110500244088

388M.Pesgensetal.

Figure1.Hierarchicalpowertraincontrol.

TheCVTanditshydraulicactuationsystemaredepictedinfigures2,3.ThehydraulicsystemnotonlyhastoguaranteegoodtrackingbehavioroftheCVTbutalsohastorealizeclampingforcesthat,ontheonehand,arehighenoughtopreventbeltslipbut,ontheotherhand,areaslowaspossibletomaximizethetransmissionefficiencyandtoreducewear.Inpractice,theclampingforceslevelsarekeptatlevelsthatavoidbeltslipatalltimes,whilestillmaintaininganacceptabledegreeoftransmissionefficiency.

ThemainfocusofthispaperisontheratiocontroloftheCVT,usingthehydraulicactuationsystemoffigure3.Thepresentedcontrolconceptisbasedontheworkof[3,4].Itenablestrackingoftheratiosetpoint,whileguaranteeingatleastoneofthetwopulleypressuresetpointstobeequaltoitslowerconstraint.Eventhoughthecontrollereffectivelychangesfromcontrollingoneofthetwopressurestotheother,noactualswitchingbetweendifferentcontrollerstakesplace.Amongtheapproachesseenintheliterature,someincorporateaswitchingalgorithm[3,5],whereasotherscontrolonlyoneofthetwo(usuallytheprimary)pressures[6,7].Althoughtheformerapproachcannotguaranteeoneofthetwopressurestobeequaltoitslowerconstraint,thelattercannotexplicitlypreventtheuncontrolledpressuretostayaboveitslowerconstraint.

Therestofthispaperisorganizedasfollows.First,amathematicalmodelisderivedforthemechanicalpartoftheCVTinsection2.Next,insection3,thehydraulicpartismodeled.Thephysicalconstraints,imposedbythehydraulicsystem,arediscussedinsection4.TheseconstraintsaretakenintoaccountbytheCVTratiocontroller,thatisdevelopedinsection5

Figure2.Variator.

HydraulicallyactuatedCVT389

Figure3.Variatorwithhydraulicsystem.

andisbasedontheearlierderivedmodelsforthemechanicalandthehydraulicCVTparts.Thetrackingperformanceofthiscontrollerisexperimentallyevaluatedinsection6.Finally,section7givessomeconcludingremarks.

2.ThepushbeltCVT

TheCVT(figure2)consideredhereisequippedwithaVanDoornemetalpushbelt.Thisbeltconsistsofalargenumber(around350)ofV-shapedsteelblockelements,heldtogetherbyanumber(between9and12)ofthinsteeltensionrings.Thebeltrunsontwopulleys,namelytheprimarypulleyattheenginesideandthesecondarypulleyatthewheelside.Eachpulleyconsistsofoneaxiallyfixedandonemoveablesheave,operatedbymeansofahydrauliccylinder.Thecylinderscanbepressurized,generatingaxialforces(clampingforcesorthrusts)onthebelt,necessaryfortransmissionoftorque(withoutmacro-slipofthebelt)andforratiochange.Herethedistinctionismadebetweenmicro-slip,neededfortorquetransferbetweenbeltandpulleys,andmacro-slip,whichshouldbeavoidedatalltimesforitsnegativeeffectonefficiencyandespeciallytheriskofseverebeltandpulleywear[8].

Theboundedtransmissionratiorcvt∈[rcvt,LOW,rcvt,OD]isdefinedhereastheratioofsecondarypulleyspeedωsoverprimarypulleyspeedωp,so:

rcvt=

ωsωp

(1)

Inderivingthevariatormodel,ithasbeenassumedthatthepulleysarerigidandperfectlyaligned,andthattheV-shapedblocksarerigidandthesteelringsareinextensible.Thebeltisassumedtoruninperfectcirclesonthepulleys.Further,ithasbeenassumedthattheclampingforcesarelargeenoughtopreventmacrobeltslip.Theeffectsofmicro-sliparerelativelysmallwithrespecttotheratiochangebehavioroftheCVT,andare,therefore,neglectedinthemodel.ThepowertransmissionbetweenthebeltandthepulleysismodeledasCoulombfriction(whichisassumedinthemajorityofCVTvariatorresearch[3]).

390M.Pesgensetal.

Usingtheseassumptions,therunningradiiRpandRsofthebeltontheprimaryandsecondarypulleysarefunctionsoftheratiorcvtonlyandarerelatedby:

Rp=rcvt·Rs

(2)

Theaxialpositionsα(α=pfortheprimarypulley,α=sforthesecondaryone)ofthemoveablepulleysheaveofpulleyαisalsocompletelydeterminedbytheratiorcvt.Denotingthetaperangleoftheconicalsheavesbyϕ(seefigure4),itiseasilyseenthatsαisgivenby:

sα=2·tan(ϕ)·(Rα−Rα,min)

(3)

Subscript‘max’(or‘min’)impliesthemaximum(orminimum)valuepossible,e.g.Xmax=max(X),unlessstatedotherwise.Differentiationwithrespecttotimeyieldstheaxialvelocitys˙αofthemoveablesheaveofpulleyα

s˙α=να(rcvt)·r˙cvt

(4)

wherethefunctionναfollowsfromthegeometryofthevariator.

Assumingthattheradialfrictionforcecomponentbetweenthepulleyandthebeltiszero,thecriticalpulleyclampingforce(equalforbothpulleys,neglectingthevariator’sefficiency)isgivenbyreferences[3,5](forpulleyα):

Fcrit=

cos(ϕ)·|Tα|2·µ·Rα

(5)

whereTαisthenettransmittedtorquebetweenbeltandpulleyandµisthecoulombfrictioncoefficientbetweenpulleysandbelt.Thefactor2appears,astherearetwofrictionsurfacesbetweenpulleyandbelt.

RadialforcesbetweenbeltandpulleyscanbemainlycontributedtocentrifugalforcesandCoriolisforces.Inthedetailedthrustforcemodelofref.[9],itisreportedthateveniftheslidingangle(andhencethefrictionforceangle)ξbetweenthebeltpathandthefrictionforcevectorchangesalongthepulleycircumference,itsvalueconvergesrapidlytowardsvalueslessthan10◦.Asaresult,theangleisassumedzero.Thefrictionforceangleξwouldenterintoequation(5)asamultiplicationfactorcos(ξ),whichrapidlyconvergesto1forsmallangles.Forthechoiceofµ,aworst-caseapproachisapplied.Itischosenasthemaximumofthetractioncurve(ofwhichseveralhavebeenpresentedinref.[10]),whichisthepointoftransitionfrommicro-sliptomacro-slip.Thelowestvalueofallthemaximafoundinref.[10],aswellasinref.[8](forbothverysimilarvariators)is0.09,thevalueofµthathasbeenusedhere.

Figure4.Pulleysheavedefinitions.

HydraulicallyactuatedCVT391

Thetorqueratioταistheratiooftransmittedtorqueandmaximallytransmittabletorquewithoutbeltslipforpulleyα:

τα=

TαTα,max

=

cos(ϕ)·Tα2·µ·Rα·Fα

(6)

Asinapracticalvehicleapplicationagoodestimateofthetorquesactingonthesecondarypulleyisnotavailable,thefollowingmodifiedtorqueratioisintroduced:

τs󰀁

ˆpcos(ϕ)·T

=

2·µ·Rp·Fs

(7)

ˆpcanbeobtainedfromthedynamicdrivelineTheestimatedprimarytransmittedtorqueT

equationstogetherwithengineandtorqueconvertercharacteristics(alsoseesection4).In

ˆp=Tp,itiseasilyseenthat(usingequation(6)):caseofaperfecttorqueestimation,i.e.T

τs󰀁=

Pp

·τsPs

(8)

withtransmittedpowerPα=Tα·ωα.AsithasbeenassumedthatPp=Ps,themodifiedtorqueratiobecomesequaltothetorqueratioforthesecondarypulley.

AnimportantpartofthemodelforthemechanicalpartoftheCVTisthesub-modelfortherateofratiochangeasafunctionof,forinstance,theclampingforces.Sub-modelsofthistypeareproposed,amongothers,byGuebelietal.[11],Ideetal.[12,13]andShafaietal.[14].TheblackboxmodelofIdeispreferredhere,asitreasonablydescribestheresultsofaseriesofexperimentswithmetalV-beltCVTs[3,4].

ThesteadystateversionofIde’smodelyieldsarelationfortheprimaryclampingforceFpthatisrequiredtomaintainagivenratiorcvtwithagivensecondaryclampingforceFsandagivenprimarytorqueTp(throughthemodifiedtorqueratioτs󰀁):

Fp=κ(rcvt,τs󰀁)·Fs

(9)

Forobviousreasons,thequantityκinequation(9)iscalledthethrustratio.Someexperimen-tallyobtainedresultsforthishighlynon-linearfunctionoftheCVTratiorcvtandthetorqueratioτs󰀁aregiveninfigure5.

Forinstationarysituations,Ide’smodelstatesthattherateofratiochanger˙cvtisafunctionoftheratiorcvt,primarypulleyspeedωp,clampingforcesFpandFsandtorqueratioτs󰀁:

r˙cvt=kr(rcvt)·|ωp|·Fshift;

Fshift=Fp−κ(rcvt,τs󰀁)·Fs

(10)

AnaxialforcedifferenceFshift,weightedbythethrustratioκresultsinaratiochange,andis

thereforecalledtheshiftforce.Theoccurrenceofωpinthemodel(10)isplausiblebecauseanincreasingshiftforceisneededfordecreasingpulleyspeedstoobtainthesamerateofratiochange.ThereasonisthatlessV-shapedblocksenterthepulleyspersecondwhenthepulleyspeeddecreases.Asaresulttheradialbelttravelperrevolutionofthepulleysmustincreaseandthisrequiresahighershiftforce.However,itisfarfromobviousthattherateofratiochangeisproportionaltoboththeshiftforceandtheprimarypulleyspeed.krisanon-linearfunctionoftheratiorcvtandhasbeenobtainedexperimentally.Experimentaldatahasbeenusedtoobtainapiecewiselinearfit,whicharedepictedinfigure6.Theestimationofkrhas

392M.Pesgensetal.

Figure5.

Contourplotofκ(rcvt,τs󰀁).

Figure6.Fitofkr(rcvt);greyed-outdotscorrespondtodatawithreducedaccuracy.

HydraulicallyactuatedCVT393

Figure7.Comparisonofshiftingspeed,Ide’smodelvs.measurement.

beenobtainedusingtheinverseIdemodel:

kr(rcvt)=

r˙cvt

|ωp|·Fshift

(11)

InthedenominatorFshiftispresent,thevalueofwhichcanbecome(closeto)zero.Obviously,theestimateisverysensitiveforerrorsinFshiftwhenitsvalueissmall.Thedominantdis-turbancesinFshiftarecausedbyhigh-frequencypumpgeneratedpressureoscillations,whichdonotaffecttheratio(duetothelow-passfrequencybehaviorofunmodeledvariatorpulleyinertias).Thestandarddeviationofthepressureoscillationsandotherhigh-frequencydistur-banceshasbeendeterminedapplyingahigh-passButterworthfiltertothedataofFshift.Toavoidhigh-frequencydisturbancesinFshiftblurringtheestimateofkr,estimatesforvaluesofFshiftsmallerthanatleastthreetimesthedisturbance’sstandarddeviationhavebeenignored(thesehavebeenplottedasgreydotsinfigure6),whereastheotherpointshavebeenplottedasblackdots.Thewhitelineistheresultingfitofthisdata.Thefewpointswithnegativevalueforkrhavebeenidentifiedaslocalerrorsinthemapofκ.TovalidatethequalityofIde’smodel,theshiftingspeedr˙cvt,recordedduringaroadexper-iment,iscomparedwiththesamesignalpredictedusingthemodel.Modelinputsarethehydraulicpulleypressures(pp,ps)andpulleyspeeds(ωp,ωs)togetherwiththeestimated

ˆp).Theresultisdepictedinfigure7.Themodeldescribestheshiftingprimarypulleytorque(T

speedwell,butforsomeupshiftsitpredictstoolargevalues.ThishappensonlyforhighCVTratios,i.e.rcvt>1.2,wherethedataofκisunreliableduetoextrapolation(seefigure5).

3.Thehydraulicsystem

ThehydraulicpartoftheCVT(seefigure3)consistsofarollervanepumpdirectlyconnectedtotheengineshaft,twosolenoidvalvesandapressurecylinderoneachofthemovingpulley

394M.Pesgensetal.

sheaves.Thevolumebetweenthepumpandthetwovalvesincludingthesecondarypulleycylinderisreferredtoasthesecondarycircuit,thevolumedirectlyconnectedtoandincludingtheprimarypulleycylinderistheprimarycircuit.Excessiveflowinthesecondarycircuitbleedsofftowardtheaccessories,whereastheprimarycircuitcanblowofftowardthedrain.Allpressuresaregagepressures,definedrelativetotheatmosphericpressure.Thedrainisatatmosphericpressure.

TheclampingforcesFpandFsarerealizedmainlybythehydrauliccylindersonthemove-ablesheavesanddependonthepressuresppandps.Asthecylindersareanintegralpartofthepulleys,theyrotatewithanoftenveryhighspeed,socentrifugaleffectshavetobetakenintoaccountandthepressureinthecylinderswillnotbehomogeneous.Therefore,theclampingforceswillalsodependonthepulleyspeedsωpandωs.Furthermore,apreloadedlinearelasticspringwithstiffnessksprisattachedtothemoveablesecondarysheave.Thisspringhastoguaranteeaminimalclampingforcewhenthehydraulicsystemfails.Togetherthisresultsinthefollowingrelationsfortheclampingforces:

2

Fp=Ap·pp+cp·ωp

2

Fs=As·ps+cs·ωs−kspr·ss+F0

(12)(13)

wherecpandcsareconstants,whereasF0isthespringforcewhenthesecondarymoveable

sheaveisatpositionss=0.Furthermore,ApandAsarethepressurizedpistonsurfaces.Inthehydraulicsystemoffigure3,theprimarypressureissmallerthanthesecondarypressureifthereisanoilflowfromthesecondarytotheprimarycircuit.Therefore,toguaranteethatinanycasetheprimaryclampingforcecanbeuptotwiceaslargeasthesecondaryclampingforce,theprimarypistonsurfaceApisapproximatelytwiceaslargeasthesecondarysurfaceAs.Itisassumedthattheprimaryandthesecondarycircuitarealwaysfilledwithoilofconstanttemperatureandaconstantairfractionof1%.Thevolumeofcircuitα(α=p,s)isgivenby:

Vα=Vα,min+Aα·sα

(14)

Vα,ministhevolumeifsα=0andAαisthepressurizedpistonsurface.

Thelawofmassconservation,appliedtotheprimarycircuit,combinedwithequation(14),resultsin:

κoil·Vp·p˙p=Qsp−Qpd−Qp,leak−Qp,V(15)Qspistheoilflowfromthesecondarytotheprimarycircuit,Qpdistheoilflowfromthe

primarycircuittothedrain,Qp,leakisthe(relativelysmall)oilflowleakingthroughnarrowgapsfromtheprimarycircuitandQp,Vistheoilflowduetoachangeintheprimarypulleycylindervolume.Furthermore,κoilisthecompressibilityofoil.TheoilflowQspisgivenby:

󰀅2

Qsp=cf·Asp(xp)··|ps−pp|·sign(ps−pp)(16)

ρwherecfisaconstantflowcoefficientandρistheoildensity.Asp,theequivalentvalveopeningareaforthisflowpath,dependsontheprimaryvalvestempositionxp.FlowQpdfollowsfrom:

󰀅2

Qpd=cf·Apd(xp)·(17)·pp

ρHere,Apdistheequivalentopeningareaoftheprimaryvalvefortheflowfromprimarycircuittothedrain.TheconstructionofthevalveimpliesthatAsp(xp)·Apd(xp)=0forallpossiblexp.

HydraulicallyactuatedCVT395

FlowQp,leakisassumedtobelaminarwithleakflowcoefficientcpl,so:

Qp,leak=cpl·pp

Theflowduetoachangeoftheprimarypulleycylindervolumeisdescribedby:

˙pQp,V=Ap·s

withs˙pgivenbyequation(4).

Applicationofthelawofmassconservationtothesecondarycircuityields

κoil·Vs·ps=Qpump−Qsp−Qsa−Qs,leak−Qs,V

(20)(19)(18)

TheflowQpump,generatedbytherollervanepump,dependsontheangularspeedωeoftheengineshaft,onthepumpmodem(m=SSforsinglesidedandm=DSfordoublesidedmode),andthepressurepsatthepumpoutlet,soQpump=Qpump(ωe,ps,m).QsaistheflowfromthesecondarycircuittotheaccessoriesandQs,leakistheleakagefromthesecondarycircuit.FlowQsaismodeledas:

󰀅2

(21)·|ps−pa|·sign(ps−pa)Qsa=cf·Asa(xs)·

ρwhereAsa,theequivalentvalveopeningofthesecondaryvalve,dependsonthevalvestempositionxs.ThelaminarleakageflowQs,leakisgivenby(withflowcoefficientcsl):

Qs,leak=csl·ps

Theflowduetoachangeofthesecondarypulleycylindervolumeis:

˙sQs,V=As·s

(23)(22)

withs˙saccordingtoequation(3).

Theaccessorycircuitcontainsseveralpassivevalves.Inpractice,thesecondarypressurepswillalwaysbelargerthantheaccessorypressurepa,i.e.nobackflowoccurs.Therelationbetweenpaandpsisapproximatelylinear,so

pa=ca0+ca1·ps

(24)

withconstantsca0>0andca1∈(0,1).

NowthatacompletemodelofthepushbeltCVTanditshydraulicsisavailable,thecontrolleranditsoperationalconstraintscanbederived.

4.Theconstraints

TheCVTratiocontroller(infact)controlstheprimaryandsecondarypressures.Severalpressureconstraintshavetobetakenintoaccountbythiscontroller:

1.thetorqueconstraintspα≥pα,torquetopreventsliponthepulleys;

2.thelowerpressureconstraintspα≥pα,lowtokeepbothcircuitsfilledwithoil.Here,fairlyarbitrary,pp,low=3[bar]ischosen.ToenableasufficientoilflowQsatotheaccessorycircuit,andforaproperoperationofthepassivevalvesinthiscircuititisnecessarythat

396M.Pesgensetal.

QsaisgreaterthanaminimumflowQsa,min.Aminimumpressureps,lowof4[bar]turnsouttobesufficient;

3.theupperpressureconstraintspα≤pα,max,topreventdamagetothehydrauliclines,cylindersandpistons.Hence,pp,max=25[bar],ps,max=50[bar];

4.thehydraulicconstraintspα≥pα,hydtoguaranteethattheprimarycircuitcanbleedofffastenoughtowardthedrainandthatthesecondarycircuitcansupplysufficientflowtowardtheprimarycircuit.Thepressurespp,torqueandps,torqueinconstraint1dependonthecriticalclampingforce

ˆpiscalculatedusingthestationaryenginetorqueFcrit,equation(5).TheestimatedtorqueT

map,torqueconvertercharacteristicsandlock-upclutchmode,togetherwithinertiaeffectsoftheengine,flywheelandprimarygearboxshaft.Asafetyfactorks=0.3withrespecttothe

ˆp,maxhasbeenintroducedtoaccountfordisturbancesonestimatedmaximalprimarytorqueT

ˆp,suchasshockloadsatthewheels.ThenthepulleyclampingforcetheestimatedtorqueT

(equalforbothpulleys,neglectingthevariatorefficiency)neededfortorquetransmissionbecomes:

Ftorque=

ˆp|+ks·Tˆp,max)cos(ϕ)·(|T

2·µ·Rp

(25)

Consequently,theresultingpressurescanbeeasilyderivedusingequations(12)and(13):

pp,torqueps,torque

󰀋1󰀊2

=Ftorque−cp·ωp

Ap

󰀇1󰀆2

Ftorque−cs·ωs=−kspr·ss−F0As

(26)(27)

Exactlythesameclampingstrategyhasbeenpreviouslyusedbyref.[3]duringteststandefficiencymeasurementsofthisgearboxandtestvehicleroadtests.Nosliphasbeenreportedduringanyofthoseexperiments.Asthemaingoalofthisworkistoanimprovedratiotrackingbehavior,theclampingstrategyhasremainedunchanged.

Afurtherelaborationofconstraints4isbasedonthelawofmassconservationfortheprimarycircuit.Firstofall,itisnotedthatforthiselaborationtheleakageflowQp,leakandthecompressibilitytermκoil·Vp·p˙pmaybeneglectedbecausetheyaresmallcomparedtotheotherterms.Furthermore,itismentionedagainthattheflowsQspandQpdcanneverbeunequaltozeroatthesametime.Finally,itischosentoreplacetherateofratiochanger˙cvtbythedesiredrateofratioshiftr˙cvt,d,thatisspecifiedbythehierarchicaldrivelinecontroller.Ifr˙cvt,d<0,thenoilhastoflowoutoftheprimarycylindertothedrain,soQpd>0andQsp=0.Constraint4withrespecttotheprimarypulleycircuitthenresultsinthefollowingrelationforthepressurepp,hyd:

pp,hyd

ρoil=·

2

󰀌

rcvt,d)Ap·νp·max(0,−˙

cf·Apd,max

󰀍2

(28)

whereApd,maxisthemaximumopeningoftheprimaryvalveintheflowpathfromtheprimarycylindertothedrain.

Inasimilarway,arelationforthesecondarypulleycircuitpressureps,hydinconstraint4canbederived.Thisconstraintisespeciallyrelevantifr˙cvt>0,i.e.iftheflowQspfromthesecondarytotheprimarycircuithastobepositiveand,asaconsequence,Qpd=0.Thisthen

HydraulicallyactuatedCVT397

resultsin:

ps,hyd

ρoil

=pp,d+·

2

󰀌

˙cvt,d)Ap·νp·max(0,r

cf·Asp,max

󰀍2

(29)

whereAsp,maxisthemaximumopeningoftheprimaryvalveintheflowpathfromthesecondarytotheprimarycircuit.

ForthedesignoftheCVTratiocontrolleritisadvantageoustoreformulatetoconstraintsintermsofclampingforcesinsteadofpressures.AssociatingaclampingforceFα,βwiththepressurepα,βandusingequations(12)and(13)thisresultsintherequirement:

Fα,min≤Fα≤Fα,max

withminimumpulleyclampingforces:

Fα,min=max(Fα,low,Fα,torque,Fα,hyd)

(31)(30)

5.Controldesign

Itisassumedinthissectionthatateachpointoftimet,theprimaryspeedωp(t),theratiorcvt(t),theprimarypressurepp(t)andthesecondarypressureps(t)areknownfrommeasurements,filteringand/orreconstruction.Furthermore,itisassumedthattheCVTismountedinavehiculardrivelineandthatthedesiredCVTratiorcvt,d(t)andthedesiredrateofratiochanger˙cvt,d(t)arespecifiedbytheoverallhierarchicaldrivelinecontroller.Thisimplies,forinstance,thatateachpointoftimetheconstraintforcescanbedetermined.

ThemaingoalofthelocalCVTcontrolleristoachievefastandaccuratetrackingofthedesiredratiotrajectory.Furthermore,thecontrollershouldalsoberobustfordisturbances.Animportantsubgoalistomaximizetheefficiency.Itisquiteplausible(andotherwisesupportedbyexperiments,[3])thattorealizethissub-goaltheclampingforcesFpandFshavetobeassmallaspossible,takingtherequirementsinequation(30)intoaccount.

Theoutputoftheratiocontrollerissubjecttotheconstraintsofequation(31).TheconstraintsFα≥Fα,mincaneffectivelyraisetheclampingforcesetpointofonepulley,resultinginanundesirableratiochange.Thiscanbecounteractedbyraisingtheoppositepulley’sclampingforceaswell,usingmodel-basedcompensatortermsintheratiocontroller.UsingIde’smodel,i.e.usingequation(10),expressionsfortheratiochangeforcesFp,ratioandFs,ratio(figure8)canbeeasilyderived:

Fp,ratio=Fshift,d+κ·Fs,minFs,ratio=

−Fshift,d+Fp,min

κ

(32)(33)

whereFshift,disthedesiredshiftingforce,basicallyaweightedforcedifferencebetweenbothpulleys.Asexplainedearlier,κdependsonτs󰀁,whichinturndependsonFs.Thisisanimplicitrelation(Fs,ratiodependsonFs),whichhasbeentackledbycalculatingκfrompressuremeasurements.

Itwillnowbeshownthatateachtime,oneofthetwoclampingforcesisequaltoFα,min,whereastheotherdeterminestheratio.Usingequations(30),(32)and(33)thedesiredprimary

398M.Pesgensetal.

Figure8.Ratiocontrollerwithconstraintscompensation

andsecondaryclampingforcesFp,dandFs,daregivenby:

󰀂

Fp,d=Fp,ratio

ifFshift,d+κ·Fs,min>Fp,min

Fs,d=Fs,min

󰀂

Fp,d=Fp,min

ifFshift,d+κ·Fs,minFs,d=Fs,ratio

(34)

(35)

Infact,theratioiscontrolledinsuchawaythattheshiftingforceFshiftbecomesequaltoFshift,d.FortheresultingshiftingforceholdsFshift=Fp,d−κ·Fs,d,so:

󰀁

Fp,ratio−κ·Fs,min=Fshift,difFshift,d+κ·Fs,min>Fp,min

Fshift=(36)

Fp,min−κ·Fs,ratio=Fshift,difFshift,d+κ·Fs,minTocompletethecontroller,Fshift,dmustbespecified.Asthedynamicsofthevariator(accord-ingtoIde’smodel)arequitenon-linear,anequivalentinputuisintroduced,usinganinverserepresentationoftheIdemodelforFshift,d:

Fshift,d=

u+r˙cvt,dkr·|ωp|(37)

Basicallyafeedback-linearizingweightingofuwiththereciprocalofboth|ωp|andkrisapplied.Thiscancelsthe(known)non-linearitiesinthevariator,see,e.g.Slotineetal.[15].Further,asetpointfeedforwardisintroduced,whichwillreducethephaselagofthecontrolledsystemresponses.

Owingtomodelinaccuraciesorduetoexternaldisturbancesunaccountedfor(liketheupperclampingforceconstraints),differencesγbetweenr˙cvtandr˙cvt,dwilloccur:

r˙cvt=r˙cvt,d+u+γ

(38)

Goodtrackingbehaviorisobtainedifucancelsγwell.Alinearfeedbackcontrollerhasbeenchosenforubasedontheknowledgethat(contrarytoequation(10)),thereareinertiasinvolved,requiringatleastasecondordercontroller.Consequently,aPIDcontrollerisused.

HydraulicallyactuatedCVT399

Theproportionalactionisnecessaryforarapidreductionoferrors,whereastheintegratingactionisneededinordertotrackrampratiosetpointswithzeroerror.Somederivativeactionprovednecessarytogainlargerstabilitymargins(andlessoscillatoryresponses).Thecontrollerisimplementedasfollows:

󰀄t

󰀉󰀈

u=P·(rcvt,d−rcvt)+I·˙cvt(39)ke·(rcvt,d−rcvt)dτ+D·r

0

whereke∈{0,1}switchestheintegratoronandoffdependingoncertainconditionsthatare

explainedfurtheron.ThederivativeactionofthecontrolleronlyactsonthemeasuredCVTratiosignaltoavoidanexcessivecontrollerresponseonstepwisechangesoftheratiosetpoint.Additionally,ahigh-frequencypolehasbeenaddedtothederivativeoperationtopreventexcessivegainsathighfrequencies.ThecontrollerparametersP,IandDhavebeentunedmanually.

Duringinstancesofactuatorsaturation(becauseofthemaximumforceconstraints),theclosedloopiseffectivelybroken(measurementrcvtdoesnotreacttochangesinuanymore).Thiswillleadtodegradedperformance,asthevalueofthecontroller’sintegratorcontinuestogrow.Thisso-calledintegratorwindupisundesirable.Aconditionalanti-windupmechanismhasbeenaddedtolimittheintegrator’svalueduringsaturation:

󰀎

1ifFp,ratio≤Fp,max∧Fs,ratio≤Fs,max

ke=(40)

0ifFp,ratio>Fp,max∨Fs,ratio>Fs,maxIfeitherpressuresaturates(pp=pp,maxorps=ps,max),theshiftingspeederrorγinevitablybecomeslarge.Theanti-windupalgorithmensuresstability,butthetrackingbehaviorwilldeteriorate.Thisisahardwarelimitationwhichcanonlybetackledbyenhancingthevariatorandhydraulicshardware.Theadvantageofaconditionalanti-windupvs.astandard(linear)algorithmisthatthelinearapproachrequirestuningforgoodperformance,whereasthecon-ditionalapproachdoesnot.Furthermore,theperformanceoftheconditionalalgorithmcloselyresemblesthatofawell-tunedlinearmechanism.

6.Experimentalresults

AstheCVTisalreadyimplementedinatestvehicle,in-vehicleexperimentsonarollerbenchhavebeenperformedtotuneandvalidatethenewratiocontroller.Topreventanon-synchronizedoperationofthrottleandCVTratio,theacceleratorpedalsignal(seefigure1)hasbeenusedastheinputforthevalidationexperiments.Thecoordinatedcontrollerwilltrackthemaximumengineefficiencyoperatingpoints.Asemikick-downactionatacruise-controlledspeedof∼50km/hfollowedbyapedalbackouthasbeenperformedinasinglereferenceexper-iment.Therecordedpedalangle(seefigure9)hasbeenappliedtothecoordinatedcontroller.Thisapproachcancelsthelimitedhumandriver’srepeatability.

Theupperplotoffigure10showstheCVTratioresponsecalculatedfromspeedmeasure-mentsusingequation(1),theplotdepictsthetrackingerror.Asthisisaquitedemandingexperiment,thetrackingisstilladequate.Muchbettertrackingperformancecanbeobtainedwithmoresmoothsetpoints,butthecharacteristicsoftheresponseswillbecomelessdistinctaswell.Figure11showstheprimaryandsecondarypulleypressures.Theinitialmainpeakintheerrorsignal(aroundt=1.5s)iscausedbysaturationofthesecondarypressure(lowerplotoffigure11),duetoapumpflowlimitation.Ifafasterinitialresponsewererequired,adaptationofthehydraulicshardwarewouldbenecessary.Aftertheinitialfastdownshift,theratioreachesitssetpoint(aroundt=7s)beforedownshiftingagain.Allchangesinshifting

400M.Pesgensetal.

Figure9.PedalinputfortheCVTpowertrain.

direction(t=1.3,t=1.6andt=7.5s)occurwitharelativelysmallamountofovershoot,whichshowsthattheintegratoranti-windupalgorithmperformswell.

Lookingattheprimarypressureinthevicinityoft=1.5s,itcanbeobservedthatthispressurepeaksrepeatedlyaboveitssetpoint.Thisbehavioriscausedbyperformancelimi-tationsoftheprimarypressurecontroller.Thedevelopedcontrollerguaranteesthatonlyonepulleypressuresetpointatthetimeisraisedaboveitslowerconstraint,andonlytorealize

Figure10.CVTratioresponseandtrackingerror,rollerbenchsemi-kickdown.

HydraulicallyactuatedCVT401

Figure11.Primaryandsecondarypulleypressures,rollerbenchsemi-kickdown.

adesiredratio.Thisisvisualizedinfigure12.HigherclampingforcescausemorelossesintheCVT[10],aslongasnomacro-slipoccurs.Themaincausesareoilpumppowerdemand(approximatelylinearwithpressure)andlossesinthebeltitself,whichbothincreasewithincreasingclampingpressure,assupportedbymeasurements[16].Hence,thiscontrollerhasapotentialforimprovingtheefficiencyofaCVT,comparedtonon-modelbasedcontrollers.

Figure12.Newcontroller’spulleypressuresetpointsminuslowerconstraints.

402M.Pesgensetal.

Lookingbacktothelowerplotoffigure10,thesecond(positive)peak(afterthefirstnegativepeakduetoactuatorsaturation)representstheovershootoftheratioresponseduetoashiftingdirectionchange.Thisquantitydescribesthetrackingperformanceofacontrollerwell,andwillbeusedtoevaluateacontroller’sperformance.Theovershootiscomputedhereasthe(positive)maximumoftheratioerror:max(rcvt,d−rcvt).Also,themeanabsoluteerror󰀃N

(1/N)0|rcvt,d−rcvt|(fortheNdatapointsinthe10sresponse)willbeusedtocompareresults.

Thesameexperimenthasbeenperformedforseveralvariationsonthecontroller.Foreachofthesevariations,allconstraintsarestillimposed,butsomeofthecompensatortermsintheratiocontrollerhavebeentemporarilyswitchedoff(theverticalarrowsinfigure8).Theresultshavebeencomparedwiththeresultsforthetotalcontrollerandaredepictedinfigure13.Thecasesthatwillbeaddressedare:1.2.3.4.5.

Allfeedforwardsandcompensatorson(‘total’).Nosetpointfeedforward(‘setpffoff’),r˙cvt,d=0inequation(37).

Nocritical(nobeltslip)torqueconstraintcompensation(‘Tcompoff’),Ftorque=0.Nohydraulicconstraintscompensation(‘hydrcompoff’),Fα,hyd=0.

Notorquetransmissionnorhydraulicconstraintscompensation(‘T,hydrcompoff’),Ftorque=0,Fα,hyd=0.

Itisimmediatelyclearthatofallalternatives,thetotalcontrollerwithallfeedforwardsandcompensatorson(‘total’)describedinthepreviousparagraphperformsbest,implyingthatallcontrollertermshaveapositivecontributiontowardsminimizingthetrackingerror.Switchingoffeitherthehydraulicconstraintscompensationterms(‘hydrcompoff’)orthetorquetrans-missioncompensator(‘Tcompoff’)doesnotseverelydegradethetrackingquality.However,switchingbothcompensatorsoff(‘T,hydrcompoff’)doesintroducelargetrackingerrors.Thisoccursbecausethemaximumoperatorofbothconstraintsistakentocalculatethecompen-satingaction,andifoneconstraintcompensatoriszero,theoutputofthemaximumoperator

Figure13.Overshootandmeanabsoluteerrorforseveralcontrolleralternatives.

HydraulicallyactuatedCVT403

willstillbenon-zeroduetothesecondconstraint.Bothcompensatorsswitchedoffsimulta-neouslyeffectivelyintroducea‘deadzone’inthecontrolleroutputu,theresultofwhichisobvious.Theresponsewiththesetpointfeedforwardswitchedoff(‘setpffoff’)increasestheerrorsduetoincreasedphaselagoftheresultingresponse.Theobtainedresultsofthetotaldevelopedcontrollershowbettertrackingbehavior(overshootandmeanabsoluteerror)andlowertransientpulleypressures(onlyduringratiochange,astheclampingstrategyisequal)comparedwithresultsobtainedwithapreviouslyadoptedcontroller,asdescribedinref.[3].ThiscouldbeanindicationforthepotentialforimprovingtheCVTefficiencyofthenewcontroller,asdescribedbefore.

Vehicletestsincludingtipshifting(featuringstepwiseratiosetpointchanges)havebeenper-formedonatesttrack,seefigure14.Thestepwisechangesintheratiosetpointaretrajectoriesthatcannotberealized.Hence,themeasuredCVTratiowillalwayslagbehind.Hence,thisexperimentdemonstratestherobustnessagainstactuatorsaturation,asthepressureofthepul-leythatcontrolstheratiowillsaturate.Astheerrorsinthefeedforwardtermsofthecontrollerwillincrease,thefeedbackcontrollerbecomesincreasinglyimportant.Alsotheanti-windupmechanismoftheratiocontrollerneedstopreventovershoot.Resultsofanexperimentdrivingatacruise-controlledspeedof50km/haredepictedinfigures15and16.Anewgearratiosetpointisgeneratedevery2s.

Atthestartoftheup-shiftratioresponsesatt=2.1sandt=4.2s,aninverseresponseispresent.Astheshiftingspeedsareindeedveryhighinthisexperiment,becauseofthelayoutofthehydraulicsystem,thesecondarycircuitneedstosupplytheprimarycircuitwithoil.Asaresult,thesecondarypressurerisesinadvancetotheprimarypressureandcausesaninitialdownshift.Aroundt=3sandt=5s,theratioinitiallyrisesapproximatelylinear,causedbythelimitedpumpflowastheoilpumprunsatenginespeed,whichislow.Upshiftingisfurthercharacterizedbysomeovershoot,whichisclearlyvisibleatt=14s.Astheprimarypressurecannotdropsufficientlyquickduetoalimitedprimaryvalveflow-throughareatowardthedrain,upshiftingcontinuesandcausesovershoot.Thesecondarypressureonlysaturatesbrieflyduetothelimitedpumpflowaftereachratiosetpointchange.Muchlessovershootispresentduringadownshift,thespeedofwhichisnotlimitedbypumpflow.Againtheprimarypressurepeaksaboveitssetpointwhenthesecondarypressureisincreasedrapidly,caused

Figure14.Experimentalvehicleduringtip-shiftsatthetesttrack.

404M.Pesgensetal.

Figure15.CVTratioresponseandtrackingerror,roadtipshifting.

bylimitationsintheprimarypressurecontroller.Thisphenomenonlowersthemaximumdownshiftspeedandisvisibleasaslight‘bump’intheratioatt=6.2sandt=8.2s.

Asthemaingoalofthepresentedexperimentsistodemonstrateanewratiocontrollerconcept,duringtheexperimentsbeltsliphasbeenavoidedusingaprovenclampingstrategyasmentionedearlier.Also,anonlinemodel-baseddetectionalgorithmwasused,verifyingthat|τs󰀁|≤1.Twomethodstodetectbeltslipoff-linefrommeasurementdata(withoutdirect

Figure16.Primaryandsecondarypulleypressures,roadtipshifting.

HydraulicallyactuatedCVT405

measurementsofthebelt’srunningradiusonthepulleystocalculatetheso-calledgeometricratio)havebeenusedaftertheexperiments.First,ithasbeenverifiediftherangeofCVTratiosgeometricallypossibleisnotexceeded(rLOW≤rcvt≤rOD).Secondly,themaximumshiftingspeedoftheCVTislimitedduetolimitedclampingforcesandvariatorspeed,seeequation(10).Thecoefficientoffrictionintheexcessive(macro-)slipregionofapush-beltdecreaseswithslipspeed[8].Thiscausesunstabledynamicbehavior,andhenceslipspeedwillincreaserapidlywhenthetorquecapacityofaV-beltisexceeded.Astheratioiscalculatedfrommeasuredpulleyspeeds,excessivelyfastratiochanges(highvaluesofr˙cvt)canindicatebeltslip.Theresultsofeachmeasurementhavebeenscrutinized,theresultofwhichdidnotshowanytracesofbeltslipeffects.

7.Conclusions

Anewratiocontrollerforametalpush-beltCVTwithahydraulicbeltclampingsystemhasbeendeveloped.Onthebasisofdynamicmodelsofthevariatorandhydraulics,compensatortermsofsystemconstraints,asetpointfeedforwardandalinearizingfeedbackcontrollerhavebeenimplemented.ThefeedbackcontrollerisaPIDcontrollerwithconditionalanti-windupprotection.Thetotalratiocontrollerguaranteesthat,atleastoneofthepressuresetpointsisalwaysminimalwithrespecttoitsconstraints,whiletheotherisraisedabovethemini-mumleveltoenableshifting.ThisapproachhaspotentialforaCVTefficiencyinprovement.Rollerbenchandroadexperimentswithavehiclebuilt-inCVTshowthatadequatetrackingisobtained.Thelargestdeviationsfromtheratiosetpointarecausedbyactuatorpressuresatura-tion.Experimentswithseveralcontrollervariationsfeaturingfeedforwardsbeingswitchedoffrevealthatallimplementedfeedforwardandconstraintcompensatortermshaveabeneficialeffectonminimizingthetrackingerror.Tipshiftexperimentsrevealedgoodrobustnessagainstactuatorsaturation.References

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