外文文献原稿和译文关于自动巡航消防下车的设计.docx
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外文文献原稿和译文关于自动巡航消防下车的设计
外文文献原稿和译文
原稿
Oscillation,Instability,andControlofStepperMotors
Abstract:
Anovelapproachtoanalyzinginstabilityinpermanent-magnetsteppermotorsispresented.Itisshownthattherearetwokindsofunstablephenomenainthiskindofmotor:
mid-frequencyoscillationandhigh-frequencyinstability.Nonlinearbifurcationtheoryisusedtoillustratetherelationshipbetweenlocalinstabilityandmid-frequencyoscillatorymotion.Anovelanalysisispresentedtoanalyzethelossofsynchronismphenomenon,whichisidentifiedashigh-frequencyinstability.Theconceptsofseparatricesandattractorsinphase-spaceareusedtoderiveaquantitytoevaluatethehigh-frequencyinstability.Byusingthisquantityonecaneasilyestimatethestabilityforhighsupplyfrequencies.Furthermore,astabilizationmethodispresented.Ageneralizedapproachtoanalyzethestabilizationproblembasedonfeedbacktheoryisgiven.Itisshownthatthemid-frequencystabilityandthehigh-frequencystabilitycanbeimprovedbystatefeedback.
Keywords:
Steppermotors;instability;nonlinearity;statefeedback
1.Introduction
Steppermotorsareelectromagneticincremental-motiondeviceswhichconvertdigitalpulseinputstoanalogangleoutputs.Theirinherentsteppingabilityallowsforaccuratepositioncontrolwithoutfeedback.Thatis,theycantrackanysteppositioninopen-loopmode,consequentlynofeedbackisneededtoimplementpositioncontrol.SteppermotorsdeliverhigherpeaktorqueperunitweightthanDCmotors;inaddition,theyarebrushlessmachinesandthereforerequirelessmaintenance.Allofthesepropertieshavemadesteppermotorsaveryattractiveselectioninmanypositionandspeedcontrolsystems,suchasincomputerharddiskdriversandprinters,XY-tables,robotmanipulators,etc.
Althoughsteppermotorshavemanysalientproperties,theysufferfromanoscillationorunstablephenomenon.Thisphenomenonseverelyrestrictstheiropen-loopdynamicperformanceandapplicableareawherehighspeedoperationisneeded.Theoscillationusuallyoccursatsteppingrateslowerthan1000pulse/s,andhasbeenrecognizedasamid-frequencyinstabilityorlocalinstability,oradynamicinstability.Inaddition,thereisanotherkindofunstablephenomenoninsteppermotors,thatis,themotorsusuallylosesynchronismathighersteppingrates,eventhoughloadtorqueislessthantheirpull-outtorque.Thisphenomenonisidentifiedashigh-frequencyinstabilityinthispaper,becauseitappearsatmuchhigherfrequenciesthanthefrequenciesatwhichthemid-frequencyoscillationoccurs.Thehigh-frequencyinstabilityhasnotbeenrecognizedaswidelyasmid-frequencyinstability,andthereisnotyetamethodtoevaluateit.
Mid-frequencyoscillationhasbeenrecognizedwidelyforaverylongtime,however,acompleteunderstandingofithasnotbeenwellestablished.Thiscanbeattributedtothenonlinearitythatdominatestheoscillationphenomenonandisquitedifficulttodealwith.
Mostresearchershaveanalyzeditbasedonalinearizedmodel.Althoughinmanycases,thiskindoftreatmentsisvalidoruseful,atreatmentbasedonnonlineartheoryisneededinordertogiveabetterdescriptiononthiscomplexphenomenon.Forexample,basedonalinearizedmodelonecanonlyseethatthemotorsturntobelocallyunstableatsomesupplyfrequencies,whichdoesnotgivemuchinsightintotheobservedoscillatoryphenomenon.Infact,theoscillationcannotbeassessedunlessoneusesnonlineartheory.
Therefore,itissignificanttousedevelopedmathematicaltheoryonnonlineardynamicstohandletheoscillationorinstability.ItisworthnotingthatTaftandGauthier,andTaftandHarnedusedmathematicalconceptssuchaslimitcyclesandseparatricesintheanalysisofoscillatoryandunstablephenomena,andobtainedsomeveryinstructiveinsightsintothesocalledlossofsynchronousphenomenon.Nevertheless,thereisstillalackofacomprehensivemathematicalanalysisinthiskindofstudies.Inthispaperanovelmathematicalanalysisisdevelopedtoanalyzetheoscillationsandinstabilityinsteppermotors.
Thefirstpartofthispaperdiscussesthestabilityanalysisofsteppermotors.Itisshownthatthemid-frequencyoscillationcanbecharacterizedasabifurcationphenomenon(Hopfbifurcation)ofnonlinearsystems.Oneofcontributionsofthispaperistorelatethemid-frequencyoscillationtoHopfbifurcation,thereby;theexistenceoftheoscillationisprovedTheoreticallybyHopftheory.High-frequencyinstabilityisalsodiscussedindetail,andanovelquantityisintroducedtoevaluatehigh-frequencystability.Thisquantityisveryeasytocalculate,andcanbeusedasacriteriatopredicttheonsetofthehigh-frequencyinstability.Experimentalresultsonarealmotorshowtheefficiencyofthisanalyticaltool.
Thesecondpartofthispaperdiscussesstabilizingcontrolofsteppermotorsthroughfeedback.Severalauthorshaveshownthatbymodulatingthesupplyfrequencythemid-frequency.Instabilitycanbeimproved.Inparticular,PickupandRussellhavepresentedadetailedanalysisonthefrequencymodulationmethod.Intheiranalysis,Jacobiserieswasusedtosolveaordinarydifferentialequation,andasetofnonlinearalgebraicequationshadtobesolvednumerically.Inaddition,theiranalysisisundertakenforatwo-phasemotor,andtherefore,theirconclusionscannotapplieddirectlytooursituation,whereathree-phasemotorwillbeconsidered.Here,wegiveamoreelegantanalysisforstabilizingsteppermotors,wherenocomplexmathematicalmanipulationisneeded.Inthisanalysis,ad–qmodelofsteppermotorsisused.Becausetwo-phasemotorsandthree-phasemotorshavethesameq–dmodelandtherefore,theanalysisisvalidforbothtwo-phaseandthree-phasemotors.Uptodate,itisonlyrecognizedthatthemodulationmethodisneededtosuppressthemid-frequencyoscillation.Inthispaper,itisshownthatthismethodisnotonlyvalidtoimprovemid-frequencystability,butalsoeffectivetoimprovehigh-frequencystability.
2.DynamicModelofStepperMotors
Thesteppermotorconsideredinthispaperconsistsofasalientstatorwithtwo-phaseorthreephasewindings,andapermanent-magnetrotor.Asimplifiedschematicofathree-phasemotorwithonepole-pairisshowninFigure1.Thesteppermotorisusuallyfedbyavoltage-sourceinverter,whichiscontrolledbyasequenceofpulsesandproducessquare-wavevoltages.Thismotoroperatesessentiallyonthesameprincipleasthatofsynchronousmotors.Oneofmajoroperatingmannerforsteppermotorsisthatsupplyingvoltageiskeptconstantandfrequencyofpulsesischangedataverywiderange.Underthisoperatingcondition,oscillationandinstabilityproblemsusuallyarise.
Figure1.Schematicmodelofathree-phasesteppermotor.
Amathematicalmodelforathree-phasesteppermotorisestablishedusingq–dframereferencetransformation.Thevoltageequationsforthree-phasewindingsaregivenby
va=Ria+L*dia/dt−M*dib/dt−M*dic/dt+dλpma/dt,
vb=Rib+L*dib/dt−M*dia/dt−M*dic/dt+dλpmb/dt,
vc=Ric+L*dic/dt−M*dia/dt−M*dib/dt+dλpmc/dt,
(1)
whereRandLaretheresistanceandinductanceofthephasewindings,andMisthemutualinductancebetweenthephasewindings._pma,_pmband_pmcaretheflux-linkagesofthephasesduetothepermanentmagnet,andcanbeassumedtobesinusoidfunctionsofrotorposition_asfollow
λpma=λ1sin(Nθ),
λpmb=λ1sin(Nθ−
),
λpmc=λ1sin(Nθ-
),
(2)
whereNisnumberofrotorteeth.Thenonlinearityemphasizedinthispaperisrepresentedbytheaboveequations,thatis,theflux-linkagesarenonlinearfunctionsoftherotorposition.
Byusingtheq;dtransformation,theframeofreferenceischangedfromthefixedphaseaxestotheaxesmovingwiththerotor(refertoFigure2).Transformationmatrixfromthea;b;cframetotheq;dframeisgivenby
(3)
Forexample,voltagesintheq;dreferencearegivenby
(4)
Inthea;b;creference,onlytwovariablesareindependent(iaCibCicD0);therefore,theabovetransformationfromthreevariablestotwovariablesisallowable.Applyingtheabovetransformationtothevoltageequations
(1),thetransferredvoltageequationintheq;dframecanbeobtainedas
vq=Riq+L1*diq/dt+NL1idω+Nλ1ω,
vd=Rid+L1*did/dt−NL1iqω,(5)
Figure2.a,b,candd,qreferenceframe.
whereL1DLCM,and!
isthespeedoftherotor.Itcanbeshownthatthemotor’storquehasthefollowingform
T=3/2Nλ1iq,(6)
Theequationofmotionoftherotoriswrittenas
J*dω/dt=3/2*Nλ1iq−Bfω–Tl,(7)
whereBfisthecoefficientofviscousfriction,andTlrepresentsloadtorque,whichisassumedtobeaconstantinthispaper.
Inordertoconstitutethecompletestateequationofthemotor,weneedanotherstatevariablethatrepresentsthepositionoftherotor.Forthispurposethesocalledloadangleisusuallyused,whichsatisfiesthefollowingequation
Dδ/dt=ω−ω0,(8)
where!
0issteady-statespeedofthemotor.Equations(5),(7),and(8)constitutethestatespacemodelofthemotor,forwhichtheinputvariablesarethevoltagesvqandvd.Asmentionedbefore,steppermotorsarefedbyaninverter,whoseoutputvoltagesarenotsinusoidalbutinsteadaresquarewaves.However,becausethenon-sinusoidalvoltagesdonotchangetheoscillationfeatureandinstabilityverymuchifcomparedtothesinusoidalcase(aswillbeshowninSection3,theoscillationisduetothenonlinearityofthemotor),forthepurposesofthispaperwecanassumethesupplyvoltagesaresinusoidal.Underthisassumption,wecangetvqandvdasfollows
vq=Vmcos(Nδ),
vd=Vmsin(Nδ),(9)