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A robust vector control for induction motor drives with an adaptive sliding mode control law

A robust vector control for induction motor

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A robust vector control for induction motor drives with an adaptive sliding mode control law

  1. 1. Journal oftheFranklinInstitute348(2011)300–314 A robustvectorcontrolforinductionmotordrives with anadaptivesliding-modecontrollaw Oscar Barambonesa,, PatxiAlkortab aDpto. Ingenier´ıa deSistemasyAutom atica, EUIdeVitoria,UniversidaddelPa´ıs Vasco,Nievescano12,01006 Vitoria, Spain bDpto. Ingenier´ıa deSistemasyAutom atica, EUIdeEibar,UniversidaddelPa´ıs Vasco,Avda.Otaola,2920600 Eibar (Gipuzkoa) Received 4January2010;receivedinrevisedform24November2010;accepted30November2010 Available online7December2010 Abstract A noveladaptivesliding-modecontrolsystemisproposedinordertocontrolthespeedofan induction motordrive.Thisdesignemploystheso-calledvector(orfieldoriented)controltheoryfor the inductionmotordrives.Thesliding-modecontrolisinsensitivetouncertaintiesandpresentsan adaptive switchinggaintorelaxtherequirementfortheboundoftheseuncertainties.Theswitching gain isadaptedusingasimplealgorithmwhichdoesnotimplyahighcomputationalload.Stability analysis basedonLyapunovtheoryisalsoperformedinordertoguaranteetheclosedloopstability. Finally, simulationresultsshownotonlythattheproposedcontrollerprovideshigh-performance dynamic characteristics,butalsothatthisschemeisrobustwithrespecttoplantparametervariations and externalloaddisturbances. 2010 TheFranklinInstitute.PublishedbyElsevierLtd.Allrightsreserved. 1. Introduction The inductionmotorisacomplexstructurethatconvertselectricalenergy intomechanical energy. Althoughinductionmachineswereintroducedmorethanahundredyearsago,the researchanddevelopmentinthisareaappearsto benever-ending.Traditionally,ACmachines with aconstantfrequencysinusoidalpower supplyhavebeenusedinconstant-speed applications, whereasDC machineswerepreferredforvariablespeeddrives,sincetheypresent www.elsevier.com/locate/jfranklin 0016-0032/$32.00 2010 TheFranklinInstitute.PublishedbyElsevierLtd.Allrightsreserved. doi:10.1016/j.jfranklin.2010.11.008 Correspondingauthor.Tel.: þ34 945013235;fax: þ34 945013270. E-mail address: ispbacao@ehu.es(O.Barambones).
  2. 2. a simplercontrol.Besides,ACmachinespresentedsomedisadvantagesincomparisonwith DC ones,ashighercost,higherrotorinertiaand maintenanceproblems.Nevertheless,inthe last twoorthreedecadeswehaveseenextensiveresearchanddevelopmenteffortsinvariable- frequency,variable-speedAC machinedrivestechnology [1], whichhaveovercomesomeofthe abovedisadvantagesoftheACmotors. The developmentoffieldorientedcontrolinthebeginningof1970smadeitfeasibleto control theinductionmotorasaseparatelyexcitedDCmotor [1–3]. Inthissense,thefield- orientedtechniqueguaranteesthedecouplingoftorqueandfluxcontrolcommandsforthe inductionmotor.Thismeansthatwhenthefluxisgovernedbymeansofcontrollingthe current id, thetorqueisnotaffected.Similarly,whenthetorqueisgovernedbycontrolling the current iq, thefluxisnotaffectedand,therefore,itcanbeachievedtransientresponse as fastasinthecaseofDCmachines. On theotherhand,whendealingwithindirectfield-orientedcontrolofinduction motors,aknowledgeofrotorspeedisrequiredinordertoorienttheinjectedstatorcurrent vector andtoestablishanadequatespeedfeedbackcontrol.Althoughtheuseofaflux estimatorindirectfieldorientedcontroleliminatestheneedofthespeedsensorinorderto orient theinjectedstatorcurrentvector,thismethodisnotpractical.Thisisbecausethe flux estimatordoesnotworkproperlyinalowspeedregion.Thefluxestimatorpresentsa pole ontheoriginofthe S plane (pureintegrator),andthereforeitisverysensitivetothe offset ofthevoltagesensorandtheparametervariations. However,thespeedorpositionsensorofinductionmotorstilllimitsitsapplicationsto somespecialenvironmentsnotonlyduetothedifficultiesofmountingthesensor,butalso becauseoftheneedoflowcostandreliablesystems.Theresearchanddevelopmentworkon a sensorlessdriverfortheACmotorisprogressinggreatly.Muchworkhasbeendoneusing thefieldorientedbasedmethodapproach [4–7]. Intheseschemesthespeedisobtainedbased onthemeasurementofstatorvoltagesandcurrents.Ontheotherhand,theinductionmotor modelcanbeobtainedusingaNeuralNetworkapproach.IntheworkofAlanisetal. [8] a discrete-timenonlinearsystemidentificationviarecurrenthighorderneuralnetworksis proposed.Inthisworkasixth-orderdiscrete-timeinductionmotormodelinthestatorfixed referenceframeiscalculatedusingtheproposedrecurrentneuralnetworksscheme. Nevertheless,therobustnesstoparametervariationsandloaddisturbancesinthe inductionmachinesstilldeservestobefurtherstudiedand,inparticular,specialattention should bepaidtothelowspeedregiontransients. Thus, theperformanceofthefieldorientedcontrolstronglydependsonuncertainties, which areusuallyduetounknownparameters,parametervariations,externalload disturbances,unmodelledandnonlineardynamics,etc.Therefore,manystudieshavebeen made onthemotordrivesinordertopreservetheperformanceundertheseparameter variationsandexternalloaddisturbances,suchasnonlinearcontrol,optimalcontrol, variablestructuresystemcontrol,adaptivecontrol,neuralcontrolandfuzzycontrol [9–13]. Recently,thegeneticalgorithmapproachhasalsobeenusedinordertocontrolthe electric motors.TheworkofMontazeri-Ghetal. [14], describestheapplicationofthe geneticalgorithmfortheoptimizationofthecontrolparametersinparallelhybridelectric vehiclesdrivenbyanelectricinductionmachine. To overcometheabovesystemuncertainties,the variablestructurecontrolstrategyusing the sliding-modehasbeenfocussedonmanystudiesandresearchforthecontroloftheAC servo drivesysteminthepastdecade [15–19]. Thesliding-modecontrolcanoffermanygood properties,suchasgoodperformanceagainstunmodelled dynamics,insensitivitytoparameter O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 301
  3. 3. variations,externaldisturbance rejection andfastdynamicresponse [20]. Theseadvantagesof the sliding-modecontrolmaybeemployedinthepositionandspeedcontrolofanACservo system. The robustpropertiesofthesliding-modesystemsarealsobeenemployedinthe observersdesign [21]. Inthisworkanobserver-basedsliding-modecontrolproblemis investigatedforaclassofuncertaindeltaoperatorsystemswithnonlinearexogenous disturbanceandthecontrolsystemstabilityisdemonstratedusingtheLyapunovstability theory. IntheworkofBoiko [22] the estimationprecisionandbandwidthofsliding-mode observersareanalyzedinthefrequencydomainfordifferentsettingsoftheobserverdesign parameters.Inthispaperanexampleofsliding-modeobserverdesignforestimationofDC motor speedfromthemeasurementsofarmaturecurrentisconsidered. A position-and-velocitysensorlesscontrolforbrushlessDCmotorsusinganadaptive sliding modeobserverisproposedinFuruhashi [23]. Inthisworkasliding-modeobserver is proposedinordertoestimatethepositionandvelocityforbrushlessDCmotors.Then, the velocityofthesystemisregulatedusingaPIcontrol.Asensorlesssliding-modetorque control forinductionmotorsusedinhybridelectricvehicleapplicationsisdevelopedin Proca etal. [24]. Thesliding-modecontrolproposedinthisworkallowsforfastandprecise torque trackingoverawiderangeofspeed.Thepaperalsopresentstheidentificationand parameterestimationofaninductionmotormodelwithvaryingparameters.Inthepaper [25] a surveyofapplicationsofsecond-ordersliding-modecontroltomechanicalsystemsis presented.Inthispaperdifferentsecond-ordersliding-modecontrollers,previously presentedintheliterature,areshownandsomechallengingcontrolproblemsinvolving mechanicalsystemsareaddressedandsolved.Arobustsliding-modesensorlessspeed- control schemeofavoltage-fedinductionmotorisproposedinRashedetal. [26]. Inthis work asecond-orderslidingmodeisproposedinordertoreducethechatteringproblem that usuallyappearsinthetraditionalsliding-modecontrollers.IntheworkofAuroraand Ferrara [27] a sliding-modecontrolalgorithmforcurrent-fedinductionmotorsis presented.Inthispaperisproposedanadaptivesecond-ordersliding-modeobserverfor speed androtorflux,andtheloadtorqueandtherotortimeconstantarealsoestimated. Thehigherorderslidingmode(HOSM)proposedinthiswork,presentsomeadvantagesover standardsliding-modecontrolschemes,oneofthemostimportantisthechatteringreduction. However intheHOSManaccurateknowledgeofrotorfluxandmachineparametersisthekey factorinordertoobtainahigh-performanceandhigh-efficiencyinduction-motorcontrol scheme. Then,thesecontrolschemesrequireamorepreciseknowledgeofthesystemparameters or theuseofestimatorsinordertocalculatethesystemparameters,whichimpliesmore computationalcostthantraditionalsliding-modecontrollers. On theotherhand,theslidingcontrolschemesrequirepriorknowledgeoftheupperbound for thesystemuncertaintiessincethisboundis employed intheswitchinggaincalculation. It shouldbenotedthatthechoiceofsuchboundmaynotbeeasilyobtainedduetothe complicatedstructureoftheuncertainties inpracticalcontrolsystems [28,29]. Moreover,this upperboundshouldbedeterminedasaccurately aspossible,becausethevaluetobe considered fortheslidinggainincreaseswiththe bound,andthereforethecontroleffortwillbe also proportionaltothisbound.Hence,ahigh upperboundforthesystemuncertainties implies morecontroleffortandtheproblemofthechatteringwillbeincreased. In ordertosurmountthisdrawback,inthispaperisproposedanadaptivelawinorder to calculatetheslidinggain.Therefore,inourproposedadaptivesliding-modecontrol scheme wedonotneedtocalculateanupperboundofthesystemuncertainties,which O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 302
  4. 4. greatlysimplifiesthecontrollerdesign.Moreover,thisupperboundcanbeunknownand can bevariablealongthetimebecausetheslidinggainisadaptedon-line. In thissense,thispaperpresentsanewsensorlessvectorcontrolschemeconsistingon the onehandofaspeedestimationalgorithmsothatthereisnoneedforaspeedsensor and ontheotherhandofanadaptativevariablestructurecontrollawwithanintegral sliding surfacethatcompensatesfortheuncertaintiesinthesystem.Intheproposed adaptivesliding-modecontrolscheme,unlikethetraditionalsliding-modecontrolschemes, the slidinggainisnotcalculatedinadvance,becauseitisestimatedon-lineinorderto compensatethepresentsystemuncertaintiesthatcanbevariablesalongthetime. Using thisvariablestructurecontrolintheinductionmotordrive,thecontrolledspeedis insensitivetovariationsinthemotorparametersandloaddisturbances.Thisvariable structurecontrolprovidesagoodtransientresponseandexponentialconvergenceofthe speed trajectorytrackingdespiteparameteruncertaintiesandloadtorquedisturbances. The closedloopstabilityoftheproposedschemeisdemonstratedusingLyapunov stabilitytheory,andtheexponentialconvergenceofthecontrolledspeedisalsoprovided. Thisreportisorganizedasfollows.Therotor speedestimationisintroducedinSection2. Then, theproposedrobustspeedcontrolwithadaptativeslidinggainispresentedinSection3. In Section4,somesimulationresultsarepresented.Finally,concludingremarksarestatedin the lastsection. 2. Rotorspeedcomputation Many schemesbasedonsimplifiedmotormodelshavebeendevisedtosensethespeedof the inductionmotorfrommeasuredterminalquantitiesforcontrolpurposes.Inorderto obtain anaccuratedynamicrepresentationofthemotorspeed,itisnecessarytobasethe calculationonthecoupledcircuitequationsofthemotor. Since themotorvoltagesandcurrentsaremeasuredinastationaryframeofreference,it is alsoconvenienttoexpresstheseequationsinthatstationaryframe. From thestatorvoltageequationsinthestationaryframeitisobtained [3]: _c dr ¼ Lr Lm vds Lr Lm Rs þ sLs d dt ids ð1Þ _c qr ¼ Lr Lm vqs Lr Lm Rs þ sLs d dt iqs ð2Þ where c is thefluxlinkage; L is theinductance; v is thevoltage; R is theresistance; i is the current and s ¼ 1L2 m=ðLrLsÞ is themotorleakagecoefficient.Thesubscripts r and s denoterespectivelytherotorandstatorvaluesreferredtothestator,andthesubscripts d and q denote the dq-axiscomponentsinthestationaryreferenceframe. The rotorfluxequationsinthestationaryframeare [3] _c dr ¼ Lm Tr idswrcqr 1 Tr cdr ð3Þ _c qr ¼ Lm Tr iqs þ wrcdr 1 Tr cqr ð4Þ where wr is therotorelectricalspeedand Tr=Lr/Rr is therotortimeconstant. O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 303
  5. 5. The angle ye of therotorfluxvector(cr ) inrelationtothe d-axisofthestationaryframe is definedasfollows: ye ¼ arctan cqr cdr ð5Þ being itsderivative: _y e ¼ we ¼ cdr _c qrcqr _c dr c2 dr þ c2 qr ð6Þ SubstitutingEqs.(3)and(4)inEq.(6)itisobtained: we ¼ wr Lm Tr cdriqscqrids c2 dr þ c2 qr ! ð7Þ Then, substitutingEq.(6)inEq.(7),andsolvingfor wr we obtain wr ¼ 1 c2 r cdr _c qrcqr _c dr Lm Tr ðcdriqscqridsÞ ð8Þ where c2 r ¼ c2 dr þ c2 qr. Therefore,givenacompleteknowledgeofthemotorparameters,theinstantaneous speed wr can becalculatedfromthepreviousequation,wherethestatormeasuredcurrent and voltages,andtherotorfluxestimationobtainedfromarotorfluxobserverbasedon Eqs. (1)and(2)havebeenemployed. 3. Variablestructurerobustspeedcontrolwithadaptiveslidinggain In general,themechanicalequationofaninductionmotorcanbewrittenas Jw_ m þ Bwm þ TL ¼ Te ð9Þ where J and B are theinertiaconstantandtheviscousfrictioncoefficientoftheinduction motorsystemrespectively; TL is theexternalload; wm is therotormechanicalspeedin angularfrequency,whichisrelatedtotherotorelectricalspeedby wm=2wr/p where p is the polenumbers,and Te denotesthegeneratedtorqueofaninductionmotor,definedas [3] Te ¼ 3p 4 Lm Lr ðce drie qsce qrie dsÞ ð10Þ where ce dr and ce qr are therotor-fluxlinkages,thesubscript‘e’denotesthatthequantityis referredtothesynchronouslyrotatingreferenceframe; iqs e and ids e are thestatorcurrents, and p is thepolenumber. The relationbetweenthesynchronouslyrotatingreferenceframeandthestationary reference frameiscomputedbytheso-calledreversePark’stransformation: xa xb xc 2 64 3 75 ¼ cosðyeÞ sinðyeÞ cosðye2p=3Þ sinðye2p=3Þ cosðye þ 2p=3Þ sinðye þ 2p=3Þ 2 64 3 75 xd xq # ð11Þ where ye is theanglepositionbetweenthe d-axis ofthesynchronouslyrotatingandthe stationaryreferenceframes,andthequantitiesareassumedtobebalanced. O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 304
  6. 6. Using thefield-orientationcontrolprinciple [3] the currentcomponent ids e is alignedin the directionoftherotorfluxvector cr, andthecurrentcomponent iqs e is alignedinthe directionperpendiculartoit.Undertheseconditions,itissatisfiedthat ce qr ¼ 0; ce dr ¼ jcrj ð12Þ Fig. 1 shows thevectorialdiagramoftheinductionmotorinthestationaryandinthe synchronouslyrotatingreferenceframes.Thesubscripts‘s’indicatesthestationaryframe and thesubscript‘e’indicatesthesynchronouslyrotatingreferenceframe. Therefore,takingintoaccountthepreviousresults,theequationofinductionmotor torque (10)issimplifiedto Te ¼ 3p 4 Lm Lr ce drie qs ¼ KT ie qs ð13Þ wherethetorqueconstant, KT, isdefinedasfollows: KT ¼ 3p 4 Lm Lr ce dr ð14Þ ce dr being thecommandrotorflux. With theabove-mentionedfieldorientation,thedynamicsoftherotorfluxisgivenby [3] dce dr dt þ ce dr Tr ¼ Lm Tr ie ds ð15Þ Then, themechanicalequation(9)becomes w_ m þ awm þ f ¼ bie qs ð16Þ Fig. 1.Vectorialdiagramoftheinductionmotor. O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 305
  7. 7. where theparametersaredefinedas a ¼ B J ; b ¼ KT J ; f ¼ TL J ð17Þ Now,wearegoingtoconsiderthepreviousmechanicalequation(16)withuncertainties as follows: w_ m ¼ ða þ DaÞwmðf þ DfÞ þ ðb þ DbÞie qs ð18Þ where theterms Da, Db and Df representtheuncertaintiesoftheterms a, b and f respectively.Itshouldbenotedthattheseuncertaintiesareunknown,andthattheprecise calculationofanupperboundis,ingeneral,ratherdifficulttoachieve. Let usdefinethetrackingspeederrorasfollows: eðtÞ ¼ wmðtÞw mðtÞ ð19Þ where wm n is therotorspeedcommand. Takingthederivativeofthepreviousequationwithrespecttotimeyields e_ðtÞ ¼ w_ mw_ m ¼ aeðtÞ þ uðtÞ þ dðtÞ ð20Þ where thefollowingtermshavebeencollectedinthesignal u(t): uðtÞ ¼ bie qsðtÞaw mðtÞf ðtÞw_ mðtÞ ð21Þ and theuncertaintytermshavebeencollectedinthesignal d(t), dðtÞ ¼ DawmðtÞDf ðtÞ þ Dbie qsðtÞ ð22Þ To compensatefortheabovedescribeduncertaintiespresentinthesystem,asliding adaptivecontrolschemeisproposed.Intheslidingcontroltheory,theswitchinggainmust be constructedsoastoattaintheslidingcondition [20,30]. Inordertomeetthisconditiona suitable choiceoftheslidinggainshouldbemadetocompensatefortheuncertainties.To select theslidinggainvector,anupperboundoftheparametervariations,unmodelled dynamics,noisemagnitudes,etc.shouldbegiven,butinpracticalapplicationsthereare situationsinwhichtheseboundsareunknown,oratleastdifficulttocalculate.Asolution could betochooseasufficientlyhighvaluefortheslidinggain,butthisapproachcould cause atoohighcontrolsignal,oratleastmorecontrolactivitythanneededinorderto achieve thecontrolobjective. One possiblewaytoovercomethisdifficultyistoestimatethegainandtoupdateitby means ofsomeadaptationlaw,sothattheslidingconditionisachieved. Now,wearegoingtoproposetheslidingvariable S(t) withanintegralcomponentas SðtÞ ¼ eðtÞ þ Z t 0 ða þ kÞeðtÞ dt ð23Þ where k is aconstantgain,and a is aparameterthatwasalreadydefinedinEq.(17). Then theslidingsurfaceisdefinedas SðtÞ ¼ eðtÞ þ Z t 0 ða þ kÞeðtÞ dt ¼ 0 ð24Þ Now, wearegoingtodesignavariablestructurespeedcontroller,thatincorporatesan adaptiveslidinggain,inordertocontroltheACmotordrive uðtÞ ¼ keðtÞ^b ðtÞg sgnðSÞ ð25Þ O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 306
  8. 8. wherethe k is thegaindefinedpreviously, ^b is theestimatedswitchinggain, g is apositive constant, S is theslidingvariabledefinedinEq.(23)andsgnðÞ is thesignfunction. The switchinggain ^b is adaptedaccordingtothefollowingupdatinglaw: _^b ¼ gjSj; ^b ð0Þ ¼ 0 ð26Þ where g is apositiveconstantthatletuschoosetheadaptationspeedfortheslidinggain. In ordertoobtainthespeedtrajectorytracking,thefollowingassumptionsshouldbe formulated: ðA1Þ The gain k must bechosensothattheterm(aþk) isstrictlypositive.Thereforethe constant k should be k4a. ðA2Þ Thereexitsanunknownfinitenon-negativeswitchinggain b such that b4dmax þ Z; Z40 where dmaxZjdðtÞj 8t and Z is apositiveconstant. Note thatthisconditiononlyimpliesthattheuncertaintiesofthesystemarebounded magnitudes. ðA3Þ The constant g must bechosensothat gZ1. Theorem 1. Consider theinductionmotorgivenbyEq. (18). Then, if assumptions ðA1Þ–ðA3Þ are verified, the controllaw (25) leads therotormechanicalspeedwm(t) so thatthespeed trackingerrore(t)=wm(t)wm n (t) tends tozeroasthetimetendstoinfinity. The proofofthistheoremwillbecarriedoutusingtheLyapunovstabilitytheory. Proof. Define theLyapunovfunctioncandidate: VðtÞ ¼ 1 2 SðtÞSðtÞ þ 1 2 ~b ðtÞ~b ðtÞ ð27Þ where S(t) istheslidingvariabledefinedpreviouslyand ~b ðtÞ ¼ ^b ðtÞb Its timederivativeiscalculatedas _V ðtÞ ¼ SðtÞ_S ðtÞ þ ~b ðtÞ _~b ðtÞ ¼ S½_e þ ða þ kÞe þ ~b ðtÞ _^b ðtÞ ¼ S½ðae þ u þ dÞ þ ðke þ aeÞ þ ~bgjSj ¼ S½u þ d þ ke þ ð^b bÞgjSj ¼ S½ke^bgsgnðSÞ þ d þ ke þ ð^b bÞgjSj ¼ S½d^ bgsgnðSÞ þ ^bgjSjbgjSj ¼ dS^ bgjSj þ ^ bgjSjbgjSj ð28Þ rjdjjSjbgjSj rjdjjSjðdmax þ ZÞgjSj ¼ jdjjSjdmaxgjSjZgjSj rZgjSj ð29Þ O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 307
  9. 9. then _V ðtÞr0 ð30Þ It shouldbenotedthatEqs.(23),(20),(25)and(26),andtheassumptions ðA2Þ and ðA3Þ have beenusedintheproof. UsingLyapunov’sdirectmethod,since V(t) isclearlypositive-definite, _V ðtÞ is negative semidefiniteand V(t) tendstoinfinityas S(t) and ~b ðtÞ tends toinfinity,thentheequilibrium at theorigin ½SðtÞ; ~b ðtÞ ¼½0; 0 is globallystable,andthereforethevariables S(t) and ~b ðtÞ are bounded.Then,since S(t) isboundedonehasthat e(t) isalsobounded. Besides,computingthederivativeofEq.(23),itisobtained: _S ðtÞ ¼ _eðtÞ þ ða þ kÞeðtÞ ð31Þ then, substitutingEq.(20)inEq.(31), _S ðtÞ ¼ aeðtÞ þ uðtÞ þ dðtÞ þ ða þ kÞeðtÞ ¼ keðtÞ þ dðtÞ þ uðtÞ ð32Þ FromEq.(32)wecanconcludethat _S ðtÞ is boundedbecause e(t), u(t) and d(t) are bounded. Now,fromEq.(28)itisdeducedthat €V ðtÞ ¼ d_S ðtÞbg d dt jSðtÞj ð33Þ which isaboundedquantitybecause _S ðtÞ is bounded. Undertheseconditions,since €V is bounded, _V is auniformlycontinuousfunction,so Barbalat’slemmaletusconcludethat _V -0 as t-1, whichimpliesthat SðtÞ-0 as t-1. Therefore S(t) tendstozeroasthetime t tendstoinfinity.Moreover,alltrajectories startingofftheslidingsurface S=0 mustreachitasymptoticallyandthenwillremainonthis surface.Thissystem’sbehavior,onceontheslidingsurfaceisusuallycalled slidingmode [20]. Whentheslidingmodeoccursontheslidingsurface(24),then SðtÞ ¼ _S ðtÞ ¼ 0, and therefore thedynamicbehaviorofthetrackingproblem(20)isequivalentlygovernedby the followingequation: _S ðtÞ ¼ 0 ) _eðtÞ ¼ ða þ kÞeðtÞ ð34Þ Then, underassumption ðA1Þ, thetrackingerror e(t) convergestozeroexponentially. It shouldbenotedthat,atypicalmotionundersliding-modecontrolconsistsofa reaching phase duringwhichtrajectoriesstartingofftheslidingsurface S=0 movetowardsitand reachit,followedbya slidingphase duringwhichthemotionisconfinedtothissurfaceand wherethesystemtrackingerror,representedbythereduced-ordermodel(34),tendstozero. Finally,thetorquecurrentcommand, iqs en(t), canbeobtaineddirectlysubstitutingEq.(25) in Eq.(21): ie qs ðtÞ ¼ 1 b ½ke^ bgsgnðSÞ þ aw m þ w_ m þ f ð35Þ Therefore,theproposedvariablestructurespeedcontrolwithadaptiveslidinggain resolves thespeedtrackingproblemfortheinductionmotor,withuncertaintiesin mechanicalparametersandloadtorque. O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 308
  10. 10. 4. Simulationresults In thissectionwewillstudythespeedregulationperformanceoftheproposedadaptive sliding-modefieldorientedcontrolversusspeedreferenceandloadtorquevariationsby means ofsimulationexamples.Inparticular,theexamplepresentedconsistofarepre- sentativespeedreferencetrackingproblem,combinedwithloadtorquevariationsduring the evolutionoftheexperimentandconsideringacertaindegreeofuncertainty,inorderto attain acompletescopeofthebehaviorofthesystem. The blockdiagramoftheproposedrobustcontrolschemeispresentedin Fig. 2. The block‘VSCcontroller’representstheproposedadaptivesliding-modecontroller,and it isimplementedbyEqs.(23),(35),and(26).Theblock‘limiter’limitsthecurrentapplied to themotorwindingssothatitremainswithinthelimitvalue,anditisimplementedbya saturationfunction.Theblock‘ dqe-abc’ makestheconversionbetweenthesynchro- nouslyrotatingandstationaryreferenceframes,andisimplementedbyEq.(11).Theblock ‘currentcontroller’consistsofathreehysteresis-bandcurrentPWMcontrol,whichis basicallyaninstantaneousfeedbackcurrentcontrolmethodofPWMwheretheactual current(iabc) continuallytracksthecommandcurrent(iabc n ) withinahysteresisband.The block‘PWMinverter’isasixIGBT-diodebridgeinverterwith780VDCvoltagesource. The block‘fieldweakening’givesthefluxcommandbasedonrotorspeed,sothatthePWM controllerdoesnotsaturate.Theblock‘ids en calculation’providesthecurrentreference ids en fromtherotorfluxreferencethroughEq.(15).Theblock‘wr estimator’representsthe proposedrotorspeedandsynchronousspeedestimator,anditisimplementedbyEqs.(8) and (6)respectively.Finally,theblock‘IM’representstheinductionmotor. The inductionmotorusedinthiscasestudyisa50HP,460V,fourpole,60Hzmotor having thefollowingparameters: Rs ¼ 0:087 O, Rr ¼ 0:228 O, Ls=35.5 mH, Lr=35.5 mH, and Lm=34.7 mH. The systemhasthefollowingmechanicalparameters: J=1.357kgm2 and B=0.05 N m s.Itisassumedthatthereisanuncertaintyaround20%inthesystemparameters,which will beovercomebytheproposedslidingcontrol. The followingvalueshavebeenchosenforthecontrollerparameters: k=45 and g ¼ 30. In thisexamplethemotorstartsfromastandstillstateandwewanttherotorspeedto follow aspeedcommandthatstartsfromzeroandacceleratesuntiltherotorspeedis Fig. 2.Blockdiagramoftheproposedadaptivesliding-modecontrol. O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 309
  11. 11. 100 rad/s,thentherotorspeedismaintainedconstantandafter,attime1.3s,therotor deceleratesuntiltherotorspeedis80rad/s.Inthissimulationexample,thesystemstarts with aninitialloadtorque TL=0 Nmandattime t=2.3 stheloadtorquestepsfrom TL=0 to200Nm,andasbefore,itisassumedthatthereisanuncertaintyaround20%in the loadtorque. Fig. 3 shows thedesiredrotorspeed(dashedline)andtherealrotorspeed(solidline). As itmaybeobserved,afteratransitorytimeinwhichtheslidinggainisadapted,therotor speed tracksthedesiredspeedinspiteofsystemuncertainties.However,attime t=2.3 sa small speederrorcanbeobserved.Thiserrorappearsbecauseatorqueincrementoccursat this time,sothatthecontrolsystemlosestheso-called‘slidingmode’becausetheactual sliding gainistoosmallinordertoovercomethenewuncertaintyintroducedinthesystem due tothenewtorque.Butthen,afterashorttimetheslidinggainisadaptedonceagainso that thisgaincancompensateforthesystemuncertaintieswhicheliminatestherotor speed error. Fig. 4 presentsthetimeevolutionoftheestimatedslidinggain.Theslidinggainstarts from zeroandthenitisincreaseduntilitsvalueishighenoughtocompensateforthe system uncertainties.Besides,theslidingremainsconstantbecausethesystemuncertainties remain constantaswell.Later,attime2.3s,thereisanincrementinthesystem uncertaintiescausedbytheincrementintheloadtorque.Therefore,theslidingshouldbe adapted onceagaininordertoovercomethenewsystemuncertainties.Asitcanbeseenin the figureaftertheslidinggainisadapted,itremainsconstantagain,sincethesystem uncertaintiesremainconstantaswell. It shouldbenotedthattheadaptiveslidinggainallowstoemployasmallerslidinggain, so thatthevalueoftheslidinggaindonothavetobechosenhighenoughtocompensate for allpossiblesystemuncertainties,becausewiththeproposedadaptiveschemethesliding gain isadapted(ifitisnecessary)whenanewuncertaintyappearsinthesysteminorderto surmount thisuncertainty. Fig. 5 shows thetimeevolutionoftheslidingvariable.Inthisfigureitcanbeseenthat the systemreachtheslidingcondition(S(t)=0) attime t=0.13 s,butthenthesystemloses 0 0.511.522.53 0 20 40 60 80 100 120 Time (s) Rotor Speed (rad/s) wm * wm Fig. 3.Referenceandrealrotorspeedsignals(wm n , wm). O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 310
  12. 12. this conditionattime t=2.3 sduetothetorqueincrementwhich,inturn,producesan incrementinthesystemuncertaintiesthatcannotbecompensatedbytheactualvalueof the slidinggain.However,afteratransitorytimeinwhichtheslidinggainisadaptedin order tocompensatethenewsystemuncertainty,thesystemreachesthesliding conditionagain. Fig. 6 showsthecurrentofonestatorwinding.Thisfigureshowsthatintheinitialstate, the currentsignalpresentsahighvaluebecauseahightorqueisnecessarytoincrementthe rotor speedduetotherotorinertia.Then,intheconstant-speedregion,themotortorque only hastocompensatethefrictionandtherefore,thecurrentislower.Finally,attime t=2.3 sthecurrentincreasesbecausetheloadtorquehasbeenincreased. 0 0.511.522.53 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Sliding Variable Fig. 5.Slidingvariable. 0 0.511.522.53 0 2 4 6 8 10 12 14 Time (s) Sliding Gain Fig. 4.Estimatedslidinggain ð^b Þ. O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 311
  13. 13. Fig. 7 shows themotortorque.Asinthecaseofthecurrent(Fig. 6), themotortorque has ahighinitialvalueinthespeedaccelerationzoneandthenthevaluedecreasesina constant region.Later,attime t=1.3 s,themotortorquedecreasesagaininorderto reduce therotorspeed.Finally,attime t=2.3 sthemotortorqueincreasesinorderto compensatetheloadtorqueincrement.Inthisfigureitmaybeseenthatinthemotor torque appearstheso-calledchatteringphenomenon,howeverthishighfrequencychanges in thetorquewillbefilteredbythemechanicalsysteminertia. 5. Conclusions In thispapersensorlessrobustvectorcontrolforinductionmotordriveswithan adaptivevariablesliding-modevectorcontrollawhasbeenpresented.Therotorspeed 0 0.511.522.53 −500 −400 −300 −200 −100 0 100 200 300 400 500 Time (s) Stator Current Fig. 6.Statorcurrent(isa). 0 0.511.522.53 −100 −50 0 50 100 150 200 250 300 Motor Torque (N) Time (s) Fig. 7.Motortorque(Te). O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 312
  14. 14. estimatorisbasedonstatorvoltageequationsandrotorfluxequationsinthestationary referenceframe.Itisproposedasavariablestructurecontrolwhichusesanintegralsliding surfacetorelaxtherequirementoftheaccelerationsignal,thatisusualinconventional sliding-modespeedcontroltechniques.Duetothenatureoftheslidingcontrolthiscontrol scheme isrobustunderuncertaintiescausedbyparametererrorsorbychangesintheload torque. Moreover,theproposedvariablestructurecontrolincorporatesandadaptive algorithmtocalculatetheslidinggainvalue.Theadaptationoftheslidinggain,ontheone hand avoidstheneedofcomputingtheupperboundofthesystemuncertainties,andon the otherhandallowstoemployassmallerslidinggainaspossibletoovercometheactual system uncertainties.Thenthecontrolsignalofourproposedvariablestructurecontrol schemes willbesmallerthanthecontrolsignalsofthetraditionalvariablestructurecontrol schemes,becauseinthesetraditionalschemestheslidinggainvalueshouldbechosenhigh enoughtoovercomeallthepossibleuncertaintiesthatcouldappearinthesystemalong the time. The closedloopstabilityofthedesignpresentedinthispaperhasbeenprovedthought Lyapunovstabilitytheory.Finally,bymeansofsimulationexamples,ithasbeenshown that theproposedcontrolschemeperformsreasonablywellinpractice,andthatthespeed trackingobjectiveisachievedunderuncertaintiesintheparametersandloadtorque. Acknowledgments The authorsareverygratefultotheBasqueGovernmentbythesupportofthiswork through theprojectS-PE09UN12andtotheUPV/EHUbyitssupportthroughproject GUI07/08. References [1] W.Leonhard,ControlofElectricalDrives,Springer,Berlin,1996. [2] P.Vas,VectorControlofACMachines,OxfordSciencePublications,Oxford,1994. [3] B.K.Bose,ModernPowerElectronicsandACDrives,PrenticeHall,NewJersey,2001. [4] R.Beguenane,M.A.Ouhrouche,A.M.Trzynadlowski,Anewschemeforsensorlessinductionmotorcontrol drives operatinginlowspeedregion,MathematicsandComputersinSimulation71(2006)109–120. [5] S.Sunter,Slipenergyrecoveryofarotor-sidefieldorientedcontrolledwoundrotorinductionmotorfedby matrix converter,JournaloftheFranklinInstitute345(2008)419–435. [6] M.Comanescu,Aninduction-motorspeedestimatorbasedonintegralsliding-modecurrentcontrol,IEEE Transactions onIndustrialElectronics56(9)(2009)3414–3423. [7] M.I.Marei,M.F.Shaaban,A.A.El-Sattar,Aspeedestimationunitforinductionmotorsbasedonadaptive linear combiner,EnergyConversionandManagement50(2009)1664–1670. [8] A.Y.Alanis,E.N.Sanchez,A.G.Loukianov,E.A.Hernandez,Discrete-timerecurrenthighorderneural networks fornonlinearidentification,JournaloftheFranklinInstitute347(2010)1253–1265. [9] T-J.Ren,T-C.Chen,Robustspeed-controlledinductionmotordrivebasedonrecurrentneuralnetwork, Electric PowerSystemResearch76(2006)1064–1074. [10] M.Montanari,S.Peresada,A.Tilli,Aspeed-sensorlessindirectfield-orientedcontrolforinductionmotors based onhighgainspeedestimation,Automatica42(2006)1637–1650. [11] R.Marino,P.Tomei,C.M.Verrelli,Anadaptivetrackingcontrolfromcurrentmeasurementsforinduction motors withuncertainloadtorqueandrotorresistance,Automatica44(2008)2593–2599. [12] J.B.Oliveira,A.D.Araujo,S.M.Dias,Controllingthespeedofathree-phaseinductionmotorusinga simplified indirectadaptiveslidingmodescheme,ControlEngineeringPractice18(2010)577–584. [13] M.A.Fnaiech,F.Betin,G.A.Capolino,F.Fnaiech,Fuzzylogicandsliding-modecontrolsappliedtosix- phase inductionmachinewithopenphases,IEEETransactionsonIndustrialElectronics57(1)(2010) 354–364. O. Barambones,P.Alkorta/JournaloftheFranklinInstitute348(2011)300–314 313
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