A robust vector control for induction motor

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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.
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