for Hysteresis Models of Magnetostrictive Actuators
James M. Nealis 1
and Ralph C. Smith 2
Department of Mathematics
Center for Research in Scientic Computation
North CarolinaState University
Raleigh,NC 27695
Abstract
Increased control demands in applications including high speedmilling and hybrid motor design have led
totheutilization ofmagnetostrictivetransducersoperatingin hystereticandnonlinearregimes. Toachievethe
high performance capabilities of these transducers, models and control laws must accommodate thenonlinear
dynamicsinamannerwhichisrobustandfacilitatesreal-timeimplementation. Thisnecessitatesthedevelopment
ofmodelsandcontrolalgorithmswhichutilizeknownphysicstothedegreepossible,areloworder,andareeasily
updatedtoaccommodatechangingoperatingconditionssuchastemperature. Weconsiderherethedevelopment
ofnonlinearadaptiveidenticationforloworder,energy-basedmodels. Weillustratethetechniquesinthecontext
ofmagnetostrictivetransducersbuttheyaresuÆcientlygeneraltobeemployedforanumberofcommonlyused
smartmaterials. Theperformanceoftheidenticationalgorithmis illustratedthroughnumericalexamples.
Keywords: Nonlinearparameterization,hysteresisandconstitutivenonlinearities,magnetostrictivematerials
1. Introduction
Piezoceramic and magnetostrictive transducers are nding increased use in high performance applications
duetotheirset pointaccuracyandhighbandwidthcapabilities. However,theyalsoexhibithysteresisand
con-stitutive nonlinearities which must be accommodatedto achievedesign specications. At low frequenciesand
moderatedrivelevels,these eectscanoftenbemitigatedthroughfeedbackloops. Athighdrivelevelsorhigh
frequencies,however,the hysteresisandnonlineardynamics mustbeincorporatedinto models andsubsequent
controldesigns. Inthispaper,weconsiderthedevelopmentofanonlinearadaptiveparameterestimation
algo-rithmforupdatingparametersin energy-basedhysteresismodels.
Toillustrate, consider the prototypicalmagnetostrictiveactuator depicted in Figure 1. Input stresses and
displacementsare provided bythe Terfenol-D rod in response to elds generated bythe surrounding solenoid.
Asdetailedin[1,2],suchactuatorshavethecapabilityofgeneratingbroadband,highforce,responses. However,
theyalsoexhibit varying degreesof hysteresisandnonlinearities in therelationbetweentheinputeld H,the
magnetizationMandstrainsintheTerfenol-Drod. Weemploythistransducerdesignasatemplatefor
devel-opingthenonlinearadaptiveestimationtechniques discussedherebutwenotethat themodelsand estimation
techniquesaresuÆcientlygeneraltopermitdirect extensiontoanalogouspiezoelectricandferroelectricmodels
ofthetypedevelopedin[7,9].
1
Email:[email protected],Telephone: (919)515-8968
2
000000
000000
111111
111111
End
Spring
Washer
Wound Wire Solenoid
Permanent Magnet
Direction of
Rod Motion
Terfenol-D Rod
Compression
Bolt
Mass
Figure 1. Prototypicalmagnetostrictivetransducer.
ThereexistanumberoftechniquesformodelinghysteresisinmagnetostrictivematerialsincludingPreisach
models[10,12]anddomainwallmodels[1,2]. Asillustratedin[11,12],Preisachmodelscan,under
approxima-tion,belinearlyparameterizedintermsofcoeÆcientstobeidentiedthuspermittingtheuseoflinearadaptive
algorithms. Thisadvantageisoftenoset,however,bythelargenumberofrequirednonphysicalparametersand
theextensionstothetheoryrequiredtoaccommodatetemperatureandfrequencydependence. Thedomainwall
modelsareloworderandhavephysicalparametersbutexhibitanonlineardependenceontheparameters. Inthis
paper,weextendthetechniquesof[3,4]toobtainnonlinearadaptiveestimationlawsforupdatingparameters
in thesedomainwallmodels.
AsignicantdiÆcultyin developinganonlinearparameteradaptationlawis thefact that gradientupdate
methods are not always suÆcient for estimating nonlinearly occurring parameters. To illustrate, consider an
errormodeloftheform
_
e= ke+f(;) f(; b
)
wheree istheerrorbetweenthedesiredandthemeasuredtrajectories. Wedenotethemeasurable statesas,
k >0 isascalar, is anonlinearlyoccurring parameter, b
is theparameterestimate and f is ascalarvalued
nonlinearfunction. Considerthegradientupdatelaw
_
b
=erf
b
:
With thestandardLyapunovfunction,V = 1
2 (e
2
+ ~
2
),where ~
= b
,weseethat
_
V = ke 2
+e h
f(;) f(; b
)+ ~
rf
b
i
:
Ife<0,itisnecessarythatrf
b
(
b
)f(;) f(; b
)whichimpliesthatf isconvex. Ife>0,thenf must
beconcaveto ensure _
V 0. We observethe gradient method does notensure stability for all b
. A gradient
method applied to a nonlinearparameterized systemmaynot onlybeinsuÆcientbut may leadto instability.
The method we discuss heredoes notstrictly rely ona gradientrule but diers depending on the signof the
error[3,4].
InSection2wesummarizethehysteresisandtransducermodel. InSection3wereviewthenonlinearadaptive
method for thescalar case. In Section 4wepresent anextension of themethod to thevectorcase. Section5
providesnumericalexamplesofboththescalarandvectorcases.
2. Transducer Model
Wesummarizeherethemodeldevelopedin[1,2]formagnetostrictivetransducersoperatinginnonlinearand
hystereticregimes. Thismodelisformulatedintwosteps: (i)quanticationofanhystereticmagnetization,M
an
and(ii)quanticationofthetotalmagnetization,M.
Onesource of hysteresis occurs in the relationship betweenan applied magnetic eld H and the resulting
−5
0
5
x 10
4
−8
−6
−4
−2
0
2
4
6
8
x 10
5
Applied Field
Anhysteretic Magnetization
−5
0
5
x 10
4
−8
−6
−4
−2
0
2
4
6
8
x 10
5
Applied Field
Magnetization
(a) (b)
Figure 2. (a)Anhysteretic magnetization;(b)Totalmagnetization.
in thematerialand pinnedat inclusions. Physically,theanhystereticmagnetization M
an
canbethoughtof as
themagnetization obtainedwhen domainwallsaretranslated acrosspinning sites withno lossofenergy. The
anhystereticmagnetizationdependsontheeectiveeld givenbyH
e
=H+M wherequantiestheeects
oftheinterdomaincoupling. Undertheassumptionofconstantstress,webalancethermalandmagnetostrictive
energyviaBoltzmann principles. TheanhystereticmagnetizationcanbegivenbyeithertheLangevinmodel
M
an =M
s
coth
H
e
a
a
H
e
(1)
ortheIsingmodel
M
an =M
s tanh
H
e
a
(2)
dependingontheassumptionswemakeontheorientationofdipoles. HereM
s
isthesaturationmagnetostriction
andaisatemperaturedependentcoeÆcient. Thetwoanhystereticmodelsareequivalenttothirdorder. Sincethe
parameteraiscloselyeectedbythetemperature,wewillchosetoupdateainthenonlinearadaptiveparameter
estimation. The ability to adapt a to varying conditions would be extremely useful since the temperature is
diÆcult to regulatein many industrial applications. Forexample, in the transducer depictedin Figure 1,the
currentin thesolenoidcancauseOhmicheatingandeect thevalueofa.
Toquantifythetotalmagnetization,weincorporate theirreversiblemagnetizationM
irr
dueto domainwall
translation. Magnetostaticprincipalsareusedtocomputetheenergyrequiredtoreorientdipoles,asdetailed in
[1,2]. Thisyieldsthedierentialequation
@M
irr
@H =
^
Æ M
an M
irr
kÆ (M
an M
irr )
: (3)
Here Æ = sign(dH) to ensure the pinning is opposite the change in magnetization and ^
Æ is 0 if dH > 0 and
M > M
an
or dH < 0 and M < M
an
and 1 otherwise. This indicator is necessary to model the physical
observationthat,afteraeldreversal,thechangesinmagnetizationarepurelyreversibleuntiltheanhysteretic
valueisreached. Theparameterkquantiestheaverageenergyrequiredto translateadomainwall.
The reversible magnetization M
rev
is due to domain wall bending. As detailed in [1, 2], the reversible
magnetismisgivenbythealgebraicrelationship
M
rev =c(M
an M
irr
) (4)
wherecisamaterialparameterwhich quantiesthereversibilityofthematerial. Thetotalmagnetismisthen
M =(1 c)M
irr +cM
an
zation,wereformulate(5)as
@M
@H
=F(H;M) (6)
M(H
0 )=M
0
where
F(H;M)=
1
1+c @
@H M
an
^
Æ M
an M
kÆ (M^
an M)
+c @
@H M
an
(7)
with^=
1 c
. Theanhystereticandtotalmagnetizationareillustratedin Figure2.
Themodel developedin (5) or (6) quanties thehysteretic relationshipbetween theimposed eldand the
resulting magnetization. Next, we need to quantify the strains, forces, and displacements generated by the
changesinmagnetization. Wedothisintwosteps: (i)quantifythefreestrainsinthematerialand(ii)quantify
the totalstrainswhich include elastic eects. Wepresent abrief overview of themodelof the full transducer
dynamics. Foramorecompletederivationsee[2].
Wecharacterizethefreestrain,ormagnetostriction,foraTerfenol-Dactuatorbythequadraticrelation
(t)= 3
s
2M 2
s M
2
(t) (8)
where
s
denotesthe saturationmagnetostriction. Toachievebidirectionalstrainsorforces, thetransducer is
biasedby asurrounding magnet orthe application ofaDC eld to thesolenoid. Forabias ofM
s
=2, thefree
strainismodeledby
(t)= 3
s
2M 2
s
M 2
(t)+2M
s M(t)
: (9)
Oncewehavequantiedthemagnetostrictionthatoccursinresponsetoanappliedeld,wemustincorporate
thematerialselasticproperties. Weassumeoneendoftherod(x=0)to bexed whiletheotherend(x=L)
is constrained by a damped oscillator and has a point mass attached (see Figure 3). The internal damping
coeÆcient,density,Young'sModulusandpointmassaregivenbyc
D
,,E,andM
`
,respectively. Thedamping
springisassumedtohavestinessk
`
andKelvin-VoigtdampingcoeÆcientc
` .
Ifweassumelinearelasticity,Kevin-Voigtdampingandsmalldisplacements,thenthestressatanypointx,
0xL,isgivenby
(t;x)=E @u
@x
(t;x)+c
D @
2
u
@x@t
(t;x) E(t) (10)
whereu(t;x)isthelongitudinaldisplacement. Weassumethatthemagnetostrictiongivenby(9)isindependent
of position. This independence is reasonable since ux shaping via the surrounding magnet can be used to
minimizeend eects in therodwhich resultsinuniform magnetostrictionalongthe rod. Forcebalancing then
yields
A @
2
u
@t 2
= @N
tot
@x
(11)
wheretheresultantisspeciedgivenby
N
tot
(t;x)=EA @u
@x
(t;x)+c
D A
@ 2
u
@x@t
(t;x) EA(t): (12)
M
k
Rod
N
C
L
d
u
d
t
L
u
L
tot
x=L
+u
N
tot
(t;L)= k
`
u(t;L) c
` @u
@t
(t;L) M
` @
2
u
@x@t (t;L):
Wetaketheinitial conditionsto beu(0;x)=0and @u
@x
(0;x)=0. WecannowuseaGalerkinniteelementto
numericallyapproximatethesolutionto thePDE(11).
Wespecied themagnetostriction (t) to be independent of spatial location due to the design of the
sur-roundingpermanentmagnet. Thisimpliesthatthedynamicsofthetransducerwhicharecurrentlymodeledby
aPDEcanbeaccuratelyapproximatedbyadampedspringmasssystem
b
G (s)=
1
ms 2
+ks+c =
w
s 2
+ ^
ks+^c
: (13)
WeexaminedthepolesandzerosofthetransferfunctionswhichresultedfromaGalerkinniteelement
approx-imationwithvaryingnumberofbasiselements. Wefoundthemodelcouldbeadequatelyapproximatedbyone
with twopoles and no zerosfor any number of basis elements. We used thepoles and gainof these transfer
functions to developed the damped spring mass model for the transducer dynamics. Theparameters for the
resultingmodel(13)weredeterminedtobew=1:372410 2
, ^
k=7:889910 3
andc^=6:425110 7
.
3. Nonlinear Adaptive Parameter Estimation
Wewishtoadaptivelyestimateandupdatethenonlinearlyoccurringparameterainthehystereticmodel(6)
tomodeltheeectsofchangingtemperatures. Toaccomplishthis,weconsiderthetheoryin[3,4]anddevelop
modicationsrequiredforthehysteresismodelemployedhere. Onecriterionforthealgorithmisthecapability
toobtainestimatesofawhicharesuÆcientlyaccuratetomaintaintolerancesspeciedforthetransducers(e.g.,
cuttingtolerancesof.001in). Furthermore,thealgorithmmustbestableandpersistentexcitationconditions
mustbeestablishedtoensureconvergence.
Thenonlinearparameterizationassumesallofthestatesareavailableandidentiesparametersforasystem
oftheform
_
y= ky+af(u(t);)
wherek>0isascalarand2R m
is anunknownparameter,2whereis theboundedregionin which
lies. Thefunction f istakento beascalarvaluednonlinearfunctionof theinputu(t). Asmotivatedby[3,4],
weconsidertheestimationalgorithm
_
b
y = kby sat( ~ y
)+af(u; b
) a
sat
~ y
_
b
= y~
~ y
= y~ sat
~ y
~
y = yb y
(14)
where>0,sat()isasaturationfunctiondened as
sat(x)= 8
>
>
<
>
>
:
1; x1
x; jxj<1
1; x1
anda
and
arethesolutionof
a
= min
2R m
max
2
g(;)
= arg min
2R m
max
2
g(;)
g(;) = asat( ~ y
)
h
f(u; b
) f(u;) T
( b
)
i
:
Themethodwillcontinuetoadapt theparametersuntilthemagnitudeoftheerrory~islessthanthegiven.
We consider the min/max algorithm (15) to handle the regions of nonconvexity of f where the gradient
method is insuÆcient. The use of a tuning error y~
rather than a tracking error y~ensures continuity of the
adaptation as does the use of a saturation function over that of a signum function [4]. We do not need the
assumptionthat theparametersand theparameterestimatesarebounded forstability, butrather tocompute
theclosedformsolutionof(15).
Ifwedene ~
= b
andx=[~y; ~
T
] T
,thenwecanshowthatthesystem(14)isstablewithx=0byproving
thatV =e 2
+a ~
2
isaLyapunovfunction. Followingtheoryoutlinedin[5],werstnotethat _
V =2~y
_ ~ y +2a ~ _ ~ .
Ifj~yjtheny~
=0whichimplies _
V =0. Wethenneedtoshowthat _
V 0ifj~yj>. Wecanexpress _
V as
_
V = 2~y
( kby sat
~ y
+af(u; b ) a sat ~ y
+ky af(u;)) 2a ~
y~
= 2ky~
~ y+2~y
(af(u;
b
) af(u;) a ~ sat ~ y a sat ~ y )
= 2ky~
~ y+2~y
h
a(f(u; b
) f(u;) ~
) sat ~ y a sat ~ y i :
Ify~>0,thensat
~ y
=1sowemusthave
a asat ~ y (f(u; b
) f(u;) ~
) sat
~ y
forall2:
Thisimpliesthatwecanlet
a
=a max
2 sat ~ y (f(u; b
) f(u;) ~
)forany
sobythedenitionof
anda
theinequalityissatisedandhence _
V 0. Ify~<0,thensat
~ y
= 1sowe
musthave a h a(f(u; b
) f(u;) ~
)+ i
forall2
or a asat ~ y h f(u; b
) f(u;) ~
i
+sat
~ y
forall2:
Wecanagainlet
a
=a max
2 sat ~ y h f(u; b
) f(u;) ~ i forany
sobythedenitionof
anda
theinequalityissatisedagainandhence _
V 0.
Toimplementthemethod proposed in thesystem(14),it isnecessaryto solvethemin/max problem(15).
Todothis,wemustconstructaconcavecoverF()andaconvexcoverF()wherethecoverssatisfy
F()f b
f F()f b
f
for b
f =f(u; b
). Thefollowingdenitions andconstructionaresummarizedfrom[4].
Denition 1: A point 0
2
c if
0
2and
rf
0
(
0
)f f 0 whererf 0 @f @ 0 andf 0
=f(; 0
i
i+1
θ
θ
F(
f(u,
F(
θ
)
)
θ
)
θ
Figure 4. Convexandconcavecoveroff(u;).
Denition 2: ~
c
c
\where
c
isthecomplementof
c
Iff is not concaveon , then ~
c
is given by ~
c =f
12
; 34
;:::; mn
gwhere ij
=[ i
; j
] arethe regions
wheref isnotconcave, j
j
. UsingDenitions1and2,theconcavecoveroff b
f oncanbeconstructed
as
F()= (
f b
f; forall2
c
ij
+c ij
; forall2 ij
2 ~
c
(16)
where
ij
= f
j
f i
j
i
; c
ij
=f i
^
f
ij
i
; f
i
=f(; i
).
Similarly,weconstructaconvexcoveroff b
f bydening
v
f
0
jrf
0(
0
)f f 0
g
~
v
v \
F()= (
f b
f; forall2
v
ij
+c ij
; forall2 ij
2 ~
v :
(17)
OncewehaveconstructedF()andF(),aclosedform solutionto themin/max problem(15)isgivenby
a
= F( b
)
= (
rf
b
; if
b
2
c
ij
; if b
2 ij
2 ~
c
ify~
>0
a
= F( b
)
= 8
<
: rf
b
; if
b
2
v
ij
; if b
2 ij
2 ~
v
ify~
<0:
(18)
Aproofthat (18)isthesolutionto(15)can befoundin[4].
Havingestablishedthestabilityof theadaptationmethodby theLyapunovfunction statedearlier, nowwe
seek suÆcient conditionswhich establish uniform asymptotic stability of the system(14). Wesummarize the
1 0 0 0 0 2 2 0 1 1 0
Z
t2+Æ0
t2 h
(t
2 )f(u;
b
(t
2
)) f(u;) i
d 2+
0 jj
~
(t
2
)jj; (19)
thentheoriginx=0isuniformasymptoticallystable.
In Theorem 1, = 1if f(u; b
) is convex and = 1 if f(u; b
) is concave. We notice several dierences
between thiscondition and thecondition fora linearparameterization. Thesign of theintegralis important.
Thesignisnotstrictlydeterminedbyf(u; b
) f(u;)butalsobytheconvexityorconcavityoff asindicated
by. This couplingarises from themin/max algorithm andis necessarybut notsuÆcient to ensurethat the
method will leave thedeadzone, jyj~ . Theintegralmust besuÆcientlylarge to leavethe deadzone,which
necessitatesthetermincorporatingontherighthandsideof(19).
Weplacedtheexcitationconditionsonf in Theorem1. Wewishto deriveconditionsonu(t)sincewehave
somefreedom when choosing u(t). Theorem 1does notgiveconditionson the inputu to satisfy the
inequal-ity(19) nor doesit guarantee that such an input exists. Inequality(19) includes two components. First,the
magnitudeoftheintegrandmustbesuÆcientlylarge. Foralargeparametererrortheinputmustbesuch that
thedierencebetweenthefunctionevaluatedattheactualparameterandtheparameterestimateisadequately
large. Wechoseaninputsignalwhich drivesthefunction f toalevelwhereachangein theparameterismost
noticeable. Secondly,theintegralmustbethesamesignas. Thiscouplingstatesthatiff isconvex,thenthe
integrandshouldbepositive. Iff isconcave,thentheintegrandshouldbenegative. Themin/maxfeatureofthe
algorithmgivesstabilitybutanacceptableinputmustbeusedtoguaranteeparameterconvergence. Parameter
convergenceisensuredbyupdatingusingthegradientinformationandwemustpickaninputsignalaccordingly.
Toensureparameterconvergence,wecansummarizetheconditionsonuaseither
(a)Forthegiven ~
, umustreversethesignoftheintegrandof(19)whilekeepingtheconvexity/concavity
off xed.
or
(b)Forthegiven ~
, umustreversetheconvexity/concavityoff,whilepreservingthesignoftheintegrand
of(19).
(see[4]).
4. Matrix Equation Case
Sincemanyphysicalsystemswithinherenthysteresisaremodeledbyhigherorderequations,weextendhere
the scalar method proposed in [3, 4] to systemsof equations. We stated previously that, due to transducers
design and eld shaping, the smart structure can be modeled to rst approximation as damped spring mass
system. Therefore,forourmagnetostrictivetransducerapplication, theidenticationmethod mustwork forat
least asecondorder system. Toutilizethe methodfor matrixequations, we mustredeneseveral variables in
thescalarcase. Wewish to usethesolutionto themin/max problem (15),so wemustensure that wedonot
alterthataspectof theformulation.
Weconsiderheretheparameteridenticationforthematrixsystem
_
y=Ay+Bf(u;):
HereweassumethatAisdiagonalwitheigenvalues
i
. Sinceoursmartsystemisstronglydamped,wehavethe
realpartoftheeigenvaluesinthelefthalfplane. Wedene
_
b
y = Aby+Bf(u; b
) C h
sat( ~ y
))+a
sat
~ y
i
~
y = Re N
X
i=1 (by y)
i
_
b
= y~
~ y
= y~ sat
~ y
whereC=[0;; 0; 1]2R ,N isthenumberofstates,anda and arethesolutionsof a = min 2R m max 2 g(;)
= arg min
2R m
max
2
g(;)
g(;) = bsat ~ y h f(u; b
) f(u;) T ( b ) i (21)
whereb= N
X
i=1 B
i
. Itisimportanttonote thatthesolutionto themin/max problem(21)isascalarmultipleof
thesolutionto (15).
Wemustprovethatthisadaptiveparameterestimationmethodisstable. WeconsidertheLyapunovcandidate
V =y~ 2 +b ~ 2 whichyields _
V =2~y
_ ~ y +2b ~ _ b with _ ~ y =Re " N X i=1
A(~y y)+B( ^
f f) C
sat( ~ y
)+a
sat ~ y # :
Thiscanbewrittenas
_ ~ y =Re " N X i=1 i (~y y)
#
+B( ^
f f) C
sat( ~ y
)+a
sat ~ y whichyields _
V = 2~y
Re " N X i=1 i (~y y)
#
+2~y
h b( ^ f f ~
) sat( ~ y ) a sat ~ y i _
V = 2~y
Re " N X i=1 i ~ y N X i=1 i y #
+2~y
h b( ^ f f ~
) sat( ~ y ) a sat ~ y i : SinceRe( i
)<0foralli,wehave
Re " N X i=1 i ~ y # " N X i=1 ~ y # ; Re " N X i=1 i y # " N X i=1 y # (22)
where=max
i Re(
i
). Usingtheseinequalitiesweobtain
_
V 2~yy~
+2~y
b( ^ f f ~
) sat( ~ y ) a sat ~ y :
Wecompletetheproofbyusingthedenitionsofa
and
asthesolutionsof(21)inamanneranalogoustothat
oftheproofinSection3. Oneitemintheproofwemustnoteisthattheinequalitiesin(22)aretrueonlyfor
spe-cicinputfunctionsu(t). Theinputsignalweuseisamonotonicallyincreasingfunction. Thereforethestatesy
Weprovideascalarandmatrixexampletodemonstratethecapabilitiesofthenonlinearadaptiveparameter
estimationmethod. Weconsiderrstthescalarmodel. Wespecifythedynamics ofthesystemby
_
y= ky+M(u;a) (23)
where M(u;a)is the solutionof thedomain wall model (5) or(6) forthe hysteretic material. Weassumethe
parameterestimatebatobeboundedsuchthat ba2 [6300; 7300]withba(0)=6800. Wetaketheactualvalueof
a tobe 7012A/m and theremainingconstantsare givenask =4000A/m, = :01, P
s
=7:6510 5
A/m,
c =:18 and
s
=1:00510 3
. One diÆculty ofthe adaptiveparameterestimation algorithm isconstructing
an input u(t) which will provide persistent excitation. Because of the condition imposed for excitation, we
use a signal that does not cause the function to change signs. Empirically, it has been established that a
monotonically increasing or saturation type input providesaccurate results and quick convergence. Wechose
theinputsignal,u(t),asanincreasinglinearfunctionwhichdrivesthehysteresistoalevelnearsaturation. This
signal provides persistent excitation aswell as evaluates the hysteresis model at levels which most noticeably
dieraccording to theparameter a. Figure 5a illustrates theintegrand of (19)for agivenvalue of ~
to show
that thesecond condition forpersistentexcitation is met. The integrandremains positivewhile switchingthe
convexity/concavityofthefunctionMasseeninFigure5b. Figure6illustratestheabilityofthescalarnonlinear
parameterestimation method to accuratelyidentify the unknownparametera. Figure 6ashowstheevolution
of the parameterestimates which converges quickly to theactual value of 7012. The speedof convergence of
the parameterestimation is a notable resultsince we canpotentiallycombine this identication method with
acontrol technique. The Figure 6bprovides agraphof thetrackingerrory.~ For agiven speciedin design
tolerances (e.g. cutting accuracy of = 0:001 in) the method is able to track within an error of given the
conditionsof persistent excitation aresatised. Wehaveempirically noticed thechoice of aects the rateof
convergence and the range of parameter estimate values which the method achieves. This givesus a design
considerationassociatedwiththetrackingaccuracyrequired.
Wenowconsider thematrixsystemparameterestimation algorithm. Thesystemis adamped springmass
systemwhichmodelsthetransducerdynamicsofthesmarttransducergivenby(13). Again,wetakethefunction
f asthehysteresismodel(6) andtheparameteraisupdatedto modelits temperaturedependence. Figure7a
illustratestheconvergenceoftheestimatetotheactualparametervalue. Figure7bdepictsthetrackingerrorof
theadaptivesystem. Wehavesuccessfullyextended the parameteridentication to matrixsystemsasseenby
theconvergenceof theparameterandthedecayofthetrackingerror.
Forthe matrixsystemthere exist avariety implementation issues. Themodel weconsider must besolved
numerically. Noimplicit method can be used because of theunknown forcing function at the nexttime step.
ThisuncertaintyrequiresthetimesteptobesuÆcientlysmalltoensureaccuratesolutionsofthemodelgivenin
(13). Anyinaccuracyofthesolutionof(13)cancausethevalueofy~tohaveadiscontinuousjumpfrompositive
tonegativevalues. This phenomenacausesthemin/maxsolutiontojump betweenutilizingtheconvexcover
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Integrand
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P
(a) (b)
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Figure6. (a)Parameterestimate;(b)Trackingerrorforscalarcase.
and concave cover. These jumps in turn cause highly oscillatory behavior in the parameter update. We also
observedtheconvergencetobemoderatelyslowerwiththematrixsystemthanthatofthescalarcase. However,
theconvergencerateisstillreasonableforalargenumberofindustrialapplications.
6. Concluding Remarks
We have formulated the nonlinear adaptive estimation technique of [3, 4] in the context of a nonlinear
hysteresismodelformagnetostrictivetransducersandhaveextendedthetheorytothevectorcasecommensurate
with these models. Numerical examplesillustrate the capability of the method for updating the temperature
dependent parameter a to simulate the eect of changing temperature in the transducer. While developed
in the context of a model for magnetostrictive materials, the unied nature of the models (see [8]) permits
direct extensionofthetechniquetohysteresismodels forpiezoelectric,relaxorferroelectric,andshapememory
compounds.
Onedirectionofcurrentresearchfocusesontheextensionofthealgorithmstosimultaneouslyidentifymultiple
parameters;e.g.=[a; k; ; P
s
; c]. Whilethemin/maxtheoryisthesame,issuesconcerningtheidentication
oftheconvexandconcaveregionsrequireresolution.
Aseconddirection ofcurrentresearchaddressesthedevelopmentofadaptiveand robustcontroltechniques
whichutilize these models andestimation algorithms. Whileadaptivecontrol techniqueshavebeen developed
formodelswithlinearparameterizations[11,12], analogousconvergencecriteriafornonlinearmodels,ofthe
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−8
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(a) (b)
implemented is basedon the useof partial of full inverse compensatorsbasedon approximate inverses to the
models[5,6]. Inthiscase,theadaptiveestimationalgorithmspresentedherewouldbeused toupdate
parame-tersinthemodelanditsinverse. Theinverseisthenemployedinahybridcontrollercomprisedoffeedbackand
feedforwardcomponents. Thispermitsanindirectadaptiveupdatingofthecontrollertoaccommodatechanging
operatingconditions.
Acknowledgments
Thisresearchwassupportedin partbytheAir ForceOÆceof ScienticResearch underthegrant
AFOSR-F49620-01-1-0107.
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