1 Intro duction
austenite
martensite
Representing Smart Material Actuators:
Identication and Approximation
H.T. Banks
Centerfor ResearchinScienticComputation
Box 8205
North Carolina State University
Raleigh,NC 27695- 820 5
e-mail: [email protected]
and
A.J. Kurdila
Dept. of AerospaceEngineering
Texas A&MUniversity
CollegeStation, TX 77843
Alargeclassofemergingactuationdevicesand
ma-terialsexhibit stronghysteresis characteristics
dur-ing their routine op eration. For example, when
piezo ceramic actuators are op erated under the
in-uenceofsubstantialelectricelds,itisknownthat
the resulting input-output b ehavior is mildly
hys-teretic. Likewise, when shap e memory alloys are
resistivelyheated to induce phase transformations,
theinput-output resp onseat thestructural levelis
also known to b e strongly hysteretic. This note
discusses mathematical issues that arise in
identi-fyingaclass ofhysteresis op eratorsthat have b een
employedfor mo delingshap e memoryalloy
actua-tion. Sp ecically,theidentiabilityofaclassof
dis-tributed hysteresis op erators that arise inthe
con-trolinuenceop eratorofaclassofsecondorder
evo-lutionequationsisinvestigated. Inthispap erwe
in-tro ducedistributed,hystereticcontrolinuence
op-erators derived from smo othed Preisach op erators
[V,M] andgeneralizedhysteresisop erators derived
from results of Krasnoselskii and Pokrovskii. For
theseclasses, theidenticationprobleminwhichwe
seektocharacterizethehysteretic control inuence
op eratorcanb eexpressedasanoutputleastsquare
minimizationover probabilitymeasures dened on
acompactsubset ofaclosed half-plane.Consistent
andconvergentapproximationmetho dsfor
identi-cation ofthe measure characterizing the hysteresis
arereadilyobtainedfromourformulations. .
Researchinactive,orsmartmaterials,forvibration
attenuation,shap econtrolandmicro-mechanical
ac-tuationispro ceedingatarapid pace. Asactuation
devicesbased ontheelectro-mechanicalb ehaviorof
piezo ceramicsandtheshap ememoryalloyshaveb
e-come more widespread, it is now well-appreciated
that this classof actuationdevices exhibits
signi-cantnonlinearhystereticresp onse.
Shap e memory alloys are a class of metal
com-p oundswhich p ossessthecapabilityto sustainand
recoverrelativelylargestrains( 10%)without
un-dergoing plastic deformation. These unique
mate-rialcharacteristics aredueinlargepartto the
ma-terials'abilitytoundergointernalcrystalline
trans-formationsinthepresenceofexternalappliedstress
and/orchangesintemp erature. These
transforma-tionsfromtheparentphase, ,atstressfree,
high temp erature conditions to several variants of
the low temp erature phase, are also a
functionofthehistory ofthematerial. Whilethese
materials arevery much in thecategory of
emerg-ingtechnologies, several ofthemarecurrently
com-merciallyavailable andhave b eenusedin
engineer-ingapplications. Amongthemost p opularare the
nickel-titaniumalloyknown as Nitinoland copp er
zinc aluminum alloys. NiTi can b e used in high
r
6
T
T
"
familyofhysteresis
curves cyclicloadsituationswithrecoverablestrainsof
ap-proximately2%.
The phenomenological resp onses in SMA are
rea-sonablywell understo o d. Ifone takes astress-free
SMA at high temp erature inthe austenite or
par-ent phaseandco ols it,agradualtransformationto
thelowtemp eratureormartensitephaseisachieved.
Severalvariantsofthemartensiteincludingmultiple
twins areobtainedinthisco oling pro cess.
Thethermallyinducedphasetransitionsarethe
ba-sisofanexplanationofthestrainrecovery features
ofSMA.IftheSMAisinthemartensitephaseanda
unidirectionalstress isapplied, atatemp erature
dep endent critical stress
crit =
crit
( )
detwin-ning of themartensite variantsb eginsand
eventu-allyresultsinasinglevariantofdetwinned
marten-site alignedwith theaxis of the stress loading. A
similar state is achieved under the loading if one
starts with an SMA in the austenite phase.
Dur-ing these phase transformationsthe internal stress
inthe SMA changes onlyslightlyand asignicant
apparently plastic strain is achieved. If is
sig-nicantlylowduring thestress induced martensite
phase transformation,alargeresidual strain
re-mains after unloading. This strain can b e
recov-eredbyheatingtheSMA;thisistermedthe\shap e
memoryeect" or SME. This recovery can b e free
(nowork is done),fullyrestrained (the SMA is
re-strainedfromrecoveryofitsoriginaldimensionand
geometrytherebypro ducinglargeinternalstresse s),
orcontrolled(the SMAisconstrainedso partial
re-covery of the residual strain is achieved but some
stress ispresenttopreventfullrecovery).
ItisthisSMEfeatureofSMAthatcanb eexploited
to develop distributed force actuators to b e used
as controllers in comp osite materials. In a
typi-cal example one employsSMA b ers reinforcing a
non-SMA comp osite structure, e.g., SMA b ers in
an elastic matrix. The SMA b ers or \wires" are
stress loaded and deformed atlowtemp eratures in
the martensitic phase. They are then unloaded to
generatesomemartensiticresidualstrain. Once
em-b eddedintheelasticmatrixtheycanb eheated
(us-inganelectricalorthermalinput)toachieve
resid-ualstrainrecovery. InNitinolemb eddedcomp osite
structures one can pro duce up to 10 psi in
con-strainedrecovery.
Tohelpmotivatethediscussionofhysteresismo
del-ingthat followsinthenextsection,weconsiderthe
results of exp eriments measuringthestructural
re-thestress induced inresistivelyheatedshap e
mem-ory alloy wires. The exp erimental setup is rather
simple. Athinaluminumb eam,1/32inchin
thick-ness,iscatileveredatoneendasdepictedinFigure
1.A\two-way"shap ememoryalloywireisattached
totherigidbase supp ortingthecantileveredb eam,
andtoanosetattachedtothetipofthefreeendof
theb eam. Athermo couple is attached inthe
cen-ter of the length of the shap e memoryalloy wire,
andastraingaugeisattachedonthesurfaceofthe
b eam,alsoatitsmidp oint.Itisclearthatwiththis
simpleexp eriment, the temp erature (input) to the
shap ememory alloywire and theoutput strain at
the surface of the b eamcan b e collected to
char-acterize a temp erature-to-strain plantmo del.
Fig-ure2 depicts the results of aseries of exp eriments
wherein thecurrent that resistivelyheats the wire
is varied. In Figure 2a, the current in the wire is
heldconstant at3ampsuntilatargettemp erature
isachieved,thenthecurrentisturnedo. Asis
ap-parentfromtheexp eriment,themajorascentlo ops
anddescent lo opsarestronglyhysteretic. InFigure
2b,thesameproto colinanotherexp erimentalrunis
followed,exceptthatthecurrentduringactivationis
2.5amps. Notonlyishysteresisevident,by
compar-ingtheresp onsestrain-versus-temp eraturecurvesto
thosedepictedinFigure2a,the
isdep endentontheinputcurrent.
It has b een understo o d for quite some time that
there are two fundamental approaches to
mathe-maticallycharacterizingthe input-output b ehavior
ofcomplexdynamicalsystems. Inoneapproach,
in-vestigators mo delagivensystemasacollection,or
evencontinuum,ofcomp onentsforwhichidealmo
d-elscanb ederivedfromtheprinciplesofphysics. In
thesecondmetho d,theoverallqualitativeb ehavior
ofthesystemasawholeisobserved,andsome\b est
representative" is selected from a class of mo dels
that exhibit desired, empirically observed, prop
er-ties. With resp ect to mo delinghysteresis inactive
materials,forexample,dierentresearchers
(includ-ingLiangandRogers[LR],Brinson[Br],Barret[Ba]
and Lagoudas [BL]) have formulated constitutive
mo delsforshap e memoryalloys. Theb o dy of
re-searchin[LR,Br,Ba]and[BL],regardedasawhole,
fallswithintherst category describ ed ab ove. On
theotherhand,HughesandWen[HW]haveutilized
the Preisach mo del in a system theoretic sense to
representastatic,hysteretictransducerthatcanb e
used to derive comp ensators for controlling shap e
memory alloy wires. The approach in this pap er
theo-i 1 2 0 0 0 S S 3 3 3 ( 8 > > < > > : Z Z 2
1 2 1 2
1 2 1 2 2 1 1 1 0 1 1 1 1 1 2 2 s i s s s m pm m m m m m m
pm ;j ;j
s pm
pm k
k k
k pm k k
s s s
S f 2 j g
0 2S
S S
2 f0 g
! 2 !
2S 2 2 2 2 M
f 0 g %
f 0 g &
2
M
M M
2
M M M
2
2 2 2
! S
2
2B S
S! 2
S
2
1 1
2 2
2 2 The generalizedsmo othed
Preisach-Krasnoselskii-Pokrovskiicontrol
outputop erator
3 Secondorderevolutionequations s R s s ;s ; s < s
r
r x r x s ; s s ;s
r ;r
s s ;s
ut
I ;
u; k u; C ;T I C ;T
s
u C ;T
u C ;T
u C ;T
u t C ;T
u ; t
;r u t s u
;r u t s u
u S ;T S
j ;T
k u ; t
u ; t
t t ;t
u ; t
k :::j:
u C ;T
u C ;T ; I t ;T
s k u; t
R
I
ut
s ;s f ;I
I M
u C ;T
P u;f
P u;f t k u;f s t d s:
B
B u;f P u;f g k u;f s t d s g
g V M
B u;f L ;T ;V :
V V
V icalsystems andismotivatedstrongly bythework
ofHughesandWen. Inthispap erweaddressseveral
mathematicalissuesthatariseinmo delingand
iden-tication of aclass of nonlinear, hysteretic control
inuence op erators insecond orderevolutionop
er-ators. Ourformulationleads toconsistentand
cov-ergentapproximationschemesfortheidentication
problem;we arecurrently usingthe resulting
com-putationalmetho ds for numerical predictions with
exp erimentalresultsforaclassofshap ememory
al-loyactuated structures.
Let = = ( ) b e the
Preisachhalfplane[V,KP ]ofthresholdparameters
used to parameterize the PKP kernels as follows.
Dene axed monotonecontinuous ridge function
tob eusedasasmo othedrelayop eratoranddene
itstranslates
( )= ( ) =( )
where
is theclosure of . Thefunctions
dene theenvelop e ofadmissiblepaths(see Figure
3)forthesmo othPKPkernelswhichdep end ofthe
parameters =( ),amonotone inputfunction
() (or piecewise monotone input) and an initial
state 1 1 . ThegeneralizedPKPkernels
aredenedbyamapping
( ) ( ): [0 ] [0 ]
parameterizedby
. Thedenitioncanb estb e
givenbypro ceedinginthreesteps:
1. Dene the action of the kernel on monotone
inputfunctions [0 ]
2. Dene the action of the kernel on piecewise
monotoneinputfunctions [0 ]
3. Extendbydensityargumentsthedenitionin
(ii)forarbitrary [0 ].
Foranymonotone function () [0 ],dene
themonotoneoutputop erator
[ ( )]()=
max ( () ) if
max ( () ) if .
monotoneinputfunction [0 ]where
isthesetofcontinuouspiecewiselinearsplineswith
knots in[0 ]. Thedenition inthis case is
de-ned recursively on each subinterval. First, we set
anddene [ ( )]()= [ ( )]() [ ] ( )( ) =1
Obviously, for any continuous piecewise linear
dis-cretization of the input, this equation suces for
computational purp oses. A typical output is
de-pictedinFigure4. Theextensionof thisdenition
toarbitrary [0 ]now followsstandard
den-sityarguments[V,KP]. Withthisdenitionofthe
hystereticcontrolinuenceop eratorinplacewecan
prove foreach [0 ] and [0 ],the
maping [ ( )]() is continuous from
to
. Thisresult playsanessential rolein
establish-ing well-p osedness of the identication pro ceedure
andtheconvergenceofapproximationschemes.
Weshouldnotethattheargument denesthe
initialstateofthedelayedrelayop erator,andaects
theoutputofthedelayedrelayop eratoronlyifthe
input () happ ens to have an initialvalue in the
op eninterval( ). Let (
) theBorel
measurablefunctionsmapping
,let
theclass ofnite signedBorelmeasures on
, and
let [0 ]. The generalized Preisach op erator
( )isdened via
[ ( )]( ) [ ( ( ))]() ( )
Ifthecontrol inuenceop erator isdened via
( ) ( ) [ ( ( ))]() ( )
where and ,then
( ) ((0 ) )
Here isthecongugatedualofaHilb ertspace .
Basedontheab ovedenitions,wepresentour
gov-erningequations in aweak formthat incorp orates
i i 1 2 1 1 Z Z Z Z ( X X ) S S 3 3 3 3 3 3 1 1 3 3 3 S S 3 1 T Z Z T k k k k K K i
i s i i i
K
K
i i i
K K s i K K K 1 0 1 0 0 1 2 2 2 0 1 2 2 2 0 1 1 1
1 2 1
1 2 1 2
1 1 1 1 1 1 1 1 =1 1 2 1 1 =1 1 1 ! ' ! ! ! Q 2
2 B S 2Q2
2
2 \
2
2
2
f j 0 j
j 0 j g
2Q
2A A
A
S S AP S
S
2P S
f j
S g
!
!1 0
! S
P S
S S
S
! !
2 S
A P S
Q2A
P j 2R
R
S R S
P P S
P
P S V , H H , V :
V , H H , V
H
H
A q A q
q
w t A q w t A q w t B u;f t
V
A ;A u
C ;T f ;I q ; M
w q ;
w L ;T ;V L ;T ;V
w L ;T ;H L ;T ;V
w L ;T ;V ;
w C ;T ;V
w C ;T ;H :
q ;
J q ; C w q ; t w
C w q ; t w dt
w ;w
J
J q ; j t;w q ; t ;w q ; t d t :
q
M
;
; inf > F F ;
F F
F
F ;
k
E E E
@E C C hd hd h C J q ;
K ; ;:::;
a a ; a ; s
K s s ;s
s
[W]. Thatis,
Each of the emb eddings and
are dense and continuous. The pivot space is
identiedwithitsdualspace bytheRiesz-map.
Following[BIW,BSW],wedenetwoparameter
de-p endent op erators ( ) and ( ) that represent
the damping and stiness op erators, resp ec tively.
The parameters areassumed to lie inacompact
metric space . In op erator form, the equations
governingthedynamicsofthesystemofinterest to
ushave theform([BI])
()+ ( )_( )+ ( ) ()=[ ( )]()(3.1)
in .
Under standard assumptions on the op erators
,wecanprove(see[BIW,BSW])thatfor
[0 ] and (
)and each ( ) ,
thereisauniquesolution ( )suchthat
((0 ) ) ((0 ) )
_ ((0 ) ) ((0 ) )
((0 ) )
andmoreover,we actuallyhave
([0 ] )
_ ([0 ] )
Equation(3.1) constitutes the equations governing
theinput/outputb ehaviorofthedynamicalsystem.
Forpurp oses ofidentifyingthe\parameters" ( )
thatcharacterizethesystem,we mustcho osea
rea-sonable measurement, or observation, error
func-tional. Forexample,the quadraticmeasureof
out-puterror ([BK])
( ) = 1 2 ( )() ~ + 1 2 _( )() ~ _
is a commonlyused error criterion. Here ~ ~
_
de-note the exp erimentallyobserved data. More
gen-erally,we mayassume hastheform
( ) ( ( )() _( )()) ()
(3.2)
(3.2),subjecttothedynamics(3.1),over and
,where isasuitableclassofmeasures.
Tosp ecifypreciselythisclass ofmeasures,weturn
to results fromthetheory ofprobabilitymeasures.
Let b eacompactsubsetof
andlet ( )
b ethespaceofprobabilitymeasures on taken
with the Prohorov metricof convergence in
distri-bution. Asdiscussed in[BF,B,EK ],theProhorov
metricisdened for ( )by
( ) 0 ( ) ( ) +
closed,
where denotes theusual neighb orho o d set for
. Then it is well known that ( ) 0 as
(i.e.,convergencein metric)isequivalent
to ( ) ( ) for all Borel sets with
( )=0. Moreover,ifweview ( )asasubset
ofthedualspace ( ) where ( )istheusual
space of continuous functions on , convergence
inthe metricisequivalenttoweak convergence:
i.e.,
ifandonlyif
for ( ).
Finally we also have that = ( ) with this
metric is a compact metric space. Under suitable
assumptions,the cost function of (3.2)is weak
lower semicontinousin ( )on . These
re-sultscanb eusedtoguaranteeexistenceofsolutions
to the identication problem (see [BKW1] for
de-tails).
Theformulationjustoutlinedalsolendsitself
read-ily to the development of computationalmetho ds
andalgorithmsfortheidenticationor\parameter"
estimationprobleminvolving(3.1),(3.2). Againwe
useresults fromprobabilitytheory(forasummary,
seeTheorem 3.3in[BF]). Dene for =1 2
thesets
^
0 =1
where isasetof rationalpairs =( )
in such that is dense in and is
theDiracmeasure with atomat . Then we have
that
^
is dense in ( ) inthe metric.
This denes a sequence ^
of nite dimensional
SMA wire
L
α
Hysteresis ID Experiment
10-13-95, Run A (Heat = 3 Amps)
0
0.2
0.4
0.6
0.8
1
1.2
20
30
40
50
Temperature (Celsius)
Strain (volts)
Hysteresis ID Experiment
10-13-95, Run B (Heat = 2.5 Amps)
0.4
0.6
0.8
1
1.2
Strain (volts)
H V
H
q ;
Q Q
Q
P
Q2A M
N
N M
K
N ;M ;K N ;M ;K
Figure 1:
Acknowledgement
References
Figure 2:
49
4
8 Hysteresis in SMAExp eriments(upp er) 3
Amps,(lower)2.5Amps
Quart. Appl.Math.
Control Theory and
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co-ecients inconcrete realizations of (3.1)) and
ap-proximating state spaces . This results
inacomputationallytractableapproximating
min-imizationproblemfor (3.2)over (3.1) restricted to
(i.e.,Galerkinapproximatesystems), and
^
. One can then prove that the resulting
solu-tions( )converge inan appropriate
sense toasolutionof theoriginalproblemof
mini-mizing(3.2)over subjecttothesystem(3.1).
Detailscanb efoundin[BKW2].Ourinitial
compu-tational eorts based on these ideas, also rep orted
in[BKW2],aremostpromising.
Exp erimentalSetup
This research was supp orted in part by the U.S.
AirForceOce ofScienticResearchunder grants
AFOSRF49620-95-1-0236andF49620-93-1-280and
in part by the U.S. Army Research Oce under
grantDAAL03-92-012.
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r(x)
x
+1
-1
s1
s2
s1 +
a
s2 +
a
r(x-s
2
)
r(x-s
1)
s1
s2
S
M
1
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