DTIC ADA439822: A Model for a Thin Magnetostrictive Actuator

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A Model for a Thin Magnetostrictive Actuator
by R. Venkataraman, P.S. Krishnaprasad
CDCSS T.R. 98-6
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A model for a thin magnetostrictive actuator
R.Venkataraman, P.S. Krishnaprasad
Department of Electrical Engineering and Institute for Systems Research
University of Maryland, College Park, MD 20742
(cid:3)
fvenkat,[email protected]
Abstract actuator. It is a low (6) dimensional model with 10 pa-
rametersandhenceissuitableforreal-timecontrol. The
In this paper, we propose a model for dynamic magne- modelincorporatesfeaturesobservedinacommercialac-
tostrictive hysteresis in a thin rod actuator. We derive tuator[1], likethe hystereticbehaviourofmagnetostric-
two equations that represent magnetic and mechanical tion as a function of the external (cid:12)eld; dependence on
dynamic equilibrium. Our model results from an appli- the rateofthe input, eddycurrentlosses,inertial e(cid:11)ects
cation of the energy balance principle. It is a dynamic and mechanical damping e(cid:11)ects. In Section 3 we ana-
model as it accounts for inertial e(cid:11)ects and mechani- lyze this model for periodic forcingfunctions. Using the
cal dissipation as the actuator deforms, and also eddy- Schauder Fixed Point Theorem, we prove that the solu-
current losses in the ferromagnetic material. tion isanasymptoticallystableperiodicorbit,when the
Wealsoshowrigorouslythatthe modeladmitsape- parameters are subject to certain constraints.
riodic solution that is asymptotically stable when a pe-
riodic forcing function is applied.
2 Thin magnetostrictive actuator model
1 Introduction
Weareinterestedindevelopingalowdimensionalmodel
There is growing interest in the design and control of foramagnetostrictiveactuator. The mainmotivation is
smart structures { systems with embedded sensors and to use it for control purposes. Therefore the starting
actuators that provide enhanced ability to program a point of our work is Jiles and Atherton’s macroscopic
desiredresponsefrom a system. Applicationsofinterest model for hysteresis in a ferromagnetic rod [2]. In our
include: (a) smart helicopter rotors with actuated (cid:13)aps model, we treat the actuator itself along with the asso-
thataltertheaerodynamicandvibrationalpropertiesof ciatedprestress,magneticpath,tobeamass-springsys-
the rotor in conjunction with evolving (cid:13)ight conditions tem with magneto-elastic coupling. As we show later,
and aerodynamic loads; (b) smart (cid:12)xed wings with ac- ourmodelisonlytechnicallyvalidwhentheinputsignal
tuators that alter airfoil shape to accomodate changing is periodic. However, this is the case in many applica-
drag/liftconditions;(c)smartmachinetoolswithactua- tionswhereoneobtains recti(cid:12)edlinearor rotarymotion
torsto compensateforstructural vibrationsundervary- byapplyingaperiodicinputatahighfrequencytothese
ing loads. In these and other examples, key technolo- actuators. For instance in our hybrid motor [3, 4], we
gies include actuators based on materials that respond produced a rotary motion using both piezoelectric and
to changing electric, magnetic, and thermal (cid:12)elds via magnetostrictivemotorsinamechanicalclampandpush
piezoelectric,magnetostrictiveandthermo-elasto-plastic arrangement.
interactions. In an earlier work, we explored the connections be-
Typically such materials exhibit complex nonlinear tween abulk ferromagnetichysteresismodel and energy
and hysteretic responses(see Figure1 for an exampleof balance principles [5]. We present in this paper, an ex-
a magnetostrictive material Terfenol-D used in a com- tension of this theory to include magnetostriction and
mercial actuator). Controlling such materials is thus a eddy current losses. This is done by equating the work
challenge. The present work is concerned with the de- done by external sources (both magnetic and mechani-
velopmentofaphysics-basedmodelformagnetostrictive cal), withthe changeinthe internalenergyofthe mate-
materialthatcaptureshystereticphenomenaandcanbe rial, changein kinetic energy,and losses in the magneti-
subject to rigorous mathematical analysis towards con- zation process and the mechanical deformation.
trol design.
In Section 2 we propose a model for magnetostric-
(cid:14)Wbat+(cid:14)Wmech =(cid:14)Wmag +(cid:14)Wmagel+(cid:14)Wel
tion that describes the dynamic behaviour of a thin rod
Changeininternalenergy
(cid:3)
ThisresearchwassupportedinpartbyagrantfromtheNational
ScienceFoundation’s EngineeringResearch Centers Program: NSFD +(cid:14)Lmag+(cid:14)Lel+| (cid:14)K{z } (1)
CDR 8803012, andby theArmy ResearchO(cid:14)ce underSmartStruc-
tures URI Contract No. DAAL03-92-G0121 and under the MURI97 losses Changeinkineticenergy
Program Grant No. DAAG55-97-1-0114 to the Center for Dynamics | {z } |{z}
andControlofSmartStructures(throughHarvardUniversity). In Equation 1, (cid:14)K is the work done in changing the
kinetic energyofthe systemconsistingofthemagnetoe-
lasticrod,(cid:14)Wmag isthechangeinthemagneticpotential
energy, (cid:14)Wmagel is the changein the magnetoelasticen- (cid:14)Wbat = (cid:22)0H dM
I
ergy,(cid:14)Wel is thechangeinthe elasticenergy, (cid:14)Lmag are
the losses due to the change in the magnetization, and Let an external force F in the x direction produce a
(cid:14)Lel are the losses due to the elastic deformation of the uniform compressive stress in the x direction (cid:27) within
rod. the actuator. The total displacement of the edge of the
The expressionfor (cid:14)Wmag will be givenshortly. The actuator rod be x. Thus the mechanical work done by
1 2
elastic energy is given by Wel = 2lx ; where x is theexternalforceinacycleofmagnetizationisgivenby
the total strain multiplied the length of the actuator. [7],
Chikazumi [6] derives an expression for the magnetoe-
lastic energy density of a three dimensional crystal. It (cid:14)Wmech = F dx
I
is of the formwhere the strain componentsmultiply the
square of the magnetization components. In our one The total work done by the battery and the external
dimensional case, we can similarly write down the fol- force is
lowingexpressionforthemagnetoelasticenergyWmagel:
(cid:14)Wbat+(cid:14)Wmech =V (cid:22)0H dM + F dx
2 I I
Wmagel =bM xV
Weseethataddingtheintegralofanyperfectdi(cid:11)erential
where b is the magneto-elastic coupling constant and V over a cycle does not change the value on the left hand
is the volume of the magnetostrictive rod. M is the av- side. Therefore,
erage magnetic moment of the rod. The expression for
the magnetic hysteresis losses (cid:14)Lmag is due to Jiles and
Atherton. The motivation for this term is the observa- (cid:14)Wbat+(cid:14)Wmech =V (cid:22)0H dM + (cid:11)M dM + F dx
(cid:18)I I (cid:19) I
tion that the hysteresislossesare due to irreversibledo-
(3)
main wall motions in a ferromagnetic solid. They arise
Equations 2 and 3 give,
from various defects in the solids and are discussed in
detail byJiles andAtherton[2]. The mathematicalcon-
2bMx
sequences of this hypothesis is discussed in detail in our V(cid:22)0 (H+(cid:11)M (cid:0) (cid:22)0 )dM (cid:1)(cid:1)(cid:1)
earlier work [5]. H 2
+ (F (cid:0)lx(cid:0)c1x_ (cid:0)meffx(cid:127)(cid:0)VbM )dx = (cid:14)Wmag
H +V ksign(H_)(1(cid:0)c)dMirr (4)
(cid:14)Lmag = Vksign(H_)(1(cid:0)c)dMirr
I H
De(cid:12)ne the e(cid:11)ective (cid:12)eld to be,
The line integral implies that the integration is carried
2bMx
outoveronefullcycleoftheinputvoltage/currentwhich He =H +(cid:11)M (cid:0)
(cid:22)0
is assumed to be periodic. The reason for this will be
discussedshortly. Thelossesduetomechanicaldamping Astheintegrationisoveronecycleofmagnetization,we
are assumed to be (cid:14)Lel = c1x_ dx: The change in the have
kinetic energy (cid:14)K = mefHf x(cid:127)dx: Therefore,
H He dM =(cid:0) M dHe
I I
(cid:14)Wbat+(cid:14)Wmech =(cid:14)Wmag + meff x(cid:127)dx ::: Itwasobservedin[5], thatifM isafunctionofHe then
I there areno lossesin one cycle. This is the situation for
(cid:14)K a paramagnetic material where M = Man is given by
2 | {z } Lang(cid:19)evin’sexpressionasafunctionofHe. Henceforthe
+V bM dx+V 2bMxdM+ lxdx
I I I lossless case, the magnetic potential energy is given by,
(cid:14)Wmagel (cid:14)Wel
(cid:14)Wmag =(cid:0)V Man dHe
| {z } | {z }
I
+V ksign(H_)(1(cid:0)c)dMirr+ c1x_ dx (2)
I I
Thus Equation 4 can be rewritten as
(cid:14)Lmag (cid:14)Lel
| {z } | {z } ksign(H_)(1(cid:0)c) dMirr
Now we obtain expressions for the left hand side of the V(cid:22)0 (Man(cid:0)M (cid:0) (cid:22)0 dHe )dHe
above equation. For a thin cylindrical magnetostrictive H 2
+ (F (cid:0)dx(cid:0)c1x_ (cid:0)meffx(cid:127)(cid:0)bM V)dx=0
actuator, with an averagemagnetic moment M, and an
H
uniform magnetic (cid:12)eld in the x direction H, the work Note thatthe aboveequationisvalid only if H; M; x; x_
done by the battery in changing the magnetization per are periodic functions of time. In other words, the tra-
unit volume, in one cycle, is given by jectory is a periodic orbit. We now make the hypothesis
thatthefollowingequationisvalidwhenwegofromone
point to another point on this periodic orbit. 2 2 2
N 8(cid:25)(cid:26) B (cid:0)A
Red = 2 2
lm B +A
ksign(H_)(1(cid:0)c) dMirr
V(cid:22)0 (Man(cid:0)M (cid:0) (cid:22)0 dHe )dHe
Theactualworkdonebythebatteryinchangingthe
R 2
+ (F (cid:0)dx(cid:0)c1x_ (cid:0)meffx(cid:127)(cid:0)bM V)dx=0 magnetizationandtoreplenishthelossesduetotheeddy
R currents in one cycle is now given by
Theaboveequationisassumedtoholdonlyon theperi-
odicorbit. SincedxanddHe areindependentvariations
arisingfromindependentcontroloftheexternalprestress (cid:14)W(cid:22)bat = (cid:14)Wbat+ Peddy dt
and applied magnetic (cid:12)eld respectively, the integrands I
must be equal to zero.
= (cid:0)V (cid:22)0M dHe+ Peddy dt
I I
ksign(H_)(1(cid:0)c) dMirr Figure 2 shows a schematic of the full model. The
Man(cid:0)M (cid:0) = 0 (5)
(cid:22)0 dHe hystereticinductorstandsforthemagnetostrictiveactu-
2
meff x(cid:127)+c1x_ +dx+bM V = F (6) ator model.
Jiles and Atherton relate the irreversible and the re-
3 Qualitative analysis of the magne-
versible magnetizations as follows [2],
tostrictive actuator model
M = Mrev + Mirr: It isveryimportantto note that the modelequations(9
{ 10) are only valid when all the M, H are periodic in
Mrev = c(Man(cid:0)Mirr):
time. What we mean, is that solution of the equations
dM dMirr dMan
= (cid:14)M(1 (cid:0) c) + c (7) represent the physics of the system under these condi-
dH dH dH
tions. But, usually in practice we do not know apriori
where (cid:14)M is de(cid:12)ned by, what state the system is in. Then can we use the above
model? Theanswerinthea(cid:14)rmativeisprovidedinthis
section. We show analytically that even if we start at
0 : H_ <0 and Man(He)(cid:0)M(H)>0
the origin in the M (cid:0)H plane (which is usually not on
(cid:14)M =8 0 : H_ >0 and Man(He)(cid:0)M(H)<0 the hysteresis loop), and apply a periodic input H_, we
< 1 : otherwise:
tend asymptotically towards this periodic solution.
: (8) Itwasshowninanearlierwork[5],thatEquation(9)
Finally after some algebraic manipulations, the equa-
has an orbitally asymptotically stable limit cycle when
tions for the magnetostriction model are given by b = 0 (no coupling) and the input is co-sinusoidal. The
situationismuchmorecomplicatedwhenthecouplingis
dM k(cid:22)0(cid:14)cddMHaen +(cid:14)M(Man(cid:0)M) dH non-zero. Itremainstobeshownthatthereexistsanor-
dt = k(cid:22)0(cid:14)(cid:0) (cid:14)M(Man(cid:0)M)+k(cid:22)0(cid:14)cddMHaen (cid:11)(cid:0)2(cid:22)b0x dt (9) bitallyasymptoticallystablelimitcycleforco-sinusoidal
meff(cid:0)x(cid:127)+c1x_ +dx+bM2V(cid:1)(cid:0)=F (cid:1) (10) inputs H_; with b6=0:
The inputs to the aboveset of equationsare ddHt and 3.1 The uncoupled model with periodic pertur-
F, while x is the mechanical displacement. bation
Amagnetostrictivematerialhas(cid:12)niteresistivity,and
Beforestudyingthefull coupledsystem,weconsiderthe
therefore there are eddy currents circulating within the
e(cid:11)ectofperiodicperturbationsontheuncoupledmodels.
rod. Using Maxwell’s equations, we can derive the fol-
De(cid:12)ne state variables,
lowingsimpleexpressionforthepowerlossesduetoeddy
currents [3].
2 2 2 x1 = H
V lm B +A
Peddy = 2 2 2 x2 = M
N 8(cid:25)(cid:26) B (cid:0)A
y1 = x
where A; B arethe inner and the outerradii of the rod,
y2 = x_
lm is its length, N is the number of turns of coil on the
rod, and V is the voltage across the coil of the induc-
Let,
tor. Hence the eddy current losses can be represented
equivalently as a resistor in parallel with the hysteretic x1+((cid:11)(cid:0) 2b(cid:22)g0(t))x2
inductor. This ideaisquitewellknownandadiscussion z = :
a
canbe foundin[6]or[3]. Fromtheaboveexpressionfor
the power lost, the value of the resistor is, Then the state equations are:
Theorem 2 Consider the system given by Equations
x_1 =u: (11) (11{14), with input given by Equation (16) and b 6= 0.
If (x1;x2)(0)=(0;0); then the (cid:10)-limit set of the system
is an asymptotically orbitally stable periodic orbit.
x_2 =f2(x1;x2;x3;x4;g(t))u (12)
where the function f2((cid:1)) is obtained by substituting the Proof
state variables in Equation (9) and g((cid:1)) for x((cid:1)). The proof is identical to the one with b=0 [5].
2
x3 =sign(u): (13)
Denote the periodic solution of the perturbed mag-
netic system (11 - 14) with perturbation g((cid:1)) and in-
1 x2 put u((cid:1)), as x(cid:22)((cid:1)). It is a two dimensional vector and
0 : x3 <0 and coth(z)(cid:0) 1z (cid:0) Mx2s >0 a T = 2!(cid:25) periodic function. De(cid:12)ne the sets B =
x4 =8 0 : x3 >0 and coth(z)(cid:0) z (cid:0) Ms <0 f(cid:30) 2 C([0;T];R) : j(cid:30)j (cid:20) (cid:12)1; j(cid:30)(t) (cid:0) (cid:30)(t(cid:22))j (cid:20)
< 1 : otherwise: M1jt(cid:0)t(cid:22)j 8 t;t(cid:22) 2 [0;T]g; D = f 2 C([0;T];R) :
: (14) j j (cid:20) (cid:12)2; j (t)(cid:0) (t(cid:22))j (cid:20) M2jt(cid:0)t(cid:22)j 8 t;t(cid:22) 2 [0;T]g;
where(cid:12)1;(cid:12)2;M1;M2arepositiveconstants. LetP1; P2 :
bV 2 2
y_ =Ay+ h (t) (15) C([0;T];R ) ! C([0;T];R) denote the projection oper-
meff
ators de(cid:12)ned by P1(f;g)=f and P2(f;g)=g:
2
T 0 1 Consider the mappings G : B ! C([0;T];R ); g((cid:1)) 7!
where y = y1 y2 ; A= d c1 : 2
(cid:20) (cid:0)meff (cid:0)meff (cid:21) x(cid:22)((cid:1)) and H : D ! C([0;T];R ); h((cid:1)) 7! y(cid:22)((cid:1)): We (cid:12)rst
g((cid:1)); h((cid:1)) ar(cid:2)e 2!(cid:25) peri(cid:3)odic functions. The input is given show G to be continuous.
by,
Theorem 3 G is a continuous map.
u(t)=U cos(!t): (16)
Proof Let the system (11 - 14) be represented by
3.1.1 Analysis of the uncoupled magnetic sys-
tem 3
x_ =f(t;x;(cid:11)~); (t;x) 2 D (cid:26) R
Theproofofexistence and uniquenessoftrajectoriesfor
2bg(t)
thesystem(11-14)isexactlyasintheearlierpaper[5], where (cid:11)~ = (cid:11)(cid:0) (cid:22)0 ; and D is an open set. The state
with the some modi(cid:12)cations. x is 2-dimensional because the discrete states x3 and x4
are functions of x1; x2 and u. Let the initial condition
Theorem 1 Consider the system of equations (11 - 14) be (x1;x2)(0)=(0;0):
with b 6= 0: Let the input be given by Equation (16). If gn ! g in the uniform norm over [0;T] where T is
Suppose the period of f; then (cid:11)~n ! (cid:11)~: Considerthe sequenceof
systems x_ =fn(t;x)=f(t;x;(cid:11)~n): As f is continuous in
jgj(cid:20)G (17) (cid:11)~, fn ! f in the uniform norm if (cid:11)~n ! (cid:11)~ (Theorem
2bG 8). The solutions of each of the systems ffng and f
and (cid:11)~ =(cid:11)(cid:0) (cid:22)0 satis(cid:12)es
exist and is unique for t 2 [0;T]: Then by Theorem 9,
thesolutions (cid:30)n(t) ofx_ =fn(t;x) convergeuniformlyto
c(cid:11)~3Mas < 1 (18) (cid:30)(t) the solution of x_ =f(t;x;(cid:11)~) for t 2 [0;T]:
k (cid:11)~cMs Consider the time interval [T;2T]: We have shown that
(cid:22)0 1(cid:0) 3a (cid:0)2(cid:11)~Ms >0: (19)
(cid:30)n(T) ! (cid:30)(T): Then again by Theorem 9, (cid:30)n(t) !
(cid:0) (cid:1)
(cid:30)(t) for t 2 [T;2T]: Thus we can keep extending the
Also
solutions(cid:30)n(t)and(cid:30)(t)andobtainuniformconvergence
overanyinterval[mT;(m+1)T]wherem>0:Therefore,
0<c<1: (20)
foreach m and(cid:15) > 0, thereexists N(m) > 0such that
(cid:15)
Then the exists a solution to the system with initial j(cid:30)n (cid:0) (cid:30)j < 3 8 n (cid:21) N(m):
condition x(0)=(0;0): Moreover this solution is unique ByTheorem2thereexistasymptoticallyorbitallystable
U U
for all time t(cid:21)0 and lies in the compact set [(cid:0)!; !](cid:2) periodic orbits x(cid:22)n of the systems x_ = fn(t;x) and x(cid:22) of
[(cid:0)Ms;Ms]: the system x_ = f(t;x;(cid:11)~): Hence for each (cid:15) > 0, there
(cid:15)
Proof The proof is very similar to that of the system e(cid:15)xists M (cid:21) 0 such that jx(cid:22)n (cid:0) (cid:30)nj < 3 and jx(cid:22) (cid:0) (cid:30)j <
with b=0 [5]. 3 8 m (cid:21) M and t 2 [mT;(m+1)T]:
2: Hence foralln (cid:21) N(M)and t 2 [mT;(m+1)T] where
m (cid:21) M; we have jx(cid:22)n (cid:0) x(cid:22)j (cid:20) jx(cid:22)n (cid:0) (cid:30)nj + j(cid:30)n (cid:0) (cid:30)j +
From now on until the end of this section, it is always
jx(cid:22) (cid:0) (cid:30)j < (cid:15): Hence G is a continuous map.
assumed that the parameters satisfy conditions (17-20).
2
3.1.2 Analysisof the uncoupled mechanical sys- Our choice of (cid:22)b > 0 ensures that if jbj (cid:20) (cid:22)b then
tem P2 (cid:14) G : B1 7! D1 and P1 (cid:14) H: D1 7! B1:
2
In this subsection, we consider the mechanical system
We now return to the dynamic model of magne-
with periodic perturbation given by Equation (15). We
tostriction (9,10) and prove the main theorem of this
assumethehomogenoussystem(thatis,(15)withh(t)=
paper.
0) to be asymptotically stable. The relevant results are
collected in the appendix. Theorem 7 Consider the dynamic model for magne-
tostriction given by Equations (11 - 16). Suppose the
Theorem 4 Consider the system (15). If the eigenval-
2(cid:25) matrix A has eigenvalues with negative real parts and
ues of A have negative real parts and h((cid:1)) is an ! peri-
2(cid:25) the parameters satisfy conditions (18-20) with the mag-
odic function, then (15) has an ! periodic solution that netostriction constant b (cid:20) (cid:22)b de(cid:12)ned in the statement of
is asymptotically orbitally stable.
Theorem6. Then thereexistsanorbitally asymptotically
Proof This follows from Lemma 10 and Theorem 11 in stable periodic orbit of the system.
the appendix.
2 Proof The sets B1 and D1 are compact and convex by
Theorem 12. Then B1 (cid:2) D1 is compact in the uniform
Theorem 5 If the eigenvalues of A have negative real product norm by Theorem 13. Obviously it is also con-
parts, then H is a continuous map. vex.
Let (cid:9) be de(cid:12)ned as, (cid:9) : B1 (cid:2) D1 ! B1 (cid:2)
Proof This againfollows from Lemma 10 and Theorem D1;(cid:9) (x2;y1) = (P1 (cid:14) H(x2);P2 (cid:14) G(y1)): Then (cid:9) is
11. continuousbecauseP2(cid:14)G and P1(cid:14)H arecontinuousby
2
Theorems 3 and 5, and the continuity of the projection
operator.
3.1.3 Analysis of the coupled magnetostriction Then by the Schauder Fixed Point Theorem (Theo-
model rem 14), there exists a limit point of the mapping (cid:9) in
the setB1(cid:2)D1: Thisgivesus the periodicity ofthe two
In this section, we prove the existence of an orbitally
statevariablesx2andy1. Ingeneral,the(cid:12)xedpointmay
asymptotically stable periodic orbit for the magne-
notbeunique,butwhentheinitialstateistheorigin,the
tostriction model.
(cid:10)limitsetisuniquebytheuniquenessofsolutions. Now,
Let D1 denote the range of P2 (cid:14) G and B1 denote the
(y1;y2) = Gx2 and by Theorem 4, (y1;y2) is anasymp-
range of P1 (cid:14) H: Thus P2 (cid:14) G : B 7! D1 and P1 (cid:14) H:
toticallystableperiodicorbit. Also(x1;x2) = Hy1 and
D 7! B1:
by Theorem 2, (x1;x2) is an asymptotically stable peri-
Theorem 6 There exists a (cid:22)b > 0 such that if jbj (cid:20) (cid:22)b odicorbit. Theotherstatevariables(x3;x4)areperiodic
then P2 (cid:14) G : B1 7! D1 and P1 (cid:14) H: D1 7! B1: because they are determined by x1; x2 and u.
2
Proof First we show that the sets B1 and D1 have the
same structure as that of B and D respectively. Then
we choose(cid:22)b so that the domains and ranges of G and H A Mathematical Preliminaries
Ms
aresuitably adjusted. Choose(cid:12)1 =Ms and M1 = 3a U
in the de(cid:12)nition of the set D: Theorem 8 If X and Y are normed linear spaces and
By Theorem 1, the elements of D1 are uniformly f is a mapping from X to Y , then f is continuous at x
bounded by Ms: Let x(cid:22)=Gg: Therefore P2 (cid:14) Gg = x(cid:22)2: if and only if for each sequence fxng in X converging to
1
Nowx(cid:22)2(t2)(cid:0)x(cid:22)2(t1) = 0 x(cid:22)_2(t1+s(t2(cid:0)t1))(t2(cid:0)t1)ds x we have ff(xn)g converging to f(x) in Y.
by the Mean Value TheRorem. As the parameters of the
Theorem 9 Suppose ffng; n=1;2;(cid:1)(cid:1)(cid:1); is a sequence
system (11 - 14) satisfy the conditions (17 - 20), the
ofuniformlyboundedfunctionsde(cid:12)nedandsatisfying the
vector (cid:12)eld f(t;x)u(t) is uniformly bounded. Therefore n+1
Carath(cid:19)eodory conditions on an open set D in R with
jx(cid:22)2(t1)(cid:0)x(cid:22)2(t2)j (cid:20) M1jt2(cid:0)t1j: Thus D1 has the same
limn!1fn = f0 uniformly on compact subsets of D.
structure of D.
Suppose (tn;xn) is a sequence of points in D converging
Lety(cid:22)=Hh:Thereforey(cid:22)1 = P1 (cid:14)Hh:Theelements
2 to (t0;x0) in D as n ! 1 and let (cid:30)n(t); n = 1;2;(cid:1)(cid:1)(cid:1),
of B1 are uniformly bounded because H is linear in h
be asolution of theequationx_ =fn(t;x) passing through
andthefunctionsh 2 Dareuniformlybounded. jy(cid:22)1j (cid:20)
jy(cid:22)j (cid:20) jP1 (cid:14) HjMs2 = (cid:12)2: We need to choose(cid:22)b so that the point (tn;xn). If (cid:30)0(t) is de(cid:12)ned on [a;b] and
2bG is unique, then there is an integer n0 such that each
(cid:11)~ = (cid:11)(cid:0) (cid:22)0 de(cid:12)ned in Theorem 1 satis(cid:12)es Conditions
(cid:30)n(t); n(cid:21)0, can be de(cid:12)ned on [a;b] and converges uni-
(18) and (19). Such a non-zero(cid:22)b obviously exists. Now
1 formly to (cid:30)0(t) uniformly on [a;b]:
y(cid:22)1(t2)(cid:0)y(cid:22)1(t1) = 0 y(cid:22)_1(t1+s(t2(cid:0)t1))(t2(cid:0)t1)dsbythe
Mean Value TheoRrem. jy(cid:22)_j (cid:20) jAj(cid:12)2 + bV(cid:12)12 = M2: Consider the homogenous linear periodic system
Therefore jy(cid:22)1(t2)(cid:0)y(cid:22)1(t1)j (cid:20) M2jt2(cid:0)t1j: Thus B1 has
the same structure of B. x_ =A(t)x (21)
and the non-homogenous system [3] R. Venkataraman, \A hybrid actuator," Master’s
thesis, University of Marylandat College Park,May
x_ =A(t)x+f(t) (22) 1995.
whereA(t+T)=A(t); T > 0andA(t)isacontinuous [4] R. Venkataraman, W. Dayawansa, and P. Krish-
n(cid:2)n real or complex matrix function of t. naprasad., \The hybrid motor prototype: Demon-
stration results." TR 98-2, Technical report of the
De(cid:12)nition 1 IfA(t)isann(cid:2)ncontinuousmatrixfunc-
Institute for Systems Research, University of Mary-
tion on ((cid:0)1;1) and D is a given class of functions
land at College Park.
which contains the zero function, the homogenous sys-
tem x_ =A(t)x is said to be noncritical with respect to D [5] R. Venkataraman and P. Krishnaprasad., \Quali-
if the only solution of Equation (21) which belongs to D tative analysis of the bulk ferromagnetic hysteresis
is the solution x = 0: Otherwise, system (21) is said to model." preprint, 1998.
be critical with respect to D:
[6] S. Chikazumi, Physics of Magnetism. John Wiley
and Sons, Inc., 1966.
The set PT denoting the set of T-periodic continuous
functions is aBanachspacewiththe sup-norm. That is, [7] J. William Fuller Brown, Magnetoelastic Interac-
jfj=sup(cid:0)1<t<1jf(t)j; f 2 PT:LetB denotetheset tions. Springer Tracts in Natural Philosophy,
n
of continuous bounded functions from R to R . Springer{Verlag,1966.
Theorem 10 [8] (a) System (21) with A(t) 2 PT is [8] J. Hale, Ordinary Di(cid:11)erential Equations. Krieger
noncritical with respect to B if and only if the character- Publishing Company, Malabar, Florida, 1980.
istic exponents of (21) have nonzero real parts.
[9] J. Pryce, Basic Methods of Linear Functional Anal-
(b) System (21) with A 2 PT is noncritical with
ysis. Hutchinson University Library, London, 1973.
respect to PT if and only if I (cid:0)X(T) is nonsingular,
when X(t); X(0) = I, is a fundamental matrix solution
of (21).
Theorem 11 Suppose A is in PT. Then the nonho-
mogenous equation (22) has a solution Kf in PT, if and
onlyifsystem(21)isnoncriticalwithrespecttoPT. Fur-
thermore, if system (21) is noncritical with respect to
PT, then Kf is the only solution of (22) in PT and is
linear and continuous in f.
m
Theorem 12 Suppose D is a compact subset of R ;
M;(cid:12) are positive constants and A is the subset of
n
C(D;R ) such that (cid:30) 2 A implies j(cid:30)j (cid:20) (cid:12); j(cid:30)(t)(cid:0)
(cid:30)(t(cid:22))j (cid:20) Mjt(cid:0)t(cid:22)j for t;t(cid:22)2 D: Then the set A is convex
and compact.
Theorem 13 [9]LetA; B becompactsubsetsofX;j(cid:1)j1)
and(Y;j(cid:1)j2)respectively. ThenA(cid:2)B iscompact(under
eitherofthestandardmetricsjj(x;y);(x(cid:19);y(cid:19))jj1 =jx(cid:0)x(cid:19)j1+
jy(cid:0)y(cid:19)j2; jj(x;y);(x(cid:19);y(cid:19))jj2 =max(jx(cid:0)x(cid:19)j1;jy(cid:0)y(cid:19)j2)):)
Theorem 14 (Schauder) [8] If A is a convex, compact
subset of a Banach space X and f : A ! A is continu-
ous, then f has a (cid:12)xed point in A:
References
[1] R. Venkataraman and P. Krishnaprasad., \Charac-
terization of the Etrema MP 50/6 magnetostrictive
actuator."TR98-1,TechnicalreportoftheInstitute
forSystemsResearch,UniversityofMarylandatCol-
lege Park.
[2] D. Jiles and D. Atherton, \Theory of ferromagnetic
hysteresis,"JournalofMagnetismandMagneticMa-
terials, vol. 61, pp. 48{66, 1986.
10 20 20 20 20 20 20 20
0 10 10 10 10
10
-10 0 0 0 0 0 0
Microns--3200 Microns---321000 Microns---321000 Microns---321000 Microns---321000 Microns-20 Microns-20 Microns-100
-40 -40 -40 -40 -40 -40 -40
-20
-50 -50 -50 -50 -50
-60-2 -1 0 1 2 -60-2 -1 0 1 2 -60-2 -1 0 1 2 -60-2 -1 0 1 2 -60-2 -1 0 1 2 -60-2 -1 0 1 2 -60-2 -1 0 1 2 -30-1.5 -1 -0.5 0 0.5 1 1.5
Amps Amps Amps Amps Amps Amps Amps Amps
(a)0.5Hz (b)1Hz (c)5Hz (d)10Hz (e)50Hz (f)100Hz (g)200Hz (h)500Hz
Figure1: ETREMAMP50/6Actuatordisplacement (Microns)vscurrent(Amps) characteristicat di(cid:11)erentdriving
frequencies .
I R
I + I
l 2
u V Red
-
Figure2: Athin magnetostrictiveactuatorinaresistive
circuit.