T R R ECHNICAL ESEARCH EPORT A Model for a Thin Magnetostrictive Actuator by R. Venkataraman, P.S. Krishnaprasad CDCSS T.R. 98-6 (ISR T.R. 98-37) + CENTER FOR DYNAMICS C D - AND CONTROL OF S SMART STRUCTURES The Center for Dynamics and Control of Smart Structures (CDCSS) is a joint Harvard University, Boston University, University of Maryland center, supported by the Army Research Office under the ODDR&E MURI97 Program Grant No. DAAG55-97-1-0114 (through Harvard University). This document is a technical report in the CDCSS series originating at the University of Maryland. Web site http://www.isr.umd.edu/CDCSS/cdcss.html Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. 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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 periodic 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 magnetoelastic 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 magnetic 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 system 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 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 magnetostriction 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 system 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

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