Approximate Inversion of the Preisach Hysteresis Operator with Application to Control of Smart Actuators

Transcription

1 1 Approximate nversion of the Preisach Hysteresis Operator with Application to Control of Smart Actators Ram V. yer, Member, EEE, Xiaobo Tan, Member, EEE, and P. S. Krishnaprasad, Fellow, EEE Abstract Hysteresis poses a challenge for control of smart actators. A fndamental approach to hysteresis control is inverse compensation. For practical implementation, it is desirable for the inpt fnction generated via inversion to have reglarity properties stronger than continity. n this paper we consider the problem of constrcting right inverses for the Preisach model for hysteresis. Under mild conditions on the density fnction, we show the existence and weak-star continity of the rightinverse, when the Preisach operator is considered to act on Hölder continos fnctions. Next, we introdce the concept of reglarization to stdy the properties of approximate inverse schemes for the Preisach operator. Then, we present the Fixed Point and Closest-Match algorithms for approximately inverting the Preisach operator. The convergence and continity properties of these two nmerical schemes are stdied. Finally, we present the reslts of an open-loop trajectory tracking experiment for a magnetostrictive actator. ndex Terms Hysteresis, Approximate inversion, Reglarization, Smart actators, Preisach operator, Magnetostriction, Piezoelectricity, Shape memory alloys, Electro-active polymers, Fixed Point iteration algorithm, Closest-Match algorithm. sed lately to model the hysteresis phenomenon in piezoelectric [5], magnetostrictive materials [6], [7], shape-memory alloys [8], [9] and electro-active polymers [7]. The Preisach operator is a model of the phenomenological type. Althogh in general, the Preisach operator does not provide physical insight into the problem, it is capable of prodcing behaviors similar to those of physical systems [4]. t is of great interest to the smart strctres and controls commnity becase of its tility in developing low order models that can be sed for designing real-time controllers. A fndamental idea in coping with hysteresis is inverse compensation (see, e.g., [5], [1], [11], [1]), as illstrated in Fig. 1. f one can constrct an approximate right inverse Ŵ 1 of the hysteresis operator W, then the otpt y of W will approximately eqal the reference trajectory y ref. y ref W^ 1 W y. NTRODUCTON SMART materials, e.g., magnetostrictives, piezoceramics, and shape memory alloys (SMAs), exhibit strong copling between applied electromagnetic/thermal fields and strains that can be exploited for actation and sensing. Hysteresis in smart materials, however, poses a significant challenge in smart material actators (also called smart actators). Models for hysteresis in smart materials can be classified into those that are physics-based and those that are phenomenology-based. Physics-based models se principles of thermodynamics to obtain constittive relationships between conjgate variables. Sch examples inclde the iles-atherton model [1] and the ferromagnetic hysteresis model [], [3], where hysteresis is considered to arise from pinning of domain walls on defect sites. The most poplar hysteresis model sed for magnetic materials has been the Preisach operator [4], and it has been This research was spported by the Army Research Office nder the ODDR&E MUR97 Program Grant No. DAAG to the Center for Dynamics and Control of Smart Strctres (throgh Harvard University). R. V. yer is with the Department of Mathematics and Statistics, Texas Tech University, Lbbock, TX 7949 USA ( rvenkata@math.tt.ed). X. Tan is with the Department of Electrical & Compter Engineering, Michigan State University, East Lansing, M 4884 USA ( xbtan@ms.ed). P. S. Krishnaprasad is with the nstitte for Systems Research and the Department of Electrical & Compter Engineering, University of Maryland, College Park, MD 74 USA ( krishna@isr.md.ed). Fig. 1. llstration of inverse compensation. This paper deals with approximate inversion of the Preisach operator Γ, where is reqired to be a Hölder continos fnction. t contains five contribtions: a) the proof of weakstar continity of the inverse acting on the space of Hölder continos fnctions, nder a mild and easily verifiable condition on the Preisach density fnction; b) the formlation of reglarization for the inversion problem; c) the development of a Fixed Point iteration algorithm and its convergence analysis; d) the development of the Closest-Match algorithm and its convergence analysis; and e) Experimental validation of the Closest-Match algorithm. These contribtions are briefly discssed next. Brokate and Sprekels [13] prove the existence and continity of the inverse of the Preisach operator when the domain is the space of continos fnctions, nder very mild conditions on the density fnction. Visintin [14] proves a theorem on the weak-star continity of the inverse, when the domain is the space of Hölder continos fnctions, nder very strong sfficient conditions on the density fnction that are not easily verifiable. Figre 7 shows an identified (in a non-parametric manner) density fnction for a magnetostrictive actator [7]. The density fnction has a vale zero on a large area of the Preisach domain and this implies that Visintin s condition will not be satisfied for this actator. We need a theorem for the weak-star continity of the inverse operator acting on

2 spaces of Hölder continos fnctions that only depends on the density conditions close to the diagonal on the Preisach plane. Sch a theorem wold be in the same spirit as Corollary.11.1 of Brokate and Sprekels [13] for the continity of the inverse operator acting on the space of continos fnctions. We present a theorem in Section that concldes the reslts of Visintin s theorem nder mild conditions on the density fnction (these conditions are still stronger than Brokate and Sprekels conditions, as expected). The tility of this theorem to control engineers is that the conditions can be easily verified. n or earlier work [15], we showed that the approximate inverse to an incrementally strictly increasing (S) Preisach operator can be compted nmerically. For 1, C[, T ], consider the ordering 1 if and only if 1 (t) (t) for all t [, T ]. Then the Preisach operator is said to be incrementally strictly increasing [16] (S) if there exist constants k 1, k > sch that k 1 ( 1 ) Γ[ 1 ] Γ[ ] k ( 1 ). This definition is different from the piecewise strictly increasing operator (PS) defined by Brokate and Sprekels. A Preisach operator is said to be piecewise strictly increasing if (Γ[](T ) Γ[]())((T ) ()) for a monotone inpt C[, T ]. Under the mild condition that the density fnction is integrable and non-zero almost everywhere on a strip of positive width along the diagonal on the Preisach plane, it is easy to show that the corresponding Preisach operator is PS. The S condition reqires very stringent conditions on the density fnction. For example, if the density fnction took a constant positive vale on the set α min β α α max in the Preisach plane, then it is S. We have shown in earlier work [15], [17], that the Fixed Point iteration: n+1 = n + 1 k (y Γ( n )) converges to a fnction that satisfies Γ[ ] = y via a contraction. Leang and Devasia [18] apply this reslt to the positioning of piezoelectric actators. n this paper, nder the (significantly) milder condition of PS Preisach operators, we show the convergence of the same scheme withot sing a contraction argment. We are led to the space of Lipschitz continos fnctions as we wold like the soltion of the inverse problem, to have reglarity properties stronger than jst continity. For example, in the case of indctors or transformers with a ferromagnetic core, the Preisach operator is sally considered to map the axial magnetic field H(t) fnction to the axial magnetization M(t) (see Mayergoyz [4]) Γ[H](t) = M(t). (1) The electro-motive force across the terminals of the indctor is then proportional to the time-derivative of B(t) = µ (H(t) + M(t)), where µ is the permittivity of free-space. Therefore, it is desirable for H(t) which is the soltion to Eqation (1) to be a differentiable fnction of time. Similar considerations apply to other sitations where one ses the Preisach operator, for example, piezoelectricity. Now, Rademacher s Theorem states that a fnction that is Lipschitz on an open sbset of R n is almost everywhere differentiable on that sbset in the sense of the Lebesge measre [19], and so it is reasonable to seek Lipschitz continos fnctions as soltions to the inverse problem. Consideration of Lipschitz fnctions is also motivated by constraints on implementation of control signals often encontered in practice. Or theorem (see Theorem.) shows that the inverse maps generic fnctions in the space of Hölder continos fnctions on [, T ] denoted by C,ν [, T ] to the space C,ν 1 [, T ] where ν 1 ν 1. This reslt implies that in general, even if the desired otpt fnction is differentiable, the inpt fnction need not be a Lipschitz continos fnction. For engineering reasons, if one wishes to obtain a Lipschitz continos fnction as the (approximate) inverse of a Hölder continos fnction, an operation called mollification [] has to be carried ot. The natral qestion that arises then is the following: if the desired otpt fnction is changed by a small amont either de to noise or by design, then how close is the reslting mollified soltion to the original mollified soltion (and in what sense)? This is a qestion of enormos engineering importance, and to discss it, we develop the notion of reglarization for solving the inversion problem in Section. Two approximate inversion algorithms for the Preisach operator are then developed. Both algorithms se the PS property of a Preisach operator. n Section V, we present the Fixed Point algorithm to approximately invert the Preisach operator, and stdy its convergence and continity properties nder the PS condition. Next, the Closest-Match Algorithm is developed and analyzed in Section V. The latter algorithm is applied to tracking control of a magnetostrictive actator, and experimental reslts are reported to demonstrate its efficacy.. THE PRESACH OPERATOR AND TS NVERSE To fix the notation and the problem setp, the Preisach operator Γ and some known reslts are reviewed first in Sbsection -A. Sbsection -B then stdies the weak-star continity of Γ 1 in the space of Hölder continos fnctions nder the weak condition. Let be a closed interval, ν (, 1], and T >. The following notation will be sed to denote different fnction spaces: C[, T ]: space of continos fnctions on [, T ]; C m [, T ]: space of monotone, continos fnctions on [, T ]; C pm [, T ]: space of piecewise monotone, continos fnctions on [, T ]; C [, T ]: space of continos fnctions taking vales in, i.e., (t), C [, T ], t [, T ]; C pm, [, T ]: C pm [, T ] C [, T ]; C,ν [, T ]: space of Hölder continos fnctions on [, T ], i.e., C,ν [, T ], t 1, t [, T ], (t ) (t 1 ) sp t 1,t T t t 1 ν C, for some constant C. Other spaces sch as C,ν [, T ] are defined analogosly to the definition of C [, T ] from C[, T ]. n this paper, the following two norms are heavily sed: = sp (t), C[, T ], and t T,ν = + sp t 1,t T (t ) (t 1 ) t t 1 ν,

3 3 for C,ν [, T ]. A. The Preisach Operator A detailed treatment on the Preisach operator can be fond in [4], [14], [13]. For a pair of thresholds (β, α) with β α, consider a delayed relay ˆγ β,α [, ] (called a Preisach hysteron), as illstrated in Fig.. For C[, T ) and an initial configration ζ { 1, 1}, v = ˆγ β,α [, ζ] is defined as, for t [, T ], v(t) = where v( ) = ζ and t = Fig.. β v if (t) < β 1 if (t) > α v(t ) if β (t) α α lim t ɛ. ɛ>,ɛ llstration of an elementary Preisach hysteron. Define the Preisach plane P = {(β, α) R : β α}, where (β, α) P is identified with ˆγ β,α. For C[, T ] and a Borel measrable configration ζ of all hysterons, ζ : P { 1, 1}, the otpt of the Preisach operator Γ is defined as Γ[, ζ ](t) = µ(β, α)ˆγ β,α [, ζ (β, α)](t)dβdα, () P for some Borel measrable fnction µ, called the Preisach density fnction. t is assmed in this paper that µ ; µ has a compact spport P; and is an integrable fnction, that is µ L 1 (P). For each t [, T ], P can be divided into two regions: P (t) P + (t) = {(β, α) P otpt of ˆγ β,α at t is 1}, = {(β, α) P otpt of ˆγ β,α at t is + 1}, so that P = P (t) P + (t). Eq. () can be rewritten as: Γ[, ζ ](t) = µ(β, α)dβdα µ(β, α)dβdα. P + (t) P (t) (3) t can be easily shown [4], [13] that each of P and P + is a connected set, and that the otpt of the Preisach operator is determined by the bondary between P and P +. The bondary is also called the memory crve, since it provides information abot the state of Γ. Ths the initial state fnction ζ can instead be replaced by a memory crve in the Preisach, plane. Using the transform: r = α β and s = α+β one can describe the memory crve as a fnction (r, ψ(r)) defined on a compact region [, r max ]. The set of admissible memory crves can then be defined as [13] Ψ = {φ φ : R+ R, φ(r) φ( r) r r, r, r, R spp (φ) < + }, where R spp (φ) = sp{r r, φ(r) }. The memory crve ψ 1 at t = is called the initial memory crve and hereafter it will be pt as the second argment of the Preisach operator. Note that ψ 1 () eqals the last inpt vale of Γ. The Preisach density will be denoted as ω(, ) in the (r, s) coordinates. n this paper both coordinate systems, (β, α) and (r, s), are sed depending on whichever is more convenient; similarly, both µ(β, α) and ω(r, s) will be sed for the Preisach density. Let the inpt signal take vales in = [ min, max ], that is, (t), t [, T ]. Define the fnction χ( ) on [, max min ]: χ (x) = inf{ Γ[, ψ 1 ](T ) Γ[, ψ 1 ][] : ψ 1 Ψ, C m [, T ], (T ) () = x}. The fnction χ( ) is continos and monotonically increasing nder or basic hypothesis on µ. t is easy to check that if χ(x) > for all x >, then Γ is PS. Let be the smallest interval that contains the otpt vales of Γ when the inpt takes vales in. t can be shown that if χ (x) >, x >, then Γ[, ψ 1 ] : C [, T ] C [, T ] is invertible and the inverse operator is also continos [14], [13]. Frthermore, Visintin [14] shows that if, x [, max min ], χ (x) Cx ν ν 1, (4) for < ν 1, ν 1, then the inverse of Γ[, ψ 1 ] maps C,ν into C,ν 1 and it is weak-star continos. B. A Milder Condition for the Weak-Star Continity of the nverse The condition (4) is strong since it needs to hold for all x [, max min ]. t is hard to verify directly also, as it is posed in terms of χ (x). n this sbsection, a weaker condition in terms of the Preisach density fnction ω(, ) is shown to lead to the weak-star continity of the inverse. Before proceeding, we sketch the constrction of the weakstar topology on C,ν [, T ], < ν < 1. A fnction f in the space C,ν (R), < ν < 1, can be expanded sing a Faber-Schader basis as described in Wojtaszczyk [1] (page 4). Ths we have a map Φ : C,ν [, T ] l given by Φ(f) = {a m,n }. ts adjoint Φ maps elements in l 1 that describe the weak-star topology on l to the dal of C,ν [, T ]. These fnctionals Φ (z), z l 1, define the weakstar topology of C,ν [, T ]. n Section, we will define a distance metric for the weak-star topology of C,ν [, T ] based on this constrction. t is well known that this topology is coarser than the norm topology on C,ν [, T ] defined sing f,ν.

4 4 The following three lemmas will be sed in proving the main reslt of this section. Lemma.1: Let ξ, ɛ >. f the Preisach density ω(r, s) Cr ξ, for some C >, for almost every (r, s) R ɛ = [, ɛ] [ min ɛ, max + ɛ], then χ (x) Kx ξ+ for x ɛ, for some K >. Proof. Let r = x. For x [, ɛ], χ (x) = inf = s [ min, max ] r r s +( x r) (x r) Cr ξ dr C (1 + ξ)( + ξ) ξ xξ+. s ( x r) ω(r, s)dsdr Lemma.: (Visintin [14]) Let X, Y, S 1, S be metric spaces sch that S 1 X and S Y with continos injections. Let f : X Y be continos and sch that it maps relatively compact sbsets of S 1 into relatively compact sbsets of S (with respect to the topologies of S 1 and S ). Then f : S 1 S is continos with respect to the topologies of S 1 and S. Lemma.3: Let = T < T 1 < < T N = T be a niform partition of [, T ] sch that i = [T i 1, T i ]; i = 1,, N, has length δ. Let f i C,ν ( i ), < ν 1, i = 1,, N, with f i,ν K and f i (T i ) = f i+1 (T i ). Then the fnction obtained by concatenation f = i f i i, where i is the indicator fnction of i, belongs in C,ν [, T ] and f,ν (1 + N 1 ν ) K. Proof. As f i,ν K, we have f i (t) K and f i (t) f i (t ) K t t ν for t, t i. This implies f(t) K, t [, T ]. Next, for t 1 and t N, f(t) f(t ) f(t) f(t 1 ) + f(t 1 ) f(t ) + + f(t N 1 f(t ) K t T 1 ν + + K T N 1 t ν. We wish to find a constant L sch that the sm a ν 1 + a ν + +a ν N L(a 1 + +a N ) ν where a 1,, a N and < ν 1. Dividing by (a 1 + +a N ) ν one obtains the following fnction on the left hand side: g(p 1,, p N ) = p ν 1 + +p ν N where p 1,, p N and N i=1 p i = 1. This fnction is maximized by p i = 1 N for all i and the maximm vale is N ( ) 1 N = N 1 ν. Ths L = N 1 ν and ν f(t) f(t ) N 1 ν K t t ν. For t and t in other intervals i, one can proceed similarly and arrive at the same ineqality. Therefore, f(t) f(t ) f,ν = f + sp t t ; t,t [,T ] t t ν K (1 + N 1 ν ). Before presenting or main theorem on Γ 1, we smmarize the continity properties of the operator Γ nder certain conditions on ω. The tility of this Theorem is that it combines reslts in Brokate and Sprekels [13], and Visintin [14] nder a common condition on the density fnction. These are the same conditions needed on ω for or main reslt. t mst be noted that these conditions are slightly stronger than those of Proposition.4.11 and Corollary.11.1 of Brokate and Sprekels [13], and weaker than Theorem 3.9 of Visintin [14]. Theorem.1: Let Γ[, ψ 1 ] be a Preisach operator with domain = [ min, max ], where ψ 1 Ψ. Assme that the density fnction ω(r, s) has compact spport; is integrable; is non-negative; and ω(r, s) Cr ξ for almost every (r, s) R ɛ = [, ɛ] [ min ɛ, max + ɛ], where C >, ξ, and ɛ >. Then: 1) Γ[, ψ 1 ] : C [, T ] C [, T ] is Lipschitz continos; ) Γ[, ψ 1 ] : C,ν [, T ] C,ν [, T ] is weak-star continos, where < ν 1; 3) Γ[, ψ 1 ] : C [, T ] C [, T ] is invertible, and its inverse can be extended to a continos operator Γ 1 [, ψ 1 ] : C [, T ] C [, T ]. Proof. By the conditions on the density, the Preisach operator Γ[, ψ 1 ] : C [, T ] C [, T ] is PS and is Lipschitz continos (by Theorem.4.11 in Brokate and Sprekels [13]). They also show that Γ maps norm-bonded sets in C,ν [, T ] to norm-bonded sets in C,ν [, T ]. As these sets are compact in the weak-star topology, Lemma. yields the weakstar continity of Γ. To show the last statement, note that χ (x) > by Lemma.1 for x (, ɛ]. As χ (x) is a continos, increasing fnction of x and so χ (x) > for all x (, b a), the proof of Theorem.11. in Brokate and Sprekels [13] applies here. Under the same conditions on ω as in the above theorem, we wold like to show the existence and continity of the inverse for the Preisach operator acting between spaces of Hölder continos fnctions. The following theorem is or main reslt. Theorem.: Assme that the Preisach density fnction ω(r, s) has compact spport, ω, and ω(r, s) Cr ξ for almost every (r, s) R ɛ = [, ɛ] [ min ɛ, max +ɛ], where C >, ξ, ɛ >. Then for any ψ 1 Ψ, Γ 1 [, ψ 1 ] is weak-star continos from C,ν [, T ] to C,ν 1 [, T ], where ν (, 1] and ν 1 = ν ξ+. Proof. Let y C,ν [, T ] with y,ν K. By Theorem.1, Γ[, ψ 1 ] is invertible and there exists C [, T ] sch that Γ[, ψ 1 ] = y. We shall show that belongs in C,ν1 [, T ]. Partition [, T ] niformly sch that = T < T 1 < T N = T and T i T i 1 δ where i = 1,, N. The choice of δ will be described shortly. Restrict y to the intervals i = [T i 1, T i ]; i = 1,, N, and obtain the fnctions y i. Similarly restricting to i one obtains i. Define the fnction: osc(v; [a, b]) = for v C[, T ] and [a, b] [, T ]. Note that max v(t) min v(t), t [a,b] t [a,b] χ (osc( i ; [t, t ])) osc(y; [t, t ]), [t, t ] i, (5)

5 5 by Lemma in [13]. As y,ν K, for t, t i, and hence y(t) y(t ) K t t ν Kδ ν, (6) which by (5) implies From Lemma.1, osc(y; i ) Kδ ν, (7) χ (osc( i ; i )) Kδ ν. (8) χ (x) C x ξ+, x [, ɛ]. (9) Now choose δ > small enogh so that: Kδ ν C (ɛ) ξ+. This together with (8), (9), and the monotone increasing property of χ ( ), implies osc( i ; i )) ɛ. Note that the choice of δ fixes the nmber of partitions N. Next, for t, t i ; i = 1, N, Eq. (5) and (9) yield: which leads to C i (t) i (t ) ξ+ χ (osc( i, [t, t ])) osc(y; [t, t ]) (by (5)) (1) K t t ν (as y,ν K), (11) i (t) i (t ) ( K C ) 1 ξ+ t t ν ξ+ = ( K C ) 1 ξ+ t t ν 1. (1) Finally, sing Lemma.3 one gets,ν1 K 1 for some K 1 >. This implies that Γ 1 [, ψ 1 ] maps norm-bonded sets in C,ν [, T ] to norm-bonded sets in C,ν1 [, T ]. As these sets are compact in the respective weak-star topologies of C,ν i [, T ], i = 1,, we apply Lemma. to Γ 1 with X = C [, T ]; Y = C [, T ]; S 1 = C,ν [, T ]; and S = C,ν 1 [, T ], to obtain the weak-star continity of Γ 1. Let < ν 1 < ν 1. As C,ν [, T ] C,ν1 [, T ], the linear fnctionals on C,ν1 [, T ] are also linear fnctionals on C,ν [, T ]. As a reslt, the weak-star topology on C,ν [, T ] (denoted by τ ) is finer than the topology (denoted by τ 1 ) inherited from the weak-star topology of C,ν 1 [, T ]. This implies that weak-star compact sets of C,ν [, T ] remain compact in the topology τ 1 []. Denote the weak-star topology of C,ν 1 [, T ] by τ. Corollary.1: Sppose that Γ is a Preisach operator with a density fnction that satisfies the conditions of Theorem.. Let U = Γ 1 [C,ν, ψ 1 ] and ν ( ) ν 1 = ξ +. Then the maps Γ 1 : C,ν [, T ], τ (U, τ), and Γ : (U, τ) ( ) C,ν [, T ], τ 1 are continos maps. Proof. By Theorem., Γ 1 : (C,ν [, T ], τ ) (U, τ) is [, T ]. To show the second statement, observe that the map Γ : (U, τ) (C,ν1 [, T ], τ) is continos by Theorem.1. Bt we mst have Γ : U C,ν [, T ] continos, as U C,ν 1 by the definition of U. So Γ : (U, τ) (C,ν [, T ], τ 1 ) is continos, by the definition of τ 1. Ths the composition ( ) ( ) Γ Γ 1 : C,ν [, T ], τ C,ν [, T ], τ 1 is continos, as τ is finer than τ 1. Note that we cannot infer a similar statement had we considered the composition Γ 1 Γ. Ths we are natrally led to the concept of right inverses of Preisach operators and fortnately, that is what is needed in applications.. REGULARZATON The objective of this section is to stdy approximate soltion methods for the operator eqation: Γ[, ψ 1 ] = y, (13) where y C[, T ]. Since the condition χ (x) > for x > garantees the existence of a continos inverse for Γ[, ψ 1 ] : C [, T ] C [, T ], theoretically there is no need for any reglarization if one is looking for jst a continos inpt fnction. However, for implementation of the inverse in nmerical and physical experiments, it is desirable that the inpt generated via inversion has certain reglarity properties, for example, Lipschitz continity. The two algorithms to be discssed later in this paper reslt in Lipschitz continos fnctions as approximate soltions to (13) for y C[, T ]. On the other hand, the proof of Theorem. shows that a piecewise strictly increasing Preisach operator has an inverse that maps generic fnctions in C,ν [, T ] to fnctions in C,ν1 [, T ] with ν 1 ν 1, which rles ot the possibility of getting a Lipschitz continos in general. This raises the isse of how to evalate an approximate inversion scheme in terms of the convergence to the exact inverse. For this prpose, it is sefl to define a norm on approximate inverses by the following procedre. As C,ν [, T ], < ν < 1, is isomorphic to l, the weakstar topology on C,ν [, T ] is defined by a contable family of semi-norms. On the other hand, C,1 [, T ] is isomorphic to L and so its weak-star topology is also defined by a contable family of semi-norms [1]. Using these semi-norms, one can define eqivalent metrics on C,ν [, T ], < ν 1 sch that convergence in any of the metrics is eqivalent to convergence in the weak-star topology (Zimmer [3], page 14). Denote any one of the metrics so obtained on C,ν i [, T ], where i = 1, and < ν 1 < ν 1, by d i (, ). A key observation is that these metrics are translation invariant, that is, d i (x + c, y + c) = d i (x, y) since they are defined sing semi-norms. One wold like to define an (indced) norm for Γ 1 in stdying the convergence of approximation schemes. Ptting the inverse operator and varios approximate inverses in a vector space wold facilitate the se of tools available to vector spaces. This can be achieved by appropriately shifting the inpt and the otpt of Γ. To be specific, considering that the inpts mst have the initial condition () = ψ 1 () and the otpts mst have the same initial vale z = Γ[; ψ 1 ](),

6 6 we define the sets Ī = {v ψ 1() v }, and = {w z w }, and the maps: Γ[, ψ 1 ] : C,ν 1[, T ] C,ν [, T ] Ī (14) ū ȳγ[ū + ψ 1 (), ψ 1 ](t) z, Γ 1 [, ψ 1 ] : C,ν [, T ] C,ν 1[, T ] Ī (15) ȳ ū = Γ 1 [ȳ + z, ψ 1 ] ψ 1 (). By translation invariance of d i, one has d 1 ( 1 ψ 1 (), ψ 1 ()) = d 1 ( 1, ), and d (y 1 z, y z ) = d (y 1, y ). t can be verified that Γ 1 [, ψ 1 ] belongs in the vector space S (with field R) of maps S : C,ν [, T ] C,ν1 [, T ] that satisfy S[θ ](t) = θ 1 t [, T ], where θ i are the zerofnctions in C,νi [, T ]; i = 1,. The zero element Θ on S is simply the element that maps all ȳ C,ν [, T ] to θ 1. On S, we can define the norm: S S = sp ȳ 1,ȳ C,ν [,T ] ȳ 1 ȳ d 1 (S[ȳ 1 ], S[ȳ ]). (16) d (ȳ 1, ȳ ) Convergence of approximate inverse schemes can be discssed sing this norm. Definition 3.1: Let Γ be defined by (14). A reglarization strategy for Γ is a family of operators sch that: R ɛ [, ψ 1 ] : C [, T ] C,ν 1 Ī [, T ], ɛ >, 1) ȳ C [, T ], ) lim Γ R ɛ [ȳ, ψ 1 ] = ȳ; (17) ɛ lim d 1(R ɛ [ȳ, ψ 1 ], Γ 1 [ȳ, ψ 1 ]) =, (18) ɛ niformly on bonded sets of C,ν [, T ]. n other words, one reqires point-wise convergence for ȳ C [, T ] and weak-star convergence for ȳ C,ν [, T ]. Obviosly, R ɛ with domain restricted to fnctions in C,ν [, T ] is in S. The following elementary lemmas hold for the family {R ɛ }. Lemma 3.1: f R ɛ Γ 1 S, as ɛ, then lim d 1(R ɛ [ȳ, ψ 1 ], Γ 1 [ȳ, ψ 1 ]) = niformly on bonded ɛ sets of C,ν [, T ]. Proof. Consider the bonded set M = {ȳ d (ȳ, ) M}. Now: d 1 (R ɛ [ȳ, ψ 1 ], Γ 1 [ȳ, ψ 1 ]) d 1 ( (Rɛ Γ 1 )[ȳ, ψ 1 ], ) (by the translation invariance of d 1 ) R ɛ Γ 1 S d (ȳ, ) (by the definition of S ) M R ɛ Γ 1 S. So given an ɛ >, there exists an ɛ > sch that: if < ɛ ɛ then d 1 (R ɛ [ȳ, ψ 1 ], Γ 1 [ȳ, ψ 1 ]) < ɛ for all y M. This lemma shows that (18) is weaker than normconvergence. Since Γ 1 : C,ν [, T ] C,ν 1 [, T ] is weakstar continos, one wold like the approximating family to have a similar property. The next lemma stdies the weak-star continity properties of the family {R ɛ }. Lemma 3.: Let Γ 1 S be bonded and {R ɛ } be a reglarization strategy for Γ. Then given ɛ > and a bonded set M, there exists an ɛ > and δ > sch that: if < ɛ ɛ; ȳ 1, ȳ M; and d (ȳ 1, ȳ ) < δ, then d 1 (R ɛ [ȳ 1, ψ 1 ], R ɛ [ȳ, ψ 1 ]) < ɛ. Proof. Let M = {ȳ d (ȳ, ) M}. Then given ɛ >, there exists ɛ > sch that for all < ɛ ɛ, we have d 1 (R ɛ [ȳ, ψ 1 ], Γ 1 [ȳ, ψ 1 ]) < ɛ 3 for all ȳ M. Therefore, for ȳ 1, ȳ M, d 1 (R ɛ [ȳ 1, ψ 1 ], R ɛ [ȳ, ψ 1 ]) d 1 (R ɛ [ȳ 1, ψ 1 ], Γ 1 [ȳ 1, ψ 1 ]) +d 1 ( Γ 1 [ȳ 1, ψ 1 ], Γ 1 [ȳ, ψ 1 ]) +d 1 (R ɛ [ȳ, ψ 1 ], Γ 1 [ȳ, ψ 1 ]) < ɛ 3 + d 1( Γ 1 [ȳ 1, ψ 1 ], Γ 1 [ȳ, ψ 1 ]) ɛ 3 + Γ 1 S d (ȳ 1, ȳ ) < ɛ, for d (ȳ 1, ȳ ) < δ, ɛ where δ > is chosen as δ = 3 Γ 1 S. This lemma shows that verifying the bondedness Γ 1 is sfficient to ensre weak-star continity-like properties of the reglarization strategy. t also shows that one shold not try to prove the weak-star continity of R ɛ for any fixed ɛ >, bt rather consider the family {R ɛ } as a whole. V. FXED-PONT TERATON-BASED APPROXMATE NVERSON n this section, an approximate inversion algorithm is proposed based on sccessive iteration. The point-wise convergence condition for a reglarization strategy (17) is proved nder the same conditions on the density fnction as in Theorems.1 and.. The second condition (18) is mch more difficlt to prove, and we will consider it in ftre research. First consider the case that the desired otpt fnction is monotone. Let C m +,[, T ] denote the space of nondecreasing, continos fnctions on [, T ] taking vales in, and C,1 m +, [, T ] denote those fnctions in C m +,[, T ] that are Lipschitz continos. We consider the eqation Γ[, ψ 1 ] = y where ψ 1 Ψ and y C m +,[, T ] (and y C,1 m +,[, T ] ) in Proposition 4.1. Analogos reslts are tre if C m,([, T ]) and C,1 m,([, T ]) (the space of nonincreasing fnctions) are considered. Proposition 4.1: Assme that the Preisach density fnction ω(r, s) has compact spport; is integrable; is non-negative; and ω(r, s) Cr ξ for almost every (r, s) R ɛ = [, ɛ] [ min ɛ, max + ɛ], where C >, ξ, and ɛ >. Let k denote the Lipschitz constant for Γ. Let ψ 1 Ψ with the corresponding otpt y. For y C m +,[, T ] with y() = y,

7 7 consider the following algorithm: { (n+1) = (n) + y Γ[(n),ψ 1 ] k, n () ψ 1 (). (19) Then: 1) For any n, (n) C m +,[, T ]; and if y C,1 m +, [, T ], (n) C,1 m +,[, T ]; ) As n, (n) converges pointwise to C m +,[, T ] with Γ[, ψ 1 ] = y; 3) For ɛ >, let N ɛ be the smallest integer satisfying N ɛ. Then k ( max min ) ɛ Γ( (Nɛ), ψ 1 ) y ɛ; 4) As n, we have (n) niformly on [, T ]. Proof. 1. Under the hypothesis on the density fnction, it is clear from Theorem.1 that Γ : C [, T ] C [, T ] is Lipschitz continos. We will first show (n) C m +,[, T ], n. Then we will show that (n) is Lipschitz continos provided y C,1 m +,[, T ]. Clearly (n) C [, T ], n. We se indction to show (n) C m +,[, T ]. Since () is a constant fnction, it is non-decreasing. Now sppose that for some n, (n) is non-decreasing. This, together with the Lipschitz continity of Γ, implies, for t 1 t T, Γ[ (n), ψ 1 ](t ) Γ[ (n), ψ 1 ](t 1 ) k ( (n) (t ) (n) (t 1 )). () Using (19), we have (n+1) (t ) (n+1) (t 1 ) = y(t ) y(t 1 ) k + (n) (t ) (n) (t 1 ) Γ[(n), ψ 1 ](t ) Γ[ (n), ψ 1 ](t 1 ) k y(t ) y(t 1 ) k (by ()), (since y C m +,[, T ]) and therefore (n+1) is non-decreasing. Next we show that (n) is Lipschitz continos for every n, if y C,1 m +, [, T ], again by indction. Note that () is Lipschitz continos and Γ : C,1 [, T ] C,1 [, T ] by Theorem.1. Hence Γ[ (), ψ 1 ] is Lipschitz continos, and by (19) (1) is Lipschitz continos. Frthermore, if we assme (n) to be Lipschitz continos, the same argments imply that (n+1) is Lipschitz continos. Ths (n) is Lipschitz continos for every n, by indction.. Consider the seqence { (n) }. As y y = Γ[ (), ψ 1 ](t), we have (1) (). n the preceding, the ineqality f g is said to be tre, if and only if f(t) g(t) for all t [, T ]. Sppose (n) (n 1) for some n 1. From (19), (n+1) = (n) + y Γ[(n), ψ 1 ] k, (1) (n) = (n 1) + y Γ[(n 1), ψ 1 ] k. () The Lipschitz continity of the operator Γ[, ψ 1 ] implies: Γ[ (n), ψ 1 ] Γ[ (n 1), ψ 1 ] k ( (n) (n 1) ). (3) Sbtracting () from (1), and sing (3), we get (n+1) (n). Note that (n) (t) > (n 1) (t) if and only if y(t) > Γ[ (n 1), ψ 1 ](t) by (19). For each t [, T ], as { (n) (t)} is a monotone increasing seqence bonded by max, the seqence (n) (t) (t) as n. Hence { (n) } converges pointwise to some. By the continity of Γ[, ψ 1 ], the seqence {Γ[ (n), ψ 1 ]} Γ[, ψ 1 ]. By (19), = + y Γ[,ψ 1] k which implies Γ[, ψ 1 ] = y. Now we have C [, T ] de to the condition on the density fnction and tem 3 of Theorem.1, and C m +,[, T ] becase each (n) is monotone and the set C m +,[, T ] is a closed sbspace of C [, T ]. 3. f for some t [, T ], y(t) Γ[ (n), ψ 1 ](t) > ɛ, then (n+1) (t) (n) (t) > ɛ k. Since y(t) Γ[ (n), ψ 1 ](t) is non-increasing with n, and (n) (t) ψ 1 () is bonded by max min, one concldes that after N ɛ iterations, y(t) Γ[ (n), ψ 1 ](t) ɛ for every t. 4. By Lemma.1 and the assmption on ω(r, s), we have χ (x) Kx ξ+ for x ɛ, for some K >. Hence y(t) Γ[ (n), ψ 1 ](t) K (n) (t) (t) ξ+. (4) From item 3 above, y Γ[ (n), ψ 1 ] as n. Eq. (4) then implies the niform convergence of { (n) } to. Based on Proposition 4.1, the following algorithm (see illstration in Fig. 3) can be sed to generate an approximate inverse ɛ C,1 [, T ] for y C ([, T ]) sch that Γ[ ɛ, ψ 1 ] y ɛ. Fixed Point Algorithm: Step 1. Pick y C,1 pm, [, T ] sch that y y ɛ = ɛ, and the variation in each monotone section of y is at least ɛ. Let = T < T 1 < T 3 < < T N 1 < T N+1 = T be the standard partition for y. We will shortly define the times T, T 4,, T N. Step. On [T, T 1 ], rn the algorithm (19) (at most N ɛ times) ntil y Γ[ (n), ψ 1 ] ɛ. Set ɛ (t) = (n) (t) for t [T, T 1 ]. Step 3. Let T T 1 be the smallest time instant sch that y (T ) = Γ[ ɛ, ψ 1 ](T 1 ). T is well defined considering Step 1. Set ɛ (t) ɛ (T 1 ) on (T 1, T ); Step 4. Rn (19) N ɛ times on [T, T 3 ] with () ɛ (T 1 ), which defines ɛ on [T, T 3 ]; Step 5. Contine Steps 3 and 4 ntil ɛ is defined p to the final time T. As in Section, for t [, T ], define ȳ(t) = y(t) y() and ū ɛ (t) = ɛ (t) (). (5) Define: R ɛ [, ψ 1 ] : C [, T ] C,1 Ī [, T ] ȳ ū ɛ, (6)

8 8 where ɛ is the reslt of the Fixed Point algorithm. Let y ɛ = Γ[ ɛ, ψ 1 ] and Γ[, ψ 1 ], Γ 1 [, ψ 1 ] be defined as in (14) and (15). Fig. 3. y ε y T 1 T T 3 ε envelope of y llstration of the fixed-point iteration-based inverse algorithm. One can establish the following reglarization-type properties for the scheme R ɛ : Theorem 4.1: Assme that the density fnction of the Preisach operator Γ satisfies the conditions of Proposition 4.1. Let ɛ >. Then: 1) ȳ C [, T ], ) and φ L 1 [, T ], ε ε lim Γ R ɛ [ȳ, ψ 1 ] = ȳ; (7) ɛ lim <R ɛ[ȳ, ψ 1 ] Γ 1 [ȳ, ψ 1 ], φ>, (8) ɛ niformly for ȳ on bonded sets of C,1 [, T ]. Proof. Given ȳ C [, T ], choose ȳ C pm, [, T ] according to Step 1. By Proposition 4.1, on the time-intervals [T n, T n+1 ] where n =, N, we have: ȳ ɛ (t) ȳ (t) = y ɛ (t) y (t) ɛ = ɛ. On the time intervals [T n+1, T n+ ] where n =, N 1, ɛ is simply a constant, and by Step 3 of the Fixed Point algorithm: ȳ ɛ (t) ȳ (t) = y ɛ (t) y (t) ɛ. Ths: ȳ ɛ ȳ = y ɛ y ɛ, and then: ȳ ɛ ȳ = y ɛ y y ɛ y + y y ɛ. Hence the scheme {R ɛ } satisfies the first condition (17) for a reglarization strategy. Next, let y C,1 [, T ], and ȳ be given by (5). Let y,1 M. Pick y C,1 pm, [, T ] sch that y y,1 δ. Then y,1 y y,1 + y,1 M + δ. The finest partition needed for all sch fnctions y is one with intervals ɛ of length M+δ. Therefore the pper bond on the nmber of iterations needed for convergence (to within ɛ in the spnorm) is (M+δ) T N ɛ ɛ. Ths we have niform convergence on bonded sets in C,1 [, T ]. By Theorem., Γ 1 : C,1 [, T ] C,ν [, T ] where Ī ν = 1 ξ+. This implies that ū = Γ 1 [ȳ, ψ 1 ] belongs in C,ν [, T ], even thogh ū Ī ɛ C,1 [, T ]. As C,ν [, T ] Ī Ī L Ī [, T ], we have L1 [, T ] C,ν [, T ], where C,ν Ī Ī,w Ī,w [, T ] denotes the weak-star dal of C,ν [, T ]. Let φ be an element Ī t of L 1 Ī [, T ]. Since ū ɛ ū as ɛ, we have: <ū ɛ ū, φ> = T (ū ɛ (t) ū (t)) φ(t) dt ū ɛ ū φ 1 as ɛ. (9) The above reslt falls slightly short of showing that R ɛ is a reglarization scheme. n order to show R ɛ is a reglarization scheme, (9) mst hold for all φ C,ν [, T ]. This is a Ī,w qestion that needs to be frther investigated in the ftre. V. DSCRETZATON-BASED APPROXMATE NVERSON n this section a discretization-based approximate inversion scheme is discssed. The discretization reslts in a discretized Preisach operator, an approximate inverse of which can be efficiently constrcted by the so called closest-match algorithm. Experimental reslts on trajectory tracking of a magnetostrictive actator based on this algorithm will also be presented. A. The Closest-Match Algorithm There are two discretization steps involved, discretization of the inpt range = [ min, max ] and discretization of the time interval [, T ]. Discretize [ min, max ] niformly into L + 1 levels and denote the reslting set of discrete inpt vales as U L = {û i, i = 1,, L + 1}, where û i = min + (i 1), and = max min L. As a conseqence of inpt discretization, the Preisach plane is discretized into cells. When restricted to inpts taking vales in U L, the Preisach operator becomes a weighted combination of a finite nmber of hysterons, where the weight of each hysteron eqals the integral of the original Preisach density fnction over the corresponding grid (see Fig. 4 for illstration). Denote this discretized Preisach operator as Γ L and its set of memory crves as Ψ L. Note that an element of Ψ L consists of L vertical or horizontal segments, each with length. ^ 1 ^ α ^ 5 ^ 4 ^ 3 ^ ^ 1 ^ 4 ^ 5 Fig. 4. llstration of the discretization scheme (L = 4), where weighting masses are located at the centers of cells. Discretization of time is performed similarly. Given N 1, the time interval [, T ] is niformly divided into N sbintervals with consective end-points denoted as {t j } N j=, where t j = j t with t = T N. β

9 9 Let D N denote the set of seqences of length N + 1 taking vales in, i.e., s D N, s[j], for j =, 1,, N. For the discretized Preisach operator Γ L, an approximate inversion problem can be formlated as follows: given ψ 1 Ψ L and s y D N, find s DU N L (set of seqences taking vales in U L ), sch that Γ L [s, ψ 1 ] s y = min Γ L (s, ψ 1 ) s y. (3) s DU N L Since Γ L : DU N L D N is not onto, only an approximate inverse s is soght in (3). Dynamic programming can be sed to solve the problem (3) [4]. However, as N and L get large, this approach becomes prohibitive in terms of comptational and storage costs. A sb-optimal scheme is to seqentially generate an inpt seqence s of length N so that at time j, Γ L [s, ψ 1 ][j] s y [j] is minimized. This decomposes the original (approximate) inverse problem of length N + 1 into N + 1 sccessive problems of length 1. To be precise, at each time instant, given the crrent memory crve ψ () (from which the crrent inpt () and otpt y () can be derived) and a desired otpt vale ŷ, find # U L, sch that Γ L [ #, ψ () ] ŷ = min U L Γ L [, ψ () ] ŷ. (31) Also the reslting memory crve ψ # shold be retrned for se at the next time instant. The following algorithm can be sed to efficiently solve the problem (31) (see Figre 5 for an illstration). As the fixed-point algorithm, it is also based on the piecewise strictly increasing property of the Preisach operator, and it flly tilizes the discrete strctre of the problem. Consider the case y () ŷ (the other case y () > ŷ is dealt with analogosly). ntitively, in this algorithm we keep increasing the inpt by one level in each iteration, ntil either a) the inpt reaches the maximm û L+1, or b) y (n) exceeds ŷ. For case a), take # = û L+1 ; for case b), take # to be (n) or (n 1) whichever yields smaller otpt error. n both cases, # so obtained solves (31). Fig. 5. t 1 t tj npt trajectory ( n) (1) () t α () ψ () ( n) ψ (1) ψ (1) Memory crve α = β ( n) β y t t 1 tj Otpt trajectory y y ( n) (1) () y llstration of the convergence of the Closest-Match Algorithm. Closest-Match Algorithm. Step. Set n=. Step 1. f (n) = û L+1, let # = (n), ψ # = ψ (n), go to Step 4; otherwise (n+1) = (n) +, ψ = ψ (n) [backp the memory crve], n = n + 1, go to Step ; Step. Evalate y (n) = Γ[ (n), ψ (n 1) ], and (at the same time) pdate the memory crve to ψ (n). Compare y (n) with ŷ: if y (n) = ŷ, let # = (n), ψ # = ψ (n), go to Step 4; if y (n) < ŷ, go to Step 1; otherwise go to Step 3; t Step 3. f y (n) ŷ y (n 1) ŷ, let # = (n), ψ # = ψ (n), go to Step 4; otherwise # = (n 1), ψ # = ψ [restore the memory crve], go to Step 4; Step 4. Exit. t s not hard to see the above algorithm yields # in at most L iterations. B. Approximate nversion Based on the Closest-Match Algorithm An algorithm to approximately solve Γ[, ψ 1 ] = y is proposed as follows: pick N 1, L 1. Step 1. Constrct ψ 1 Ψ L from ψ 1 Ψ based on the inpt discretization rles (i.e., approximating the given ψ 1 Ψ by an element in Ψ L ); Step. For y C ([, T ]), constrct s y D N via s y [j] = y(j t ); Step 3. Obtain s DU N L by applying the Closest-Match algorithm described above; Step 4. Constrct N,L C,1 [, T ] sing linear splines based on s, i.e., N,L (t) = τs [j] + (1 τ)s [j + 1], if t = (j + τ) t, j =,, N 1, and τ 1. Analogos to (5) and (6), denote ū N,L (t) = N,L (t) N,L (), ȳ(t) = y(t) y(), and define R N,L [, ψ 1 ] : C [, T ] C,1 Ī [, T ] ȳ ū N,L. (3) Similar to Proposition 4.1, for y C m +,[, T ], we have the following convergence reslts for the closest-match algorithmbased inversion scheme: Proposition 5.1: Assme that the density fnction of the Preisach operator Γ satisfies the conditions of Proposition 4.1. Let k denote the Lipschitz constant for Γ. Then for any ψ 1 Ψ, y C m+, [, T ], 1) For any N, L 1, N,L C,1 m +,[, T ]; ) As L, N, Γ( N,L, ψ 1 ) y, (33) 3) As N, L, we have N,L niformly on [, T ], where = Γ 1 [y, ψ 1 ], and C m +,[, T ]. Proof. 1. As N,L is constrcted sing linear splines, it is clear that N,L C,1 [, T ]. As y is monotone nondecreasing, N,L is also monotone and non-decreasing by the non-negativity condition on the density fnction.. Note that by the constrction of Γ L, it is also Lipschitz continos with the same Lipschitz constant k for Γ. Hence if the inpt at any instant t is increased (or decreased) by, the otpt of Γ L at time t is increased (or decreased) by no more than k. From the Closest-Match algorithm, s y [j] s y [j] < k, j =,, N, (34) where s y = Γ L [s, ψ 1 ]. By the constrction of ψ 1, it is within the -neighborhood of ψ 1 (see Visintin[14], page

10 1 113, for the definition of neighborhood of a memory crve), and hence by the Lipschitz continity of Γ, Γ[ N,L, ψ 1 ] Γ[ N,L, ψ 1 ] k. (35) Noting Γ L [s, ψ 1 ][j] = Γ[ N,L, ψ 1 ](t j ), we get from (34) and (35), for j =,, N, Γ[ N,L, ψ 1 ](t j ) s y [j] Γ[ N,L, ψ 1 ](t j ) Γ[ N,L, ψ 1 ](t j ) + Γ[ N,L, ψ 1 ](t j ) s y [j] = Γ[ N,L, ψ 1 ](t j ) Γ[ N,L, ψ 1 ](t j ) + s y [j] s y [j] k + k = k. Since both y and Γ[ N,L, ψ 1 ] are monotone, nondecreasing on [t j, t j+1 ], for t [t j, t j+1 ]; j =,, N, if Γ[ N,L, ψ 1 ](t) y(t), one has Γ[ N,L, ψ 1 ](t) y(t) Γ[ N,L, ψ 1 ](t j ) y(t j+1 ) Γ[ N,L, ψ 1 ](t j ) y(t j ) + y(t j ) y(t j+1 ) k + ρ y ( t ), (36) where ρ y ( ) is the continity modls of y. Same ineqality can be obtained if Γ[ N,L, ψ 1 ](t) y(t). Therefore, for each t : lim N,L Γ[ N,L, ψ 1 ](t) y(t) =. As y C [, T ], y is niformly continos on [, T ]. Ths the right hand side of (36) is independent of t. Therefore, Γ[ N,L, ψ 1 ] y k + ρ y ( t ). (37) Eq. (33) follows, since ρ y ( t )) as t. 3. Let = Γ 1 [y, ψ 1 ]. Then C [, T ] as Γ 1 : C [, T ] C [, T ]. The fnction is also monotone, by the non-negativity condition on the density fnction and by y C m+, [, T ]. From tem 3 of Theorem.1, Γ 1 : C [, T ] C [, T ] is continos, and hence we get from (33) that N,L as N, L. Again let Γ[, ψ 1 ] and Γ 1 [, ψ 1 ] be defined by (14) and (15), and ȳ defined by (5). The following theorem shows a continity property of R, t similar to that for the Fixed Point iteration method. Theorem 5.1: Assme that the density fnction of the Preisach operator Γ satisfies the conditions of Proposition 4.1. Then: 1) ȳ C [, T ], ) φ L 1 [, T ], lim Γ R, t [ȳ, ψ 1 ] = ȳ; (38) N,L lim <R, t [ȳ, ψ 1 ] Γ 1 [ȳ, ψ 1 ], φ>=, (39) N,L niformly for ȳ on bonded sets of C,1 [, T ]. Proof. The first item follows by simply repeating the proof of Proposition 5.1. Other than the monotonicity of (defined to be Γ 1 [ȳ, ψ 1 ]), the rest of the proof applies to this case. The proof of the second statement is exactly analogos to that of Theorem 4.1, and tilizes the convergence in the norm of the fnctions ū N,L to ū. C. Experimental Reslts on Tracking Control The above inversion algorithm is applied to tracking control of a magnetostrictive actator (made of Terfenol-D). Magnetostriction is the phenomenon of strong copling between magnetic properties and mechanical properties of some ferromagnetic materials: strains are generated in response to an applied magnetic field, while conversely, mechanical stresses in the materials prodce measrable changes in magnetization. By varying the crrent in the coil srronding the Terfenol-D rod, one can vary the magnetic field inside the rod and ths control the displacement otpt of the actator. The actator sed in this stdy is an AA-5H series Terfenol-D actator manfactred by Etrema. The displacement of the actator is measred with a LVDT sensor (Schaevitz 5MHR). Fig. 6 shows the hysteretic relationship between the crrent inpt and the displacement otpt. Fig. 6. Displacement (µ m) Crrent inpt (A) A typical hysteresis crve of the Terfenol-D actator. When the inpt crrent is qasi-static, the hysteretic behavior of the magnetostrictive actator can be modeled as [17]: H = c M = Γ[H, ψ 1 ], (4) z = l rodλ s M M s where H and M are the magnetic field and the blk magnetization along the rod direction, respectively, is the crrent inpt, z is the displacement otpt, c is the coil factor, l rod is the rod length, λ s is the satration magnetostriction, and M s is the satration magnetization. n (4) the magnetostrictive hysteresis is essentially captred by the ferromagnetic hysteresis between M and H, which is modeled by the Preisach operator Γ. For a discretization level of L, the weighting masses for Γ L can be identified throgh a constrained least sqares algorithm [7], [5]. Here L has been chosen to be 5, and t = T N = 1ms. Figre 7 shows the identified density fnction. As can be observed, the density fnction is non-zero along the β = α line, which is the same as the line r = in (r, s) coordinates (recall that the variables (α, β) and (r, s) are related according to r = α β and s = α+β ). Therefore, the key condition of Theorem. and Proposition 4.1 is satisfied,

11 11 and both Theorems 4.1 and 5.1 can be applied to this actator to find an approximate right-inverse. 3 µ(β,α) α 1 Fig. 7. The identified Preisach density fnction for a commercial magnetostrictive actator. An open loop tracking experiment has been condcted based on the Closest-Match inversion algorithm. Fig. 8 shows the comparison of the desired trajectory and the actal trajectory, together with the tracking error. The desired trajectory is chosen to vary in both amplitde freqency. The tracking error is small (nder 3 µm), which shows the inversion algorithm is effective. An extension of this approach to the closed-loop l 1 control of the magnetostrictive actator over a Hz range can be fond in Tan and Baras [6]. 1 V. CONCLUSON The Preisach operator is a poplar tool for hysteresis modeling in varios smart materials. nversion of the Preisach operator plays a fndamental role in effective control of these materials. This paper dealt with approximately inverting the Preisach operator, in sch a way that the reslting fnctions Displacement ( µ m) Tracking error ( µ m) Desired trajectory Measred trajectory Time (sec.) Time (sec.) Fig. 8. Trajectory tracking of a magnetostrictive actator based on the approximate inversion. 3 β 4 5 have some reglarity properties. We first presented a weak and easily verifiable condition that garantees the weak-star continity of the inverse operator. Motivated by this reslt, the notion of a reglarization strategy was proposed for the inversion problem. n practice, exact inversion of the Preisach operator is generally not possible de to nmerical limitations. Two inversion schemes were developed in this paper, both of which flly tilized the piecewise strictly increasing property of the Preisach operator (nder some mild conditions on the density fnction). Both algorithms yield Lipschitz continos inpts. They were shown to satisfy the first condition for a reglarization strategy. Both schemes also enjoy a continity property that is similar to bt weaker than that of a reglarization strategy. An interesting direction for ftre work is to investigate whether the two schemes satisfy the second condition (Eq. (18)) for a reglarization strategy. ACKNOWLEDGEMENT The first athor wold like to acknowledge discssions with Prof. V. Shbov on the constrction of the weak-star topology on spaces of Hölder continos fnctions. We also wold like to sincerely thank the anonymos referees for providing insightfl comments that helped s improve on the original manscript. REFERENCES [1] D.C. iles and D.L. Atherton, Theory of ferromagnetic hysteresis,. Magn. Magn. Mater., vol. 61, pp. 48 6, [] R. Venkataraman and P.S. Krishnaprasad., A model for a thin magnetostrictive actator, in Proceedings of the Conference on nformation Sciences and Systems, Princeton University, March 18-, 1998, Also pblished as a technical report, TR 98-37, nstitte for Systems Research, University of Maryland at College Park. [3] R. Venkataraman, Modeling and Adaptive Control of Magnetostrictive Actators, Ph. D. thesis, University of Maryland, College Park., 1999, Available at Year 1999, Ph.D. thesis section. [4].D. Mayergoyz, Mathematical models of Hysteresis, Springer, [5] D. Hghes and.t. Wen, Preisach modeling and compensation for smart material hysteresis, in Active Materials and Smart Strctres, 1994, vol. 47, pp [6] A.A. Adly,.D. Mayergoyz, and A. Bergvist, Preisach modeling of magnetostrictive hysteresis, ornal of Applied Physics, vol. 69, no. 8, pp , [7] R. V. yer and M. E. Shirley, Hysteresis parameter identification with limited experimental data, EEE Transactions on Magnetics, vol. 4, no. 5, pp , Sep 4. [8] R.B. Gorbet, D.W.L. Wang, and K.A. Morris, Preisach model identification of a two-wire SMA actator, in Proceedings of EEE nnernational Conference on Robotics and Atomation, 1998, pp [9] A. Ktena, D.. Fotiadis, P. D. Spanos, and C. V. Massalas, A Preisach model identification procedre and simlation of hysteresis in ferromagnets and shape-memory alloys, Physica B,, no. 36, pp. 84 9, 1. [1] G. Tao and P.V. Kokotović, Adaptive control of plants with nkown hystereses, EEE. Trans. Atomat. Contr., vol. 4, no., pp. 1, [11] R.C. Smith, nverse compensation for hysteresis magnetostrictive transdcers, Tech. Rep. CRSC-TR98-36, North Carolina State University, 1998, Technical report of the Comptational for Research in Scientific Comptation. [1] W.S. Galinaitis and R.C. Rogers, Control of a hysteretic actator sing inverse hysteresis compensation, in Mathematics and Control in Smart Strctres, V.V. Varadhan, Ed., 1998, vol. 333, pp