Recursive Estimation of the Preisach Density function for a Smart Actuator

Transcription

1 Recursive Estimation of the Preisach Density function for a Smart Actuator Ram V. Iyer Deartment of Mathematics and Statistics, Texas Tech University, Lubbock, TX ABSTRACT The Preisach oerator and its variants have been successfully used in the modeling of a hysical system with hysteresis. In an alication, one has to determine the density function describing the Preisach oerator from measurements. In our earlier work, we described a regularization method to obtain an aroximation to the density function with limited measurements. In this aer, we describe methods for recursively comuting the aroximate density function. These methods can be imlemented in real-time controllers for smart actuators. Keywords: Hysteresis, Preisach Oerator, Recursive Estimation, Density function Identification, Magnetostriction, Piezoelectric materials, Smart materials, EAPs. 1. INTRODUCTION The Preisach oerator is a mathematical tool that has been used to model the henomena of hysteresis occurring in ferromagnetism, 1 magnetostriction, 2 shae memory alloys 3 and iezoelectricity 4 for many years. In each henomenon, the constitutive relation between conjugate variables, for examle, magnetic field intensity H and magnetization M in ferromagnetism or electric field intensity E and olarization P in ferroelectricity, is described by a hysteresis oerator, usually the Preisach oerator or a variant. Part of what is needed to describe the Preisach oerator for a articular henomenon is a non-negative and integrable, density function defined on a subset of IR 2 called the Preisach domain. Several researchers starting with Mayergoyz 1 have studied the roblem of identifying the density function. A synosis of the various aroaches can be found in our earlier work. 5, 6 In our earlier work, 6 the identification roblem was systematically studied and it was shown to be an illosed roblem. This was shown using the fact that the Preisach oerator with the density function as the indeendent variable, is a comact oerator and, hence, has an unbounded inverse. The unboundedness of the inverse oerator indicates a need for regularization, so that one obtains a sequence of bounded aroximations. A non-arametric, SVD truncation-based regularization was roosed and numerical results were resented for magnetostrictive actuators and electro-active olymers. 6 Collocation in time and discretization of the inut signal levels were used, along with truncation of singular directions corresonding to very small singular values. It was found that finer discretizations lead to much larger comutation times 6 (which is to be exected). In this aer, we roose iterative methods with a view to (a) reduce comutation times, (b) roduce a sequence of aroximate density functions that can be used in a real-time controller. The second item is more imortant from an imlementation oint of view, and slow changes in the density function can be accommodated by eriodic resetting or using short recurrence sequences. As far as we are aware, adative identification of the Preisach density function and its use in the inverse control of hysteresis has only been studied by Tan and Baras. 7 They roosed a recursive least squares algorithm to identify a discretized density function. They accommodated the non-negativity condition for the density function by a simle rojection at each iteration. It is well-known that a simle rojection into the feasible region does not minimize a convex cost function. Figure 1 shows the level surfaces of a quadratic cost function in two variables - three cases are considered, with the unconstrained minimum in the second, third and fourth quadrants. If the minimum has to satisfy a non-negativity constraint, then it is clear that in each case, the rojection of the unconstrained minimum x into the first quadrant does not yield the desired solution x min. Furthermore, the imlementation of the algorithm requires the inversion of a matrix, which leads to the ersistency of excitation condition in Tan and Baras. The fulfilment of this condition would require an enormous amount of data for fine mesh selections. Our method in this aer is to numerically obtain the best aroximation of the density function

2 at each iteration, even in the absence of ersistency of excitation - or in other words, when only limited data is available. This is achieved by alying a rimal-dual interior oint method after a artial SVD comutation. As the SVD comutation is indeendent of the interior oint method, convergence follows from standard results for each of these methods. (a) Case 1: Unconstrained minimum in second quadrant (b) Case 2: Unconstrained minimum in third quadrant (c) Case 3: Unconstrained minimum in fourth quadrant Figure 1. Examles of simle rojection after unconstrained minimization. 2. PRELIMINARIES In this section, we briefly recall the Preisach oerator and the need for regularization. A detailed treatment on the Preisach oerator can be found in. 1, 8, 9 For a air of thresholds (, ) with, consider a relay R, [, ] (called a Preisach hysteron), as illustrated in Fig. 2. For u C[, T ] 1 if u(t) < v, (t) = 1 if u(t) >, v, (t ) if u(t) where v( ) = ζ and t = lim t ɛ. ɛ>,ɛ R, [u] f +1 d e u a b 1 c Figure 2. Inut-outut relationshi for the relay. To construct the Preisach oerator from the elementary hysterons when the inut is u( ), denote v, ( ) = R, [u]( ). The Preisach oerator s inut is u( ), and the outut is given by 1 :

3 ȳ(t) = µ(, )R, [u](t) dd (1) where µ(, ) L 2 (K) where K is a comact region in the (, ) lane with and suort(µ) = K. Thus for a given µ, we can define a Preisach oerator to be a ma Γ µ : C m [, T ] C m [, T ], where C m [, T ] denotes the sace of iecewise monotone continuous functions on [, T ]. For comatibility with exerimental evidence, we restrict µ(, ) to be a non-negative function. During the identification exeriments, the fixed inut u C m [, T ] usually only affects a ortion of the set K in the Preisach lane. Without loss of generality, we can restrict attention to this ortion. In the following, the set K u is the subset of K affected by the inut u. Define the set of density functions: K u := {µ L 2 (K u ) µ }. (2) Due to the assumtion on K u, the outut at time given by Γ[u]() is the same for all density functions in K u. An imortant class of Preisach oerators from the alications oint of view, are the iecewise, strictly increasing (PSI) Preisach oerators. Definition 2.1. A Preisach oerator is said to be iecewise strictly increasing (PSI) if (Γ µ [u](t ) Γ µ [u]())(u(t ) u()) > for a monotone inut u C[, T ] with u() u(t ).. Under the mild condition that the density function is continuous, and greater than zero along the diagonal =, it is easy to show that the Preisach oerator is PSI. Clearly, the definition of PSI is for a fixed density function and varying monotone increasing inut functions u. However, in the identification roblem the inut function is fixed while the density function is the indeendent variable. For a non-constant u C m [, T ], let = T < T 1 < < T N = T be the standard artition (see Brokate and Srekels 9 ). As u is non-constant, u(t i 1 ) u(t i ) for i = 1 N. Denote i = [T i 1, T i ] for i = 1 N. Define the set of density functions: K u,p SI := {} { µ L 2 (K) µ ; and (Γ[u](t 1 ) Γ[u](t 2 ))(u(t 1 ) u(t 2 )) > for t 1, t 2 i, t 1 t 2 ; i = 1,, N.} (3) The idea is that the density function for a PSI Preisach oerator (that can be identified) should come from this set. The sets K u and K u,p SI are clearly non-emty with K u,p SI K u. To further study their roerties, recall that a convex set C is a subset in a vector sace, such that if x 1, x 2 C, then θ x 1 + (1 θ) x 2 C for θ 1 (Luenberger 1 ). A cone with vertex at the origin is a set C in a vector sace with the roerty that if x C, then θ x C for all θ. A convex cone is defined as a set that is both convex and a cone. Closed convex sets are imortant from the oint of view of numerical methods for constrained otimization roblems. The following lemma was roved in Iyer and Shirley. 6 Lemma 2.2. The set K u is a closed convex cone with vertex at the origin, while the set K u,p SI is a convex cone with vertex at the origin that is not closed. Now, for all µ in K u, the initial outut w = Γ[u]() is fixed. For a given u( ) C m [, T ], define the oerator: Φ u : K u L 2 [, T ] µ(, ) y = Φ u µ( ) = Γ µ [u]( ) w (4) The oerator Φ u is a linear oerator between L 2 (K u ) and L 2 [, T ] which is not true for the Preisach oerator Γ! The key is that Φ u mas the function µ = to the function y =. The following imortant theorem concerning the identification roblem for a PSI Preisach oerator was roved in Iyer and Shirley. 6 Theorem 2.3. Let u be a non-constant function in C m [, T ] and Φ u : K u L 2 [, T ] be as defined in (4). Then, 1. {µ K u Φ u µ(t) = } = {};

4 2. Φ u : K u L 2 [, T ] is injective; 3. Range [Φ u ] L 2 [, T ]; 4. Suose that y Range[Φ u ] and Γ is PSI. Then there exists a unique µ K u,p SI such that Φ u µ = y. Remarks We conclude from the above theorem and Lemma 2.2 that: Even though µ K u,p SI for a PSI Preisach oerator, we have to carry out the identification over the set K u, as K u,p SI is not a closed convex set. The set K u is a closed convex cone, and as Φ u is a linear injective oerator on this set, it is well-suited from a numerical oint of view. However, as the closure of the range of Φ u is not all of L 2 [, T ], we need to construct a regularization scheme to counter-act noise. The following lemma roved in Iyer and Shirley 6 shows that the density identification roblem is an ill-osed one! Lemma 2.4. The oerator Φ u : L 2 (K u ) L 2 [, T ] is a comact linear oerator. Due to this lemma, a regularization scheme based on (i) the SVD of a finite dimensional aroximation of Φ u, and (ii) truncation of singular directions corresonding to small singular values was roosed in Iyer and Shirley, 6 and is described below. Let = t 1 < < t n = T be a discretization of time, so that we have : } (Φ u µ)(t j ) = y(t j ), K R j = 1,, n. (5),[u](t j ) µ(, ) dd = y(t j ), where µ(, ) is unknown, while u( ) and y(t j ) are known. The time instants t 1,, t n are known as collocation oints. Once a set of collocation oints have been determined, one can discretize the inut values so that a uniform grid is established in the region K u in the Preisach lane. Then the equation Φ u µ = y can be written as a linear equation linear equation Y = AX + ɛ, (6) as described in Shirley and Venkataraman. 5 Each element of X (excet the last) denotes the area under the density function for a articular grid element. The last element denotes the initial outut value (at time ). To account for noise, we estimate this value also. We need is a way to solve for X that will best fit the data, but at the same time kee x i, since X reresents the integral of a density function over a grid element. where We would like to minimize the function f(x) = 1 2 AX Y 2 (7) A : IR n IR m, X IR n, Y IR m, with the inequality constraint g(x) = X. Let rank(a) = q min {m, n}. If we erform a singular value decomosition on A T A, then we get: A T A = V SV T, (8) where S is an n n diagonal matrix with rank q < n, and V T V = I n n (see age 54, Bellman 11 ). The n singular values of A T A are the diagonal elements of S and be ordered as σ 1 σ 2 σ n. By the remarks following Picard s theorem, 12 we see that small singular values should be discarded as they contribute to an amlification of the noise in the solution. We erform the following stes:

5 1. Pick a tolerance value ɛ >, and set those singular values that are less than ɛ to. Call the resulting matrix that is obtained from S as S Remove the rows and columns of S 1 that are identically zero, and also remove the corresonding columns of U and V. By this rocedure, we obtain a q q diagonal matrix Ŝ, and a n q matrices ˆV that satisfies ˆV T ˆV = I q q. (9) The oerator norm of the matrix A T A ˆV Ŝ ˆV T can be seen to be: A T A ˆV Ŝ ˆV T = A T A V S 1 V T = V (S S 1 ) V T = S S 1 < ɛ. We seek solutions of the tye X = ˆV Z for the Problem (7). Using Equations (8) and (9), the cost function can be transformed to Z T ŜZ Y T A ˆV Z. Thus we form the constrained otimization roblem: minimize f(z) = 1 2 ZT ŜZ Y T A ˆV Z, (1) subject to g(z) = ˆV Z. (11) Once we have a minimizer Z to the above roblem, then the desired solution is given X = ˆV Z. For a commercial magnetostrictive actuator made by ETREMA Inc., the relation between the average Magnetic field Intensity H and average Magnetization M along the axis of the rod was modeled by a classical Preisach oerator in Shirley and Venkataraman 5 and Iyer and Shirley. 6 Figure 3 reroduces the density functions obtained by the above mentioned rocedure for different discretizations (2 Oe, 1 Oe, 6.25Oe) of the inut magnetic field. Figure 4 shows a lot of the singular values of the A matrix for each discretization level. The discretization level of 2 Oe leads to a matrix A of full rank, while for the other two discretization levels, the matrix A is rank-deficient. Figure 5 comares the density functions obtained for the discretization level of 1 Oe, using two different truncation levels for the singular values: σ 1 3 and σ 2. Unfortunately, using a truncation level of σ 4 (corresonding to 27 singular values) did not yield a good result, which imlies that the roblem is only mildly ill-osed. 3. ITERATIVE COMPUTATION OF THE PREISACH DENSITY FUNCTION In this aer, we consider a iterative version of the rocedure in Section 2. The roblem now is to minimize the function (at time t ) where f (X) = 1 2 A X Y 2 (12) A : IR n IR m, X IR n, Y IR m, with the inequality constraint g(x) = X. In the roblem (12), the row dimension of A and Y increases by one after samling of the inut and outut signals is erformed. We assume that rank(a ) min {m, n}. The solution rocedure outlined in Section 2 is comosed of two arts: (a) singular value decomosition of the matrix A T A and truncation of singular directions corresonding to small singular values; (b) solution of the constrained minimization roblem (11). To construct an iterative scheme, we need iterative methods to accomlish both arts. An iterative method for SVD is the double-bracket flow due to Brockett. 13 This method leads to the comutation of all the singular values of the matrix A, which is not necessary due to truncation. In light of

6 µ(,) 5 µ(,) (a) Discretization = 2 Oe (b) Discretization = 1 Oe µ(,) (c) Discretization = 6.25 Oe Figure 3. Identification exeriments for a commercial magnetostrictive actuator with different magnetic field discretizations the raid decay in the singular values (both for the full-rank and rank-deficient cases) as seen in Figure 4, we only need to carry out a artial SVD by comuting the significant singular values. This can be accomlished in several ways - one of them being the Lanczos iteration method, which seems to require fewer comutations for the same level of accuracy than the subsace-iteration method. 14 Other ossibilities for sequential SVD can be found in Strobach. 15 Note that ideally, one would like to comute only one iteration of the artial SVD algorithm er time-ste. Once the significant singular values and directions have been comuted (resulting in the matrices Ŝ and ˆV ), the next task is to minimize the cost function f(z) in (11). One way to accomlish this is to use a line-search Newton-Rahson algorithm due to Krishnarasad and Barakat. 16 Another method is to use a Primal-Dual 17, 18 Interior oint method. Ideally, one would like to aly only one iteration of (either) solution method at time t. At time t, the Kuhn-Tucker theorem yields the existence of a λ IR n, such that the air (Z, λ ) satisfy the necessary conditions (see Luenberger 1 ): Ŝ Z + ˆV T The solution to this set is found using Newton s method. Let (Z (i) λ ˆV T A T Y = (13) λ T ˆV Z = (14) λ (15), λ (i) ) be the values at the i-th iteration of

7 Discretization level = 2 Oe 35 3 Discretization level = 1 Oe σ 2 σ (a) Discretization= 2 Oe (b) Discretization = 1 Oe Discretization level = 6.25 Oe 3 σ (c) Discretization = 6.25 Oe Figure 4. Plot of singular values of the A matrix the Newton s method. Then, the values at the i + 1-th ste are found using the udate direction (δ Z, δ λ) (i) ) given by the solution to the equation: [ Ŝ The udate is: where (i) (λ (i) ) T ˆV (Z (i) ˆV T ) T ˆV T (Z, λ) (i+1) is chosen to ensure that λ (i+1) (i) = arg min ν ] = = (Z, λ) (i) [ Ŝ Z (i) + ˆV T (λ (i) λ (i) ˆV T ) T ˆV Z (i) A T Y ]. (16) + (i) ) (δ Z, δ λ) (i), (17). Obviously, it has to be chosen to be: λ (i) (k) + ν δ λ (i) (k) k = 1 n. In the above equation, λ (i) (k) denotes the k-th comonent of λ (i). Convergence analysis: We outline the convergence analysis for our method. Note that the SVD comutation is indeendent of the interior oint method, which simlifies the convergence analysis. Furthermore, the rank of the A matrix can only increase with, and that too only in finite stes. As rank(a ) is monotone increasing and bounded

8 Truncation level = 2 Number of singular values used = 33 4 µ(,) (a) Density function using 515 singular values (b) Density function using 33 singular values 8 x 15 8 x Truncation level = Truncation level = 2 Number of singular values used = 515 Number of singular values used = Result of Identification Result of Exeriment x 1 4 Result of identification Result of Exeriment x 1 4 (c) H versus M, using 515 singular values (d) H versus M, using 33 singular values Figure 5. Comarison of different SVD truncation schemes for inut discretization of 1 Oe. above by n, the artial SVD comutation is convergent by Theorem 4.1 of Vogel and Wade 14 assuming that A T A converges to a matrix Σ as. That the interior-oint method is convergent, is a standard result. 18 Numerical results will be addressed in a future ublication. 4. CONCLUSION In this aer, we examined the roblem of recursively determining the Preisach density function given inutoutut data, in a arameter-free manner. This roblem has two features:(a) ill-osedness and (b) a non-negativity constraint on the solution. We address both issues in our recursive imlementation. Ill-osedness of the oerator equation is addressed by a artial SVD iteration, and the non-negativity constraint is addressed by a Primal- Dual interior oint method. As the SVD iteration is indeendent of the Interior oint method comutation, convergence of our roosed method follows from standard results for each of these methods.

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