Identification and rational L2 approximation A gradient algorithm

Baratchart, Laurent; Cardelli, Michel; Olivi, Martine

doi:10.1016/0005-1098(91)90092-g

Cited by 76 publications

(39 citation statements)

References 2 publications

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“…The stable discrete‐time time‐invariant linear control system corresponds to the Hardy space approximation. The system identification by using the AFD algorithms are treated in Mi and Qian and Mi et al, as well as in other papers . A system function, as Z ‐transform of a sequence of impulses, is not necessary to be the square integrable on the unit circle, as in the Hardy space, it would well be a function in some WHSs without square integrability.…”

Section: Application Of Poafd On Wbss In Time‐varying System Identifimentioning

confidence: 99%

Reproducing kernel approximation in weighted Bergman spaces: Algorithm and applications

Dang

2019

Math Methods in App Sciences

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In this paper, we present algorithms of preorthogonal adaptive Fourier decomposition (POAFD) in weighted Bergman spaces. POAFD, as has been studied, gives rise to sparse approximations as linear combinations of the corresponding reproducing kernels. It is found that POAFD is unavailable in some weighted Hardy spaces that do not enjoy the boundary vanishing condition; as a result, the maximal selections of the parameters are not guaranteed. We overcome this difficulty with two strategies. One is to utilize the shift operator while the other is to perform weak POAFD. In the cases when the reproducing kernels are rational functions, POAFD provides rational approximations. This approximation method may be used to 1D signal processing. It is, in particular, effective to some Hardy Hp space functions for p not being equal to 2. Weighted Bergman spaces approximation may be used in system identification of discrete time‐varying systems. The promising effectiveness of the POAFD method in weighted Bergman spaces is confirmed by a set of experiments. A sequence of functions as models of the weighted Hardy spaces, being a wider class than the weighted Bergman spaces, are given, of which some are used to illustrate the algorithm and to evaluate its effectiveness over other Fourier type methods.

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Section: Application Of Poafd On Wbss In Time‐varying System Identifimentioning

confidence: 99%

Reproducing kernel approximation in weighted Bergman spaces: Algorithm and applications

Dang

2019

Math Methods in App Sciences

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“…Let us briefly review some of the results that were obtained. Existence and generic uniqueness of a best approximant, asymptotic properties, as well as an index theorem that gives a global constraint on the set of critical points, have been established in [1,4,6]; a gradient algorithm converging to a local minimum is also described in [3]. From the index theorem, uniqueness of a critical point (hence of a local and global minimum) is derived for some classes of functions, like Markov functions or exponentials; see [8] and the bibliography therein.…”

Section: Mathematicsmentioning

confidence: 99%

Weighted $H^2$ rational approximation and consistency

Leblond

Saff

Wielonsky

2002

Numerische Mathematik

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We investigate consistency properties of rational approximation of prescribed type in the weighted Hardy space H 2 − (µ) for the exterior of the unit disk, where µ is a positive symmetric measure on the unit circle T. The question of consistency, which is especially significant for gradient algorithms that compute local minima, concerns the uniqueness of critical points in the approximation criterion for the case when the approximated function is itself rational. In addition to describing some basic properties of the approximation problem, we prove for measures µ having a rational function distribution (weight) with respect to arclength on T, that consistency holds only under rather restricted conditions. (1991): 41A20, 93B30, 30E10, 68W25 Notations T, D, D unit circle, open unit disk, complement in C of the closed unit disk P n space of real polynomials of degree at most n (if n < 0, P n = {0}) M 1 n monic real polynomials of degree n having all their roots in D Mathematics Subject Classification

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“…given by (11) satisfy constraint (9) for any of full column rank and any of full row rank, respectively. It is therefore interesting to consider the following modified problem: minimize over subject to stability of Remark 2.2: Admittedly, the modified problem represents an approximation to the original problem as the new set of reduced-order models over which the model-reduction cost is minimized is a subset of the original set.…”

Section: Problem Formulationmentioning

confidence: 99%

“…So far, the most commonly taken approach to -optimal model reduction problem is to work with first-order necessary conditions for optimality, which were developed and simplified in one way or another by Meier and Luenberger [6], Wilson [7], Hyland and Bernstein [8], Halevi [9], Bryson and Carrier [10], Baratchart et al. [11], and more recently Spanos et al [12]. Accordingly, they proposed their respective algorithms to seek a solution satisfying the conditions expressed in terms of nonlinear matrix equations.…”

Section: Introductionmentioning

confidence: 99%

“…Many of the algorithms lack the proof of convergence and mathematical rigor, and some of them may even become divergent for certain initial conditions or converge to a maximum. Though Baratchart et al [11] and Spanos et al [12] established the convergence of their respective algorithms under certain conditions, the algorithms are only applicable to the singleinput/single-output (SISO) case.…”

Section: Introductionmentioning

confidence: 99%

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An approximate approach to H/sup 2/ optimal model reduction

Yan

Lam

1999

IEEE Trans. Automat. Contr.

131

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This paper deals with the problem of computing an H H H 2 optimal reduced-order model for a given stable multivariable linear system. By way of orthogonal projection, the problem is formulated as that of minimizing the H H H 2 model-reduction cost over the Stiefel manifold so that the stability constraint on reduced-order models is automatically satisfied and thus totally avoided in the new problem formulation. The closed form expression for the gradient of the cost over the manifold is derived, from which a gradient flow results as an ordinary differential equation (ODE). A number of nice properties about such a flow are established. Furthermore, two explicit iterative convergent algorithms are developed from the flow; one has a constant step-size and the other has a varying step-size and is much more efficient. Both of them inherit the properties that the iterates remain on the manifold starting from any orthogonal initial point and that the model reduction cost is decreasing to minima along the iterates. A procedure for closing the gap between the original and modified problem is proposed. In the symmetric case, the two problems are shown to be equivalent. Numerical examples are presented to illustrate the effectiveness of the proposed algorithms as well as convergence. . His current research interests include the areas of control and signal processing. James Lam (S'87-M'88-SM'99) received the first class B.Sc. degree in mechanical engineering from the University of Manchester in 1983. He then obtained the M.Phil. and Ph.D. degrees in the area of control engineering from the University of Cambridge in 1985 and 1988, respectively.

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Identification and rational L2 approximation A gradient algorithm

Cited by 76 publications

References 2 publications

Reproducing kernel approximation in weighted Bergman spaces: Algorithm and applications

Reproducing kernel approximation in weighted Bergman spaces: Algorithm and applications

Weighted $H^2$ rational approximation and consistency

An approximate approach to H/sup 2/ optimal model reduction

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