Spectral factorization of rectangular rational matrix functions with application to discrete Wiener-Hopf equations

AbstractThe properties of a discrete Wiener-Hopf equation are closely related to the factorization of the symbol of the equation. We give a necessary and sufficient condition for existence of a canonical Wiener-Hopf factorization of a possibly nonregular rational matrix function W relative to a contour which is a positively oriented boundary of a region in the finite complex plane. The condition involves decomposition of the state space in a minimal realization of W and, if it is satisfied, we give explicit formulas for the factors. The results are generalized by means of centered realizations to arbitrary rational matrix functions. The proposed approach can be used to solve discrete Wiener-Hopf equations whose symbols are rational matrix functions which admit canonical factorization relative to the unit circle

An explicit Wiener-Hopf factorization algorithm for matrix polynomials and its exact realizations within ExactMPF package

We discuss an explicit algorithm for solving the Wiener–Hopf factorization problem for matrix polynomials. By an exact solution of the problem, we understand the one constructed by a symbolic computation. Since the problem is, generally speaking, unstable, this requirement is crucial to guarantee that the result following from the explicit algorithm is indeed a solution of the original factorization problem. We prove that a matrix polynomial over the field of Gaussian rational numbers admits the exact Wiener–Hopf factorization if and only if its determinant is exactly factorable. Under such a condition, we adapt the explicit algorithm to the exact calculations and develop the ExactMPF package realized within the Maple Software. The package has been extensively tested. Some examples are presented in the paper, while the listing is provided in the electronic supplementary material. If, however, a matrix polynomial does not admit the exact factorization, we clarify a notion of the numerical (or approximate) factorization that can be constructed by following the explicit factorization algorithm. We highlight possible obstacles on the way and discuss a level of confidence in the final result in the case of an unstable set of partial indices. The full listing of the package ExactMPF is given in the electronic supplementary material

On a homeomorphism between orbit spaces of linear systems and matrix polynomials

AbstractThe orbit space of controllable systems under system similarity and the orbit space of matrix polynomials with determinant degree equal to the order of the state matrix under right equivalence are proved to be homeomorphic when the quotient compact–open topology is considered in the latter. As a consequence, the variation of the finite and left Wiener–Hopf structures under small perturbations of matrix polynomials with fixed degree for their determinants is described

Factorization of a class of matrix-functions with stable partial indices

A new effective method for factorization of a class of nonrational $n\times n$ matrix-functions with \emph{stable partial indices} is proposed. The method is a generalization of the one recently proposed by the authors which was valid for the canonical factorization only. The class of considered matrices is motivated by problems originated from applications. The properties and details of the asymptotic procedure are illustrated by examples. The efficiency of the procedure is highlighted by numerical results.Comment: 22 pages, 7 figures. arXiv admin note: text overlap with arXiv:1402.212

An asymptotic method of factorization of a class of matrix-functions

A novel method of asymptotic factorization of $n \times n$ matrix functions is proposed. Considered class of matrices is motivated by certain problems originated in the elasticity theory. An example is constructed to illustrate effectiveness of the proposed procedure. Further applications of the method is discussed.Comment: 23 pages, 3 figure