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On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials

Abdon E. Choque-Rivero

Abstract

Every matrix polynomial $\mathbf{f}_n$ can be written in the form \[ \mathbf{f}_n(z)=\mathbf{h}(z^2)+z\,\mathbf{g}_n(z^2). \] The matrix polynomial $\mathbf{f}_{2m}$ is said to be of Hurwitz type if the expression $\mathbf{g}_{2m}(z)\mathbf{h}_{2m}^{-1}(z)$ admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial $\mathbf{f}_{2m+1}$ is of Hurwitz type if $\frac{1}{z}\mathbf{h}_{2m+1}(z)\mathbf{g}_{2m+1}^{-1}(z)$ has the same property. We derive an explicit form of the Bezoutian associated with Hurwitz-type matrix polynomials. Using this explicit form, we provide an explicit proof that Hurwitz-type matrix polynomials are Hurwitz matrix polynomials. In [52], the Hurwitzness of Hurwitz-type matrix polynomials was also studied. Finally, we propose an extension of the class of Hurwitz-type matrix polynomials by adding to a non--Hurwitz-type matrix polynomial another matrix polynomial so that the resulting matrix polynomial is of Hurwitz type.

On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials

Abstract

Every matrix polynomial can be written in the form The matrix polynomial is said to be of Hurwitz type if the expression admits a representation as a finite continued fraction with positive definite matrix coefficients. Similarly, the odd-degree matrix polynomial is of Hurwitz type if has the same property. We derive an explicit form of the Bezoutian associated with Hurwitz-type matrix polynomials. Using this explicit form, we provide an explicit proof that Hurwitz-type matrix polynomials are Hurwitz matrix polynomials. In [52], the Hurwitzness of Hurwitz-type matrix polynomials was also studied. Finally, we propose an extension of the class of Hurwitz-type matrix polynomials by adding to a non--Hurwitz-type matrix polynomial another matrix polynomial so that the resulting matrix polynomial is of Hurwitz type.

Paper Structure

This paper contains 10 sections, 13 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. HTM polynomials and associated Markov parameters
  4. Stieltjes orthogonal matrix polynomials
  5. The Bezoutian and inertia for matrix polynomials
  6. Bezoutian and Hurwitz-type matrix polynomials
  7. Hurwitness of Hurwitz-type matrix polynomials
  8. Completing a matrix polynomial to an HTM polynomial
  9. Additional remarks
  10. Conclusion

Key Result

Proposition 2.5

a) The polynomials $P_{1,j}$ and $P_{2,j}$ are OMP with respect to $\sigma$ and $t\sigma$, respectively. More precisely (as seen in abmad), b) The following identities hold (as seen in ablaa and ablaa): c) For $k=1,2$, the following identity holds (as seen in ablaa)

Theorems & Definitions (40)

  • Definition 1.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Corollary 2.6
  • Remark 2.7
  • Lemma 3.1
  • proof
  • ...and 30 more