Perturbation theory of transfer function matrices

TL;DR

This paper develops a structured perturbation theory for simple zeros of transfer function matrices, providing insights into their sensitivity and comparing structured and unstructured condition numbers, supported by numerical experiments.

Contribution

It introduces a structured condition number for simple eigenvalues of polynomial system matrices, extending to rational matrices and analyzing their zeros with root vectors.

Findings

Structured condition number can be significantly smaller than unstructured ones.

Analysis applies to all zeros, including poles and non-poles.

Numerical experiments validate the theoretical results.

Abstract

Zeros of rational transfer function matrices $R(\lambda)$ are the eigenvalues of associated polynomial system matrices $P(\lambda)$, under minimality conditions. In this paper we define a structured condition number for a simple eigenvalue $\lambda_0$ of a (locally) minimal polynomial system matrix $P(\lambda)$, which in turn is a simple zero $\lambda_0$ of its transfer function matrix $R(\lambda)$. Since any rational matrix can be written as the transfer function of a polynomial system matrix, our analysis yield a structured perturbation theory for simple zeros of rational matrices $R(\lambda)$. To capture all the zeros of $R(\lambda)$, regardless of whether they are poles or not, we consider the notion of root vectors. As corollaries of the main results, we pay particular attention to the special case of $\lambda_0$ being not a pole of $R(\lambda)$ since in this case the results get simpler and can be useful in practice. We also compare our structured condition number with Tisseur's unstructured condition number for eigenvalues of matrix polynomials, and show that the latter can be unboundedly larger. Finally, we corroborate our analysis by numerical experiments.