Perturbation theory of transfer function matrices

by   Vanni Noferini, et al.

Zeros of rational transfer function matrices R(λ) are the eigenvalues of associated polynomial system matrices P(λ), under minimality conditions. In this paper we define a structured condition number for a simple eigenvalue λ_0 of a (locally) minimal polynomial system matrix P(λ), which in turn is a simple zero λ_0 of its transfer function matrix R(λ). 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(λ). To capture all the zeros of R(λ), 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 λ_0 being not a pole of R(λ) 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.


page 1

page 2

page 3

page 4


Strongly minimal self-conjugate linearizations for polynomial and rational matrices

We prove that we can always construct strongly minimal linearizations of...

Fiedler Linearizations of Multivariable State-Space System and its Associated System Matrix

Linearization is a standard method in the computation of eigenvalues and...

Wilkinson's bus: Weak condition numbers, with an application to singular polynomial eigenproblems

We propose a new approach to the theory of conditioning for numerical an...

Local Minimizers of the Crouzeix Ratio: A Nonsmooth Optimization Case Study

Given a square matrix A and a polynomial p, the Crouzeix ratio is the no...

Local Linearizations of Rational Matrices with Application to Rational Approximations of Nonlinear Eigenvalue Problems

This paper presents a definition for local linearizations of rational ma...

Recovering a perturbation of a matrix polynomial from a perturbation of its linearization

A number of theoretical and computational problems for matrix polynomial...

Algorithms for Modifying Recurrence Relations of Orthogonal Polynomial and Rational Functions when Changing the Discrete Inner Product

Often, polynomials or rational functions, orthogonal for a particular in...

Please sign up or login with your details

Forgot password? Click here to reset