Linearizations for Rosenbrock system polynomials and rational matrix functions

TL;DR

This paper introduces structured linearizations for Rosenbrock system polynomials and rational matrix functions, enabling efficient computation of system zeros and eigenstructures through Fiedler-like pencils and state-space methods.

Contribution

It develops Rosenbrock linearizations and Fiedler-like pencils for system polynomials, and extends linearization concepts to rational matrix functions using state-space realizations.

Findings

Fiedler-like pencils are Rosenbrock linearizations of system polynomials.

The proposed linearizations preserve eigenstructure of rational matrix functions.

State-space framework enables minimal dimension linearizations for rational functions.

Abstract

Our aim in this paper is two-fold: First, for computing zeros of a linear time-invariant (LTI) system $\Sigma$ in {\em state-space form}, we introduce a "trimmed structured linearization", which we refer to as {\em Rosenbrock linearization}, of the Rosenbrock system polynomial $\mathcal{S}(\lam)$ associated with $\Sigma.$ We also introduce Fiedler-like matrices for $\mathcal{S}(\lam)$ and describe constructions of Fiedler-like pencils for $\mathcal{S}(\lam).$ We show that the Fiedler-like pencils of $\mathcal{S}(\lam)$ are Rosenbrock linearizations of the system polynomial $\mathcal{S}(\lam).$ Second, with a view to developing a direct method for solving rational eigenproblems, we introduce "linearization" of a rational matrix function. We describe a state-space framework for converting a rational matrix function $G(\lam)$ to an "equivalent" matrix pencil $\mathbb{L}(\lam)$ of smallest dimension such that $G(\lam)$ and $\mathbb{L}(\lam)$ have the same "eigenstructure" and we refer to such a pencil $\mathbb{L}(\lam)$ as a "linearization" of $G(\lam).$ Indeed, by treating $G(\lam)$ as the transfer function of an LTI system $\Sigma_G$ in state-space form via state-space realization, we show that the Fiedler-like pencils of the Rosenbrock system polynomial associated with $\Sigma_G$ are "linearizations" of $G(\lam)$ when the system $\Sigma_G$ is both controllable and observable.