Linearizations for Rosenbrock system polynomials and rational matrix functions

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