Linearizations for Rosenbrock system polynomials and rational matrix functions
Abstract
Our aim in this paper is two-fold: First, for computing zeros of a linear time-invariant (LTI) system Σ in state-space form, we introduce a "trimmed structured linearization", which we refer to as Rosenbrock linearization, of the Rosenbrock system polynomial S() associated with Σ. We also introduce Fiedler-like matrices for S() and describe constructions of Fiedler-like pencils for S(). We show that the Fiedler-like pencils of S() are Rosenbrock linearizations of the system polynomial S(). 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() to an "equivalent" matrix pencil L() of smallest dimension such that G() and L() have the same "eigenstructure" and we refer to such a pencil L() as a "linearization" of G(). Indeed, by treating G() as the transfer function of an LTI system ΣG in state-space form via state-space realization, we show that the Fiedler-like pencils of the Rosenbrock system polynomial associated with ΣG are "linearizations" of G() when the system ΣG is both controllable and observable.
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