2015/05/14 by Rafikul Alam, Alam, Rafikul, Namita Behera +1
Computer Science · Engineering · Mathematics · #Matrix Theory and Algorithms #Stability and Control of Uncertain Systems #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1505.03636
Our aim in this paper is two-fold: First, for computing zeros of a linear time-invariant (LTI) system Σ in \em state-space form, we introduce a "trimmed structured linearization", which we refer to as \em Rosenbrock linearization, of the Rosenbrock system polynomial S(\lam) associated with Σ. We also introduce Fiedler-like matrices for S(\lam) and describe constructions of Fiedler-like pencils for S(\lam). We show that the Fiedler-like pencils of S(\lam) are Rosenbrock linearizations of the system polynomial 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 \mathbbL(\lam) of smallest dimension such that G(\lam) and \mathbbL(\lam) have the same "eigenstructure" and we refer to such a pencil \mathbbL(\lam) as a "linearization" of G(\lam). Indeed, by treating G(\lam) 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(\lam) when the system ΣG is both controllable and observable.