Analytic matrix descriptions with application to time-delay systems

Abstract

A polynomial matrix description(PMD) of a rational matrix $G(\lambda)$ is a matrix polynomial of the form $$ \mathbf{P}(\lambda) := \left[\begin{array}{c|c} A(\lambda) & B(\lambda) \\ \hline -C(\lambda) & D(\lambda)\end{array}\right] \text{ such that } G(\lambda) = D(\lambda) + C(\lambda) A(\lambda)^{-1} B(\lambda),$$ where $A(\lambda)$ is regular and called the state matrix. PMDs have been studied extensively in the context of higher order linear time-invariant systems. An analytic matrix description(AMD) of a meromorphic matrix $ M(\lambda)$ is a holomorphic matrix (i.e., a holomorphic matrix-valued function) of the form $$ \mathbf{H}(\lambda) := \left[\begin{array}{c|c} A(\lambda) & B(\lambda) \\ \hline -C(\lambda) & D(\lambda)\end{array}\right] \text{ such that } M(\lambda) = D(\lambda) + C(\lambda) A(\lambda)^{-1} B(\lambda),$$ where $A(\lambda)$ is regular and called the state matrix. AMDs arise in the study of linear time-invariant time-delay systems (TDS). Our aim is to develop a framework for analysis of AMDs analogous to the framework for PMDs and discuss the extent to which results of PMDs can be generalized to AMDs. We show that important results (e.g., coprime matrix-fraction descriptions (MFDs), least order PMDs, equivalence of PMDs, canonical forms, characterization of PMDs by transfer functions, structural indices of zeros and poles) which hold for PMDs can be generalized to AMDs which in turn can be utilized to analyze time-delay systems.