Realization theory of discrete-time linear switched systems

TL;DR

This paper develops a realization theory for discrete-time linear switched systems, providing conditions for their representation, minimality, and algorithms for system realization from input-output data, using non-commutative formal power series.

Contribution

It introduces necessary and sufficient rank conditions for realization, characterizes minimality, and presents algorithms for system conversion and realization from data.

Findings

Finite rank conditions characterize realizability.

Minimal realizations are unique up to isomorphism.

Algorithms for minimal realization from input-output data are provided.

Abstract

The paper presents realization theory of discrete-time linear switched systems. A discrete-time linear switched system is a hybrid system, such that the continuous sub-system associated with each discrete state is linear. In this paper we present necessary and sufficient conditions for an input-output map to admit a discrete-time linear switched state-space realization. The conditions are formulated as finite rank conditions of a generalized Hankel-matrix. In addition, we present a characterization of minimality of discrete-time linear switched systems in terms of reachability and observable.Further, we prove that minimal realizations are unique up to isomorphism. We also discuss procedures for converting a linear switched system to a minimal one and we present an algorithm for constructing a state-space representation from input-output data.The paper uses the theory rational formal power series in non-commutative variables. The latter theory was successfully applied to bilinear and state-affine systems in the past.