Minimal Realization Problems for Jump Linear Systems

TL;DR

This paper develops methods to determine the minimal state space dimension and the number of modes in jump linear systems using input-output data, without prior knowledge of mode switches, through rank-based matrix analysis.

Contribution

It introduces novel rank-based techniques to characterize minimal realization and mode count in jump linear systems from input-output data.

Findings

Hankel-like matrix rank determines state space dimension.

Minimal number of modes corresponds to minimal rank of a positive semi-definite matrix.

Methods do not require prior knowledge of mode switches.

Abstract

This paper addresses two fundamental problems in the context of jump linear systems (JLS). The first problem is concerned with characterizing the minimal state space dimension solely from input-output pairs and without any knowledge of the number of mode switches. The second problem is concerned with characterizing the number of discrete modes of the JLS. For the first problem, we develop a linear system theory based approach and construct an appropriate Hankel-like matrix. The rank of this matrix gives us the state space dimension. For the second problem we show that minimal number of modes corresponds to the minimal rank of a positive semi-definite matrix obtained via a non--convex formulation.