New Gramians for Linear Switched Systems: Reachability, Observability, and Model Reduction

TL;DR

This paper introduces new algebraic Gramians for continuous-time linear switched systems that encode reachability and observability, enabling Gramian-based analysis and model reduction with stability guarantees.

Contribution

The work presents novel Gramians satisfying generalized Lyapunov equations, linking them to system properties and developing a model reduction method with stability preservation.

Findings

Gramians encode reachability and observability spaces.

A Gramian-based criterion for reachability and observability is established.

Numerical examples demonstrate the effectiveness of the proposed method.

Abstract

In this paper, we propose new algebraic Gramians for continuous-time linear switched systems, which satisfy generalized Lyapunov equations. The main contribution of this work is twofold. First, we show that the ranges of those Gramians encode the reachability and observability spaces of a linear switched system. As a consequence, a simple Gramian-based criterion for reachability and observability is established. Second, a balancing-based model order reduction technique is proposed and, under some sufficient conditions, stability preservation and an error bound are shown. Finally, the efficiency of the proposed method is illustrated by means of numerical examples.