Balanced truncation for linear switched systems

TL;DR

This paper extends balanced truncation model reduction techniques to linear switched systems, providing error bounds, system-theoretic interpretations, and stability conditions that depend solely on input-output behavior.

Contribution

It introduces a theoretical framework for balanced truncation in linear switched systems, including error bounds and stability analysis based on input-output maps.

Findings

Provides an L2 error bound for continuous-time systems

Defines controllability and observability grammians for stable systems

Shows stability and gain estimates depend only on input-output maps

Abstract

In this paper, we present a theoretical analysis of the model reduction algorithm for linear switched systems. This algorithm is a reminiscence of the balanced truncation method for linear parameter varying systems. Specifically in this paper, we provide a bound on the approximation error in L2 norm for continuous-time and l2 norm for discrete-time linear switched systems. We provide a system theoretic interpretation of grammians and their singular values. Furthermore, we show that the performance of bal- anced truncation depends only on the input-output map and not on the choice of the state-space representation. For a class of stable discrete-time linear switched systems (so called strongly stable systems), we define nice controllability and nice observability grammians, which are genuinely related to reachability and controllability of switched systems. In addition, we show that quadratic stability and LMI estimates of the L2 and l2 gains depend only on the input-output map.