Reachable and observable sets for switched systems via generalized Lyapunov equations: application to switched descriptor systems

TL;DR

This paper demonstrates that solutions to generalized Lyapunov equations can effectively characterize the reachable and observable sets of switched systems, facilitating model reduction for switched differential-algebraic equations.

Contribution

It explains why generalized Lyapunov equations are suitable for model order reduction by linking their solutions to the system's reachable and observable sets.

Findings

GLE solutions enclose reachable sets

GLE solutions enclose observable sets

Applicable to switched DAE systems

Abstract

In a recent work [Manucci, Unger, ArXiv e-print 2404.10511, 2024], the authors propose using two generalized Lyapunov equations (GLEs) to derive a balancing-based model order reduction~(MOR) method for a general class of switched differential-algebraic equations (DAEs). This work explains why these GLEs provide solutions suitable for MOR by showing that the image set of the solutions of the two GLEs always encloses the reachable and observable set of a suitably defined switched system with the same input to output map of the switched DAE system.