A Numerical Method to Compute Stability Margins of Switching Linear Systems

TL;DR

This paper introduces a numerical method using homogeneous polynomial Lyapunov functions to compute less conservative stability margins and identify periodic trajectories in switching linear systems.

Contribution

It presents an algorithm for generating bounds on stability margins and for finding worst-case switching schemes using polynomial Lyapunov functions.

Findings

Reduced conservatism in stability margin estimation.

Algorithm effectively bounds system stability margins.

Demonstrated with three numerical examples.

Abstract

Stability margins for linear time-varying (LTV) and switched-linear systems are traditionally computed via quadratic Lyapunov functions, and these functions certify the stability of the system under study. In this work, we show how the more general class of homogeneous polynomial Lyapunov functions is used to compute stability margins with reduced conservatism, and we show how these Lyapunov functions aid in the search for periodic trajectories for marginally stable LTV systems. Our work is premised on the recent observation that the search for a homogeneous polynomial Lyapunov function for some LTV systems is easily encoded as the search for a quadratic Lyapunov function for a related LTV system, and our main contribution is an intuitive algorithm for generating upper and lower bounds on the system's stability margin. We show also how the worst-case switching scheme - which draws an LTV system closest to a periodic orbit - is generated. Three numerical examples are provided to aid the reader and demonstrate the theoretical contributions of the work.