Black-box stability analysis of hybrid systems with sample-based multiple Lyapunov functions

TL;DR

This paper introduces a data-driven probabilistic framework using multiple Lyapunov functions to assess the stability of unknown constrained switching linear systems with automaton constraints, providing bounds on their joint spectral radius.

Contribution

It develops a novel probabilistic method to estimate stability of constrained switching systems from finite observations, including bounds on the joint spectral radius.

Findings

Provides probabilistic upper bounds on the CJSR from data.

Derives deterministic lower bounds on the CJSR.

Enables stability assessment with limited data.

Abstract

We present a framework based on multiple Lyapunov functions to find probabilistic data-driven guarantees on the stability of unknown constrained switching linear systems (CSLS), which are switching linear systems whose switching signal is constrained by an automaton. The stability of a CSLS is characterized by its constrained joint spectral radius (CJSR). Inspired by the scenario approach and previous work on unconstrained switching systems, we characterize the number of observations needed to find sufficient conditions on the (in-)stability of a CSLS using the notion of CJSR. More precisely, our contribution is the following: we derive a probabilistic upper bound on the CJSR of an unknown CSLS from a finite number of observations. We also derive a deterministic lower bound on the CJSR. From this we obtain a probabilistic method to characterize the stability of an unknown CSLS.