Data-driven stability analysis of switched linear systems with Sum of Squares guarantees

TL;DR

This paper introduces a data-driven approach to probabilistically verify the stability of black-box switched linear systems using Lyapunov functions, with explicit sample complexity bounds and improvements via Sum of Squares methods.

Contribution

It develops a novel sensitivity analysis-based method for quadratic Lyapunov functions and extends it to Sum of Squares Lyapunov functions, enhancing stability guarantees.

Findings

Improved bounds for stability verification with fewer samples.

Extension of the method to Sum of Squares Lyapunov functions.

Numerical example demonstrating the effectiveness of the approach.

Abstract

We present a new data-driven method to provide probabilistic stability guarantees for black-box switched linear systems. By sampling a finite number of observations of trajectories, we construct approximate Lyapunov functions and deduce the stability of the underlying system with a user-defined confidence. The number of observations required to attain this confidence level on the guarantee is explicitly characterized. Our contribution is twofold: first, we propose a novel approach for common quadratic Lyapunov functions, relying on sensitivity analysis of a quasi-convex optimization program. By doing so, we improve a recently proposed bound. Then, we show that our new approach allows for extension of the method to Sum of Squares Lyapunov functions, providing further improvement for the technique. We demonstrate these improvements on a numerical example.