Guaranteed Control of Sampled Switched Systems using Semi-Lagrangian Schemes and One-Sided Lipschitz Constants

TL;DR

This paper introduces a control synthesis method for sampled switched systems that guarantees safety over a finite time horizon by combining semi-Lagrangian schemes with one-sided Lipschitz constants to ensure trajectories remain within a safe set.

Contribution

It presents a novel approach that combines finite gridding, Euler-based trajectory approximation, and one-sided Lipschitz constants to guarantee safety for controlled switched systems.

Findings

Method successfully ensures safety for sampled switched systems.

Applicable to stochastic systems with safety guarantees.

Provides sufficient conditions linking approximate and exact trajectories.

Abstract

In this paper, we propose a new method for ensuring formally that a controlled trajectory stay inside a given safety set S for a given duration T. Using a finite gridding X of S, we first synthesize, for a subset of initial nodes x of X , an admissible control for which the Euler-based approximate trajectories lie in S at t $\in$ [0,T]. We then give sufficient conditions which ensure that the exact trajectories, under the same control, also lie in S for t $\in$ [0,T], when starting at initial points 'close' to nodes x. The statement of such conditions relies on results giving estimates of the deviation of Euler-based approximate trajectories, using one-sided Lipschitz constants. We illustrate the interest of the method on several examples, including a stochastic one.