Distributionally Robust Control Synthesis for Stochastic Systems with Safety and Reach-Avoid Specifications

TL;DR

This paper develops a framework for synthesizing control policies that are robust against distributional uncertainties in stochastic systems, ensuring safety and reachability with high probability.

Contribution

It introduces a novel distributionally robust control synthesis method using Wasserstein ambiguity sets and control barrier certificates, with efficient optimization techniques.

Findings

Distributionally robust policies outperform non-robust ones in uncertain environments.

The proposed approach guarantees satisfaction probabilities under distributional ambiguity.

Experimental results confirm improved safety and reliability during online execution.

Abstract

We investigate the problem of synthesizing distributionally robust control policies for stochastic systems under safety and reach-avoid specifications. Using a game-theoretical framework, we consider the setting where the probability distribution of the disturbance at each time step is selected from an ambiguity set defined by the Wasserstein distance. The goal is to synthesize a distributionally robust control policy that ensures the satisfaction probability exceeds a specified threshold under any distribution within the ambiguity set. First, for both safety and reach-avoid specifications, we establish the existence of optimal policies by leveraging the dynamic programming principles. Then we demonstrate how the associated optimization problem can be efficiently solved using the dual representation of Wasserstein distributionally robust optimization. Furthermore, for safety specifications in particular, we introduce a novel concept of distributionally robust control barrier certificates and show how these enable the efficient synthesis of controllers through sum-of-squares programming techniques. Finally, our experimental results reveal that incorporating distributional robustness during the synthesis phase significantly improves the satisfaction probability during online execution, even with limited statistical knowledge of the disturbance distribution.