Distributionally Robust Control Synthesis for Stochastic Systems with Safety and Reach-Avoid Specifications
Yu Chen, Yuda Li, Shaoyuan Li, Xiang Yin
TL;DR
This paper develops distributionally robust control synthesis for discrete-time stochastic systems with safety and reach-avoid specifications under Wasserstein ambiguity sets. It provides theoretical guarantees via dynamic programming, including existence of optimal policies and a dual reformulation that reduces infinite-dimensional optimization to finite-dimensional programs, complemented by a safety-focused DR-CBC framework solved with SOS for polynomial dynamics. The approach is validated through three case studies (room-temperature reach-avoid, 1-D and 4-D DR-CBC examples), showing improved online satisfaction probabilities under limited knowledge of disturbance distributions. The work bridges formal verification with distributionally robust optimization, offering scalable synthesis tools and practical robustness benefits for safety-critical cyber-physical systems.
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.