Efficient Strategy Synthesis for Switched Stochastic Systems with Distributional Uncertainty

TL;DR

This paper presents a novel framework for designing control strategies for switched stochastic systems with uncertain noise distributions, ensuring high probability of reaching targets despite distributional ambiguity.

Contribution

It introduces a distributionally robust control synthesis method using finite abstractions and linear programming, handling Wasserstein ambiguity sets for the first time in this context.

Findings

Effective control strategies for uncertain stochastic systems demonstrated.

Framework applicable to both linear and nonlinear systems.

Reduction of complex synthesis to linear programs enhances computational efficiency.

Abstract

We introduce a framework for the control of discrete-time switched stochastic systems with uncertain distributions. In particular, we consider stochastic dynamics with additive noise whose distribution lies in an ambiguity set of distributions that are $\varepsilon-$close, in the Wasserstein distance sense, to a nominal one. We propose algorithms for the efficient synthesis of distributionally robust control strategies that maximize the satisfaction probability of reach-avoid specifications with either a given or an arbitrary (not specified) time horizon, i.e., unbounded-time reachability. The framework consists of two main steps: finite abstraction and control synthesis. First, we construct a finite abstraction of the switched stochastic system as a \emph{robust Markov decision process} (robust MDP) that encompasses both the stochasticity of the system and the uncertainty in the noise distribution. Then, we synthesize a strategy that is robust to the distributional uncertainty on the resulting robust MDP. We employ techniques from optimal transport and stochastic programming to reduce the strategy synthesis problem to a set of linear programs, and propose a tailored and efficient algorithm to solve them. The resulting strategies are correctly refined into switching strategies for the original stochastic system. We illustrate the efficacy of our framework on various case studies comprising both linear and non-linear switched stochastic systems.