Wasserstein Tube MPC with Exact Uncertainty Propagation

Abstract: We study model predictive control (MPC) problems for stochastic LTI systems, where the noise distribution is unknown, compactly supported, and only observable through a limited number of i.i.d. noise samples. Building upon recent results in the literature, which show that distributional uncertainty can be efficiently captured within a Wasserstein ambiguity set, and that such ambiguity sets propagate exactly through the system dynamics, we start by formulating a novel Wasserstein Tube MPC (WT-MPC) problem, with distributionally robust CVaR constraints. We then show that the WT-MPC problem: (1) is a direct generalization of the (deterministic) Robust Tube MPC (RT-MPC) to the stochastic setting; (2) through a scalar parameter, it interpolates between the data-driven formulation based on sample average approximation and the RT-MPC formulation, allowing us to optimally trade between safety and performance; (3) admits a tractable convex reformulation; and (4) is recursively feasible. We conclude the paper with a numerical comparison of WT-MPC and RT-MPC.