On the Inherent Distributional Robustness of Stochastic and Nominal Model Predictive Control

We define a notion of distributional robustness, via the Wasserstein metric, for closed-loop systems subject to errors in the disturbance distribution used to construct the controller. We then establish sufficient conditions for stochastic model predictive control (SMPC) to satisfy this definition of distributional robustness and establish a similar notion of distributional robustness for economic applications of SMPC. These results address incorrectly or unmodeled disturbances, demonstrate the efficacy of scenario optimization as a means to approximate and solve the SMPC problem, and unify the descriptions of robustness for stochastic and nominal model predictive control. This definition of distributional robustness for closed-loop systems is general and can be applied to other stochastic optimal control algorithms and, potentially, the developing field of distributionally robust control.