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This paper proposes a distributionally robust approach to regret optimalcontrol of discrete-time linear dynamical systems with quadratic costs subjectto a stochastic additive disturbance on the state process. The underlyingprobability distribution of the disturbance process is unknown, but assumed tolie in a given ball of distributions defined in terms of the type-2 Wassersteindistance. In this framework, strictly causal linear disturbance feedbackcontrollers are designed to minimize the worst-case expected regret. The regretincurred by a controller is defined as the difference between the cost itincurs in response to a realization of the disturbance process and the costincurred by the optimal noncausal controller which has perfect knowledge of thedisturbance process realization at the outset. Building on a well-establishedduality theory for optimal transport problems, we derive a reformulation of theminimax regret optimal control problem as a tractable semidefinite program.Using the equivalent dual reformulation, we characterize a worst-casedistribution achieving the worst-case expected regret in relation to thedistribution at the center of the Wasserstein ball. We compare the minimaxregret optimal control design method with the distributionally robust optimalcontrol approach using an illustrative example and numerical experiments.

A Distributionally Robust Approach to Regret Optimal Control using the Wasserstein Distance