Distributionally Robust Regret Optimal Control Under Moment-Based Ambiguity Sets

In this paper, we consider a class of finite-horizon, linear-quadratic stochastic control problems, where the probability distribution governing the noise process is unknown but assumed to belong to an ambiguity set consisting of all distributions whose mean and covariance lie within norm balls centered at given nominal values. To address the distributional ambiguity, we explore the design of causal affine control policies to minimize the worst-case expected regret over all distributions in the given ambiguity set. The resulting minimax optimal control problem is shown to admit an equivalent reformulation as a tractable convex program that corresponds to a regularized version of the nominal linear-quadratic stochastic control problem. While this convex program can be recast as a semidefinite program, semidefinite programs are typically solved using primal-dual interior point methods that scale poorly with the problem size in practice. To address this limitation, we propose a scalable dual projected subgradient method to compute optimal controllers to an arbitrary accuracy. Numerical experiments are presented to benchmark the proposed method against state-of-the-art data-driven and distributionally robust control design approaches.