Distributional robustness in minimax linear quadratic control with Wasserstein distance

To address the issue of inaccurate distributions in practical stochastic systems, a minimax linear-quadratic control method is proposed using the Wasserstein metric. Our method aims to construct a control policy that is robust against errors in an empirical distribution of underlying uncertainty, by adopting an adversary that selects the worst-case distribution. The opponent receives a Wasserstein penalty proportional to the amount of deviation from the empirical distribution. A closed-form expression of the finite-horizon optimal policy pair is derived using a Riccati equation. The result is then extended to the infinite-horizon average cost setting by identifying conditions under which the Riccati recursion converges to the unique positive semi-definite solution to an algebraic Riccati equation. Our method is shown to possess several salient features including closed-loop stability, and an out-of-sample performance guarantee. We also discuss how to optimize the penalty parameter for enhancing the distributional robustness of our control policy. Last but not least, a theoretical connection to the classical $H_\infty$-method is identified from the perspective of distributional robustness.