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We investigate the Distributionally Robust Regret-Optimal (DR-RO) control ofdiscrete-time linear dynamical systems with quadratic cost over an infinitehorizon. Regret is the difference in cost obtained by a causal controller and aclairvoyant controller with access to future disturbances. We focus on theinfinite-horizon framework, which results in stability guarantees. In this DRsetting, the probability distribution of the disturbances resides within aWasserstein-2 ambiguity set centered at a specified nominal distribution. Ourobjective is to identify a control policy that minimizes the worst-caseexpected regret over an infinite horizon, considering all potential disturbancedistributions within the ambiguity set. In contrast to prior works, whichassume time-independent disturbances, we relax this constraint to allow fortime-correlated disturbances, thus actual distributional robustness. While weshow that the resulting optimal controller is non-rational and lacks afinite-dimensional state-space realization, we demonstrate that it can still beuniquely characterized by a finite dimensional parameter. Exploiting this fact,we introduce an efficient numerical method to compute the controller in thefrequency domain using fixed-point iterations. This method circumvents thecomputational bottleneck associated with the finite-horizon problem, where thesemi-definite programming (SDP) solution dimension scales with the timehorizon. Numerical experiments demonstrate the effectiveness and performance ofour framework.

Wasserstein Distributionally Robust Regret-Optimal Control in the Infinite-Horizon