  We investigate the Distributionally Robust Regret-Optimal (DR-RO) control of
discrete-time linear dynamical systems with quadratic cost over an infinite
horizon. Regret is the difference in cost obtained by a causal controller and a
clairvoyant controller with access to future disturbances. We focus on the
infinite-horizon framework, which results in stability guarantees. In this DR
setting, the probability distribution of the disturbances resides within a
Wasserstein-2 ambiguity set centered at a specified nominal distribution. Our
objective is to identify a control policy that minimizes the worst-case
expected regret over an infinite horizon, considering all potential disturbance
distributions within the ambiguity set. In contrast to prior works, which
assume time-independent disturbances, we relax this constraint to allow for
time-correlated disturbances, thus actual distributional robustness. While we
show that the resulting optimal controller is non-rational and lacks a
finite-dimensional state-space realization, we demonstrate that it can still be
uniquely characterized by a finite dimensional parameter. Exploiting this fact,
we introduce an efficient numerical method to compute the controller in the
frequency domain using fixed-point iterations. This method circumvents the
computational bottleneck associated with the finite-horizon problem, where the
semi-definite programming (SDP) solution dimension scales with the time
horizon. Numerical experiments demonstrate the effectiveness and performance of
our framework.
