  We study the infinite-horizon distributionally robust (DR) control of linear
systems with quadratic costs, where disturbances have unknown, possibly
time-correlated distribution within a Wasserstein-2 ambiguity set. We aim to
minimize the worst-case expected regret-the excess cost of a causal policy
compared to a non-causal one with access to future disturbance. Though the
optimal policy lacks a finite-order state-space realization (i.e., it is
non-rational), it can be characterized by a finite-dimensional parameter.
Leveraging this, we develop an efficient frequency-domain algorithm to compute
this optimal control policy and present a convex optimization method to
construct a near-optimal state-space controller that approximates the optimal
non-rational controller in the $\mathit{H}_\infty$-norm. This approach avoids
solving a computationally expensive semi-definite program (SDP) that scales
with the time horizon in the finite-horizon setting.
