  We present a novel approach for the control of uncertain, linear
time-invariant systems, which are perturbed by potentially unbounded, additive
disturbances. We propose a \emph{doubly robust} data-driven state-feedback
controller to ensure reliable performance against both model mismatch and
disturbance distribution uncertainty. Our controller, which leverages the
System Level Synthesis parameterization, is designed as the solution to a
distributionally robust finite-horizon optimal control problem. The goal is to
minimize a cost function while satisfying constraints against the worst-case
realization of the uncertainty, which is quantified using distributional
ambiguity sets. The latter are defined as balls in the Wasserstein metric
centered on the predictive empirical distribution computed from a set of
collected trajectory data. By harnessing techniques from robust control and
distributionally robust optimization, we characterize the distributional shift
between the predictive and the actual closed-loop distributions, and highlight
its dependency on the model mismatch and the uncertainty about the disturbance
distribution. We also provide bounds on the number of samples required to
achieve a desired confidence level and propose a tractable approximate
formulation for the doubly robust data-driven controller. To demonstrate the
effectiveness of our approach, we present a numerical example showcasing the
performance of the proposed algorithm.
