  We study control of constrained linear systems with only partial statistical
information about the uncertainty affecting the system dynamics and the sensor
measurements. Specifically, given a finite collection of disturbance
realizations drawn from a generic distribution, we consider the problem of
designing a stabilizing control policy with provable safety and performance
guarantees despite the mismatch between the empirical and true distributions.
We capture this discrepancy using Wasserstein ambiguity sets, and we formulate
a distributionally robust (DR) optimal control problem, which provides
guarantees on the expected cost, safety, and stability of the system. To solve
this problem, we first present new results for DR optimization of quadratic
objectives using convex programming, showing that strong duality holds under
mild conditions. Then, by combining our results with the system-level
parametrization of linear feedback policies, we show that the design problem
can be reduced to a semidefinite program. We present numerical simulations to
validate the effectiveness of our approach and to highlight the value of
empirical distributions for control design.
