  In this paper we introduce a novel approach to distributionally robust
optimal control that supports online learning of the ambiguity set, while
guaranteeing recursive feasibility. We introduce conic representable risk,
which is useful to derive tractable reformulations of distributionally robust
optimization problems. Specifically, to illustrate the techniques introduced,
we utilize risk measures constructed based on data-driven ambiguity sets,
constraining the second moment of the random disturbance. In the optimal
control setting, such moment-based risk measures lead to tractable optimal
controllers when combined with affine disturbance feedback. Assumptions on the
constraints are given that guarantee recursive feasibility. The resulting
control scheme acts as a robust controller when little data is available and
converges to the certainty equivalent controller when a large sample count
implies high confidence in the estimated second moment. This is illustrated in
a numerical experiment.
