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This paper studies optimal control problems of unknown linear systems subject to stochastic disturbances of uncertain distribution. Uncertainty about the stochastic disturbances is usually described via ambiguity sets of probability measures or distributions. Typically, stochastic optimal control requires knowledge of underlying dynamics and is as such challenging. Relying on a stochastic fundamental lemma from data-driven control and on the framework of polynomial chaos expansions, we propose an approach to reformulate distributionally robust optimal control problems with ambiguity sets as uncertain conic programs in a finite-dimensional vector space. We show how to construct these programs from previously recorded data and how to relax the uncertain conic program to numerically tractable convex programs via appropriate sampling of the underlying distributions. The efficacy of our method is illustrated via a numerical example.

Distributionally robust uncertainty quantification via data-driven stochastic optimal control