  We present a novel data-driven distributionally robust Model Predictive
Control formulation for unknown discrete-time linear time-invariant systems
affected by unknown and possibly unbounded additive uncertainties. We use
off-line collected data and an approximate model of the dynamics to formulate a
finite-horizon optimization problem. To account for both the uncertainty
related to the dynamics and the disturbance acting on the system, we resort to
a distributionally robust formulation that optimizes the cost expectation while
satisfying Conditional Value-at-Risk constraints with respect to the worst-case
probability distributions of the uncertainties within an ambiguity set defined
using the Wasserstein metric. Using results from the distributionally robust
optimization literature we derive a tractable finite-dimensional convex
optimization problem with finite-sample guarantees for the class of convex
piecewise affine cost and constraint functions. The performance of the proposed
algorithm is demonstrated in closed-loop simulation on a simple numerical
example.
