  Distributionally robust control (DRC) aims to effectively manage
distributional ambiguity in stochastic systems. While most existing works
address inaccurate distributional information in fully observable settings, we
consider a partially observable DRC problem for discrete-time linear systems
using the Wasserstein metric. For a tractable solution, we propose a novel
approximation method exploiting the Gelbrich bound of Wasserstein distance.
Using techniques from modern distributionally robust optimization, we derive a
closed-form expression for the optimal control policy and a tractable
semidefinite programming problem for the worst-case distribution policy in both
finite-horizon and infinite-horizon average-cost settings. The proposed method
features several salient theoretical properties, such as a guaranteed cost
property and a probabilistic out-of-sample performance guarantee, demonstrating
the distributional robustness of our controller. Furthermore, the resulting
controller is shown to ensure the closed-loop stability of the mean-state
system. The empirical performance of our method is tested through numerical
experiments on a power system frequency control problem.
