  This paper presents a novel Wasserstein distributionally robust control and
state estimation algorithm for partially observable linear stochastic systems,
where the probability distributions of disturbances and measurement noises are
unknown. Our method consists of the control and state estimation phases to
handle distributional ambiguities of system disturbances and measurement
noises, respectively. Leveraging tools from modern distributionally robust
optimization, we consider an approximation of the control problem with an
arbitrary nominal distribution and derive its closed-form optimal solution. We
show that the separation principle holds, thereby allowing the state estimator
to be designed separately. A novel distributionally robust Kalman filter is
then proposed as an optimal solution to the state estimation problem with
Gaussian nominal distributions. Our key contribution is the combination of
distributionally robust control and state estimation into a unified algorithm.
This is achieved by formulating a tractable semidefinite programming problem
that iteratively determines the worst-case covariance matrices of all
uncertainties, leading to a scalable and efficient algorithm. Our method is
also shown to enjoy a guaranteed cost property as well as a probabilistic
out-of-sample performance guarantee. The results of our numerical experiments
demonstrate the performance and computational efficiency of the proposed
method.
