Distributionally Robust Linear Quadratic Control

Linear-Quadratic-Gaussian (LQG) control is a fundamental control paradigm
that is studied in various fields such as engineering, computer science,
economics, and neuroscience. It involves controlling a system with linear
dynamics and imperfect observations, subject to additive noise, with the goal
of minimizing a quadratic cost function for the state and control variables. In
this work, we consider a generalization of the discrete-time, finite-horizon
LQG problem, where the noise distributions are unknown and belong to
Wasserstein ambiguity sets centered at nominal (Gaussian) distributions. The
objective is to minimize a worst-case cost across all distributions in the
ambiguity set, including non-Gaussian distributions. Despite the added
complexity, we prove that a control policy that is linear in the observations
is optimal for this problem, as in the classic LQG problem. We propose a
numerical solution method that efficiently characterizes this optimal control
policy. Our method uses the Frank-Wolfe algorithm to identify the
least-favorable distributions within the Wasserstein ambiguity sets and
computes the controller's optimal policy using Kalman filter estimation under
these distributions.