Optimality of Linear Policies for Distributionally Robust Linear Quadratic Gaussian Regulator with Stationary Distributions

TL;DR

This paper demonstrates that linear policies remain optimal for distributionally robust LQG control under certain stationary noise distribution assumptions, and characterizes worst-case distributions and Nash equilibria.

Contribution

It proves the optimality of linear policies under Wasserstein ambiguity sets and establishes a Nash equilibrium framework for the distributionally robust LQG problem.

Findings

Linear policies are optimal under Wasserstein ambiguity sets.

Existence of Nash equilibrium between control and adversary.

Quasi closed-form solutions for worst-case distributions.

Abstract

We prove that output-feedback linear policies remain optimal for solving the Linear Quadratic Gaussian regulation problem in the face of worst-case process and measurement noise distributions when these are independent, stationary, and known to be within a radius (in the Wasserstein sense) to some reference zero-mean Gaussian noise distributions. Additionally, we establish the existence of a Nash equilibrium of the zero-sum game between a control engineer, who minimizes control cost, and a fictitious adversary, who chooses the noise distributions that maximize this cost. For general (possibly non-Gaussian) reference noise distributions, we establish a quasi closed-form solution for the worst-case distributions against linear policies. Our work provides a less conservative alternative compared to recent work in distributionally robust control.