Wasserstein Distributionally Robust Control of Partially Observable Linear Stochastic Systems

TL;DR

This paper introduces a novel Wasserstein distributionally robust control approach for partially observable linear stochastic systems, providing a tractable solution with stability guarantees and demonstrated effectiveness in power system frequency control.

Contribution

It develops a new approximation method using the Gelbrich bound for partially observable DRC, deriving a closed-form control policy and a semidefinite program for worst-case distribution, with theoretical guarantees.

Findings

Guaranteed cost property of the controller

Probabilistic out-of-sample performance guarantee

Effective empirical performance in power system control

Abstract

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.