Wasserstein Distributionally Robust Regret-Optimal Control in the Infinite-Horizon

TL;DR

This paper develops a Wasserstein distributionally robust control framework for infinite-horizon linear systems, minimizing worst-case regret under correlated disturbances, with a novel finite-dimensional characterization and efficient frequency domain computation.

Contribution

It introduces a new infinite-horizon DR-RO control method allowing for correlated disturbances, with a finite-dimensional controller characterization and a scalable frequency domain solution.

Findings

Controller can be uniquely characterized by finite parameters.

Frequency domain fixed-point iteration efficiently computes the controller.

Numerical experiments confirm the approach's effectiveness.

Abstract

We investigate the Distributionally Robust Regret-Optimal (DR-RO) control of discrete-time linear dynamical systems with quadratic cost over an infinite horizon. Regret is the difference in cost obtained by a causal controller and a clairvoyant controller with access to future disturbances. We focus on the infinite-horizon framework, which results in stability guarantees. In this DR setting, the probability distribution of the disturbances resides within a Wasserstein-2 ambiguity set centered at a specified nominal distribution. Our objective is to identify a control policy that minimizes the worst-case expected regret over an infinite horizon, considering all potential disturbance distributions within the ambiguity set. In contrast to prior works, which assume time-independent disturbances, we relax this constraint to allow for time-correlated disturbances, thus actual distributional robustness. While we show that the resulting optimal controller is non-rational and lacks a finite-dimensional state-space realization, we demonstrate that it can still be uniquely characterized by a finite dimensional parameter. Exploiting this fact, we introduce an efficient numerical method to compute the controller in the frequency domain using fixed-point iterations. This method circumvents the computational bottleneck associated with the finite-horizon problem, where the semi-definite programming (SDP) solution dimension scales with the time horizon. Numerical experiments demonstrate the effectiveness and performance of our framework.