The Distributionally Robust Infinite-Horizon LQR

TL;DR

This paper develops a novel distributionally robust infinite-horizon LQR control method that accounts for uncertainty in disturbance distributions within a Wasserstein ball, providing a computationally feasible solution with frequency domain computation and rational approximation.

Contribution

It introduces a new optimality condition for the DR controller, shows it is non-rational, and proposes a fixed-point iteration algorithm for its computation in the frequency domain.

Findings

The DR controller is non-rational and optimal within the Wasserstein ambiguity set.

A fixed-point iteration algorithm computes the controller in the frequency domain.

Rational approximation methods enable finite-order, time-domain implementation.

Abstract

We explore the infinite-horizon Distributionally Robust (DR) linear-quadratic control. While the probability distribution of disturbances is unknown and potentially correlated over time, it is confined within a Wasserstein-2 ball of a radius $r$ around a known nominal distribution. Our goal is to devise a control policy that minimizes the worst-case expected Linear-Quadratic Regulator (LQR) cost among all probability distributions of disturbances lying in the Wasserstein ambiguity set. We obtain the optimality conditions for the optimal DR controller and show that it is non-rational. Despite lacking a finite-order state-space representation, we introduce a computationally tractable fixed-point iteration algorithm. Our proposed method computes the optimal controller in the frequency domain to any desired fidelity. Moreover, for any given finite order, we use a convex numerical method to compute the best rational approximation (in $H_\infty$-norm) to the optimal non-rational DR controller. This enables efficient time-domain implementation by finite-order state-space controllers and addresses the computational hurdles associated with the finite-horizon approaches to DR-LQR problems, which typically necessitate solving a Semi-Definite Program (SDP) with a dimension scaling with the time horizon. We provide numerical simulations to showcase the effectiveness of our approach.