Data-Driven Distributionally Robust System Level Synthesis

TL;DR

This paper introduces a data-driven, distributionally robust control method for uncertain linear systems, ensuring reliable performance by accounting for model mismatch and disturbance distribution uncertainty using Wasserstein ambiguity sets.

Contribution

It develops a novel doubly robust, distributionally robust control framework leveraging System Level Synthesis and Wasserstein ambiguity sets, with theoretical guarantees and a tractable approximation.

Findings

Effective control under distributional ambiguity demonstrated in numerical example.

Bounds on sample complexity for desired confidence levels provided.

Tractable approximate formulation enables practical implementation.

Abstract

We present a novel approach for the control of uncertain, linear time-invariant systems, which are perturbed by potentially unbounded, additive disturbances. We propose a \emph{doubly robust} data-driven state-feedback controller to ensure reliable performance against both model mismatch and disturbance distribution uncertainty. Our controller, which leverages the System Level Synthesis parameterization, is designed as the solution to a distributionally robust finite-horizon optimal control problem. The goal is to minimize a cost function while satisfying constraints against the worst-case realization of the uncertainty, which is quantified using distributional ambiguity sets. The latter are defined as balls in the Wasserstein metric centered on the predictive empirical distribution computed from a set of collected trajectory data. By harnessing techniques from robust control and distributionally robust optimization, we characterize the distributional shift between the predictive and the actual closed-loop distributions, and highlight its dependency on the model mismatch and the uncertainty about the disturbance distribution. We also provide bounds on the number of samples required to achieve a desired confidence level and propose a tractable approximate formulation for the doubly robust data-driven controller. To demonstrate the effectiveness of our approach, we present a numerical example showcasing the performance of the proposed algorithm.