Distributionally Robust Model Predictive Control: Closed-loop Guarantees and Scalable Algorithms

TL;DR

This paper develops closed-loop performance guarantees and scalable algorithms for distributionally robust model predictive control (DRMPC) in linear systems, ensuring stability and robustness against uncertainties with practical computational methods.

Contribution

It introduces a scalable Newton-type algorithm for DRMPC and establishes closed-loop guarantees, including stability and performance independence from ambiguity set choices.

Findings

The proposed algorithm empirically achieves superlinear convergence.

Closed-loop guarantees ensure stability and performance bounds.

The origin input-to-state stability is achieved with Riccati-based terminal costs.

Abstract

We establish a collection of closed-loop guarantees and propose a scalable optimization algorithm for distributionally robust model predictive control (DRMPC) applied to linear systems, convex constraints, and quadratic costs. Via standard assumptions for the terminal cost and constraint, we establish distribtionally robust long-term and stage-wise performance guarantees for the closed-loop system. We further demonstrate that a common choice of the terminal cost, i.e., via the discrete-algebraic Riccati equation, renders the origin input-to-state stable for the closed-loop system. This choice also ensures that the exact long-term performance of the closed-loop system is independent of the choice of ambiguity set for the DRMPC formulation. Thus, we establish conditions under which DRMPC does not provide a long-term performance benefit relative to stochastic MPC. To solve the DRMPC optimization problem, we propose a Newton-type algorithm that empirically achieves superlinear convergence and guarantees the feasibility of each iterate. We demonstrate the implications of the closed-loop guarantees and the scalability of the proposed algorithm via two examples. To facilitate the reproducibility of the results, we also provide open-source code to implement the proposed algorithm and generate the figures.