Adaptive Stochastic MPC under Unknown Noise Distribution

TL;DR

This paper develops an adaptive stochastic MPC method for linear systems with unknown noise distribution, ensuring constraint satisfaction and improved control performance through online noise statistics learning.

Contribution

It introduces a novel adaptive SMPC scheme that learns noise statistics online and guarantees constraint satisfaction with high probability, enhancing robustness and domain of attraction.

Findings

Guarantees high-probability constraint satisfaction over time

Improves control performance as noise estimates improve

Enlarges domain of attraction in tracking problems

Abstract

In this paper, we address the stochastic MPC (SMPC) problem for linear systems, subject to chance state constraints and hard input constraints, under unknown noise distribution. First, we reformulate the chance state constraints as deterministic constraints depending only on explicit noise statistics. Based on these reformulated constraints, we design a distributionally robust and robustly stable benchmark SMPC algorithm for the ideal setting of known noise statistics. Then, we employ this benchmark controller to derive a novel robustly stable adaptive SMPC scheme that learns the necessary noise statistics online, while guaranteeing time-uniform satisfaction of the unknown reformulated state constraints with high probability. The latter is achieved through the use of confidence intervals which rely on the empirical noise statistics and are valid uniformly over time. Moreover, control performance is improved over time as more noise samples are gathered and better estimates of the noise statistics are obtained, given the online adaptation of the estimated reformulated constraints. Additionally, in tracking problems with multiple successive targets our approach leads to an online-enlarged domain of attraction compared to robust tube-based MPC. A numerical simulation of a DC-DC converter is used to demonstrate the effectiveness of the developed methodology.