Stability and Performance Analysis of Model Predictive Control of Uncertain Linear Systems

TL;DR

This paper analyzes the stability and performance of model predictive control for uncertain linear systems, providing theoretical conditions and bounds that guide controller design under model mismatch.

Contribution

It introduces stability and performance bounds for MPC applied to uncertain systems with parametric mismatch, using a theoretical framework based on relaxed dynamic programming.

Findings

MPC can stabilize uncertain linear systems under certain conditions.

Derived a performance bound relating prediction horizon and modeling errors.

Validated theoretical results through numerical simulations.

Abstract

Model mismatch often poses challenges in model-based controller design. This paper investigates model predictive control (MPC) of uncertain linear systems with input constraints, focusing on stability and closed-loop infinite-horizon performance. The uncertainty arises from a parametric mismatch between the true and the estimated system under the matrix Frobenius norm. We examine a simple MPC controller that exclusively uses the estimated system model and establishes sufficient conditions under which the MPC controller can stabilize the true system. Moreover, we derive a theoretical performance bound based on relaxed dynamic programming, elucidating the impact of prediction horizon and modeling errors on the suboptimality gap between the MPC controller and the Oracle infinite-horizon optimal controller with knowledge of the true system. Simulations of a numerical example validate the theoretical results. Our theoretical analysis offers guidelines for obtaining the desired modeling accuracy and choosing a proper prediction horizon to develop certainty-equivalent MPC controllers for uncertain linear systems.