Beyond inherent robustness: strong stability of MPC despite plant-model mismatch

TL;DR

This paper proves that Model Predictive Control (MPC) can be strongly stable even with plant-model mismatch, extending the understanding of MPC robustness to include asymptotic stability under certain conditions.

Contribution

The paper introduces new theoretical results demonstrating asymptotic stability of MPC despite plant-model mismatch, including for systems with general cost functions.

Findings

Exponential stability for quadratic cost MPC with differentiable dynamics

Asymptotic stability for general cost MPC using comparison functions

Numerical validation on the upright pendulum problem

Abstract

In this technical report, we establish the asymptotic stability of MPC under plant-model mismatch for problems where the origin remains a steady state despite mismatch. This class of problems includes, but is not limited to, inventory management, path-planning, and control of systems in deviation variables. Our results differ from prior results on the inherent robustness of MPC, which guarantee only convergence to a neighborhood of the origin, the size of which scales with the magnitude of the mismatch. For MPC with quadratic costs, continuous differentiability of the system dynamics is sufficient to demonstrate exponential stability of the closed-loop system despite mismatch. For MPC with general costs, a joint comparison function bound and scaling condition guarantee asymptotic stability despite mismatch. The results are illustrated in numerical simulations, including the classic upright pendulum problem. The tools developed to establish these results can address the stability of offset-free MPC, an open and interesting question in the MPC research literature.