A Lyapunov Function for the Combined System-Optimizer Dynamics in Inexact Model Predictive Control

TL;DR

This paper develops a Lyapunov-based stability proof for inexact nonlinear model predictive control, demonstrating that stability can be achieved with limited optimizer iterations and short sampling times, bridging theory and practice.

Contribution

It introduces a Lyapunov function for combined system-optimizer dynamics, extending stability results to broader classes of optimization methods in MPC.

Findings

Asymptotic stability proven for inexact MPC with limited optimizer iterations.

Explicit Lyapunov function constructed for combined dynamics.

Stability guaranteed under short sampling times and Lipschitz continuity.

Abstract

In this paper, an asymptotic stability proof for a class of methods for inexact nonlinear model predictive control is presented. General Q-linearly convergent online optimization methods are considered and an asymptotic stability result is derived for the setting where a limited number of iterations of the optimizer are carried out per sampling time. Under the assumption of Lipschitz continuity of the solution, we explicitly construct a Lyapunov function for the combined system-optimizer dynamics, which shows that asymptotic stability can be obtained if the sampling time is sufficiently short. The results constitute an extension to existing attractivity results which hold in the simplified setting where inequality constraints are either not present or inactive in the region of attraction considered. Moreover, with respect to the established results on robust asymptotic stability of suboptimal model predictive control, we develop a framework that takes into account the optimizer's dynamics and does not require decrease of the objective function across iterates. By extending these results, the gap between theory and practice of the standard real-time iteration strategy is bridged and asymptotic stability for a broader class of methods is guaranteed.