Superconvergence of Online Optimization for Model Predictive Control

TL;DR

This paper introduces a novel online MPC algorithm that achieves superconvergence, with errors decreasing exponentially as the horizon shifts increase, validated through theoretical analysis and numerical experiments.

Contribution

The paper presents a one-Newton-step-per-horizon online MPC algorithm demonstrating superconvergence with exponential error decay, based on recent sensitivity analysis.

Findings

Error decreases exponentially with horizon shift order

Algorithm achieves quadratic convergence per Newton's method

Numerical results confirm theoretical superconvergence

Abstract

We develop a one-Newton-step-per-horizon, online, lag-$L$, model predictive control (MPC) algorithm for solving discrete-time, equality-constrained, nonlinear dynamic programs. Based on recent sensitivity analysis results for the target problems class, we prove that the approach exhibits a behavior that we call superconvergence; that is, the tracking error with respect to the full horizon solution is not only stable for successive horizon shifts, but also decreases with increasing shift order to a minimum value that decays exponentially in the length of the receding horizon. The key analytical step is the decomposition of the one-step error recursion of our algorithm into algorithmic error and perturbation error. We show that the perturbation error decays exponentially with the lag between two consecutive receding horizons, while~the algorithmic error, determined by Newton's method, achieves quadratic convergence instead. Overall this approach induces our local exponential convergence result in terms of the receding horizon length for suitable values of $L$. Numerical experiments validate our theoretical findings.