Contraction Analysis of Continuation Method for Suboptimal Model Predictive Control

TL;DR

This work addresses the contraction properties of nonlinear systems controlled by suboptimal MPC implemented via the continuation method. It introduces a contraction metric $M$ tailored to the continuation’s hierarchical dynamics and derives a pair of matrix inequalities that bound the suboptimality’s impact on contraction toward the optimal trajectory. A numerical example verifies that the suboptimal closed-loop trajectories converge to the optimal MPC trajectory, supporting applicability beyond stabilization to economic MPC. The results provide a principled framework to certify stability and performance for suboptimal MPC with continuation-based updates.

Abstract

This letter analyzes the contraction property of the nonlinear systems controlled by suboptimal model predictive control (MPC) using the continuation method. We propose a contraction metric that reflects the hierarchical dynamics inherent in the continuation method. We derive a pair of matrix inequalities that elucidate the impact of suboptimality on the contraction of the optimally controlled closed-loop system. A numerical example is presented to verify our contraction analysis. Our results are applicable to other MPCs than stabilization, including economic MPC.

Figures (2)

• Figure 1: Trajectories of $x_4$ and $u_2$ under three different initial condition.
• Figure 2: Norm of $\zeta(U^{(1)}(t), x^{(1)}(t), t)$.

Theorems & Definitions (14)

• Definition 1
• Remark 1
• Remark 2
• Lemma 1
• proof
• Remark 3
• Theorem 1
• Corollary 1
• proof
• Lemma 2