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Continuation Method for Nonsmooth Model Predictive Control Using Proximal Technique

Ryotaro Shima, Ryuta Moriyasu, Teruki Kato

TL;DR

The paper addresses online nonsmooth model predictive control by applying a continuation method together with a proximal operator to convert the nonsmooth first-order inclusion into a linear equality system $F(z,x)=0$, enabling real-time tracking of the optimal solution. It derives a prox-based reformulation, establishes differentiability and constraint-qualification conditions to guarantee well-posedness and unique Lagrange multipliers, and demonstrates the method on a sparse MPC example where it yields explicit sparsity and smaller residuals compared to a differentiable approximation baseline. The approach facilitates solving a linear equation system for the control updates, without introducing slack variables for nonsmoothness, and can leverage efficient linear solvers such as CG/GMRES. The results indicate improved sparsity, stable performance, and competitive computation times, highlighting practical impact for real-time sparse MPC in nonlinear settings.

Abstract

This paper presents a novel framework for the continuation method of model predictive control based on optimal control problem with a nonsmooth regularizer. Via the proximal operator, the first-order optimality inclusion relation is reformulated into an equation system, to which the continuation method is applicable. In addition, we present constraint qualifications that ensure the well-posedness of the proposed equation system. A numerical example is also presented that demonstrates the effectiveness of our approach.

Continuation Method for Nonsmooth Model Predictive Control Using Proximal Technique

TL;DR

The paper addresses online nonsmooth model predictive control by applying a continuation method together with a proximal operator to convert the nonsmooth first-order inclusion into a linear equality system , enabling real-time tracking of the optimal solution. It derives a prox-based reformulation, establishes differentiability and constraint-qualification conditions to guarantee well-posedness and unique Lagrange multipliers, and demonstrates the method on a sparse MPC example where it yields explicit sparsity and smaller residuals compared to a differentiable approximation baseline. The approach facilitates solving a linear equation system for the control updates, without introducing slack variables for nonsmoothness, and can leverage efficient linear solvers such as CG/GMRES. The results indicate improved sparsity, stable performance, and competitive computation times, highlighting practical impact for real-time sparse MPC in nonlinear settings.

Abstract

This paper presents a novel framework for the continuation method of model predictive control based on optimal control problem with a nonsmooth regularizer. Via the proximal operator, the first-order optimality inclusion relation is reformulated into an equation system, to which the continuation method is applicable. In addition, we present constraint qualifications that ensure the well-posedness of the proposed equation system. A numerical example is also presented that demonstrates the effectiveness of our approach.
Paper Structure (16 sections, 5 theorems, 23 equations, 5 figures, 1 algorithm)

This paper contains 16 sections, 5 theorems, 23 equations, 5 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Continuation Method continuationbookcgmres
  4. Nonsmooth Calculus hilbert
  5. Proximal Operator proxnagahilbert
  6. NONSMOOTH MODEL PREDICTIVE CONTROL
  7. Nonsmooth Optimal Control Problem
  8. Nonsmooth MPC by Continuation Method
  9. Well-posedness of Linear Equation System
  10. Differentiability of $F$
  11. Regularity of Jacobi matrix
  12. NUMERICAL EXAMPLE: SPARSE MPC
  13. Problem Setting
  14. Conventional Method for Comparison
  15. Results of Control and Discussion
  16. ...and 1 more sections

Key Result

Lemma 1

Consider the following nonsmooth optimization problem: where $f$ is a differentiable function and $r$ is nonsmooth function. Then, the optimal solution $x^\ast$ satisfies the following first-order optimality condition:

Figures (5)

  • Figure 1: Inputs calculated by the proposed method.
  • Figure 2: Inputs calculated by the conventional method.
  • Figure 3: Residual of the equation system.
  • Figure 4: Trajectory of states controlled by the proposed method.
  • Figure : Example of nonsmooth MPC using continuation method.

Theorems & Definitions (15)

  • Definition 1: closed convex proper function
  • Definition 2: subderivative
  • Example 1
  • Lemma 1: Theorem 16.3 in hilbert
  • Definition 3: proximal operator
  • Example 2
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • ...and 5 more