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A sparse ADMM-based solver for linear MPC subject to terminal quadratic constraint

Pablo Krupa, Rim Jaouani, Daniel Limon, Teodoro Alamo

TL;DR

The paper introduces a sparse ADMM-based solver for linear MPC with a terminal ellipsoidal constraint, avoiding SOC reformulations and extra slack variables. By designing the ADMM equality constraints to incorporate the ellipsoid via a forward-looking transform with $P^{1/2}$, the projection step becomes explicit, preserving the sparsity and enabling efficient embedded implementations. The method achieves favorable performance against SOC-based and general solvers in case studies and demonstrates viability on a low-resource embedded platform. This work provides a practical, memory-efficient approach for MPC with ellipsoidal terminal constraints and broadens applicability in robust and large-scale systems. The solver is openly accessible in the SPCIES toolbox, supporting embedded optimization applications.

Abstract

Model Predictive Control (MPC) typically includes a terminal constraint to guarantee stability of the closed-loop system under nominal conditions. In linear MPC this constraint is generally taken on a polyhedral set, leading to a quadratic optimization problem. However, the use of an ellipsoidal terminal constraint may be desirable, leading to an optimization problem with a quadratic constraint. In this case, the optimization problem can be solved using Second Order Cone (SOC) programming solvers, since the quadratic constraint can be posed as a SOC constraint, at the expense of adding additional slack variables and possibly compromising the simple structure of the solver ingredients. In this paper we present a sparse solver for linear MPC subject to a terminal ellipsoidal constraint based on the alternating direction method of multipliers algorithm in which we directly deal with the quadratic constraints without having to resort to the use of a SOC constraint nor the inclusion of additional decision variables. The solver is suitable for its use in embedded systems, since it is sparse, has a small memory footprint and requires no external libraries. We compare its performance against other approaches from the literature.

A sparse ADMM-based solver for linear MPC subject to terminal quadratic constraint

TL;DR

The paper introduces a sparse ADMM-based solver for linear MPC with a terminal ellipsoidal constraint, avoiding SOC reformulations and extra slack variables. By designing the ADMM equality constraints to incorporate the ellipsoid via a forward-looking transform with , the projection step becomes explicit, preserving the sparsity and enabling efficient embedded implementations. The method achieves favorable performance against SOC-based and general solvers in case studies and demonstrates viability on a low-resource embedded platform. This work provides a practical, memory-efficient approach for MPC with ellipsoidal terminal constraints and broadens applicability in robust and large-scale systems. The solver is openly accessible in the SPCIES toolbox, supporting embedded optimization applications.

Abstract

Model Predictive Control (MPC) typically includes a terminal constraint to guarantee stability of the closed-loop system under nominal conditions. In linear MPC this constraint is generally taken on a polyhedral set, leading to a quadratic optimization problem. However, the use of an ellipsoidal terminal constraint may be desirable, leading to an optimization problem with a quadratic constraint. In this case, the optimization problem can be solved using Second Order Cone (SOC) programming solvers, since the quadratic constraint can be posed as a SOC constraint, at the expense of adding additional slack variables and possibly compromising the simple structure of the solver ingredients. In this paper we present a sparse solver for linear MPC subject to a terminal ellipsoidal constraint based on the alternating direction method of multipliers algorithm in which we directly deal with the quadratic constraints without having to resort to the use of a SOC constraint nor the inclusion of additional decision variables. The solver is suitable for its use in embedded systems, since it is sparse, has a small memory footprint and requires no external libraries. We compare its performance against other approaches from the literature.

Paper Structure

This paper contains 10 sections, 1 theorem, 30 equations, 2 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Model predictive control formulation
  3. Sparse ADMM-based solver
  4. Alternating Direction Method of Multipliers
  5. Solving the quadratically-constrained MPC using ADMM
  6. Case study
  7. Comparison with state-of-the-art solvers
  8. Implementation on an embedded system
  9. Conclusions
  10. Proof of Proposition \ref{['theo:P:projection']}

Key Result

Proposition 1

Let $a \in{\mathbb{R}}^{n}$, $P \in\mathbb{S}_{+}^{n}$, $c \in{\mathbb{R}}^{n}$, $r \in {\mathbb{R}} > 0$. Then, the solution $v^*$ of the convex optimization problem is given by $v^* = $

Figures (2)

  • Figure 1: Chain of three masses.
  • Figure 2: Closed-loop results on the reactors and separator chemical system. In subfigures (c) and (d), the results for the equMPC are presented in the right-hand-side axis.

Theorems & Definitions (4)

  • Remark 1
  • Proposition 1
  • Remark 2
  • Remark 3