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$π$MPC: A Parallel-in-horizon and Construction-free NMPC Solver

Liang Wu, Bo Yang, Xu Yang, Yilin Mo, Yang Shi, Ján Drgoňa

TL;DR

The paper tackles NMPC for long horizons on embedded hardware by addressing the limitations of existing ADMM-based solvers, which are either general QP solvers or sequential Riccati-based methods. It introduces πMPC, a parallel-in-horizon, construction-free ADMM solver that uses a velocity-based formulation and a novel variable-splitting to enable horizon-wise parallel updates with closed-form steps, augmented by an accelerated ADMM with restart. The approach operates directly on system matrices without explicit MPC-to-QP construction and is validated through AFTI-16, scalability, and nonlinear CSTR benchmarks, showing competitive per-iteration efficiency and strong GPU scalability. The construction-free design reduces online software complexity and deployment barriers, enabling efficient long-horizon NMPC on embedded and GPU platforms, with future work aimed at integrating differentiable πMPC as a layer in deep learning models.

Abstract

The alternating direction method of multipliers (ADMM) has gained increasing popularity in embedded model predictive control (MPC) due to its code simplicity and pain-free parameter selection. However, existing ADMM solvers either target general quadratic programming (QP) problems or exploit sparse MPC formulations via Riccati recursions, which are inherently sequential and therefore difficult to parallelize for long prediction horizons. This technical note proposes a novel \textit{parallel-in-horizon} and \textit{construction-free} nonlinear MPC algorithm, termed $π$MPC, which combines a new variable-splitting scheme with a velocity-based system representation in the ADMM framework, enabling horizon-wise parallel execution while operating directly on system matrices without explicit MPC-to-QP construction. Numerical experiments and accompanying code are provided to validate the effectiveness of the proposed method.

$π$MPC: A Parallel-in-horizon and Construction-free NMPC Solver

TL;DR

The paper tackles NMPC for long horizons on embedded hardware by addressing the limitations of existing ADMM-based solvers, which are either general QP solvers or sequential Riccati-based methods. It introduces πMPC, a parallel-in-horizon, construction-free ADMM solver that uses a velocity-based formulation and a novel variable-splitting to enable horizon-wise parallel updates with closed-form steps, augmented by an accelerated ADMM with restart. The approach operates directly on system matrices without explicit MPC-to-QP construction and is validated through AFTI-16, scalability, and nonlinear CSTR benchmarks, showing competitive per-iteration efficiency and strong GPU scalability. The construction-free design reduces online software complexity and deployment barriers, enabling efficient long-horizon NMPC on embedded and GPU platforms, with future work aimed at integrating differentiable πMPC as a layer in deep learning models.

Abstract

The alternating direction method of multipliers (ADMM) has gained increasing popularity in embedded model predictive control (MPC) due to its code simplicity and pain-free parameter selection. However, existing ADMM solvers either target general quadratic programming (QP) problems or exploit sparse MPC formulations via Riccati recursions, which are inherently sequential and therefore difficult to parallelize for long prediction horizons. This technical note proposes a novel \textit{parallel-in-horizon} and \textit{construction-free} nonlinear MPC algorithm, termed MPC, which combines a new variable-splitting scheme with a velocity-based system representation in the ADMM framework, enabling horizon-wise parallel execution while operating directly on system matrices without explicit MPC-to-QP construction. Numerical experiments and accompanying code are provided to validate the effectiveness of the proposed method.
Paper Structure (16 sections, 5 theorems, 34 equations, 4 figures, 2 tables, 1 algorithm)

This paper contains 16 sections, 5 theorems, 34 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Contribution
  4. Problem Formulation
  5. Preliminaries: Previous ADMM for MPC
  6. General ADMM algorithm
  7. Structure-exploited ADMM via Riccati Recursion
  8. Parallel-in-horizon ADMM Algorithm
  9. Closed-form ADMM Sub-steps via Velocity-based Formulation
  10. Residuals for Stopping Criteria
  11. Accelerated ADMM with Restart
  12. Numerical Example
  13. AFTI-16 Aircraft System
  14. Randomly Generated System
  15. Nonlinear CSTR System
  16. ...and 1 more sections

Key Result

Theorem 1

The update Eqn. eqn_ADMM2_v_z_update admits a closed-form solution as follows,

Figures (4)

  • Figure 1: AFTI-16 closed-loop trajectory tracking performance
  • Figure 2: Convergence iterations to $10^{-6}$ residual at each MPC step for AFTI-16 (capped at 10,000)
  • Figure 3: Per-iteration computation time as a function of (a) prediction horizon, where system dimension is $(n, m) = (100, 30)$ and (b) system dimension, where the prediction horizon is $N = 500$
  • Figure 4: CSTR trajectory tracking with inlet temperature disturbance

Theorems & Definitions (13)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 5
  • Lemma 1
  • ...and 3 more