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A Parallel-in-Time Newton's Method for Nonlinear Model Predictive Control

Casian Iacob, Hany Abdulsamad, Simo Särkkä

TL;DR

This work tackles the heavy computational burden of constrained nonlinear MPC over long planning horizons by introducing parallel-in-time Newton methods. By embedding three associative scans (co-state, value function, and state propagation) within a unified Newton iteration and coupling them with primal log-barrier IP or ADMM, the authors achieve logarithmic-time scaling with horizon length $O(\log N)$ on GPUs. Numerical results on a torque-limited pendulum and a force-limited cart-pole demonstrate substantial speedups over sequential methods, with additional validation via an MPC simulation showing real-time feasibility. The approach broadens the practical applicability of nonlinear MPC to high-frequency or long-horizon tasks on massively parallel hardware, enabling faster planning and improved stability in constrained dynamical systems.

Abstract

Model predictive control (MPC) is a powerful framework for optimal control of dynamical systems. However, MPC solvers suffer from a high computational burden that restricts their application to systems with low sampling frequency. This issue is further amplified in nonlinear and constrained systems that require nesting MPC solvers within iterative procedures. In this paper, we address these issues by developing parallel-in-time algorithms for constrained nonlinear optimization problems that take advantage of massively parallel hardware to achieve logarithmic computational time scaling over the planning horizon. We develop time-parallel second-order solvers based on interior point methods and the alternating direction method of multipliers, leveraging fast convergence and lower computational cost per iteration. The parallelization is based on a reformulation of the subproblems in terms of associative operations that can be parallelized using the associative scan algorithm. We validate our approach on numerical examples of nonlinear and constrained dynamical systems.

A Parallel-in-Time Newton's Method for Nonlinear Model Predictive Control

TL;DR

This work tackles the heavy computational burden of constrained nonlinear MPC over long planning horizons by introducing parallel-in-time Newton methods. By embedding three associative scans (co-state, value function, and state propagation) within a unified Newton iteration and coupling them with primal log-barrier IP or ADMM, the authors achieve logarithmic-time scaling with horizon length on GPUs. Numerical results on a torque-limited pendulum and a force-limited cart-pole demonstrate substantial speedups over sequential methods, with additional validation via an MPC simulation showing real-time feasibility. The approach broadens the practical applicability of nonlinear MPC to high-frequency or long-horizon tasks on massively parallel hardware, enabling faster planning and improved stability in constrained dynamical systems.

Abstract

Model predictive control (MPC) is a powerful framework for optimal control of dynamical systems. However, MPC solvers suffer from a high computational burden that restricts their application to systems with low sampling frequency. This issue is further amplified in nonlinear and constrained systems that require nesting MPC solvers within iterative procedures. In this paper, we address these issues by developing parallel-in-time algorithms for constrained nonlinear optimization problems that take advantage of massively parallel hardware to achieve logarithmic computational time scaling over the planning horizon. We develop time-parallel second-order solvers based on interior point methods and the alternating direction method of multipliers, leveraging fast convergence and lower computational cost per iteration. The parallelization is based on a reformulation of the subproblems in terms of associative operations that can be parallelized using the associative scan algorithm. We validate our approach on numerical examples of nonlinear and constrained dynamical systems.
Paper Structure (18 sections, 54 equations, 3 figures, 4 algorithms)

This paper contains 18 sections, 54 equations, 3 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Constrained Nonlinear Optimal Control Problem
  4. Primal Log-Barrier Interior Point Method
  5. Alternating Direction Method of Multipliers
  6. Parallel Computation via Prefix-Sum Operations
  7. Parallel Dynamic Programming Realization of Newton's Method
  8. Augmented Costs and Lagrangians for IP and ADMM
  9. Parallel Co-State Pass
  10. Parallel Value Function Pass
  11. Parallel State Propagation Pass
  12. Parallel Newton's Method for Optimal Control
  13. Convergence and Feasibility
  14. Numerical Results
  15. Torque-Limited Pendulum Swing-Up Problem
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Runtime average of parallel and sequential realization of Newton-type ADMM and interior point methods applied to a torque-constrained pendulum swing-up problem solved for various planning horizons.
  • Figure 2: Runtime average of parallel and sequential realization of Newton-type ADMM and interior point methods applied to a force-constrained cart-pole swing-up problem solved for various planning horizons.
  • Figure 3: Model predictive control of constrained pendulum (left) and constrained cart-pole (right).