[Paper Review] Preconditioned Continuation Model Predictive Control

Andrew Knyazev, Yuta Fujii+1 more|arXiv (Cornell University)|Jun 8, 2015

Matrix Theory and Algorithms11 references4 citations

TL;DR

This paper proposes a preconditioned continuation model predictive control (CNMPC) framework for nonlinear optimal control, enhancing convergence of iterative solvers via efficient preconditioning and replacing GMRES with MINRES. It achieves significantly faster online computation and improved control quality in minimum-time problems, with parallelizable preconditioner setup enabling real-time applicability on modern hardware.

ABSTRACT

Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. A Continuation/GMRES Method for NMPC, suggested by T. Ohtsuka in 2004, uses the GMRES iterative algorithm to solve a forward difference approximation $Ax=b$ of the Continuation NMPC (CNMPC) equations on every time step. The coefficient matrix $A$ of the linear system is often ill-conditioned, resulting in poor GMRES convergence, slowing down the on-line computation of the control by CNMPC, and reducing control quality. We adopt CNMPC for challenging minimum-time problems, and improve performance by introducing efficient preconditioning, utilizing parallel computing, and substituting MINRES for GMRES.

Motivation & Objective

• To extend Continuation NMPC to handle minimum-time optimal control problems with terminal constraints and design parameters.
• To address slow convergence of GMRES in ill-conditioned linear systems arising in CNMPC by introducing effective preconditioning.
• To improve computational efficiency and reduce memory usage by replacing GMRES with MINRES in the iterative solution process.
• To enable real-time implementation through parallel computation of the preconditioner, suitable for multi-core, GPU, or FPGA architectures.
• To demonstrate that preconditioning enhances control quality with minimal computational overhead in challenging nonlinear optimal control scenarios.

Proposed method

• Extends CNMPC formulation to include terminal constraints and optimization parameters, enabling application to time-optimal control problems.
• Uses forward difference approximation to form the linear system $Ax = b$ at each time step, solving via iterative Krylov subspace methods.
• Introduces a preconditioner for GMRES that reduces the condition number of the system matrix, accelerating convergence.
• Replaces GMRES with MINRES as the iterative solver, reducing memory requirements and improving convergence speed in symmetric systems.
• Proposes a concurrent, parallelizable algorithm for preconditioner setup, minimizing online computational cost.
• Employs a time-varying preconditioner recalculated at intervals $t_p$, balancing accuracy and computational load.

Experimental results

Research questions

• RQ1 Can preconditioning significantly accelerate convergence of iterative solvers in Continuation NMPC for nonlinear optimal control?
• RQ2 How does replacing GMRES with MINRES affect convergence speed and memory usage in CNMPC applications?
• RQ3 What is the impact of parallel preconditioner construction on online computation time and real-time control feasibility?
• RQ4 Can the proposed method effectively solve minimum-time optimal control problems with terminal constraints and inequality constraints?
• RQ5 How does the frequency of preconditioner re-computation affect solution quality and computational efficiency?

Key findings

• Preconditioning reduces the number of GMRES iterations from the maximum allowed (10) to as low as 1 or 2, significantly accelerating convergence.
• With $k_{ ext{max}} = 1$ and preconditioner updated every 0.2 seconds, the method achieves near-optimal performance, demonstrating strong robustness.
• Increasing $k_{ ext{max}}$ from 2 to 10 yields only marginal improvements in residual and $\|F\|$, indicating diminishing returns beyond moderate iteration counts.
• The preconditioned GMRES with $t_p = 0.4$ sec and $k_{ ext{max}} = 2$ reduces iterations by a factor of five compared to non-preconditioned GMRES, with similar or better solution quality.
• The use of MINRES instead of GMRES reduces memory requirements and improves convergence speed, particularly beneficial for large-scale or real-time systems.
• Parallel preconditioner setup enables efficient implementation on multi-core, GPU, or FPGA platforms, making the method viable for high-speed control applications.