Accurate Solutions to Optimal Control Problems via a Flexible Mesh and Integrated Residual Transcription

TL;DR

This paper introduces a novel method combining flexible mesh design with integrated residual transcription to enhance the accuracy of numerical solutions in optimal control problems, especially for non-smooth or stiff dynamics.

Contribution

It presents the first numerical implementation of a flexible mesh with integrated residual transcription, improving convergence and accuracy over traditional fixed mesh methods.

Findings

Over two times more accurate than fixed mesh collocation at the same computational cost.

Flexible meshing captures discontinuities exactly.

Method benefits real-time control applications.

Abstract

We propose joining a flexible mesh design with an integrated residual transcription in order to improve the accuracy of numerical solutions to optimal control problems. This approach is particularly useful when state or input trajectories are non-smooth, but it may also be beneficial when dynamics constraints are stiff. Additionally, we implement an initial phase that will ensure a feasible solution is found and can be implemented immediately in real-time controllers. Subsequent iterations with warm-starting will improve the solution until optimality is achieved. Optimizing over the mesh node locations allows for discontinuities to be captured exactly, while integrated residuals account for the approximation error in-between the nodal points. First, we numerically show the improved convergence order for the flexible mesh. We then present the feasibility-driven approach to solve control problems and show how flexible meshing and integrated residual methods can be used in practice. The presented numerical examples demonstrate for the first time the numerical implementation of a flexible mesh for an integrated residual transcription. The results show that our proposed method can be more than two times more accurate than conventional fixed mesh collocation for the same computational time and more than three times more accurate for the same problem size.