Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Reliable Solution to Dynamic Optimization Problems using Integrated Residual Regularized Direct Collocation

Yuanbo Nie, Eric C. Kerrigan

TL;DR

The paper tackles dynamic optimization problems where direct collocation struggles, especially in the presence of singular arcs. It introduces Integrated Residual Regularized Direct Collocation (IRR-DC), which augments direct transcription with an integrated residual penalty and allows dynamic constraints to be enforced as equalities or inequalities, balanced by a tunable regularization parameter. Across Goddard Rocket and Ventilator Control examples, IRR-DC suppresses singular-arc oscillations, yields substantially lower error metrics, and accelerates mesh refinement compared to standard DC and other regularization approaches, while remaining easy to configure for off-the-shelf solvers. This work demonstrates that IRR-DC delivers higher reliability and accuracy without sacrificing DC’s simplicity or computational efficiency, making it a practical option for challenging dynamic optimization problems with high-index DAEs or algebraic constraints.

Abstract

Direct collocation is a widely used method for solving dynamic optimization problems (DOPs), but its implementation simplicity and computational efficiency are limited for challenging problems like those involving singular arcs. In this paper, we introduce the direct transcription method of integrated residual regularized direct collocation (IRR-DC). This method enforces dynamic constraints through a combination of explicit constraints and penalty terms within discretized DOPs. This method retains the implementation simplicity of direct collocation while significantly improving both solution accuracy and efficiency, particularly for challenging problem types. Through the examples, we demonstrate that for difficult problems where traditional direct collocation results in excessive fluctuations or large errors between collocation points, IRR-DC effectively suppresses oscillations and yields solutions with greater accuracy (several magnitudes lower in various error metrics) compared to other regularization alternatives.

Reliable Solution to Dynamic Optimization Problems using Integrated Residual Regularized Direct Collocation

TL;DR

The paper tackles dynamic optimization problems where direct collocation struggles, especially in the presence of singular arcs. It introduces Integrated Residual Regularized Direct Collocation (IRR-DC), which augments direct transcription with an integrated residual penalty and allows dynamic constraints to be enforced as equalities or inequalities, balanced by a tunable regularization parameter. Across Goddard Rocket and Ventilator Control examples, IRR-DC suppresses singular-arc oscillations, yields substantially lower error metrics, and accelerates mesh refinement compared to standard DC and other regularization approaches, while remaining easy to configure for off-the-shelf solvers. This work demonstrates that IRR-DC delivers higher reliability and accuracy without sacrificing DC’s simplicity or computational efficiency, making it a practical option for challenging dynamic optimization problems with high-index DAEs or algebraic constraints.

Abstract

Direct collocation is a widely used method for solving dynamic optimization problems (DOPs), but its implementation simplicity and computational efficiency are limited for challenging problems like those involving singular arcs. In this paper, we introduce the direct transcription method of integrated residual regularized direct collocation (IRR-DC). This method enforces dynamic constraints through a combination of explicit constraints and penalty terms within discretized DOPs. This method retains the implementation simplicity of direct collocation while significantly improving both solution accuracy and efficiency, particularly for challenging problem types. Through the examples, we demonstrate that for difficult problems where traditional direct collocation results in excessive fluctuations or large errors between collocation points, IRR-DC effectively suppresses oscillations and yields solutions with greater accuracy (several magnitudes lower in various error metrics) compared to other regularization alternatives.

Paper Structure

This paper contains 16 sections, 21 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Numerical Solution of Dynamic Optimisation Problems
  3. Direct Transcription methods
  4. Handing of dynamic constrains
  5. Implementing dynamic relationships as equality constraints in the DDOP
  6. Implementing dynamic relationships as inequality constraints or as penalty terms in the DDOP
  7. Error measures in direct transcription methods
  8. Characteristics of Direct Collocation and Integrated Residual Approaches
  9. Integrated Residual Regularized Direct Collocation
  10. Example Problems
  11. Singular Control Example: Goddard Rocket
  12. Suppression of singular arc fluctuations
  13. Mesh refinement performance
  14. Comparison with other commonly used regularization mechanisms
  15. Algebraic Constraint Example: Ventilator Control
  16. ...and 1 more sections

Figures (3)

  • Figure 1: Numerical solution to the Goddard rocket problem (Hermite-Simpson discretisation)
  • Figure 2: Comparison of different regularization schemes for the suppression of singular arc fluctuations. Value for the dot-dashed line is from betts2010practical.
  • Figure 3: Comparison of residual error for the algebraic constraint in the ventilator control problem