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Singular Arcs in Optimal Control: Closed-loop Implementations without Workarounds

Nikilesh Ramesh, Ross Drummond, Pablo Rodolfo Baldivieso Monasterios, Yuanbo Nie

TL;DR

The paper tackles the difficulty of solving optimal control problems with singular arcs in online closed-loop settings. It advocates Integrated Residual Methods (IRM), particularly IRR-DC, as a direct transcription-based approach that suppresses singular-arc fluctuations by minimizing the integral of residuals, avoiding the need for problem-specific treatments. Through benchmark problems including a second-order singular control task and a Single Machine Infinite Bus system, the authors show that IRR-DC yields closed-loop solutions that closely match analytic optima, reduces fluctuations compared to direct collocation, and remains robust without prior knowledge of singular arc structure. The findings suggest IRM-based EMPC offers practical and reliable performance for large-scale online OCPs, reducing the need for iterative retuning or structural analyses. Future work aims to dissect the influence of MPC configurations on closed-loop cost and singular-arc suppression.

Abstract

Singular arcs emerge in the solutions of Optimal Control Problems (OCPs) when the optimal inputs on some finite time intervals cannot be directly obtained via the optimality conditions. Solving OCPs with singular arcs often requires tailored treatments, suitable for offline trajectory optimization. This approach can become increasingly impractical for online closed-loop implementations, especially for large-scale engineering problems. Recent development of Integrated Residual Methods (IRM) have indicated their suitability for handling singular arcs; the convergence of error measures in IRM automatically suppresses singular arc-induced fluctuations and leads to non-fluctuating solutions more suitable for practical problems. Through several examples, we demonstrate the advantages of solving OCPs with singular arcs using {IRM} under an economic model predictive control framework. In particular, the following observations are made: (i) IRM does not require special treatment for singular arcs, (ii) it solves the OCPs reliably with singular arc fluctuation suppressed, and (iii) the closed-loop results closely match the analytic optimal solutions.

Singular Arcs in Optimal Control: Closed-loop Implementations without Workarounds

TL;DR

The paper tackles the difficulty of solving optimal control problems with singular arcs in online closed-loop settings. It advocates Integrated Residual Methods (IRM), particularly IRR-DC, as a direct transcription-based approach that suppresses singular-arc fluctuations by minimizing the integral of residuals, avoiding the need for problem-specific treatments. Through benchmark problems including a second-order singular control task and a Single Machine Infinite Bus system, the authors show that IRR-DC yields closed-loop solutions that closely match analytic optima, reduces fluctuations compared to direct collocation, and remains robust without prior knowledge of singular arc structure. The findings suggest IRM-based EMPC offers practical and reliable performance for large-scale online OCPs, reducing the need for iterative retuning or structural analyses. Future work aims to dissect the influence of MPC configurations on closed-loop cost and singular-arc suppression.

Abstract

Singular arcs emerge in the solutions of Optimal Control Problems (OCPs) when the optimal inputs on some finite time intervals cannot be directly obtained via the optimality conditions. Solving OCPs with singular arcs often requires tailored treatments, suitable for offline trajectory optimization. This approach can become increasingly impractical for online closed-loop implementations, especially for large-scale engineering problems. Recent development of Integrated Residual Methods (IRM) have indicated their suitability for handling singular arcs; the convergence of error measures in IRM automatically suppresses singular arc-induced fluctuations and leads to non-fluctuating solutions more suitable for practical problems. Through several examples, we demonstrate the advantages of solving OCPs with singular arcs using {IRM} under an economic model predictive control framework. In particular, the following observations are made: (i) IRM does not require special treatment for singular arcs, (ii) it solves the OCPs reliably with singular arc fluctuation suppressed, and (iii) the closed-loop results closely match the analytic optimal solutions.

Paper Structure

This paper contains 20 sections, 20 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Singular arcs
  3. Illustrative example: Aly-Chan problem
  4. Contributions
  5. Numerical Methods for Solving Optimal Control Problem
  6. Indirect method
  7. Direct methods
  8. Solving OCPs with Singular Arcs
  9. Overview of commonly used methods
  10. Indirect methods for solving singular arcs
  11. Direct methods for solving singular arcs
  12. Direct Collocation
  13. Integral Residual Method
  14. Suitability analysis for closed-loop implementation
  15. Numerical Examples
  16. ...and 5 more sections

Figures (4)

  • Figure 1: Solutions to the Aly-Chan problem ALY01041973. All methods implemented with a fixed mesh of 100 nodes and Hermite-Simpson discretisation. The state plot has $[x_1, x_2,x_3]^\top$.
  • Figure 2: Comparison of different solutions to the second order singular control problem. Closed-loop implementations use a time step of $0.01s$ and a horizon of $5s$.
  • Figure 3: Solutions to the SMIB problem with different methods. The state plot represents $[\frac{x_1}{C_1}, \frac{x_2}{C_2}]^\top$ over time.
  • Figure 4: Phase portrait showing the optimal trajectories of the SMIB system for various initial conditions computed using receding horizon implementation of IRR-DC method.

Theorems & Definitions (2)

  • Definition 1: Singular interval
  • Definition 2: Singular arc