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A Comparative Study of Sensitivity Computations in ESDIRK-Based Optimal Control Problems

Anders Hilmar Damm Andersen, John Bagterp Jørgensen

TL;DR

This work addresses efficient gradient computation for OCPs solved with fixed-step ESDIRK integrators by comparing iterated IND, which differentiates the discretization and reuses Newton-related structures, against direct IND, which uses exact stage solves. The authors benchmark these approaches on a quadruple-tank system within a multiple-shooting NLP solved by SQP, evaluating metrics such as SQP/QP iterations, KKT violations, and Jacobian updates. Results show that iterated IND delivers performance near a heavily refactorized base-case while reducing LU factorizations and updates, and generally outperforms direct IND, particularly for longer control-intervals; direct IND may converge only under favorable tolerances or shorter horizons. The findings guide practitioners in selecting sensitivity computation strategies for ESDIRK-based OCPs, balancing accuracy, efficiency, and robustness in gradient-based optimization.

Abstract

In this paper, we compare the impact of iterated and direct approaches to sensitivity computation in fixed-step explicit singly diagonally-implicit Runge-Kutta (ESDIRK) methods when applied to optimal control problems (OCPs). We use the principle of internal numerical differentiation (IND) strictly for the iterated approach, i.e., reusing the iteration matrix factorizations, the number of Newton-type iterations, and Newton iterates, to compute the sensitivities. The direct method computes the sensitivities without using the Newton schemes. We compare the impact of the iterated and direct sensitivity computations in OCPs for the quadruple tank system. We benchmark the iterated and direct approaches with a base case. This base case is an OCP that applies an ESDIRK method that refactorizes the iteration matrix in every Newton iteration and uses a direct approach for sensitivity computations. In these OCPs, we vary the number of integration steps between control intervals and we evaluate the performance based on the number of SQP and QPs iterations, KKT violations, and the total number of function evaluations, Jacobian updates, and iteration matrix factorizations. The results indicate that the iterated approach outperforms the direct approach but yields similar performance to the base case.

A Comparative Study of Sensitivity Computations in ESDIRK-Based Optimal Control Problems

TL;DR

This work addresses efficient gradient computation for OCPs solved with fixed-step ESDIRK integrators by comparing iterated IND, which differentiates the discretization and reuses Newton-related structures, against direct IND, which uses exact stage solves. The authors benchmark these approaches on a quadruple-tank system within a multiple-shooting NLP solved by SQP, evaluating metrics such as SQP/QP iterations, KKT violations, and Jacobian updates. Results show that iterated IND delivers performance near a heavily refactorized base-case while reducing LU factorizations and updates, and generally outperforms direct IND, particularly for longer control-intervals; direct IND may converge only under favorable tolerances or shorter horizons. The findings guide practitioners in selecting sensitivity computation strategies for ESDIRK-based OCPs, balancing accuracy, efficiency, and robustness in gradient-based optimization.

Abstract

In this paper, we compare the impact of iterated and direct approaches to sensitivity computation in fixed-step explicit singly diagonally-implicit Runge-Kutta (ESDIRK) methods when applied to optimal control problems (OCPs). We use the principle of internal numerical differentiation (IND) strictly for the iterated approach, i.e., reusing the iteration matrix factorizations, the number of Newton-type iterations, and Newton iterates, to compute the sensitivities. The direct method computes the sensitivities without using the Newton schemes. We compare the impact of the iterated and direct sensitivity computations in OCPs for the quadruple tank system. We benchmark the iterated and direct approaches with a base case. This base case is an OCP that applies an ESDIRK method that refactorizes the iteration matrix in every Newton iteration and uses a direct approach for sensitivity computations. In these OCPs, we vary the number of integration steps between control intervals and we evaluate the performance based on the number of SQP and QPs iterations, KKT violations, and the total number of function evaluations, Jacobian updates, and iteration matrix factorizations. The results indicate that the iterated approach outperforms the direct approach but yields similar performance to the base case.
Paper Structure (16 sections, 28 equations, 5 figures, 2 tables)

This paper contains 16 sections, 28 equations, 5 figures, 2 tables.

Table of Contents

  1. INTRODUCTION
  2. THE ESDIRK METHODS FOR ODES
  3. Computation of initial guesses for Newton iterations
  4. SENSITIVITY ANALYSIS
  5. Iterated IND
  6. State sensitivity
  7. Input sensitivity
  8. Implementations details
  9. Direct IND
  10. Comments on iteration matrix refactorization strategies
  11. OPTIMAL CONTROL PROBLEM
  12. NUMERICAL EXPERIMENTS
  13. Varying integration step-size between control intervals
  14. Low SQP tolerance and short control intervals
  15. Comments on the numerical experiments
  16. ...and 1 more sections

Figures (5)

  • Figure 1: A solution of the OCP applying ESDIRK12 with $N=10$ and iterated IND sensitivities.
  • Figure 2: OCP statistics using ESDIRK12.
  • Figure 3: OCP statistics using ESDIRK23.
  • Figure 4: OCP statistics using ESDIRK34.
  • Figure 5: Comparison of ESDIRK-based OCP algorithms with direct and iterated sensitivity computations. All methods apply 10 integration steps between control intervals..