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Numerical optimal control for distributed delay differential equations: A simultaneous approach based on linearization of the delayed variables

Tobias K. S. Ritschel

Abstract

Time delays are ubiquitous in industrial processes, and they must be accounted for when designing control algorithms because they have a significant effect on the process dynamics. Therefore, in this work, we propose a simultaneous approach for numerical optimal control of delay differential equations with distributed time delays. Specifically, we linearize the delayed variables around the current time, and we discretize the resulting implicit differential equations using Euler's implicit method. Furthermore, we transcribe the infinite-dimensional optimal control problem into a finite-dimensional nonlinear program, which we solve using Matlab's fmincon. Finally, we demonstrate the efficacy of the approach using a numerical example involving a molten salt nuclear fission reactor.

Numerical optimal control for distributed delay differential equations: A simultaneous approach based on linearization of the delayed variables

Abstract

Time delays are ubiquitous in industrial processes, and they must be accounted for when designing control algorithms because they have a significant effect on the process dynamics. Therefore, in this work, we propose a simultaneous approach for numerical optimal control of delay differential equations with distributed time delays. Specifically, we linearize the delayed variables around the current time, and we discretize the resulting implicit differential equations using Euler's implicit method. Furthermore, we transcribe the infinite-dimensional optimal control problem into a finite-dimensional nonlinear program, which we solve using Matlab's fmincon. Finally, we demonstrate the efficacy of the approach using a numerical example involving a molten salt nuclear fission reactor.

Paper Structure

This paper contains 14 sections, 58 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Optimal control problem
  3. Steady state and stability
  4. Delay linearization
  5. Steady state and stability
  6. Simultaneous approach
  7. Jacobian of residual functions
  8. Gradient of the objective function
  9. Implementation
  10. Distributed time delays
  11. Hagen-Poiseuille flow
  12. Molten salt nuclear reactor
  13. Numerical example
  14. Conclusions

Figures (3)

  • Figure 1: Optimal ramping of the thermal energy generation based on four different time-varying setpoints. Top row: The generated thermal energy and the temperature in the reactor core. Middle row: The thermal reactivity and the temperature in the heat exchanger. Bottom row: The optimal external reactivity and the average velocity corresponding to the optimal pressure difference.
  • Figure 2: The difference between the generated thermal energy obtained with 1) the DDEs where the velocity profile is approximated as described in the Appendix and 2) the approximate system \ref{['eq:ide']} where the delayed variable is linearized.
  • Figure 3: The concentrations of the neutron precursor groups for the simulation shown in Fig. \ref{['fig:optimal:control:comparison']} where the thermal energy generation is increased to $10$ MW.