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

Tobias K. S. Ritschel, Søren Stange

Abstract

Time delays are ubiquitous in industry, and they must be accounted for when designing control strategies. However, numerical optimal control (NOC) of delay differential equations (DDEs) is challenging because it requires specialized discretization methods and the time delays may depend on the manipulated inputs or state variables. Therefore, in this work, we propose to linearize the delayed states around the current time. This results in a set of implicit differential equations, and we compare the steady states and the corresponding stability criteria of the DDEs and the approximate system. Furthermore, we propose a simultaneous approach for NOC of DDEs based on the linearization, and we discretize the approximate system using Euler's implicit method. Finally, we present a numerical example involving a molten salt nuclear fission reactor.

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

Abstract

Time delays are ubiquitous in industry, and they must be accounted for when designing control strategies. However, numerical optimal control (NOC) of delay differential equations (DDEs) is challenging because it requires specialized discretization methods and the time delays may depend on the manipulated inputs or state variables. Therefore, in this work, we propose to linearize the delayed states around the current time. This results in a set of implicit differential equations, and we compare the steady states and the corresponding stability criteria of the DDEs and the approximate system. Furthermore, we propose a simultaneous approach for NOC of DDEs based on the linearization, and we discretize the approximate system using Euler's implicit method. Finally, we present a numerical example involving a molten salt nuclear fission reactor.
Paper Structure (14 sections, 35 equations, 4 figures, 1 table)

This paper contains 14 sections, 35 equations, 4 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. Discretization
  8. Nonlinear program
  9. Jacobian of residual functions
  10. Gradient of the objective function
  11. Implementation
  12. NUCLEAR FISSION
  13. NUMERICAL EXAMPLE
  14. Conclusions

Figures (4)

  • Figure 1: The roots of the characteristic function in \ref{['eq:dde:stability']} for the original DDEs (blue circles), and the roots of the characteristic function in \ref{['eq:ide:stability']} for the approximate system (red circles). The roots $-2.33$, $-4.80$, and $-20.2$ for the approximate system are not shown. The dashed and solid lines indicate the roots of the real and imaginary parts of the characteristic function in \ref{['eq:dde:stability']}, respectively, and their intersections indicate the actual roots.
  • Figure 2: Optimal tracking of four different time-varying setpoints for the thermal energy generation using the simultaneous approach described in Section \ref{['sec:simultaneous:approach']}. The top and middle rows show the generated energy, the core and heat exchanger temperatures, and the thermal reactivity when the optimal external reactivity and flow velocity (bottom row) are used to simulate the original system of DDEs presented in Section \ref{['sec:nuclear:fission']}.
  • Figure 3: The difference between the generated thermal energy obtained with the original DDEs presented in Section \ref{['sec:nuclear:fission']} and with the approximate system.
  • Figure 4: The concentrations of the neutron precursor groups corresponding to the simulation shown in Fig. \ref{['fig:optimal:control:comparison']} where the setpoint increases to $10$ MW.