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An algorithm for distributed time delay identification based on a mixed Erlang kernel approximation and the linear chain trick

Tobias K. S. Ritschel, John Wyller

TL;DR

This work tackles identifying distributed time delays in delay differential equations by approximating the memory kernel with a finite mixed Erlang mixture $\hat{\alpha}^{(M)}(t)$ and transforming the resulting DDEs into an ODE system via the linear chain trick. It then casts kernel and state estimation as a dynamical least-squares problem solved with a single-shooting approach and forward sensitivities, using standard Matlab tools. The key contributions are the practical kernel-approximation scheme, the ODE reformulation enabling conventional simulation/optimization, and the demonstration that the method accurately identifies distributed delays in both a logistic growth model and a stiff point reactor kinetics model. The approach offers a scalable, accessible route to incorporate distributed delays in model-based control and optimization without requiring specialized DDE solvers, with potential applicability across engineering and physical systems.

Abstract

Time delays are ubiquitous in industry and nature, and they significantly affect both transient dynamics and stability properties. Consequently, it is often necessary to identify and account for the delays when, e.g., designing a model-based control strategy. However, identifying delays in differential equations is not straightforward and requires specialized methods. Therefore, we propose an algorithm for identifying distributed delays in delay differential equations (DDEs) that only involves simulation of ordinary differential equations (ODEs). Specifically, we 1) approximate the kernel in the DDEs (also called the memory function) by the probability density function of a mixed Erlang distribution and 2) use the linear chain trick (LCT) to transform the resulting DDEs into ODEs. Finally, the parameters in the kernel approximation are estimated as the solution to a dynamical least-squares problem, and we use a single-shooting approach to approximate this solution. We demonstrate the efficacy of the algorithm using numerical examples that involve the logistic equation and a point reactor kinetics model of a molten salt nuclear fission reactor.

An algorithm for distributed time delay identification based on a mixed Erlang kernel approximation and the linear chain trick

TL;DR

This work tackles identifying distributed time delays in delay differential equations by approximating the memory kernel with a finite mixed Erlang mixture and transforming the resulting DDEs into an ODE system via the linear chain trick. It then casts kernel and state estimation as a dynamical least-squares problem solved with a single-shooting approach and forward sensitivities, using standard Matlab tools. The key contributions are the practical kernel-approximation scheme, the ODE reformulation enabling conventional simulation/optimization, and the demonstration that the method accurately identifies distributed delays in both a logistic growth model and a stiff point reactor kinetics model. The approach offers a scalable, accessible route to incorporate distributed delays in model-based control and optimization without requiring specialized DDE solvers, with potential applicability across engineering and physical systems.

Abstract

Time delays are ubiquitous in industry and nature, and they significantly affect both transient dynamics and stability properties. Consequently, it is often necessary to identify and account for the delays when, e.g., designing a model-based control strategy. However, identifying delays in differential equations is not straightforward and requires specialized methods. Therefore, we propose an algorithm for identifying distributed delays in delay differential equations (DDEs) that only involves simulation of ordinary differential equations (ODEs). Specifically, we 1) approximate the kernel in the DDEs (also called the memory function) by the probability density function of a mixed Erlang distribution and 2) use the linear chain trick (LCT) to transform the resulting DDEs into ODEs. Finally, the parameters in the kernel approximation are estimated as the solution to a dynamical least-squares problem, and we use a single-shooting approach to approximate this solution. We demonstrate the efficacy of the algorithm using numerical examples that involve the logistic equation and a point reactor kinetics model of a molten salt nuclear fission reactor.
Paper Structure (17 sections, 1 theorem, 54 equations, 3 figures, 2 tables)

This paper contains 17 sections, 1 theorem, 54 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. System
  3. Mixed Erlang approximation
  4. A note on convergence
  5. The linear chain trick
  6. Algorithm
  7. Dynamical least-squares problem
  8. Single-shooting
  9. Sensitivities
  10. Implementation
  11. Numerical examples
  12. The logistic equation
  13. Absolute delay
  14. Nuclear fission
  15. Conclusions
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\alpha:[0, \infty) \rightarrow [0, \infty)$ be a kernel that satisfies Assumption ass:alpha:cont:norm and define $\beta:[0, \infty) \rightarrow [0, 1]$ as Next, define $\hat{\beta}^{(M)}: [0, \infty) \rightarrow [0, 1]$ as where $c_m = \beta(t_{m+1}) - \beta(t_m)$, $a = 1/\Delta t$, and $t_m = m \Delta t$. Note that $\hat{\beta}_m$ depends on the rate parameter $a$. Then, $\hat{\beta}^{(\in

Figures (3)

  • Figure 1: Estimation results for the logistic equation with a distributed delay. Left column: The true (hardly visible) and estimated kernels (top), the corresponding absolute error (middle), and the estimates of the coefficients $\{c_m\}_{m=0}^M$ (bottom). Right column: The population density for the true parameters (top), the absolute difference in population density for the true and estimated parameters (middle), and the true (when applicable) and estimated values of $a$, $N_0$, and $\kappa$ (bottom). The colors are consistent across the figure.
  • Figure 2: Estimation results for the logistic equation with an absolute delay. Left column: The true delay and the estimated kernels (top) and the estimates of the coefficients $\{c_m\}_{m=0}^M$ (bottom). Right column: The population density for the true parameters (top), the absolute difference in population density for the true and the estimated parameters (middle), and the true (when applicable) and estimated values of $a$, $\tau$, $N_0$, and $\kappa$ (bottom). The colors are consistent across the figure.
  • Figure 3: Estimation results for the point reactor kinetics model. Left: True and estimated kernel (top), the corresponding absolute error (second from top), the estimates of the coefficients $\{c_m\}_{m=0}^M$ (third from top), and the true (when applicable) and estimated values of $a$, $\{C_{i, 0}\}_{i=1}^n$, $\rho_0$, and $\kappa$ (bottom). Right: The concentrations (top and second from the top) and reactivity relative to $\beta$ (third from the top) for the true parameters and the maximum relative absolute difference (max. rel. err.) obtained with the estimated parameters (bottom). The colors are consistent across the left column.

Theorems & Definitions (7)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6