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

TL;DR

This paper introduces a novel algorithm that identifies distributed delays in delay differential equations by approximating the delay kernel with a mixed Erlang distribution and transforming the problem into ODEs for easier analysis.

Contribution

It presents a new method combining mixed Erlang kernel approximation with the linear chain trick to identify delays using only ODE simulations.

Findings

Effective delay identification demonstrated on logistic and nuclear reactor models.

The approach simplifies delay differential equations to ODEs for easier analysis.

Numerical results show accurate delay parameter estimation.

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.