[Paper Review] Time response of a scalar dynamical system with multiple delays via Lambert W functions

Shuo-Tsung Chen, Shun‐Pin Hsu+2 more|arXiv (Cornell University)|Sep 7, 2016

Sports Dynamics and Biomechanics3 references3 citations

TL;DR

This paper presents an analytical method to compute the time response of scalar dynamical systems with multiple discrete delays using the Lambert W function. By applying the Laplace transform and expressing eigenvalues via infinite branches of the Lambert W function, the solution is derived as a sum of exponentials, enabling exact analytical response computation for initial conditions and inputs, with numerical validation against standard solvers like dde23.

ABSTRACT

In this work, we establish the response of scalar systems with multiple discrete delays based on the Laplace transform. The time response function is expressed as the sum of infinite series of exponentials acting on eigenvalues inside countable branches of the Lambert W functions. Eigenvalues in each branch of Lambert W function are computed by a numerical iteration. Numerical examples are presented to illustrate the results obtained.

Motivation & Objective

• To develop an analytical formula for the time response of scalar dynamical systems with multiple discrete delays.
• To overcome the limitation of purely numerical methods like dde23 by providing a closed-form solution.
• To extend the use of Lambert W functions beyond single-delay systems to multiple delays.
• To enable precise controller design for delay systems through an exact response representation.

Proposed method

• Apply the Laplace transform to the delay differential equation (DDE) with multiple discrete delays.
• Derive the state transition function as a sum of exponentials indexed by eigenvalues from countable branches of the Lambert W function.
• Compute eigenvalues iteratively within each Lambert W branch using numerical methods.
• Express the total time response as a combination of initial condition response, preshape function response, and convolution with the input function.
• Use partial fraction expansion in the Laplace domain to obtain coefficients for each exponential term.
• Approximate the infinite series using a finite number of eigenvalue branches (e.g., k = -11 to 10) for numerical computation.

Experimental results

Research questions

• RQ1 Can the time response of a scalar delay system with multiple discrete delays be expressed analytically using the Lambert W function?
• RQ2 How can eigenvalues of a multi-delay system be systematically computed across all branches of the Lambert W function?
• RQ3 What is the structure of the state transition function in terms of Lambert W branches for multiple delays?
• RQ4 How accurate is the analytical response approximation when compared to numerical solvers like dde23?
• RQ5 Can the method be generalized to include arbitrary initial conditions and input functions?

Key findings

• The time response of a multi-delay scalar system is expressed as an infinite series of exponentials, each associated with an eigenvalue from a distinct branch of the Lambert W function.
• Eigenvalues are computed numerically via iterative solution within each branch of the Lambert W function, enabling practical implementation.
• The analytical solution for the initial condition response and input-driven response is derived using partial fraction expansion in the Laplace domain.
• Numerical results show near-identical responses to dde23, validating the accuracy of the Lambert W-based analytical method.
• The method successfully computes responses for systems with up to three delays, including complex conjugate eigenvalue pairs and their contributions.
• The use of 32 eigenvalues (k = -11 to 10) provides a highly accurate approximation of the infinite series, with convergence observed in the time domain.