Analysis of higher order time delay systems using Lambert W function

TL;DR

This paper reassesses the Lambert W function approach for analyzing higher order time delay systems, extending recent results to nth order systems, characterizing a new class, and demonstrating controller design and stability analysis.

Contribution

It extends Lambert W function analysis to nth order systems, introduces a new class of systems, and applies the method for controller design and stability assessment.

Findings

Eigenvalues are associated with two Lambert W branches.

The method is validated through an example.

A new class of systems is characterized for this analysis.

Abstract

In this note, analysis of time delay systems using Lambert W function approach is reassessed. A common canonical form of time delay systems is defined. We extended the recent results of [6] for second order into nth order system. The eigenvalues of a time delay system are either real or complex conjugate pairs and therefore, the whole eigenspectrum can be associated with only two real branches of the Lambert W function. A new class of time delay systems is characterized to extend the applicability of the above said method. A state variable transformation is used to transform the proposed class of systems into the common canonical form. Moreover, this approach has been exploited to design a controller which places a subset of eigenvalues at desired locations. Stability is analyesed by the help of Nyquist plot. The approach is validated through an example.