Time-Transformation-Based Analysis of Systems with Periodic Delay via Perturbative Expansion

TL;DR

This paper introduces a perturbative method to explicitly approximate time-transformations for analyzing the stability of systems with small periodic delays, simplifying the complex analysis of time-varying delays.

Contribution

It develops a perturbative expansion technique to compute explicit time-transformations for systems with small periodic delays, addressing the non-uniqueness and numerical challenges.

Findings

Explicit approximate time-transformations are derived for systems with small periodic delays.

The method simplifies stability analysis by converting systems to a form with constant delay.

Numerical example demonstrates the effectiveness of the approach.

Abstract

It is difficult to analyze the stability of systems with time-varying delays. One approach is to construct a time-transformation that converts the system into a form with a constant delay but with a time-varying scalar appearing in the system matrices. The stability of this transformed system can then be analyzed using methods to bound the effect of the time-varying scalar. One issue is that this transformation is non-unique and requires the solution of an Abel equation. A specific time-transformation typically must be computed numerically. We address this issue by computing an explicit, although approximate, time-transformation for systems where the delay has a constant plus small periodic term. We use a perturbative expansion to construct our explicit solutions. We provide a simple numerical example to illustrate the approach. We also demonstrate the use of this time-transformation to analyze stability of the system with this class of periodic delays.