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arXiv · 2603.15961

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

Jungbae Chun·Sengiyumva Kisole·Matthew M. Peet·Peter Seiler
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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.

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Jungbae Chun, Sengiyumva Kisole, Matthew M. Peet, Peter Seiler. 2026-03-16. Time-Transformation-Based Analysis of Systems with Periodic Delay via Perturbative Expansion. https://arxiv.org/abs/2603.15961

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