Time-Transformation-Based Analysis of Systems with Periodic Delay via Perturbative Expansion
This work addresses a specific problem in control theory for systems with periodic delays, but it is incremental as it builds on existing transformation methods.
The paper tackles the difficulty of analyzing stability in systems with time-varying delays by developing an explicit, approximate time-transformation for delays with a constant plus small periodic term, using perturbative expansion and demonstrating its application through a numerical example.
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.