Additional dynamics in transformed time-delay systems

In studying the stability of time-delay systems, many published results use a transformation to transform a system with single time-delay to a system with distributed delay. In this article, the inherent limitations of such approaches are studied. Specifically, it is shown that such a transformation incurs additional dynamics which can be characterized by appropriate additional eigenvalues. The critical delay values when such additional eigenvalues cross the imaginary axis can be explicitly calculated. If the smallest of such delays is the smallest of the stability delay limit of the original system, then any stability criteria obtained using such transformation will be conservative. Some examples are also included.