Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems

TL;DR

This paper introduces a method to parameterize seed functions for transforming time-varying delay systems into equivalent fixed delay systems, aiming to improve stability analysis and control by reducing parameter variation.

Contribution

It proposes a basis-based parameterization of seed functions, enabling systematic selection for better transformations compared to ad-hoc methods.

Findings

Parameterization using $L_2$ basis improves seed function selection.

Better seed functions lead to smaller parameter variations in transformations.

Numerical examples demonstrate the impact of seed function choice on system boundedness.

Abstract

Abel's classic transformation shows that any well-posed system with time-varying delay is equivalent to a parameter-varying system with fixed delay. The existence of such a parameter-varying constant delay representation then simplifies the problems of stability analysis and optimal control. Unfortunately, the method for construction of such transformations has been ad-hoc -- requiring an iterative time-stepping approach to constructing the transformation beginning with a seed function subject to boundary-value constraints. Moreover, a poor choice of seed function often results in a constant delay representation with large time-variations in system parameters -- obviating the benefits of such a representation. In this paper, we show how the set of all feasible seed functions can be parameterized using a basis for $L_2$. This parameterization is then used to search for seed functions for which the corresponding time-transformation results in smaller parameter variation. The parameterization of admissible seed functions is illustrated with numerical examples that contrast how well-chosen and poorly chosen seed functions affect the boundedness of a time transformation.