Parameterization of Seed Functions for Equivalent Representations of Time-Varying Delay Systems
This work addresses a bottleneck in stability analysis and control for time-varying delay systems, offering a systematic method to replace ad-hoc approaches, though it is incremental in improving parameter variation.
The paper tackles the problem of constructing equivalent constant-delay representations for time-varying delay systems by parameterizing all feasible seed functions using an L2 basis, enabling a search for seed functions that minimize parameter variation. Numerical examples demonstrate how seed function choice affects the boundedness of the time transformation.
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.