Structure-Preserving Low-Rank Model Reduction for Second-Order Time-Delay Systems

This paper introduces two model order-reduction techniques for second-order time-delay systems. The first method involves converting the second-order system into a first-order form, along with a set of related structure-preserving algorithms. The second method avoids converting the original model into a first-order form and uses direct projection to produce the reduced system, which can also retain the structure of the original one. The key idea of the proposed methods is to utilize low-rank Gramian approximations to construct reduced-order models. The time-delay Gramians are decomposed into low-rank approximations using a recurrence formula directly based on the expansion coefficient vectors of Laguerre functions. Then, we employ the low-rank square root method to create a low-dimensional system that closely approximates the original system. Ultimately, two numerical illustrations are provided to validate the precision and effectiveness of our proposed algorithms.