Balanced truncation for bilinear time-delay systems with approximate Gramians
Abstract
A balanced truncation (BT) procedure is presented to achieve a structure-preserving model order reduction (MOR) for bilinear time-delay systems in this paper. We attempt to define Gramians for bilinear time-delay systems based on Volterra series theory. The controllability and observability Gramians in the frequency domain are given explicitly, which align with the ones of bilinear systems and time-delay systems. Based on the derived Gramians, a BT procedure is provided to produce reduced order models with the same structure. We also propose a numerical quadrature rule based on the truncated Laguerrre expansion to give an accurate approximation to Gramians. The resulting low-rank structure of approximate Gramians benefits a lot the efficient execution of the whole MOR procedure. Two numerical examples are simulated to showcase the efficiency of our approach.
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BibTeX
@misc{wang2026balanced,
title = {{Balanced truncation for bilinear time-delay systems with approximate Gramians}},
author = {Xiaolong Wang and Xin Su and Xuerong Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06939},
eprint = {2609.06939},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06939},
}
RIS
TY - UNPB
TI - Balanced truncation for bilinear time-delay systems with approximate Gramians
AU - Xiaolong Wang
AU - Xin Su
AU - Xuerong Yang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.06939
UR - https://arxiv.org/abs/2609.06939
AB - A balanced truncation (BT) procedure is presented to achieve a structure-preserving model order reduction (MOR) for bilinear time-delay systems in this paper. We attempt to define Gramians for bilinear time-delay systems based on Volterra series theory. The controllability and observability Gramians in the frequency domain are given explicitly, which align with the ones of bilinear systems and time-delay systems. Based on the derived Gramians, a BT procedure is provided to produce reduced order models with the same structure. We also propose a numerical quadrature rule based on the truncated Laguerrre expansion to give an accurate approximation to Gramians. The resulting low-rank structure of approximate Gramians benefits a lot the efficient execution of the whole MOR procedure. Two numerical examples are simulated to showcase the efficiency of our approach.
ER -