Stable Rational Approximation of PDE Transfer Functions with H∞ Error Bounds
Abstract
Transfer functions of Partial Differential Equations (PDEs) are irrational and difficult to obtain. Unlike for rational transfer functions, only a few methods are available for the control and analysis of irrational transfer functions. Thus, for robust analysis and control of PDEs, we need to use a rational approximation of the transfer function, preferably with provable H∞ error bounds. In this paper, we extend Krylov subspace model reduction to infinite-dimensional systems modeled by Partial Integral Equations (PIEs) -- an equivalent representation of PDEs. First, we propose a state transformation of the PIE system that allows us to compute a controllability matrix. Then, to obtain a rational approximation, we project the PIE states onto the vector space spanned by the truncated controllability matrix. Next, we show that sufficient conditions for exponential stability and well-posedness of the PIE system guarantee the exponential stability of the reduced-order system and provide constructive finite-frequency H∞ error bounds. Finally, we provide four illustrative numerical examples to demonstrate the performance of the proposed method.
Related
- Lower bounds in H2-rational approximation to Blaschke products
- Wasserstein bounds for invariant measures in the Smoluchowski-Kramers approximation with stable noise
- Campana Rational Connectedness and Weak Approximation of Del Pezzo Orbifolds
- Applications of AAA rational approximation
- Weighted Laplace Spaces for Spectral Measures and Rational Approximation
BibTeX
@misc{talitckii2026stable,
title = {{Stable Rational Approximation of PDE Transfer Functions with $H_\infty$ Error Bounds}},
author = {Aleksandr Talitckii and Matthew M. Peet},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08497},
eprint = {2610.08497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08497},
}
RIS
TY - UNPB
TI - Stable Rational Approximation of PDE Transfer Functions with H_∞ Error Bounds
AU - Aleksandr Talitckii
AU - Matthew M. Peet
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08497
UR - https://arxiv.org/abs/2610.08497
AB - Transfer functions of Partial Differential Equations (PDEs) are irrational and difficult to obtain. Unlike for rational transfer functions, only a few methods are available for the control and analysis of irrational transfer functions. Thus, for robust analysis and control of PDEs, we need to use a rational approximation of the transfer function, preferably with provable H_∞ error bounds. In this paper, we extend Krylov subspace model reduction to infinite-dimensional systems modeled by Partial Integral Equations (PIEs) -- an equivalent representation of PDEs. First, we propose a state transformation of the PIE system that allows us to compute a controllability matrix. Then, to obtain a rational approximation, we project the PIE states onto the vector space spanned by the truncated controllability matrix. Next, we show that sufficient conditions for exponential stability and well-posedness of the PIE system guarantee the exponential stability of the reduced-order system and provide constructive finite-frequency H_∞ error bounds. Finally, we provide four illustrative numerical examples to demonstrate the performance of the proposed method.
ER -