Quantitatively hyper-positive real rational functions IV: The canonical case
Abstract
In the linear time-invariant framework, passive systems are modeled by positive real functions. Dissipative systems can be modeled by the subset of quantita- tively Hyper-positive real functions, related through nested inclusions. This family was introduced and studied in our three previous works. Here, we further focus our attention on the proper subset of canonical Hyper-Positive functions. Although this family is "small", its exploration is well motivated: First, this set turns to be associated with absolute stability (the Lurie problem). Then, a systematic parametrization of all canonical Hyper-Positive functions is introduced. Moreover each canonical Hyper-Positive function can be viewed as an extreme point of the convex set of Hyper-Positive functions. Specifically a convex combination of canonical Hyper-Positive is Hyper-Positive, but not canonical. These observations hold in both frameworks: of analytic functions and of state-space realization arrays. Technically, some of the analysis is facilitated by employing Quadratic Matrix Inclusions of both, matrices and of matrix-valued rational functions.
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BibTeX
@misc{alpay2026quantitatively,
title = {{Quantitatively hyper-positive real rational functions IV: The canonical case}},
author = {Daniel Alpay and Izchak Lewkowicz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03398},
eprint = {2609.03398},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03398},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Quantitatively hyper-positive real rational functions IV: The canonical case
AU - Daniel Alpay
AU - Izchak Lewkowicz
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03398
UR - https://arxiv.org/abs/2609.03398
AB - In the linear time-invariant framework, passive systems are modeled by positive real functions. Dissipative systems can be modeled by the subset of quantita- tively Hyper-positive real functions, related through nested inclusions. This family was introduced and studied in our three previous works. Here, we further focus our attention on the proper subset of canonical Hyper-Positive functions. Although this family is "small", its exploration is well motivated: First, this set turns to be associated with absolute stability (the Lurie problem). Then, a systematic parametrization of all canonical Hyper-Positive functions is introduced. Moreover each canonical Hyper-Positive function can be viewed as an extreme point of the convex set of Hyper-Positive functions. Specifically a convex combination of canonical Hyper-Positive is Hyper-Positive, but not canonical. These observations hold in both frameworks: of analytic functions and of state-space realization arrays. Technically, some of the analysis is facilitated by employing Quadratic Matrix Inclusions of both, matrices and of matrix-valued rational functions.
ER -