The geometry of Darlington synthesis (in memory of W. Cauer)

Patrick Dewilde

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 6, page 1379-1386
  • ISSN: 1641-876X

Abstract

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We revisit the classical problem of 'Darlington synthesis', or Darlington embedding. Although traditionally it is solved using analytic means, a more natural way to approach it is to use the geometric properties of a well-chosen Hankel map. The method yields surprising results. In the first place, it allows us to formulate necessary and sufficient conditions for the existence of the embedding in terms of systems properties of the transfer operation to be embedded. In addition, the approach allows us to extend the solution to situations where no analytical transform is available. The paper has a high review content, as all the results presented have been obtained during the last twenty years and have been published. However, we make a systematic attempt at formulating them in a geometric way, independent of an accidental parametrization. The benefit is clarity and generality.

How to cite

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Dewilde, Patrick. "The geometry of Darlington synthesis (in memory of W. Cauer)." International Journal of Applied Mathematics and Computer Science 11.6 (2001): 1379-1386. <http://eudml.org/doc/207560>.

@article{Dewilde2001,
abstract = {We revisit the classical problem of 'Darlington synthesis', or Darlington embedding. Although traditionally it is solved using analytic means, a more natural way to approach it is to use the geometric properties of a well-chosen Hankel map. The method yields surprising results. In the first place, it allows us to formulate necessary and sufficient conditions for the existence of the embedding in terms of systems properties of the transfer operation to be embedded. In addition, the approach allows us to extend the solution to situations where no analytical transform is available. The paper has a high review content, as all the results presented have been obtained during the last twenty years and have been published. However, we make a systematic attempt at formulating them in a geometric way, independent of an accidental parametrization. The benefit is clarity and generality.},
author = {Dewilde, Patrick},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {time-varying systems; coprime factorization; contractive operators; Darlington synthesis; Hankel operator; state space methods; time-varying case},
language = {eng},
number = {6},
pages = {1379-1386},
title = {The geometry of Darlington synthesis (in memory of W. Cauer)},
url = {http://eudml.org/doc/207560},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Dewilde, Patrick
TI - The geometry of Darlington synthesis (in memory of W. Cauer)
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 6
SP - 1379
EP - 1386
AB - We revisit the classical problem of 'Darlington synthesis', or Darlington embedding. Although traditionally it is solved using analytic means, a more natural way to approach it is to use the geometric properties of a well-chosen Hankel map. The method yields surprising results. In the first place, it allows us to formulate necessary and sufficient conditions for the existence of the embedding in terms of systems properties of the transfer operation to be embedded. In addition, the approach allows us to extend the solution to situations where no analytical transform is available. The paper has a high review content, as all the results presented have been obtained during the last twenty years and have been published. However, we make a systematic attempt at formulating them in a geometric way, independent of an accidental parametrization. The benefit is clarity and generality.
LA - eng
KW - time-varying systems; coprime factorization; contractive operators; Darlington synthesis; Hankel operator; state space methods; time-varying case
UR - http://eudml.org/doc/207560
ER -

References

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  8. Dewilde P. (1999): Generalized Darlington synthesis. — IEEE Trans. Circ. Syst. – I: Fund. Theory Appl., Vol.45, No.1, pp.41–58. 
  9. Dewilde P. and van der Veen A.-J. (1998): Time-Varying Systems and Computations. — Dordrecht: Kluwer. 
  10. Fuhrmann P.A. (1981): Linear Systems and Operators in Hilbert Space. — New York: McGraw-Hill. Zbl0456.47001
  11. Helson H. (1964): Lectures on Invariant Subspaces. — New York: Academic Press. Zbl0119.11303
  12. Newcomb R. (1966): Linear Multiport Synthesis. — New York: McGraw Hill. 
  13. Masani P. and Wiener N. (1957): The prediction theory of multivariable stochastic processes. — Acta Math., Vol.98, No.1, pp.111–150. Zbl0080.13002
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