Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics

Boe-Shong Hong

doi:doi:10.11648/j.acm.20140305.14

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Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics

Boe-Shong Hong

Published in Applied and Computational Mathematics (Volume 3, Issue 5)

Received: 20 August 2014 Accepted: 10 September 2014 Published: 20 September 2014

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Abstract

This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-time domain and the mode-frequency domain are developed, whereof the H∞-norm is extended to be 2D-H∞ norm in mode-frequency domain. On grounds of this small-gain theorem, robust performance of any Sturm-Liouville plant can be formulated as robust stability of a feedback connection, whereupon feedback syntheses can be performed via modal-spectral μ-loopshaping.

Published in	Applied and Computational Mathematics (Volume 3, Issue 5)
DOI	10.11648/j.acm.20140305.14
Page(s)	217-224
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Small Gain Theorem, Distributed Control, Robust Control, nD Transfer Function

References

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Boe-Shong Hong. (2014). Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics. Applied and Computational Mathematics, 3(5), 217-224. https://doi.org/10.11648/j.acm.20140305.14

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Boe-Shong Hong. Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics. Appl. Comput. Math. 2014, 3(5), 217-224. doi: 10.11648/j.acm.20140305.14

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Boe-Shong Hong. Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics. Appl Comput Math. 2014;3(5):217-224. doi: 10.11648/j.acm.20140305.14

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@article{10.11648/j.acm.20140305.14,
  author = {Boe-Shong Hong},
  title = {Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {5},
  pages = {217-224},
  doi = {10.11648/j.acm.20140305.14},
  url = {https://doi.org/10.11648/j.acm.20140305.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140305.14},
  abstract = {This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-time domain and the mode-frequency domain are developed, whereof the H∞-norm is extended to be 2D-H∞ norm in mode-frequency domain. On grounds of this small-gain theorem, robust performance of any Sturm-Liouville plant can be formulated as robust stability of a feedback connection, whereupon feedback syntheses can be performed via modal-spectral μ-loopshaping.},
 year = {2014}
}

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TY  - JOUR
T1  - Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics
AU  - Boe-Shong Hong
Y1  - 2014/09/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140305.14
DO  - 10.11648/j.acm.20140305.14
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JF  - Applied and Computational Mathematics
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EP  - 224
PB  - Science Publishing Group
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AB  - This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-time domain and the mode-frequency domain are developed, whereof the H∞-norm is extended to be 2D-H∞ norm in mode-frequency domain. On grounds of this small-gain theorem, robust performance of any Sturm-Liouville plant can be formulated as robust stability of a feedback connection, whereupon feedback syntheses can be performed via modal-spectral μ-loopshaping.
VL  - 3
IS  - 5
ER  -

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Author Information

Boe-Shong Hong

Department of Mechanical Engineering, National Chung Cheng University, Chia-Yi 62012, Taiwan

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APA Style

Boe-Shong Hong. (2014). Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics. Applied and Computational Mathematics, 3(5), 217-224. https://doi.org/10.11648/j.acm.20140305.14

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ACS Style

Boe-Shong Hong. Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics. Appl. Comput. Math. 2014, 3(5), 217-224. doi: 10.11648/j.acm.20140305.14

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AMA Style

Boe-Shong Hong. Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics. Appl Comput Math. 2014;3(5):217-224. doi: 10.11648/j.acm.20140305.14

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@article{10.11648/j.acm.20140305.14,
  author = {Boe-Shong Hong},
  title = {Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics},
  journal = {Applied and Computational Mathematics},
  volume = {3},
  number = {5},
  pages = {217-224},
  doi = {10.11648/j.acm.20140305.14},
  url = {https://doi.org/10.11648/j.acm.20140305.14},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20140305.14},
  abstract = {This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-time domain and the mode-frequency domain are developed, whereof the H∞-norm is extended to be 2D-H∞ norm in mode-frequency domain. On grounds of this small-gain theorem, robust performance of any Sturm-Liouville plant can be formulated as robust stability of a feedback connection, whereupon feedback syntheses can be performed via modal-spectral μ-loopshaping.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Small Gain Theorem for Distributed Feedback Control of Sturm-Liouville Dynamics
AU  - Boe-Shong Hong
Y1  - 2014/09/20
PY  - 2014
N1  - https://doi.org/10.11648/j.acm.20140305.14
DO  - 10.11648/j.acm.20140305.14
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 217
EP  - 224
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20140305.14
AB  - This paper constructs the small-gain theorem upon a general class of Sturm-Liouville systems. It appears that the feedback connection of two Sturm-Liouville sub-systems is guaranteed of well-posedness, Hurwitz, dissipativity and passivity in L2-spaces provided the loop gain is less than 1. To construct the theorem, spatiotemporal transfer-function and geometrical isomorphism between the space-time domain and the mode-frequency domain are developed, whereof the H∞-norm is extended to be 2D-H∞ norm in mode-frequency domain. On grounds of this small-gain theorem, robust performance of any Sturm-Liouville plant can be formulated as robust stability of a feedback connection, whereupon feedback syntheses can be performed via modal-spectral μ-loopshaping.
VL  - 3
IS  - 5
ER  -

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