# Argument increment stability criterion for linear delta models

• Volume: 13, Issue: 4, page 485-491
• ISSN: 1641-876X

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## Abstract

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Currently used stability criteria for linear sampled-data systems refer to the standard linear difference equation form of the system model. This paper presents a stability criterion based on the argument increment rule modified for the delta operator form of the sampled-data model. For the asymptotic stability of this system form it is necessary and sufficient that the roots of the appropriate characteristic equation lie inside a circle in the left half of the complex plane, the radius of which is inversely proportional to the sampling period. Therefore the argument increment of the system characteristic polynomial of an asymptotically stable delta model has to increase by 2πn if this circle has been run around in the counter-clockwise direction. The criterion developed based on this principle permits not only the proof of the system stability itself, but also the approximation of the dominant roots of its characteristic equation.

## How to cite

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Hofreiter, Milan, and Zítek, Pavel. "Argument increment stability criterion for linear delta models." International Journal of Applied Mathematics and Computer Science 13.4 (2003): 485-491. <http://eudml.org/doc/207660>.

@article{Hofreiter2003,
abstract = {Currently used stability criteria for linear sampled-data systems refer to the standard linear difference equation form of the system model. This paper presents a stability criterion based on the argument increment rule modified for the delta operator form of the sampled-data model. For the asymptotic stability of this system form it is necessary and sufficient that the roots of the appropriate characteristic equation lie inside a circle in the left half of the complex plane, the radius of which is inversely proportional to the sampling period. Therefore the argument increment of the system characteristic polynomial of an asymptotically stable delta model has to increase by 2πn if this circle has been run around in the counter-clockwise direction. The criterion developed based on this principle permits not only the proof of the system stability itself, but also the approximation of the dominant roots of its characteristic equation.},
author = {Hofreiter, Milan, Zítek, Pavel},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {delta model; stability criterion; conformal mapping; delta operator; argument increment rule; sampled data system; dominant roots estimation; damping ratio; natural frequency},
language = {eng},
number = {4},
pages = {485-491},
title = {Argument increment stability criterion for linear delta models},
url = {http://eudml.org/doc/207660},
volume = {13},
year = {2003},
}

TY - JOUR
AU - Hofreiter, Milan
AU - Zítek, Pavel
TI - Argument increment stability criterion for linear delta models
JO - International Journal of Applied Mathematics and Computer Science
PY - 2003
VL - 13
IS - 4
SP - 485
EP - 491
AB - Currently used stability criteria for linear sampled-data systems refer to the standard linear difference equation form of the system model. This paper presents a stability criterion based on the argument increment rule modified for the delta operator form of the sampled-data model. For the asymptotic stability of this system form it is necessary and sufficient that the roots of the appropriate characteristic equation lie inside a circle in the left half of the complex plane, the radius of which is inversely proportional to the sampling period. Therefore the argument increment of the system characteristic polynomial of an asymptotically stable delta model has to increase by 2πn if this circle has been run around in the counter-clockwise direction. The criterion developed based on this principle permits not only the proof of the system stability itself, but also the approximation of the dominant roots of its characteristic equation.
LA - eng
KW - delta model; stability criterion; conformal mapping; delta operator; argument increment rule; sampled data system; dominant roots estimation; damping ratio; natural frequency
UR - http://eudml.org/doc/207660
ER -

## References

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2. Chemodanov B.K. (1977): Mathematical Fundations of Automatic ControlTheory. - Moscow: Vysha Shkola, (in Russian).
3. Feuer A. and Goodwin G.C. (1996): Sampling in Digital Processing and Control. - Boston: Birkhauser. Zbl0864.93011
4. Middleton R.H. and Goodwin G.C. (1989): Digital Control and Estimation: A Unified Approach. - Englewood Cliffs: Prentice-Hall. Zbl0754.93053
5. Ogata K. (1995): Discrete-Time Control Systems. - Englewood Cliffs: Prentice-Hall.
6. Zítek P. (1990): Rate of stability criterion for discrete dynamic systems. - Acta Technica CSAV, Vol. 35, No. 1, pp. 83-92.
7. Zítek P. (2001): Mathematical and Simulation Models in Complex Domain. - Prague: CTU.
8. Zítek P. and Petrova R. (2001): Discrete approximation of an isochronic systems using delta transform. - Proc. Conf. Information Engineering and Process Control, IEPC, Prague: Masaryk Academy of Work, pp. 71-72.

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