# Quantitative L^{P} stability analysis of a class of linear time-varying feedback systems

• Volume: 13, Issue: 2, page 179-184
• ISSN: 1641-876X

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## Abstract

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The L^{P} stability of linear feedback systems with a single time-varying sector-bounded element is considered. A sufficient condition for L^{P} stability, with 1 ≤ p ≤ ∞, is obtained by utilizing the well-known small gain theorem. Based on the stability measure provided by this theorem, quantitative results that describe output-to-input relations are obtained. It is proved that if the linear time-invariant part of the system belongs to the class of proper positive real transfer functions with a single pole at the origin, the upper bound on the output-to-input ratio is constant. Thus, an explicit closed-form calculation of this bound for some simple particular case provides a powerful generalization for the more complex cases. The importance of the results is illustrated by an example taken from missile guidance theory.

## How to cite

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Gurfil, Pini. "Quantitative L^{P} stability analysis of a class of linear time-varying feedback systems." International Journal of Applied Mathematics and Computer Science 13.2 (2003): 179-184. <http://eudml.org/doc/207633>.

@article{Gurfil2003,
abstract = {The L^\{P\} stability of linear feedback systems with a single time-varying sector-bounded element is considered. A sufficient condition for L^\{P\} stability, with 1 ≤ p ≤ ∞, is obtained by utilizing the well-known small gain theorem. Based on the stability measure provided by this theorem, quantitative results that describe output-to-input relations are obtained. It is proved that if the linear time-invariant part of the system belongs to the class of proper positive real transfer functions with a single pole at the origin, the upper bound on the output-to-input ratio is constant. Thus, an explicit closed-form calculation of this bound for some simple particular case provides a powerful generalization for the more complex cases. The importance of the results is illustrated by an example taken from missile guidance theory.},
author = {Gurfil, Pini},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {time-varying Lur'e systems; functional analysis; L^\{P\}-stability},
language = {eng},
number = {2},
pages = {179-184},
title = {Quantitative L^\{P\} stability analysis of a class of linear time-varying feedback systems},
url = {http://eudml.org/doc/207633},
volume = {13},
year = {2003},
}

TY - JOUR
AU - Gurfil, Pini
TI - Quantitative L^{P} stability analysis of a class of linear time-varying feedback systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2003
VL - 13
IS - 2
SP - 179
EP - 184
AB - The L^{P} stability of linear feedback systems with a single time-varying sector-bounded element is considered. A sufficient condition for L^{P} stability, with 1 ≤ p ≤ ∞, is obtained by utilizing the well-known small gain theorem. Based on the stability measure provided by this theorem, quantitative results that describe output-to-input relations are obtained. It is proved that if the linear time-invariant part of the system belongs to the class of proper positive real transfer functions with a single pole at the origin, the upper bound on the output-to-input ratio is constant. Thus, an explicit closed-form calculation of this bound for some simple particular case provides a powerful generalization for the more complex cases. The importance of the results is illustrated by an example taken from missile guidance theory.
LA - eng
KW - time-varying Lur'e systems; functional analysis; L^{P}-stability
UR - http://eudml.org/doc/207633
ER -

## References

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1. Desoer C.A. and Vidyasagar M. (1975): Feedback Systems: Input-Output Properties. - New York: Academic Press. Zbl0327.93009
2. Doyle J.C., Francis B.A. and Tannenbaum A.R. (1992): Feedback ControlTheory. - New York: Macmillan Publishing.
3. Gurfil P., Jodorkovsky M. and Guelman M. (1998): Finite time stability approach to proportional navigation systems analysis. - J. Guid. Contr. Dynam., Vol. 21, No. 6, pp. 853-861.
4. Mossaheb S. (1982): The circle criterion and the L^P stability of feedback systems. - SIAM J. Contr. Optim., Vol. 20, No. 1, pp. 144-151. Zbl0473.93057
5. Sandberg I.W. (1964): A frequency domain condition for the stability of feedback systems containing a single time-varying nonlinear element.- Bell Syst. Tech. J., Vol. 43, pp. 1601-1608. Zbl0131.31704
6. Sandberg I.W. (1965): Some results on the theory of physical systems governed by nonlinear functional equations. - Bell Syst. Tech. J., Vol. 44, No. 5, pp. 871-898. Zbl0156.15804
7. Sandberg I.W. and Johnson K.K. (1990): Steady state errors and the circle criterion. - IEEE Trans. Automat. Contr., Vol. 35, No. 1, pp. 530-534.
8. Shinar J. (1976): Divergence range of homing missiles. - Israel J. Technol., Vol. 14, pp. 47-55.
9. Vidyasagar M. (19): Nonlinear Systems Analysis, 2nd Ed.. -New Jersey: Upper Saddle River. Zbl0759.93001
10. Zames G. (1990): On input-output stability of time-varying nonlinear feedback systems-Part II: Conditions involving circles in the frequency planeand sector nonlinearities. - IEEE Trans. Automat. Contr., Vol. AC-11, No. 2, pp. 465-476.
11. Zarchan P. (1990): Tactical and Strategic Missile Guidance. -Washington: AIAA.

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