Generalized practical stability analysis of discontinuous dynamical systems

Guisheng Zhai; Anthony Michel

International Journal of Applied Mathematics and Computer Science (2004)

  • Volume: 14, Issue: 1, page 5-12
  • ISSN: 1641-876X

Abstract

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In practice, one is not only interested in the qualitative characterizations provided by the Lyapunov stability, but also in quantitative information concerning the system behavior, including estimates of trajectory bounds, possibly over finite time intervals. This type of information has been ascertained in the past in a systematic manner using the concept of practical stability. In the present paper, we give a new definition of generalized practical stability (abbreviated as GP-stability) and establish some sufficient conditions concerning GP-stability for a wide class of discontinuous dynamical systems. As in the classical Lyapunov theory, our results constitute a Direct Method, making use of auxiliary scalar-valued Lyapunov-like functions. These functions, however, have properties that differ significantly from the usual Lyapunov functions. We demonstrate the applicability of our results by means of several specific examples.

How to cite

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Zhai, Guisheng, and Michel, Anthony. "Generalized practical stability analysis of discontinuous dynamical systems." International Journal of Applied Mathematics and Computer Science 14.1 (2004): 5-12. <http://eudml.org/doc/207679>.

@article{Zhai2004,
abstract = {In practice, one is not only interested in the qualitative characterizations provided by the Lyapunov stability, but also in quantitative information concerning the system behavior, including estimates of trajectory bounds, possibly over finite time intervals. This type of information has been ascertained in the past in a systematic manner using the concept of practical stability. In the present paper, we give a new definition of generalized practical stability (abbreviated as GP-stability) and establish some sufficient conditions concerning GP-stability for a wide class of discontinuous dynamical systems. As in the classical Lyapunov theory, our results constitute a Direct Method, making use of auxiliary scalar-valued Lyapunov-like functions. These functions, however, have properties that differ significantly from the usual Lyapunov functions. We demonstrate the applicability of our results by means of several specific examples.},
author = {Zhai, Guisheng, Michel, Anthony},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {discontinuous dynamical system; Lyapunov-like function; generalized practical stability (GP-stability); quantitative analysis},
language = {eng},
number = {1},
pages = {5-12},
title = {Generalized practical stability analysis of discontinuous dynamical systems},
url = {http://eudml.org/doc/207679},
volume = {14},
year = {2004},
}

TY - JOUR
AU - Zhai, Guisheng
AU - Michel, Anthony
TI - Generalized practical stability analysis of discontinuous dynamical systems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2004
VL - 14
IS - 1
SP - 5
EP - 12
AB - In practice, one is not only interested in the qualitative characterizations provided by the Lyapunov stability, but also in quantitative information concerning the system behavior, including estimates of trajectory bounds, possibly over finite time intervals. This type of information has been ascertained in the past in a systematic manner using the concept of practical stability. In the present paper, we give a new definition of generalized practical stability (abbreviated as GP-stability) and establish some sufficient conditions concerning GP-stability for a wide class of discontinuous dynamical systems. As in the classical Lyapunov theory, our results constitute a Direct Method, making use of auxiliary scalar-valued Lyapunov-like functions. These functions, however, have properties that differ significantly from the usual Lyapunov functions. We demonstrate the applicability of our results by means of several specific examples.
LA - eng
KW - discontinuous dynamical system; Lyapunov-like function; generalized practical stability (GP-stability); quantitative analysis
UR - http://eudml.org/doc/207679
ER -

References

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  1. DeCarlo R., Branicky M., Pettersson S. and Lennartson B. (2000): Perspectives and results on the stability and stabilizability of hybrid systems. Proc. IEEE, Vol. 88, No.7, pp.1069-1082. . 
  2. Hespanha J.P. and Morse A.S. (1999): Stability of switched systems with average dwell-time. Proc. 38th IEEE Conf. Decision and Control, Phoenix, pp. 2655-2660. 
  3. Michel A.N. (1999): Recent trends in the stability analysis of hybrid dynamical systems. IEEE Trans. Circ. Syst. I: Fund. Theory Applic., Vol.45, No.1, pp.120-134. Zbl0981.93055
  4. Michel A.N. (1970): Quantitative analysis of simple and interconnected systems: Stability, boundedness and trajectory behavior. IEEE Trans. Circ. Theory, Vol.17, No.3, pp.292-301. 
  5. Michel A.N. and Porter D.W. (1971): Analysis of discontinuous large-scale systems: Stability, transient behaviour and trajectory bounds. Int. J. Syst. Sci., Vol. 2, No. 1, pp.77-95. Zbl0217.58203
  6. Michel A.N., Wang K. and Hu B. (2000): Qualitative Theory of Dynamical Systems, 2nd Ed. New York: Marcel Dekker. 
  7. Lakshmikantham V., Leela S. and Martynyuk A.A. (1991): Practical Stability of Nonlinear Systems. Singapore: World Scientific. 
  8. Weiss L. and Infante E.F. (1967): Finite time stability under perturbing forces and on product spaces. IEEE Trans. Automat. Contr., Vol. AC-12, No.1, pp. 54-59. Zbl0168.33903
  9. Zhai G. and Michel A.N. (2002): On practical stability of switched systems. Proc. 41st IEEE Conf. Decision and Control, Las Vegas, pp.3488-3493. 
  10. Zhai G., Hu B., Yasuda K. and Michel A.N. (2000): Piecewise Lyapunov functions for switched systems with average dwell time. Asian J. Contr., Vol. 2, No.3, pp.192-197. 
  11. Zhai G., Hu B., Yasuda K. and Michel A.N. (2001): Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach. Int. J. Syst. Sci., Vol. 32, No.8, pp.1055-1061. Zbl1022.93043
  12. Zhai G., Hu B., Yasuda K. and Michel A.N. (2002): Stability and gain analysis of discrete-time switched systems. Trans. Inst. Syst. Contr. Inf. Eng., Vol.15, No.3, pp.117-125. 

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