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2004 | 14 | 1 | 5-12

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Generalized practical stability analysis of discontinuous dynamical systems

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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.








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  • Department of Opto-Mechatronics, Wakayama University, 930 Sakaedani, Wakayama 640-8510, Japan
  • Department of Electrical Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA


  • 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. .
  • 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.
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  • 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.
  • 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.
  • Michel A.N., Wang K. and Hu B. (2000): Qualitative Theory of Dynamical Systems, 2nd Ed. New York: Marcel Dekker.
  • Lakshmikantham V., Leela S. and Martynyuk A.A. (1991): Practical Stability of Nonlinear Systems. Singapore: World Scientific.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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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