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Extension of Higher Order Derivatives of Lyapunov Functions in Stability Analysis of Nonlinear Systems | ||
| AUT Journal of Modeling and Simulation | ||
| مقاله 4، دوره 41، شماره 1، تیر 2009، صفحه 25-33 اصل مقاله (202.45 K) | ||
| نوع مقاله: Research Article | ||
| شناسه دیجیتال (DOI): 10.22060/miscj.2009.220 | ||
| چکیده | ||
| The Lyapunov stability method is the most popular and applicable stability analysis tool of nonlinear dynamic systems. However, there are some bottlenecks in the Lyapunov method, such as need for negative definiteness of the Lyapunov function derivative in the direction of the system’s solutions. In this paper, we develop a new theorem to dispense the need for negative definite-ness of Lyapunov function derivative. We introduce new sufficient conditions for asymptotic stability of equilibrium states of nonlinear systems considering some inequalities for the higher order time derivatives of Lyapunov function. If the above-mentioned inequalities are found, then the stability analysis of an equilibrium state is reduced to check the characteristic equation for a controllable canonical form LTI co-system. The poles of co-system are required to be negative real ones. Some examples are presented to demonstrate the approach. | ||
| کلیدواژه‌ها | ||
| Nonlinear dynamic systems؛ Lyapunov methods؛ Stability Analysis | ||
| عنوان مقاله [English] | ||
| Extension of Higher Order Derivatives of Lyapunov Functions in Stability Analysis of Nonlinear Systems | ||
| چکیده [English] | ||
| The Lyapunov stability method is the most popular and applicable stability analysis tool of nonlinear dynamic systems. However, there are some bottlenecks in the Lyapunov method, such as need for negative definiteness of the Lyapunov function derivative in the direction of the system’s solutions. In this paper, we develop a new theorem to dispense the need for negative definite-ness of Lyapunov function derivative. We introduce new sufficient conditions for asymptotic stability of equilibrium states of nonlinear systems considering some inequalities for the higher order time derivatives of Lyapunov function. If the above-mentioned inequalities are found, then the stability analysis of an equilibrium state is reduced to check the characteristic equation for a controllable canonical form LTI co-system. The poles of co-system are required to be negative real ones. Some examples are presented to demonstrate the approach. | ||
| کلیدواژه‌ها [English] | ||
| Nonlinear dynamic systems, Lyapunov methods, Stability Analysis | ||
| مراجع | ||
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[1] M. Vidyasagar, “Nonlinear Systems Analysis”, Prentice Hall, 2’nd Ed, 1993. [2] H. K. Khalil, “Nonlinear systems”, Prentice Hall, Englewood Cliffs, NJ, Third ed., 2002. [3] K.S Narendra, and A.M. Annaswamy, “Persistent excitation in adaptive systems”, Internat. J. Control, vol. 45, pp. 127-160, 1987. [4] D. Aeyels and J. Peuteman, “A new asymptotic stability criterion for nonlinear time-variant differential equations”, IEEE Trans. Automat. Control, vol. 43, pp. 968-971, 1998. [5] R. Bellman, “Vector Lyapunov functions,” SIAM J. Control, vol. 1, pp. 32–34, 1962. [6] V. M. Matrosov, “Method of vector Liapunov functions of interconnected systems with distributed parameters (survey)” (in Russian), Avtomatika i Telemekhanika, vol. 33, pp. 63–75, 1972. [7] A.A. Martynyuk, “Stability analysis by comparison technique”, Nonlinear Analysis 62 (2005) 629-641. [8] S. G. Nersesov, W.M. Haddad, “On the stability and control of nonlinear dynamical systems via vector Lyapunov functions”, IEEE Transaction on Automatic Control, vol. 51, no. 2, 2006. [9] M. B. Kudaev, “A study of the behavior of the trajectories of systems of differential equations by means of Lyapunov functions”, Dokl. Akad. Nauk. SSSR 147, pp. 1285-1287, 1962. J. A. Yorke, “A theorem on Liapunov functions using ϋ”, Theory of Computing Systems, Springer, vol. 4, no 1, p | ||
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