J. L. Mata–Machuca¹, R. Martínez–Guerra¹, R. Aguilar–López*²

¹ Department of Automatic Control, CINVESTAV–IPN, Mexico D.F., Mexico. jmata@ctrl.cinvestav.mx, rguerra@ctrl.cinvestav.mx

² Department of Biotechnology and Bioengineering, CINVESTAV–IPN, Mexico D.F., Mexico. Phone(55) 57473733 Fax (55) 57473982. *E–mail: raguilar@cinvestav.mx

ABSTRACT

In this paper, we consider the synchronization problem via a nonlinear observer design. A new exponential polynomial observer for a class of nonlinear oscillators is proposed, which is robust against output noises. A sufficient condition for synchronization is derived analytically with the help of the Lyapunov stability theory. The proposed technique has been applied to synchronize chaotic systems (Rikitake and Rössler systems) by numerical simulation.

Keywords: synchronization, polynomial observer, Lipschitz system, algebraic observability condition.

RESUMEN

En este trabajo se considera el problema de sincronización por medio del diseño de un observador no lineal. Se propone un nuevo observador polinomial exponencial para una clase de osciladores no lineales. La condición suficiente para lograr la sincronización es desarrollada analíticamente con la ayuda de la teoría de estabilidad de Lyapunov. La técnica propuesta ha sido aplicada para sincronizar sistemas caóticos (los sistemas de Rikitake y Rössler) empleando simulaciones numéricas.

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References

[1] A. Fradkov. Cybernetical physics: from control of chaos to quantum control. Berlin: Springer, 2007.         [ Links ]

[2] M. Chen, D. Zhou and Y. Shang, "A sliding mode observer based secure communication scheme," Chaos, Solitons Fractals, vol. 25, pp. 573–8, 2005.         [ Links ]

[3] M. Feki, "Observer–based exact synchronization of ideal and mismatched chaotic systems," Physics Letters A, vol. 309, pp. 53–60, 2003.         [ Links ]

[4] O. Morgül and M. Feki, "A chaotic masking scheme by using synchronized chaotic systems," Physics Letter A, vol. 251, pp. 169–176, 1999.         [ Links ]

[5] R. Guo, "A simple adaptive controller for chaos and hyperchaos synchronization," Physics Letters A, vol. 372, pp. 5593–5597, 2008.         [ Links ]

[6] R. Martínez–Guerra, W. Yu and E. Cisneros–Saldaña, "A new model–free sliding observer to synchronization problem," Chaos Solitons Fractals, vol. 36, pp. 1141–1156, 2008.         [ Links ]

[7] R. Aguilar–Lopez and R. Martínez–Guerra, "Synchronization of a Class of Chaotic signals via robust observer design," Chaos, Solitons Fractals, vol. 37, pp. 581–587, 2008.         [ Links ]

[8] C. Hua and X. Guan, "Synchronization of chaotic systems based on PI observer design," Physics Letters A, vol. 334, pp. 382–389, 2005.         [ Links ]

[9] R. Martínez–Guerra, J. Cruz, R. Gonzalez, R. Aguilar, "A new reduced–order Observer design for the synchronization of Lorenz systems," Chaos, Solitons Fractals, vol. 28, pp. 511–7, 2006.         [ Links ]

[10] O. Morgül and E. Solak, "Observed based synchronization of chaotic systems," Phys. Rev. E, vol. 54, pp. 4803–4811, 1996.         [ Links ]

[11] R. Femat and G. Solís–Perales, Robust synchronization of chaotic systems via feedback, Springer Verlag, 2008.         [ Links ]

[12] F. Wang and C. Liu, "A new criterion for chaos and hyperchaos synchronization using linear feedback control," Physics Letters A, vol. 360, pp. 274–278, 2006.         [ Links ]

[13] L. Pecora, T. Caroll, "Synchronization in chaotic systems," Phys. Rev. Lett., vol. 64, pp. 821–4, 1990.         [ Links ]

[14] S. Raghavan and J. Hedrick, "Observer design for a class of nonlinear systems," Int Journal Control, vol. 59, pp. 515–528, 1994.         [ Links ]

[15] F. Thau, "Observing the state of non–linear dynamic systems," Int Journal Control, vol. 17, pp. 471–479, 1973.         [ Links ]

[16] R. Rajamani, "Observers for Lipschitz nonlinear systems," IEEE Trans. Aut. Control, vol. 43, no.3, pp. 397–401, 1998.         [ Links ]

[17] S. Diop and R. Martínez–Guerra, "An algebraic and data derivative information approach to nonlinear system diagnosis," in Proceedings of the European Control Conference (ECC), Porto, Portugal, 2001, pp. 23342339.         [ Links ]

[18] R. Horn, C. Johnson, Matrix analysis, Cambridge University Press, New York, 1985, pp. 176.         [ Links ]

[19] O. Rössler, "An Equation for Continuous Chaos," Phys. Lett., vol. 57 A, pp. 397–398, 1976.         [ Links ]

[20] R. Martínez–Guerra, A. Poznyak and V. Díaz, "Robustness of high–gain observers for closed–loop nonlinear systems: theoretical study and robotics control application," Int Journal Systems Science, vol. 31, pp. 1519–1529, 2000.         [ Links ]

[21] T. Rikitake, "Oscillations of a system of disk dynamos". Proc Cambridge Philos Soc, vol. 54, pp. 89105, 1958.         [ Links ]