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Adaptive Control and Synchronization of the Harb System

Dr.V. Sundarapandian

Abstract


This paper investigates the adaptive control and synchronization of the uncertain Harb system with unknown parameters. The Harb system is one of the important and simple paradigms of three-dimensional chaotic systems discovered by Harb and Zohdy in 2002. Adaptive control is a popular method in the control systems literature which is used whenever the parameters of the systems are not available. In adaptive control method, we modify the control strategies with estimates of the parameters and using Lyapunov stability theory, we derive update laws for the estimates of the system parameters. In this paper, we first design an adaptive controller so as to stabilize the uncertain Harb system to its unstable equilibrium at the origin based on the adaptive control theory and Lyapunov stability theory. Then we derive adaptive control laws to achieve global chaos synchronization of identical uncertain Harb systems with unknown parameters. Numerical simulations are shown to demonstrate the effectiveness of the proposed adaptive control and synchronization schemes for the uncertain Harb system.

Keywords


Adaptive Control, Stabilization, Chaos, Synchronization, Harb System

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References


K.T. Alligood, T. Sauer and J.A. Yorke, Chaos: An Introduction to Dynamical Systems, New York: Springer-Verlag, 1997.

E.N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci., vol. 20, pp. 130-141, 1963.

S.S. Ge, C. Wang and T.H. Lee, “Adaptive backstepping control of a class of chaotic systems,” Internat. J. Bifurcation and Chaos, vol. 10, pp. 1149-1156, 2000.

X. Wang, L. Tian and L. Yu, “Adaptive control and slow manifold analysis of a new chaotic system,” Internat. J. Nonlinear Science, vol. 21, pp. 43-49, 2000.

M. Sun, L. Tian, S. Jiang and J. Xun, “Feedback control and adaptive control of the energy resource chaotic system,” Chaos, Solitons and Fractals, vol. 32, pp. 168-180, 2007.

V. Sundarapandian, “Adaptive control and synchronization of hyperchaotic Liu system,” Internat. J. Computer Science, Engineering and Information Technology, vol. 1, no. 2, pp. 29-40, 2011.

L.M. Pecora and T.L. Carroll, “Synchronization in chaotic systems,” Physical Review Letters, vol. 64, pp. 821-824, 1990.

M. Lakshmanan and K. Murali, Chaos in Nonlinear Oscillators: Controlling and Synchronization, World Scientific, Singapore, 1996.

S.K. Han, C. Kerrer and Y. Kuramoto, “D-phasing and bursting in coupled neural oscillators,” Physical Review Letters, vol. 75, pp. 3190-3193, 1995.

B. Blasius, A. Huppert and L. Stone, “Complex dynamics and phase synchronization in spatially extended ecological system,” Nature, Vol. 399, pp. 354-359, 1999.

K. Murali and M. Lakshmanan, “Secure communication using a compound signal from generalized synchronizable systems,” Physical Review Letters A, vol. 241, pp. 303-310, 1998.

M. Feki, “An adaptive chaos synchronization scheme applied to secure communication,” Chaos, Solitons and Fractals, vol. 18, pp. 141-148, 2003.

T. Yang, “A survey of chaotic secure communication systems,” Internat. J. Computational Cognition, vol. 2, no. 2, pp. 81-130, 2004.

T. Yang and L.O. Chua, “Control of chaos using sampled-data feedback control,” Internat. J. Bifurcat. Chaos, vol. 9, pp. 215-219, 1999.

E. Ott, C. Grebogi and J.A. Yorke, “Controlling chaos,” Physical Review Letters, vol. 64, pp. 1196-1199, 1990.

J.H. Park and O.M. Kwon, “A novel criterion for delayed feedback control,” Internat. J. Bifurcat. Chaos, vol. 9, pp. 215-219, 1999.

Y.G. Yu and S.C. Zhang, “Adaptive backstepping synchronization of uncertain chaotic systems,” Chaos, Solitons and Fractals, vol. 27, pp. 1369-1375, 2006.

J.H. Park, “Chaos synchronization of chaotic system via nonlinear control,” Chaos, Solitons and Fractals, vol. 25, no. 3, pp. 579-584, 2005.

U.E. Vincent, “Chaos synchronization using active control and backstepping control,” Nonlinear Analysis: Modelling and Control, vol. 13, pp. 253-261, 2008.

V. Sundarapandian and R. Karthikeyan, “Global chaos synchronization of hyperchaotic Liu and hyperchaotic Chen systems by active nonlinear control,” CIIT International Journal of Digital Signal Processing, vol. 3, no. 3, pp. 134-139, 2011.

V. Sundarapandian and R. Karthikeyan, “Global chaos synchronization of Chen and Cai systems by active nonlinear control,” CIIT International Journal of Digital Signal Processing, vol. 3, no. 3, pp. 140-144, 2011.

T.L. Liao and S.H. Tsai, “Adaptive synchronization of chaotic systems and its applications to secure communications”, Chaos, Solitons and Fractals, vol. 11, pp. 1387-1396, 2000.

J.H. Park, S.M. Lee and O.M. Kwon, “Adaptive synchronization of Genesio-Tesi chaotic system via a novel feedback control,” Physics Letters A, vol. 371, pp. 263-270, 2007.

V. Sundarapandian, “Adaptive stabilization and synchronization of hyperchaotic Qi system,” Computer Science Engineering: An International Journal, vol. 1, no. 2, pp. 14-25, 2011.

V. Sundarapandian and R. Karthikeyan, “Adaptive synchronization of uncertain Li and T chaotic systems,” Internat. J. Engineering Science and Technology, vol. 3, no. 5, pp. 4272-4283, 2011.

V.I. Utkin, “Sliding mode control design principles and applications to electric drives,” IEEE Transactions on Industrial Electronics, vol. 40, pp. 23-36, 1993.

V. Sundarapandian and S. Sivaperumal, “Sliding mode control based global chaos synchronization of four-scroll attractors,” CIIT International J. Programmable Device Circuits and Systems, vol. 3, no. 6, pp. 297-302, 2011.

V. Sundarapandian and S. Sivaperumal, “Global chaos synchronization of hyperchaotic Chen system by sliding mode control,” Internat. J. Engineering Science and Technology, vol. 3, no. 5, pp. 4265-4271, 2011.

A.M. Harb and M.M. Zohdy, “Chaos and bifurcation control using nonlinear recursive controller,” Nonlinear Analysis: Modelling and Control, vol. 7, pp. 37-43, 2002.

W. Hahn, The Stability of Motion, Springer, Berlin, 1967.


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