Stabilization of the Lorenz Chaotic Equations by Fuzzy Controller
In this paper, a fuzzy controller is designed for
stabilization of the Lorenz chaotic equations. A simple Mamdani
inference method is used for this purpose. This method is very simple
and applicable for complex chaotic systems and it can be
implemented easily. The stability of close loop system is investigated
by the Lyapunov stabilization criterion. A Lyapunov function is
introduced and the global stability is proven. Finally, the
effectiveness of this method is illustrated by simulation results and it
is shown that the performance of the system is improved.
[1] C. Wang, T. Lin, T. Lee, H. Liu, "Adaptive hybrid intelligent control for
uncertain nonlinear dynamical systems", IEEE Trans. on Systems, Man
and Cybernetics- part B: Cybernetics, vol. 32, no. 5, Oct. 2003.
[2] C. Park, C. Lee, M. Park, "Design of an adaptive fuzzy model based
controller for chaotic dynamics in Lorenz systems with uncertainty", J.
Information Sciences, vol. 147, pp. 245-266, 2002.
[3] S.Hu, Y. Liu, "Robust H∞ control of multiple time-delay uncertain
nonlinear system using fuzzy model and adaptive neural network", J.
Fuzzy Sets and Systems, pp. 1-18, 2003.
[4] W. Jiang, Q. Dong, D. Bin, "Observer-based robust adaptive variable
universe fuzzy control for chaotic system", J. Chaos, Solitons and
Fractals, vol. 23, pp. 1013-1032, 2005.
[5] X. Wang, Adaptive Fuzzy System and Control, Prentice-Hall, First
Edition, New Jercy, 1994.
[6] K. Tanaka, T. Ikeda, H .O. Wang, "Controlling chaos via model-based
fuzzy control system design", in Proc. of the 36th Conf. on Decision &
Control, Sun Diego, California USA, Dec.1997.
[7] K. Tanaka, T. Ikeda, H.O. Wang, "A unified approach to controlling
chaos Via an LMI-based fuzzy control system Design", IEEE Trans. On
Circuits & Systems, vol. 45, no. 10, pp. 1021-1040, Oct.1998.
[8] L. Chen and G. Chen, "fuzzy modeling, prediction, and control of
uncertain chaotic systems based on time series", IEEE Trans. On
Circuits & System, vol. 47, no. 10, pp. 29-45, Oct. 2000.
[9] Z. Li, J. B. Park, Y. H. Joo, "Chaotifying continuous-time TS fuzzy
systems via discretization", IEEE Trans. On Circuits & System, vol. 48,
no. 10, pp. 1122-1134, Oct. 2001.
[10] G. Feng, G. Chen, "Adaptive control of discrete-time chaotic systems: a
fuzzy control approach", J. Chaos, Solitons and Fractals, vol. 23, pp.
459-467, 2005.
[11] O. Castillo, P. Mellin, Soft Computing for Control of Nonlinear
Dynamical Systems, First Edition, Springer-Verlag Ltd, New York,
2001.
[12] E. N. Lorenz, "Deterministic Non-Periodic Flow", J. Automata , vol.
12, pp. 130-141, 1963.
[1] C. Wang, T. Lin, T. Lee, H. Liu, "Adaptive hybrid intelligent control for
uncertain nonlinear dynamical systems", IEEE Trans. on Systems, Man
and Cybernetics- part B: Cybernetics, vol. 32, no. 5, Oct. 2003.
[2] C. Park, C. Lee, M. Park, "Design of an adaptive fuzzy model based
controller for chaotic dynamics in Lorenz systems with uncertainty", J.
Information Sciences, vol. 147, pp. 245-266, 2002.
[3] S.Hu, Y. Liu, "Robust H∞ control of multiple time-delay uncertain
nonlinear system using fuzzy model and adaptive neural network", J.
Fuzzy Sets and Systems, pp. 1-18, 2003.
[4] W. Jiang, Q. Dong, D. Bin, "Observer-based robust adaptive variable
universe fuzzy control for chaotic system", J. Chaos, Solitons and
Fractals, vol. 23, pp. 1013-1032, 2005.
[5] X. Wang, Adaptive Fuzzy System and Control, Prentice-Hall, First
Edition, New Jercy, 1994.
[6] K. Tanaka, T. Ikeda, H .O. Wang, "Controlling chaos via model-based
fuzzy control system design", in Proc. of the 36th Conf. on Decision &
Control, Sun Diego, California USA, Dec.1997.
[7] K. Tanaka, T. Ikeda, H.O. Wang, "A unified approach to controlling
chaos Via an LMI-based fuzzy control system Design", IEEE Trans. On
Circuits & Systems, vol. 45, no. 10, pp. 1021-1040, Oct.1998.
[8] L. Chen and G. Chen, "fuzzy modeling, prediction, and control of
uncertain chaotic systems based on time series", IEEE Trans. On
Circuits & System, vol. 47, no. 10, pp. 29-45, Oct. 2000.
[9] Z. Li, J. B. Park, Y. H. Joo, "Chaotifying continuous-time TS fuzzy
systems via discretization", IEEE Trans. On Circuits & System, vol. 48,
no. 10, pp. 1122-1134, Oct. 2001.
[10] G. Feng, G. Chen, "Adaptive control of discrete-time chaotic systems: a
fuzzy control approach", J. Chaos, Solitons and Fractals, vol. 23, pp.
459-467, 2005.
[11] O. Castillo, P. Mellin, Soft Computing for Control of Nonlinear
Dynamical Systems, First Edition, Springer-Verlag Ltd, New York,
2001.
[12] E. N. Lorenz, "Deterministic Non-Periodic Flow", J. Automata , vol.
12, pp. 130-141, 1963.
@article{"International Journal of Electrical, Electronic and Communication Sciences:50662", author = "Behrooz Rezaie and Zahra Rahmani Cherati and Mohammad Reza Jahed Motlagh and Mohammad Farrokhi", title = "Stabilization of the Lorenz Chaotic Equations by Fuzzy Controller", abstract = "In this paper, a fuzzy controller is designed for
stabilization of the Lorenz chaotic equations. A simple Mamdani
inference method is used for this purpose. This method is very simple
and applicable for complex chaotic systems and it can be
implemented easily. The stability of close loop system is investigated
by the Lyapunov stabilization criterion. A Lyapunov function is
introduced and the global stability is proven. Finally, the
effectiveness of this method is illustrated by simulation results and it
is shown that the performance of the system is improved.", keywords = "Chaotic system, Fuzzy control, Lorenz equation.", volume = "2", number = "10", pages = "2134-4", }