Lorenz System Stabilization Using Fuzzy Controllers

Authors

  • Radu-Emil Precup "Politehnica" University of Timisoara Department of Automation and Applied Informatics Bd. V. Parvan 2, RO-300223 Timisoara, Romania
  • Marius L. Tomescu "Aurel Vlaicu" University Computer Science Faculty Complex Universitar M, Str. Elena Dragoi 2, RO-310330 Arad, Romania
  • Stefan Preitl "Politehnica" University of Timisoara Department of Automation and Applied Informatics Bd. V. Parvan 2, RO-300223 Timisoara, Romania

Keywords:

chaotic systems, fuzzy control, Lyapunov functions, nonlinear equations and systems

Abstract

The paper suggests a Takagi Sugeno (TS) fuzzy logic controller (FLC) designed to stabilize the Lorentz chaotic systems. The stability analysis of the fuzzy control system is performed using Barbashin-Krasovskii theorem. This paper proves that if the derivative of Lyapunov function is negative semi-definite for each fuzzy rule then the controlled Lorentz system is asymptotically stable in the sense of Lyapunov. The stability theorem suggested here offers sufficient conditions for the stability of the Lorenz system controlled by TS FLCs. An illustrative example describes the application of the new stability analysis method.

References

Calvo O., Cartwright J. H. E., Fuzzy control of chaos, International Journal of Bifurcation and Chaos, Vol. 8, Number 8, pp. 1743-1747, 1998. http://dx.doi.org/10.1142/S0218127498001443

Khalil H. K., Nonlinear Systems, 3rd Edition, Prentice Hall, Englewood Cliffs, NJ, 2002.

Lima R., Pettini M., Suppression of chaos by resonant parametric perturbations, Physical Letter A, Vol. 41, pp. 726-733, 1990. http://dx.doi.org/10.1103/physreva.41.726

Lorenz E. N., The Essence of Chaos, University of Washington Press, 1993. http://dx.doi.org/10.4324/9780203214589

Ott E., Grebogi, C., Yorke, J. A, Controlling chaos, Physical Review Letter, Vol. 64, pp. 1196-1199, 1990. http://dx.doi.org/10.1103/PhysRevLett.64.1196

Precup R.-E., Preitl S., Optimisation criteria in development of fuzzy controllers with dynamics, Engineering Applications of Artificial Intelligence, Vol. 17, No. 6, pp. 661-674, 2004. http://dx.doi.org/10.1016/j.engappai.2004.08.004

Pyragas K, Continuous control of chaos by self-controlling feedback, Physical Letter A, Vol. 170, pp. 421-427, 1992. http://dx.doi.org/10.1016/0375-9601(92)90745-8

Pyragas K., Tamaevièius A., Experimental control of chaos by delayed self-controlling feedback, Physical Letter A, Vol. 180, pp. 99-102, 1993. http://dx.doi.org/10.1016/0375-9601(93)90501-P

Schuster H. G., Handbook of Chaos Control: Foundations and Applications, Wiley-VCH Verlag GmbH, 1999. http://dx.doi.org/10.1002/3527607455

Wang H. O., Tanaka K., Fuzzy Modeling and Control of Chaotic Systems, in Integration of Fuzzy Logic and Chaos Theory, Springer-Verlag, Berlin, Heidelberg, 2006. http://dx.doi.org/10.1007/3-540-32502-6_3

Zhong L., Halang W. A., Chen, G. (Eds.), Integration of Fuzzy Logic and Chaos Theory, Springer- Verlag, Berlin, Heidelberg, 2006.

Published

2007-09-01

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