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Modified Lorenz Chaos Synchronization Via Active Sliding Mode Controller

능동 슬라이딩 모드 제어기를 이용한 변형된 Lorenz 카오스 동기화

  • Ryu, Ki-Tak (Offshore Training Team, Korea Institute of Maritime and Fisheries Technology) ;
  • Lee, Yun-Hyung (Education Management Team, Korea Institute of Maritime and Fisheries Technology)
  • 류기탁 (한국해양수산연수원 해양플랜트교육팀) ;
  • 이윤형 (한국해양수산연수원 교육운영팀)
  • Received : 2018.04.25
  • Accepted : 2018.07.06
  • Published : 2018.07.31

Abstract

Chaos is one of the most significant topics in nonlinear science, and has been intensively studied since the Lorenz system was introduced. One characteristic of a chaotic system is that the signals produced by it do not synchronize with any other system. It therefore seems impossible for two chaotic systems to synchronize with each other, but if the two systems exchange information in just the right way, they can synchronize. This paper addresses the problem of synchronization in a modified Lorenz chaotic system based on active control, sliding mode control, and the Lyapunov stability theory. The considered synchronization scheme consists of identical drive and response generalized systems coupled with linear state error variables. For this, a brief overview of the modified Lorenz chaotic system is given. Then, control rules are derived for chaos synchronization via active control and slide mode control theory, with a strategy for solving the chattering problem. The asymptotic stability of the overall feedback system is established using the Lyapunov stability theory. A set of computer simulation works is presented graphically to confirm the validity of the proposed method.

References

  1. E. N. Lorenz, "Deterministic Nonperiodic Flow", Journal of Atmospheric Sciences, vol. 20, pp. 130-141, 1963. DOI: https://doi.org/10.1175/1520-0469(1963)020<0130:DNF> 2.0.CO;2 https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
  2. G. Chen and T. Ueta, "Yet Another Chaotic Attractor," International Journal of Bifurcation and Chaos, vol. 9, no. 7, pp. 1465-1466, 1999. DOI: https://doi.org/10.1142/S0218127499001024 https://doi.org/10.1142/S0218127499001024
  3. C. Liu, T. Liu, L. Liu, and K. Liu, "A new Chaotic Attractor," Chaos, Solitons and Fractals, vol. 22, no. 5, pp. 1031-1038, 2004. DOI: https://doi.org/10.1016/j.chaos.2004.02.060 https://doi.org/10.1016/j.chaos.2004.02.060
  4. O. E. Rossler, "An Equation for Continuous Chaos," Physics Letters A, vol. 57, no. 5, pp. 397-398, 1976. DOI: https://doi.org/10.1016/0375-9601(76)90101-8 https://doi.org/10.1016/0375-9601(76)90101-8
  5. J. Lu and G. Chen, "A New Chaotic Attractor Coined," International Journal of Bifurcation and Chaos, vol. 12, no. 3, pp. 659-661, 2002. DOI: https://doi.org/10.1142/S0218127402004620 https://doi.org/10.1142/S0218127402004620
  6. J. C. Sprott, "Some Simple Chaotic Flows," Physical Review E, vol. 50, pp. R647-R650, 1994. DOI: https://doi.org/10.1103/PhysRevE.50.R647 https://doi.org/10.1103/PhysRevE.50.R647
  7. S. Celikovsky and G. Chen, "On a Generalized Lorenz Canonical Form of Chaotic Systems," International Journal of Bifurcation and Chaos, vol. 12, no. 8, pp. 1789-1812, 2002. DOI: https://doi.org/10.1142/S0218127402005467 https://doi.org/10.1142/S0218127402005467
  8. M. Feki, "An Adaptive Chaos Synchronization Scheme Applied to Secure Communication," Chaos, Solitons and Fractals, vol. 18, no. 1, pp. 141-148, 2003. DOI: https://doi.org/10.1016/S0960-0779(02)00585-4 https://doi.org/10.1016/S0960-0779(02)00585-4
  9. A. A. Zaher and A. Abu-Rezq, "On the Design of Chaos-based Secure Communication Systems," Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3721-3737, 2011. DOI: https://doi.org/10.1016/j.cnsns.2010.12.032 https://doi.org/10.1016/j.cnsns.2010.12.032
  10. R. Rhouma and S. Belghith, "Cryptoanalysis of a Chaos Based Cryptosystem on DSP," Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 2, pp. 876-884, 2011. DOI: https://doi.org/10.1016/j.cnsns.2010.05.017 https://doi.org/10.1016/j.cnsns.2010.05.017
  11. J. E. Skinner, "Low-dimensional Chaos in Biological Systems," Nature Biotechnology, vol. 12, pp. 596-600, 1994. DOI: https://doi.org/10.1038/nbt0694-596 https://doi.org/10.1038/nbt0694-596
  12. Y. Nakamura and A. Sekiguchi, "Chaotic Mobile Robot," IEEE Transactions on Robotics and Automation, vol. 17, no. 6, pp. 898-904, 2001. DOI: https://doi.org/10.1109/70.976022 https://doi.org/10.1109/70.976022
  13. C. K. Volos, I. M. Kyripianidisb and I. N. Stouboulosb, "A Chaotic Path Planning Generator for Autonomous Mobile Robots," Robotics and Autonomous Systems, vol. 60, no. 4, pp. 651-656, 2012. DOI: https://doi.org/10.1016/j.robot.2012.01.001 https://doi.org/10.1016/j.robot.2012.01.001
  14. X. S. Yang and Q. Yuan, "Chaos and Transient Chaos in Simple Hopfield Neural Networks," Neurocomputing, vol. 69, no. 1-3, pp. 232-241, 2005. DOI: https://doi.org/10.1016/j.neucom.2005.06.005 https://doi.org/10.1016/j.neucom.2005.06.005
  15. L. M. Pecora and T. L. Carroll, "Synchronization in Chaotic Systems," Physical Review Letters, vol. 64, pp. 821-824, 1990. DOI: https://doi.org/10.1103/PhysRevLett.64.821 https://doi.org/10.1103/PhysRevLett.64.821
  16. Park, J.H. & Kwon, O.M., "A novel criterion for delayed feedback control of time-delay chaotic systems," Chaos, Solitons & Fractals, vol. 23, no. 2, pp 495-501, 2005. DOI: https://doi.org/10.1016/j.chaos.2004.05.023 https://doi.org/10.1016/j.chaos.2004.05.023
  17. L. Lu, C. Zhang, and Z. A. Guo, "Synchronization Between Two Different Chaotic Systems with Nonlinear Feedback Control," Chinese Physics, vol. 16, no. 6, pp. 1603-07, 2007. DOI: https://doi.org/10.1088/1009-1963/16/6/019 https://doi.org/10.1088/1009-1963/16/6/019
  18. E. W. Bai and K. E. Lonngren, "Synchronization of Two Lorenz Systems using Active Control", Chaos, Solitons and Fractals, vol. 8, no. 1, pp. 51-58, 1997. DOI: https://doi.org/10.1016/S0960-0779(96)00060-4 https://doi.org/10.1016/S0960-0779(96)00060-4
  19. H. N. Agiza and M. T. Yassen, "Synchronization of Rossler and Chen Chaotic Dynamical Systems Using Active Control", Physical Letters A, vol. 278, no. 4, pp. 191-197, 2001. DOI: https://doi.org/10.1016/S0375-9601(00)00777-5 https://doi.org/10.1016/S0375-9601(00)00777-5
  20. M. C. Ho and Y. C. Hung, "Synchronization of Two Different Systems by Using Generalized Active Control", Physical Letters. A, vol. 301, no. 5-6, pp. 424 -428, 2002. DOI: https://doi.org/10.1016/S0375-9601(02)00987-8 https://doi.org/10.1016/S0375-9601(02)00987-8
  21. X. Wu and J. Lu, "Parameter Identification and Backstepping Control of Uncertain Lü System," Chaos, Solitons and Fractals, vol. 18, no. 4, pp. 721-729, 2003. DOI: https://doi.org/10.1016/S0960-0779(02)00659-8 https://doi.org/10.1016/S0960-0779(02)00659-8
  22. U. E. Vincent, "Chaos Synchronization Using Active Control and Backstepping Control: A Comparative Analysis," Nonlinear Analysis: Modelling and Control, vol. 13, no. 2, pp. 253-261, 2008.
  23. J. Yan and M. Hung, T. Chiang, and Y. Yang, "Robust Synchronization of Chaotic Systems via Adaptive Sliding Mode Control," Physics Letters A, vol. 356, no. 3, pp. 220-225, 2006. DOI: https://doi.org/10.1016/j.physleta.2006.03.047 https://doi.org/10.1016/j.physleta.2006.03.047
  24. V. Sundarapandian, "Sliding mode controller design for synchronization of Shimizu-Morioka chaotic systems," International Journal of Information Sciences and Techniques, vol. 1, no. 1, pp. 20-29, 2011.
  25. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, "Determining Lyapunov Exponents from a Time Series," Physica D, vol. 16, pp. 285-317, 1985. DOI: https://doi.org/10.1016/0167-2789(85)90011-9 https://doi.org/10.1016/0167-2789(85)90011-9