Journal of the Korea Academia-Industrial cooperation Society (한국산학기술학회논문지)

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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
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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.

카오스는 비선형 과학 분야에서 매우 중요한 주제 중의 하나이며, Lorenz가 처음으로 소개한 이후 집중적으로 연구되어지고 있다. 카오스 시스템의 한 특성은 카오스 시스템에 의해 생성된 신호는 다른 어떤 시스템과 동기화되지 않는다는 것이다. 따라서 두 카오스 시스템은 서로 동기화되는 것이 불가능한 것처럼 보이지만, 만약 두 시스템이 적절한 방법으로 정보를 교환한다면 이 두 시스템은 동기화가 가능하다. 본 논문에서는 능동 제어와 슬라이딩 모드 제어, 그리고 리아프노프 안정도 이론을 기반으로 하는 변형된 Lorenz 카오스 시스템의 동기화 문제에 대해 다룬다. 동기화를 위해 고려한 기법은 선형상태 오차 변수에 의해 짝을 이룬 구동시스템과 응답시스템으로 구성된다. 이를 위해 우선 대상 카오스 시스템에 대해 간단히 살펴본다. 다음으로 능동제어, 슬라이딩 모드 제어 기법을 이용한 카오스 시스템의 동기화와 채터링 문제를 해결하기 위한 제어 방법을 도출한다. 전체 폐루프 시스템의 점근적 안정도는 리아프노프 안정도 이론에 의해 증명한다. 컴퓨터 시뮬레이션은 제안한 방법의 타당성을 확인하기 위해 그래픽으로 제시한다.

Keywords

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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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
  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