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Chaos control and modified projective synchronization of unknown heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive backstepping control

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Abstract

This paper proposes the chaos control and the modified projective synchronization methods for unknown heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive backstepping control. Because of the nonlinear terms of the gyroscope system, the system exhibits chaotic motions. Occasionally, the extreme sensitivity to initial states in a system operating in chaotic mode can be very destructive to the system because of unpredictable behavior. In order to improve the performance of a dynamic system or avoid the chaotic phenomena, it is necessary to control a chaotic system with a regular or periodic motion beneficial for working with a particular condition. As chaotic signals are usually broadband and noise-like, synchronized chaotic systems can be used as cipher generators for secure communication. Obviously, the importance of obtaining these objectives is specified when the dynamics of gyroscope system are unknown. In this paper, using the neural backstepping control technique, control laws are established which guarantees the chaos control and the modified projective synchronization of unknown chaotic gyroscope system. In the neural backstepping control, Gaussian radial basis functions are utilized to on-line estimate the system dynamic functions. Also, the adaptation laws of the on-line estimators are derived in the sense of Lyapunov function. Thus, the unknown chaotic gyroscope system can be guaranteed to be asymptotically stable. Also, the control objectives have been achieved.

The proposed method allows us to arbitrarily adjust the desired scaling by controlling the slave system. It is not necessary to calculate the Lyapunov exponents and the eigenvalues of the Jacobian matrix, which makes it simple and convenient. Also, it is a systematic procedure for modified projective synchronization of chaotic systems and it can be applied to a variety of chaotic systems no matter whether it contains external excitation or not. Notice that it needs only one controller to realize modified projective synchronization no matter how much dimensions the chaotic system contains and the controller is easy to be implemented. It seems that the proposed method can be useful for practical applications of chaotic gyroscope systems in the future.

Numerical simulations are presented to verify the proposed control and synchronization methods.

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References

Chaos, chaos control and chaos synchronization

  1. Chen, G.: Controlling Chaos and Bifurcations in Engineering Systems. CRC Press, Boca Raton (1999)

    Google Scholar 

  2. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64, 1196–1199 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, G., Dong, X.: From Chaos to Order: Perspectives, Methodologies and Applications. World Scientific, Singapore (1988)

    Google Scholar 

  4. Feng, J., Xu, C., Tang, J.: Controlling Chen’s chaotic attractor using two different techniques based on parameter identification. J. Chaos Solitons Fractals 32, 1413–1418 (2007)

    Article  MATH  Google Scholar 

  5. Harb, A.M., Zahar, A.A., Al Qaisia, A.A., Zohdy, M.A.: Recursive backstepping control of chaotic doffing oscillators. J. Chaos Solitons Fractals 34, 639–645 (2007)

    Article  Google Scholar 

  6. Farivar, F., Shoorehdeli, M.A., Nekoui, M.A., Teshnehlab, M.: Gaussian radial basis adaptive backstepping control for a class of nonlinear systems. J. Appl. Sci. 9(2), 284–257 (2009)

    Google Scholar 

  7. Femat, R., Jauregui-Ortiz, R., Solis-Perales, G.A.: Chaos-based communication scheme via robust asymptotic feedback. IEEE Trans. Circuits Syst. I 48, 1161–1169 (2001)

    Article  Google Scholar 

  8. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MathSciNet  Google Scholar 

  9. Yang, J.Z., Hu, G., Xiao, J.H.: Chaos synchronization in coupled chaotic oscillators with multiple positive Lyapunov exponents. Phys. Rev. Lett. 80, 496 (1998)

    Article  Google Scholar 

  10. Shahverdiev, E.M.: Synchronization in systems with multiple time delays. Phys. Rev. E 70, 067202 (2004)

    Article  Google Scholar 

  11. Xu, J.F., Min, L.Q., Chen, G.R.: A chaotic communication scheme based on generalized synchronization and hash functions. Chin. Phys. Lett. 21, 1445 (2004)

    Article  Google Scholar 

  12. Rulkov, N.F., Sushchik, M.M., Tsimring, L.S.: Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51, 980 (1995). Abarbanel HDL

    Article  Google Scholar 

  13. Rosenblutn, M.G., Pikovsky, A.S., Kurths, J.: Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804 (1996)

    Article  Google Scholar 

  14. Cao, L.Y., Lai, Y.C.: Anti-phase synchronism in chaotic systems. Phys. Rev. E 58, 382–386 (1998)

    Article  Google Scholar 

  15. Mainieri, R., Rehacek, J.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82, 3042–3045 (1999)

    Article  Google Scholar 

  16. Wen, G.L., Xu, D.L.: Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems. J. Chaos Solitons Fractals 26, 71 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. Wen, G.L., Xu, D.L.: Observer-based control for full-state projective synchronization of a general class of chaotic maps in any dimension. Phys. Lett. A 333, 420 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  18. Yan, J., Li, C.: Generalized projective synchronization of a unified chaotic system. J. Chaos Solitons Fractals 26, 1119–1124 (2005)

    Article  MATH  Google Scholar 

  19. Yan, J.P., Li, C.P.: Generalized projective synchronization for the chaotic Lorenz system and the chaotic Chen system. J. Shanghai Univ. 10, 299 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Farivar, F., Shoorehdeli, M.A., Nekoui, M.A., Teshnehlab, M.: Generalized projective synchronization for chaotic systems via Gaussian radial basis adaptive backstepping control. J. Chaos Solitons Fractals 42, 826–839 (2009)

    Article  MATH  Google Scholar 

  21. Li, G.H.: Modified projective synchronization of chaotic system. J. Chaos Solitons Fractals 32(5), 1786–1790 (2007)

    Article  MATH  Google Scholar 

Chaotic gyroscope system

  1. Leipnik, R.B., Newton, T.A.: Double strong attractors in rigid body motion. Phys. Lett. A 86, 63–67 (1981)

    Article  MathSciNet  Google Scholar 

  2. Chen, H.K.: Chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping. J. Sound Vib. 255, 719–740 (2002)

    Article  Google Scholar 

  3. Ge, Z.-M.: Chaos Control for Rigid Body Systems. Gau Lih Book, Taipei (2002)

    Google Scholar 

  4. Chen, H.K.: Chaos and chaos synchronization of a symmetric gyro with linear-pulse-cubic damping. J. Sound Vib. 255(4), 719–740 (2002)

    Article  Google Scholar 

  5. Van Dooren, R.: Comments on chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping. J. Sound Vib. 268, 632–634 (2003)

    Article  Google Scholar 

  6. Chen, H.K.: Author’s reply. J. Sound Vib. 268, 635–636 (2003)

    Article  Google Scholar 

  7. Chen, H.K., Ge, Z.-M.: Bifurcations and chaos of a two-degree-of-freedom dissipative gyroscope. J. Chaos Solitons Fractals 24, 125–136 (2005)

    MATH  MathSciNet  Google Scholar 

  8. Chen, H.K., Ge, Z.M.: Bifurcations and chaos of a two-degree-of-freedom dissipative gyroscope. J. Chaos Solitons Fractals 24, 125–136 (2005)

    MATH  MathSciNet  Google Scholar 

  9. Idowu, B.A., Vincent, U.E., Njah, A.N.: Control and synchronization of chaos in nonlinear gyros via backstepping design. Int. J. Nonlinear Sci. 5(1), 11–19 (2008)

    MathSciNet  Google Scholar 

  10. Yan, J.J., Hung, M.L., Lin, J.S., Liao, T.L.: Controlling chaos of a chaotic nonlinear gyro using variable structure control. J. Mech. Syst. Signal Process. 21, 2515–2522 (2007)

    Article  Google Scholar 

  11. Lei, Y., Xu, W., Zheng, H.: Synchronization of two chaotic nonlinear gyros using active control. Phys. Lett. A 343, 153–158 (2005)

    Article  MATH  Google Scholar 

  12. Salarieh, H.: Comment on: Synchronization of two chaotic nonlinear gyros using active control [Physics Letter A, Vol. 343, p. 153, 2005]. Phys. Lett. A 372, 2539–2540 (2008)

    Article  MATH  Google Scholar 

  13. Idowu, B.A., Vincent, U.E., Njah, A.N.: Generalized adaptive backstepping synchronization for non-identical parametrically excited systems. Nonlinear Anal. Model. Control 14(2), 165–176 (2009)

    MATH  MathSciNet  Google Scholar 

  14. Yan, J.J., Hung, M.L., Liao, T.L.: Adaptive sliding mode control for synchronization of chaotic gyros with fully unknown parameters. J. Sound Vib. 298, 298–306 (2006)

    Article  MathSciNet  Google Scholar 

  15. Yau, H.T.: Nonlinear rule-based controller for chaos synchronization of two gyros with linear-plus-cubic damping. J. Chaos Solitons Fractals 34, 1357–1365 (2007)

    Article  MATH  Google Scholar 

  16. Yau, H.T.: Chaos synchronization of two uncertain chaotic nonlinear gyros using fuzzy sliding mode control. J. Mech. Syst. Signal Process. 22, 408–418 (2008)

    Article  Google Scholar 

  17. Farivar, F., Shoorehdeli, M.A., Nekoui, M.A., Teshnehlab, M.: Chaos synchronization of uncertain nonlinear gyros via hybrid control. In: IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp. 1365–1370 (2009)

    Chapter  Google Scholar 

  18. Farivar, F., Nekoui, M.A., Teshnehlab, M., Shoorehdeli, M.A.: Neural sliding mode control for chaos synchronization of uncertain nonlinear. Adv. Appl. Math. Sci. 4(1), 41–56 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Salarieh, H., Alasty, A.: Chaos synchronization of nonlinear gyros in presence of stochastic excitation via sliding mode control. J. Sound Vib. 313, 760–771 (2008)

    Article  Google Scholar 

  20. Hung, M.L., Yan, J.J., Liao, T.L.: Generalized projective synchronization of chaotic nonlinear gyros coupled with dead-zone input. J. Chaos Solitons Fractals 35, 181–187 (2008)

    Article  MathSciNet  Google Scholar 

  21. Mohseni, S.A., Mohseni, S.M.: Fuzzy neural networks controller for a chaotic nonlinear gyro using sliding mode surfaces. In: International Symposium on Industrial Electronics (ISIE), Cambridge, pp. 1150–1153 (2008)

    Google Scholar 

  22. Ge, Z.M., Lee, J.K.: Chaos synchronization and parameter identification for gyroscope system. Appl. Math. Comput. 163, 667–682 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yau, H.T.: Generalized projective chaos synchronization of gyroscope systems subjected to dead-zone nonlinear inputs. Phys. Lett. A 372, 2380–2385 (2008)

    Article  MATH  Google Scholar 

  24. Yau, H.T.: Synchronization and anti-synchronization coexist in two-degree-of-freedom dissipative gyroscope with nonlinear inputs. Nonlinear Anal., Real World Appl. 9, 2253–2261 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Backstepping control and neural backstepping control

  1. Yassen, M.T.: Chaos control of chaotic dynamical systems using backstepping design. J. Chaos Solitons Fractals 27, 537–548 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Khalil, H.: Nonlinear Sytems, 3rd edn. Prentice-Hall, New York (2002)

    Google Scholar 

  3. Peng, Y.F., Lin, M.H., Chiu, C.H., Lin, C.M.: Development of adaptive intelligent backstepping tracking control for uncertain chaotic systems. In: 6th International Conference on Machine Learning and Cybernetics, pp. 2037–2043 (2007)

    Chapter  Google Scholar 

  4. Kwan, C., Lewis, F.L.: Robust backstepping control of nonlinear systems using neural networks. IEEE Trans. Syst. Man Cybern. A 30(5), 753–766 (2000)

    Article  Google Scholar 

  5. Lewis, F.L., Yesildirek, A., Liu, K.: Robust backstepping control of induction motor using neural networks. IEEE Trans. Neural Netw. 11(6), 1178–1187 (2000)

    Google Scholar 

  6. Zhang, Y., Peng, P.Y., Jiang, Z.P.: Stable neural controller design for unknown nonlinear systems using backstepping. IEEE Trans. Neural Netw. 11(5), 1347–1359 (2000)

    Article  Google Scholar 

  7. Chen, B., Liu, X., Tong, S.: Adaptive fuzzy output tracking control of MIMO nonlinear uncertain systems. IEEE Trans. Fuzzy Syst. 15(2), 287–300 (2007)

    Article  Google Scholar 

  8. Hsu, C.F., Lin, C.M., Lee, T.T.: Wavelet adaptive backstepping control for a class of nonlinear systems. IEEE Trans. Neural Netw. 17(5), 1175–1183 (2006)

    Article  Google Scholar 

  9. He, P., Jagannathan, S.: Neuroemission controller for reducing cyclic dispersion in lean combustion spark ignition engines. J. Autom. 41, 1133–1142 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Jagannathan, S.: Discrete-time adaptive control of feedback linearizable nonlinear systems. Proc. IEEE Conf. Decis. Control 3, 1704–1709 (1996)

    Google Scholar 

  11. Jagannathan, S., Lewis, F.L.: Discrete-time neural net controller for a class of nonlinear dynamical systems. IEEE Trans. Autom. Control 41(11), 1693–1699 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jagannathan, S., Lewis, F.L.: Multilayer discrete-time neural-net controller with guaranteed performance. IEEE Trans. Neural Netw. 7(1), 107–130 (1996)

    Article  Google Scholar 

  13. Jagannathan, S., Commuri, S., Lewis, F.L.: Robust backstepping control of robotic systems using neural networks. J. Intell. Robot. Syst. 23, 105–128 (1998)

    Article  MATH  Google Scholar 

  14. Jagannathan, S.: Control of a class of nonlinear discrete-time systems using multilayer neural networks. IEEE Trans. Neural Netw. 12(5), 1113–1120 (2001)

    Article  MathSciNet  Google Scholar 

  15. Lewis, F.L., Jagannathan, S., Yesildirek, A.: Neural Network Control of Robot Manipulators and Nonlinear Systems. Taylor & Francis, London (1999)

    Google Scholar 

  16. Alanis, A.Y., Sanchez, E.N., Loukianov, A.G.: Discrete-time adaptive backstepping nonlinear control via high-order neural networks. IEEE Trans. Neural Netw. 18(4), 1185–1195 (2007)

    Article  Google Scholar 

  17. Polycarpou, M.M.: Stable adaptive neural control scheme for nonlinear systems. IEEE Trans. Autom. Control 41(3), 447–451 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. Lin, F.J., Wai, R.J., Chen, H.P.: A PM synchronous servo motor drive with an on-line trained fuzzy neural network controller. IEEE Trans. Energy Convers. 13(4), 319–325 (1998)

    Article  Google Scholar 

  19. Wang, H., Wang, Y.: Neural-network-based fault-tolerant control of unknown nonlinear systems. Proc. IEE Conf. Control Theory Appl. 146, 389–398 (1999)

    Article  Google Scholar 

  20. Lin, C.M., Hsu, C.F.: Neural-network-based adaptive control for induction servomotor drive system. IEEE Trans. Ind. Electron. 49(1), 115–123 (2002)

    Article  Google Scholar 

  21. Lin, C.M., Hsu, C.F.: Neural network hybrid control for antilock braking systems. IEEE Trans. Neural Netw. 14(2), 351–359 (2003)

    Article  Google Scholar 

  22. Wang, D., Huang, J.: Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback from. IEEE Trans. Neural Netw. 16(1), 195–202 (2005)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mechatronics Engineering, Science and Research Branch, Islamic Azad University, P.O. Box 14515/775, Tehran, Iran

    Faezeh Farivar

  2. Faculty of Electrical Engineering, Department of Mechatronics Engineering, K.N. Toosi University of Technology, Tehran, Iran

    Mahdi Aliyari Shoorehdeli

  3. Faculty of Electrical Engineering, Department of Control Engineering, K.N. Toosi University of Technology, Tehran, Iran

    Mohammad Ali Nekoui & Mohammad Teshnehlab

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  1. Faezeh Farivar
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  2. Mahdi Aliyari Shoorehdeli
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  3. Mohammad Ali Nekoui
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  4. Mohammad Teshnehlab
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Correspondence to Faezeh Farivar.

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Farivar, F., Aliyari Shoorehdeli, M., Nekoui, M.A. et al. Chaos control and modified projective synchronization of unknown heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive backstepping control. Nonlinear Dyn 67, 1913–1941 (2012). https://doi.org/10.1007/s11071-011-0118-z

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