Nonlinear Dynamics

, Volume 80, Issue 4, pp 1731–1744 | Cite as

Synchronization and stabilization of fractional second-order nonlinear complex systems

  • Mohammad Pourmahmood AghababaEmail author
Original Paper


Chaos control and synchronization of second-order nonautonomous fractional complex chaotic systems are discussed in this paper. A novel fractional nonsingular terminal sliding surface which is suitable for second-order fractional systems is proposed. It is proved that once the state trajectories of the system reach to the proposed sliding surface, they will be converged to the origin within a given finite time. After establishing the desired terminal sliding surface, a novel robust single sliding mode control law is introduced to force the system trajectories to reach the terminal sliding surface over a finite time. The stability and robustness of the proposed method are proved using the latest version of the fractional Lyapunov stability theorem. The proposed method is implemented for synchronization of two uncertain different fractional chaotic systems to confirm the theoretical results. Moreover, the fractional-order gyro system is stabilized using the proposed fractional sliding mode control scheme. It is worth noticing that the proposed fractional sliding mode approach is still a general control method and can be applied for control of second- order uncertain nonautonomous/autonomous fractional systems.


Fractional sliding mode Second-order chaotic system Synchronization Finite-time stability 


  1. 1.
    Chen, H.K.: Chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping. J. Sound Vib. 255, 719–740 (2002)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ge, Z.M., Yu, T.C., Chen, Y.S.: Chaos synchronization of a horizontal platform system. J. Sound Vib. 268, 731–749 (2003)CrossRefGoogle Scholar
  3. 3.
    Bowong, S., Kakmeni, M., Koina, R.: Chaos synchronization and duration time of a class of uncertain chaotic systems. Math. Comput. Simul. 71, 212–228 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Jiang, G.P., Zheng, W.X., Tang, W.K.S., Chen, G.: Integral-observer-based chaos synchronization. IEEE Trans. Circuits Syst. 53, 110–114 (2006)CrossRefGoogle Scholar
  5. 5.
    Yau, H.-T.: Design of adaptive sliding mode controller for chaos synchronization with uncertainties. Chaos Soliton Fract. 22, 341–347 (2004)CrossRefzbMATHGoogle Scholar
  6. 6.
    Gong, C.Y., Li, Y.M., Sun, X.H.: Backstepping control of synchronization for biomathematical model of muscular blood vessel. J. Appl. Sci. 24, 604–607 (2006)Google Scholar
  7. 7.
    Aghababa, M.P.: A novel adaptive finite-time controller for synchronizing chaotic gyros with nonlinear inputs. Chin. Phys. B 20, 090505 (2011)CrossRefGoogle Scholar
  8. 8.
    Aghababa, M.P., Aghababa, H.P.: Synchronization of mechanical horizontal platform systems in finite time. Appl. Math. Model 36, 4579–4591 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Aghababa, M.P., Aghababa, H.P.: Finite-time stabilization of uncertain non-autonomous chaotic gyroscopes with nonlinear inputs. Appl. Math. Mech. 33, 155–164 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Lu, J.G., Chen, G.: A note on the fractional-order Chen system. Chaos Soliton Fract. 27, 685–688 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Lu, J.G.: Chaotic dynamics of the fractional-order Lu system and its synchronization. Phys. Lett. A 354, 305–311 (2006)CrossRefGoogle Scholar
  12. 12.
    Li, C., Chen, G.: Chaos and hyperchaos in the fractional-order Rossler equations. Phys. A Stat. Mech. Appl. 341, 55–61 (2004)CrossRefGoogle Scholar
  13. 13.
    Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos Soliton Fract. 26, 1125–1133 (2005)CrossRefzbMATHGoogle Scholar
  14. 14.
    Yang, Q., Zeng, C.: Chaos in fractional conjugate Lorenz system and its scaling attractors. Commun. Nonlinear Sci. Numer. Simul. 15, 4041–4051 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Wu, X., Li, Ji, Chen, Gu: Chaos in the fractional order unified system and its synchronization. J. Frank. Inst. 345, 392–401 (2008)CrossRefzbMATHGoogle Scholar
  16. 16.
    Guo, L.J.: Chaotic dynamics and synchronization of fractional-order Genesio-Tesi systems. Chin. Phys. B 14, 1517–1521 (2005)CrossRefGoogle Scholar
  17. 17.
    Zhu, H., Zhou, S., Zhang, J.: Chaos and synchronization of the fractional-order Chua’s system. Chaos Soliton Fract. 39, 1595–1603 (2009)CrossRefzbMATHGoogle Scholar
  18. 18.
    Ge, Z.-M., Ou, C.-Y.: Chaos in a fractional order modified Duffing system. Chaos Soliton Fract. 34, 262–291 (2007)CrossRefzbMATHGoogle Scholar
  19. 19.
    Hu, J., Han, Y., Zhao, L.: Synchronizing chaotic systems using control based on a special matrix structure and extending to fractional chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 15, 115–123 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Peng, G.: Synchronization of fractional order chaotic systems. Phys. Lett. A 363, 426–432 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Wang, J.-W., Zhang, Y.-B.: Synchronization in coupled nonidentical incommensurate fractional-order systems. Phys. Lett. A 374, 202–207 (2009)CrossRefzbMATHGoogle Scholar
  22. 22.
    Lu, J.G.: Nonlinear observer design to synchronize fractional-order chaotic systems via a scalar transmitted signal. Phys. A 359, 107–118 (2006)CrossRefGoogle Scholar
  23. 23.
    Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Phys. A 387, 57–70 (2008)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Hosseinnia, S.H., Ghaderi, R., Ranjbar, A., Mahmoudian, M., Momani, S.: Sliding mode synchronization of an uncertain fractional order chaotic system. Comput. Math. Appl. 59, 1637–1643 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Aghababa, M.P.: A novel terminal sliding mode controller for a class of non-autonomous fractional-order systems. Nonlinear Dyn. 73, 679–688 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Aghababa, M.P.: No-chatter variable structure control for fractional nonlinear complex systems. Nonlinear Dyn. 73, 2329–2342 (2013)Google Scholar
  27. 27.
    Aghababa, M.P.: Finite-time chaos control and synchronization of fractional-order chaotic (hyperchaotic) systems via fractional nonsingular terminal sliding mode technique. Nonlinear Dyn. 69, 247–261 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)zbMATHGoogle Scholar
  29. 29.
    Li, Y., Chen, Y.Q., Podlubny, I.: Stability of fractional order nonlinear dynamic systems: Lapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Utkin, V.I.: Sliding modes in control optimization. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  31. 31.
    Yu, X., Man, Z.: Multi-input uncertain linear systems with terminal sliding-mode control. Automatica 34, 389–392 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Yu, X., Man, Z.: Fast terminal sliding-mode control design for nonlinear dynamical systems. IEEE Trans. Circuit Syst. 49, 261–264 (2002)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Jesus, I.S., Machado, J.A.T.: Fractional control of heat diffusion systems. Nonlinear Dyn. 54, 263–282 (2008)CrossRefzbMATHGoogle Scholar
  34. 34.
    Rapaić, M.R., Jeličić, Z.D.: Optimal control of a class of fractional heat diffusion systems. Nonlinear Dyn. 62, 39–51 (2010)Google Scholar
  35. 35.
    Tavazoei, M.S., Haeri, M., Bolouki, S., Siami, M.: Using fractional-order integrator to control chaos in single-input chaotic systems. Nonlinear Dyn 55, 179–190 (2009)Google Scholar
  36. 36.
    Deng, W., Li, C., Lu, J.: Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Lee, S.M., Choi, S.J., Ji, D.H., Park, J.H., Won, S.C.: Synchronization for chaotic Lur’e systems with sector restricted nonlinearities via delayed feedback control. Nonlinear Dyn. 59, 277–288 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Kwon, O.M., Park, J.H., Lee, S.M.: Secure communication based on chaotic synchronization via interval time-varying delay feedback control. Nonlinear Dyn. 63, 239–252 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Aghababa, M.P., Aghababa, H.P.: Synchronization of nonlinear chaotic electromechanical gyrostat systems with uncertainties. Nonlinear Dyn. 67, 2689–2701 (2012)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Aghababa, M.P., Aghababa, H.P.: Synchronization of chaotic systems with uncertain parameters and nonlinear inputs using finite-time control technique. Nonlinear Dyn. 69, 1903–1914 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Wu, X., Wang, H.: A new chaotic system with fractional order and its projective synchronization. Nonlinear Dyn. 61, 407–417 (2010) Google Scholar
  42. 42.
    Petráš, I.: Chaos in the fractional-order Volta’s system: modeling and simulation. Nonlinear Dyn. 57, 157–170 (2009)CrossRefzbMATHGoogle Scholar
  43. 43.
    Zeng, C., Yang, Q., Wang, J.: Chaos and mixed synchronization of a new fractional-order system with one saddle and two stable node-foci. Nonlinear Dyn. 65, 457–466 (2011)Google Scholar
  44. 44.
    Wang, Z., Sun, Y., Qi, G., van Wyk, B.J.: The effects of fractional order on a 3-D quadratic autonomous system with four-wing attractor. Nonlinear Dyn. 62, 139–150 (2010)CrossRefzbMATHGoogle Scholar
  45. 45.
    Balochian, S., Sedigh, A.K., Haeri, M.: Stabilization of fractional order systems using a finite number of state feedback laws. Nonlinear Dyn. 66, 141–152 (2011)Google Scholar
  46. 46.
    Hamamci, S.E.: Stabilization using fractional-order PI and PID controllers. Nonlinear Dyn. 51, 329–343 (2008)CrossRefzbMATHGoogle Scholar
  47. 47.
    Odibat, Z.M.: Adaptive feedback control and synchronization of non-identical chaotic fractional order systems. Nonlinear Dyn. 60, 479–487 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    Jesus, I.S., Machado, J.A.T.: Development of fractional order capacitors based on electrolyte processes. Nonlinear Dyn. 56, 45–55 (2009)CrossRefzbMATHGoogle Scholar
  49. 49.
    Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  50. 50.
    Ge, Z.-M., Hsu, M.-Y.: Chaos excited chaos synchronizations of integral and fractional order generalized Van der Pol systems. Chaos Soliton Fract. 36, 592–604 (2008)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Electrical Engineering DepartmentUrmia University of TechnologyUrmiaIran

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