Nonlinear Dynamics

, Volume 92, Issue 3, pp 935–959 | Cite as

Adaptive synchronization control based on QPSO algorithm with interval estimation for fractional-order chaotic systems and its application in secret communication

Original Paper
  • 77 Downloads

Abstract

In this paper, the synchronization problem and its application in secret communication are investigated for two fractional-order chaotic systems with unequal orders, different structures, parameter uncertainty and bounded external disturbance. On the basis of matrix theory, properties of fractional calculus and adaptive control theory, we design a feedback controller for realizing the synchronization. In addition, in order to make it better apply to secret communication, we design an optimal controller based on optimal control theory. In the meantime, we propose an improved quantum particle swarm optimization (QPSO) algorithm by introducing an interval estimation mechanism into QPSO algorithm. Further, we make use of QPSO algorithm with interval estimation to optimize the proposed controller according to some performance indicator. Finally, by comparison, numerical simulations show that the controller not only can achieve the synchronization and secret communization well, but also can estimate the unknown parameters of the systems and bounds of external disturbance, which verify the effectiveness and applicability of the proposed control scheme.

Keywords

Fractional-order chaotic system Adaptive synchronization control QPSO algorithm with interval estimation Secret communication 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grants 61522302.

References

  1. 1.
    Hilfer, R.: Application of Fractional Calculus in Physics. World Scientific, Singapore (2000)CrossRefMATHGoogle Scholar
  2. 2.
    Wen, X.J., Wu, Z.M., Lu, J.G.: Stability analysis of a class of nonlinear fractional-order systems. IEEE Trans. Circuits Syst.-I 55, 1178–1182 (2008)Google Scholar
  3. 3.
    Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)MATHGoogle Scholar
  4. 4.
    Hua, C., Zhang, T., Li, Y., Guan, X.: Robust output feedback control for fractional order nonlinear systems with time-varying delays. IEEE/CAA J. Autom. Sin. 3, 477–482 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Lim, Y.-H., Oh, K.-K., Ahn, H.-S.: Stability and stabilization of fractional-order linear systems subject to input saturation. IEEE Trans. Autom. Control 5, 1062–1067 (2013)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Qustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst. I 47, 25–39 (2000)CrossRefGoogle Scholar
  7. 7.
    Wang, Q., Qi, D.-L.: Synchronization for fractional order chaotic systems with uncertain parameters. Int. J. Control Autom. Syst. 14, 211–216 (2016)CrossRefGoogle Scholar
  8. 8.
    Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, 034101 (2003)CrossRefGoogle Scholar
  9. 9.
    Chen, C.G., Chen, G.R.: Chaos and hyperchaos in the fractional-order Rössler equations. Phys. A 341, 55–61 (2004)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, C.G., Chen, G.R.: Chaos in the fractional order Chen system and its control. Chaos Solitons Fract. 3, 549–554 (2004)CrossRefMATHGoogle Scholar
  11. 11.
    Deng, W.H., Li, C.P.: Chaos synchronization of the fractional Lü system. Phys. A 353, 61–72 (2005)CrossRefGoogle Scholar
  12. 12.
    Petras, I.: A note on the fractional-order Chua’s system. Chaos Solitons Fract. 38, 140–147 (2008)CrossRefGoogle Scholar
  13. 13.
    Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chuas system. IEEE Trans. Circuits Syst. I 42, 485–490 (1995)CrossRefGoogle Scholar
  14. 14.
    Gao, X., Yu, J.B.: Chaos in the fractional order periodically forced complex Duffings oscillators. Chaos Solitons Fract. 24, 1097–1104 (2005)CrossRefMATHGoogle Scholar
  15. 15.
    Vaseghi, B., Pourmina, M.A., Mobayen, S.: Secure communication in wireless sensor networks based on chaos synchronization using adaptive sliding mode control. Nonlinear Dyn. 89, 1689–1704 (2017)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Muthukumar, P., Balasubramaniam, P., Ratnavelu, K.: Synchronization of a novel fractional order stretch-twist-fold (stf) flow chaotic system and its application to a new authenticated encryption scheme (aes). Nonlinear Dyn. 77, 1547–1559 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Xu, Y., Wang, H., Li, Y., Pei, B.: Image encryption based on synchronization of fractional chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 19, 3735–3744 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    N’Doye, I., Voos, H., Darouach, M.: Observer-based approach for fractional-order chaotic synchronization and secure communication. IEEE J. Emerg. Sel. Top. Circuits Syst. 3, 442–450 (2013)CrossRefGoogle Scholar
  19. 19.
    Muthukumar, P., Balasubramaniam, P., Ratnavelu, K.: Sliding mode control for generalized robust synchronization of mismatched fractional order dynamical systems and its application to secure transmission of voice messages. ISA Trans. (2017).  https://doi.org/10.1016/j.isatra.2017.07.007 Google Scholar
  20. 20.
    Podlubny, I.: Fractional-order systems and PID-controllers. IEEE Trans. Autom. Control 44, 208–214 (1999)CrossRefMATHGoogle Scholar
  21. 21.
    Odibet, Z., Corson, N., Aziz-Alaoui, M.: Synchronization of fractional order chaotic systems via linear control. Int. J. Bifurc. Chaos 20, 81–97 (2010)CrossRefMATHGoogle Scholar
  22. 22.
    Zhu, H., Zhou, S.B., Zhang, J.: Chaos and synchronization of the fractional-order Chua’s system. Chaos Solitons Fract. 39, 1595–1603 (2009)CrossRefMATHGoogle Scholar
  23. 23.
    Taghvafard, H., Erjaee, G.H.: Phase and anti-phase synchronization of fractional order chaotic systems via active control. Commun. Nonlinear Sci. Numer. Simul. 16, 4079–4088 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Wu, X.J., Lu, H.T., Shen, S.L.: Synchronization of a new fractional-order hyperchaotic system. Phys. Lett. A 373, 2329–2337 (2009)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Li, C., Wang, J., Lu, J., Ge, Y.: Observer-based stabilisation of a class of fractional order non-linear systems for \(0<\alpha <2\) case. IET Control Theory Appl. 8, 1238–1246 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wang, X.Y., Zhang, X.P., Ma, C.: Modified projective synchronization of fractional-order chaotic systems via active sliding mode control. Nonlinear Dyn. 69, 511–517 (2012)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Li, C.L., Su, K.L., Wu, L.: Adaptive sliding mode control for synchronization of a fractional-order chaotic system. J. Comput. Nonlinear Dyn. 8, 031005 (2013)CrossRefGoogle Scholar
  28. 28.
    Aghababa, M.P.: Design of hierarchical terminal sliding mode control scheme for fractional-order systems. IET Sci. Meas. Technol. 9, 122–133 (2014)CrossRefGoogle Scholar
  29. 29.
    Ma, T.D., Jiang, W.B., Fu, J.: Impulsive synchronization of fractional order hyperchaotic systems based on comparison system. Acta Phys. Sin. 61, 090503 (2012)MATHGoogle Scholar
  30. 30.
    Maione, G.: Continued fractions approximation of the impulse response of fractional-order dynamic systems. IET Control Theory Appl. 2, 564–572 (2008)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Liu, J.-G.: A novel study on the impulsive synchronization of fractional-order chaotic systems. Chin. Phys. B 22, 060510 (2013)CrossRefGoogle Scholar
  32. 32.
    Odibat, Z.: A note on phase synchronization in coupled chaotic fractional order systems. Nonlinear Anal. 13, 779–789 (2012)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Muthukumar, P., Balasubramaniam, P., Ratnavelu, K.: Synchronization and an application of a novel fractional order king cobra chaotic system, Chaos: an Interdisciplinary. J. Nonlinear Sci. 24, 033105 (2014)MATHGoogle Scholar
  34. 34.
    Pan, L., Guan, Z., Zhou, L.: Chaos multiscale-synchronization between two different fractional-order hyperchaotic systems based on feedback control. Int. J. Bifurc. Chaos 23, 1350146 (2013)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Maheri, M., Arifin, N.M.: Synchronization of two different fractional-order chaotic systems with unknown parameters using a robust adaptive nonlinear controller. Nonlinear Dyn. 85, 825–838 (2016)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Lu, J.G., Chen, Y.Q.: Robust stability and stabilization of fractional-order interval systems with the fractional order \(\alpha \): the \(0<\alpha <1\) case. IEEE Trans. Autom. Control 55, 152–158 (2010)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Lu, J.G., Chen, G.: Robust stability and stabilization of fractional-order interval systems: an LMI Approach. IEEE Trans. Autom. Control 54, 1294–1299 (2009)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Pan, G., Wei, J.: Design of an adaptive sliding mode controller for synchronization of fractional-order chaotic systems. Acta Phys. Sin. 64, 040505 (2015)Google Scholar
  39. 39.
    Yin, C., Dadras, S., Zhong, S., Chen, Y.: Control of a novel of class of fractional-order chaotic systems via adaptive sliding mode control approach. Appl. Math. Model. 37, 2469–2483 (2013)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Zhou, P., Zhu, P.: A practical synchronization approach for fractional-order chaotic systems. Nonlinear Dyn. 89, 1719–1726 (2017)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Zhang, R.X., Yang, S.P.: Synchronization of fractional-order chaotic systems with different structures. Acta Phys. Sin. 57, 6852–6858 (2008)MATHGoogle Scholar
  42. 42.
    Pan, L., Zhou, W., Zhou, L., Sun, K.: Chaos synchronization between two different fractional-order hyperchaotic systems. Commun. Nonlinear Sci. Numer. Simul. 16, 2628–2640 (2011)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Zhang, R., Yang, S.: Robust synchronization of different fractional-order chaotic systems with unknown parameters using adaptive sliding mode approach. Nonlinear Dyn. 71, 269–278 (2013)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Wang, Z., Huang, X., Zhao, Z.: Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives. Nonlinear Dyn. 69, 999–1007 (2012)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Chen, M., Shao, S.Y., Shi, P., Shi, Y.: Disturbance-observer-based robust synchronization control for a class of fractional-order chaotic systems. IEEE Trans. Circuits Syst. II 64, 417–421 (2017)CrossRefGoogle Scholar
  46. 46.
    Kilbas, A.A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, New York (2006)MATHGoogle Scholar
  47. 47.
    Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19, 2951–2957 (2014)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Adomian, G.: A review of the decomposition method and some recent results for nonlinear equations. Math. Comput. Model. 13, 17–43 (1990)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Ray, S.S., Bera, R.K.: An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method. Appl. Math. Comput. 167, 561–571 (2005)MathSciNetMATHGoogle Scholar
  50. 50.
    Fatoorehchi, H., Abolghasemi, H., Zarghami, R., Rach, R.: Feedback control strategies for a cerium-catalyzed Belousov–Zhabotinsky chemical reaction system. Can. J. Chem. Eng. 93, 1212–1221 (2015)CrossRefMATHGoogle Scholar
  51. 51.
    Fatoorehchi, H., Rach, R., Sakhaeinia, H.: Explicit Frost-Kalkwarf type equations for calculation of vapour pressure of liquids from triple to critical point by the Adomian decomposition method. Can. J. Chem. Eng. 95, 2199–2208 (2017)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical EngineeringBeihang University (Beijing University of Aeronautics and Astronautics)BeijingChina

Personalised recommendations