Nonlinear Dynamics

, Volume 95, Issue 2, pp 1661–1672 | Cite as

Dynamic programming strategy based on a type-2 fuzzy wavelet neural network

  • Ardashir Mohammadzadeh
  • Weidong ZhangEmail author
Original Paper


In this paper, an optimal control scheme, based on dynamic programming strategy, is presented for synchronization of uncertain fractional-order chaotic/hyperchaotic systems. In the scheme, a type-2 fuzzy wavelet neural network (T2FWNN) is proposed for estimation of the unknown functions in dynamics of system. For solving the fractional optimal control problem, fractional-order derivative is approximated by using Oustaloup recursive approximation method. Simulation studies verify the effectiveness of the proposed control scheme and the proposed T2FWNN.


Dynamic programming Oustaloup recursive approximation Type-2 fuzzy wavelet neural network Fractional-order hyperchaotic systems 



This paper is partly supported by the National Science Foundation of China (61473183, 61627810, U1509211).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. 1.
    Huang, C., Cao, J.: Active control strategy for synchronization and anti-synchronization of a fractional chaotic financial system. Stat. Mech. Appl. Phys. A 473, 262–275 (2017)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Velmurugan, G., Rakkiyappan, R.: Hybrid projective synchronization of fractional-order chaotic complex nonlinear systems with time delays. J. Comput. Nonlinear Dyn. 11(3), 031016 (2016)zbMATHGoogle Scholar
  3. 3.
    Zouari, F., Boulkroune, A., Ibeas, A.: Neural adaptive quantized output-feedback control-based synchronization of uncertain time-delay incommensurate fractional-order chaotic systems with input nonlinearities. Neurocomputing 237, 200–225 (2017)Google Scholar
  4. 4.
    Vaidyanathan, S.: Adaptive integral sliding mode controller design for the control and synchronization of a novel jerk chaotic system. In: Vaidyanathan, S., Lien, C.H. (eds.) Applications of Sliding Mode Control in Science and Engineering, pp. 393–417. Springer, Cham (2017)Google Scholar
  5. 5.
    Hsu, C.-F., Tsai, J.-Z., Chiu, C.-J.: Chaos synchronization of nonlinear gyros using self-learning PID control approach. Appl. Soft Comput. 12(1), 430–439 (2012)Google Scholar
  6. 6.
    Lin, T.-C., Kuo, C.-H.: \({H_\infty }\) synchronization of uncertain fractional order chaotic systems: adaptive fuzzy approach. ISA Trans. 50(4), 548–556 (2011)Google Scholar
  7. 7.
    Lin, T.-C., Lee, T.-Y., Balas, V.E.: Adaptive fuzzy sliding mode control for synchronization of uncertain fractional order chaotic systems. Chaos Solitons Fractals 44(10), 791–801 (2011)zbMATHGoogle Scholar
  8. 8.
    Lin, T.-C., Lee, T.-Y.: Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control. IEEE Trans. Fuzzy Syst. 19(4), 623–635 (2011)Google Scholar
  9. 9.
    Lin, T.-C., Chen, M.-C., Roopaei, M.: Synchronization of uncertain chaotic systems based on adaptive type-2 fuzzy sliding mode control. Eng. Appl. Artif. Intell. 24(1), 39–49 (2011)Google Scholar
  10. 10.
    Qi-Shui, Z., Jing-Fu, B., Yong-Bin, Y., Xiao-Feng, L.: Impulsive control for fractional-order chaotic systems. Chin. Phys. Lett. 25(8), 2812–2815 (2008)Google Scholar
  11. 11.
    Li, C., Liao, X., Wong, K.: Lag synchronization of hyperchaos with application to secure communications. Chaos Solitons Fractals 23(1), 183–193 (2005)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chen, H.-Y., Liang, J.-W.: Adaptive wavelet neural network controller for active suppression control of a diaphragm-type pneumatic vibration isolator. Int. J. Control Autom. Syst. 15(3), 1456–1465 (2017)MathSciNetGoogle Scholar
  13. 13.
    Samadianfard, S., Asadi, E., Jarhan, S., Kazemi, H., Kheshtgar, S., Kisi, O., Sajjadi, S., Abdul Manaf, A.: Wavelet neural networks and gene expression programming models to predict short-term soil temperature at different depths. Soil Tillage Res. 175, 37–50 (2018)Google Scholar
  14. 14.
    Song, D., Ding, Y., Li, X., Zhang, B., Mingyu, X.: Ocean oil spill classification with RADARSAT-2 SAR based on an optimized wavelet neural network. Remote Sens. 9(8), 799 (2017)Google Scholar
  15. 15.
    Abiyev, R.H., Kaynak, O.: Fuzzy wavelet neural networks for identification and control of dynamic plants: a novel structure and a comparative study. IEEE Trans. Ind. Electron. 55(8), 3133–3140 (2008)Google Scholar
  16. 16.
    Yilmaz, S., Oysal, Y.: Fuzzy wavelet neural network models for prediction and identification of dynamical systems. IEEE Trans. Neural Netw. 21(10), 1599–1609 (2010)Google Scholar
  17. 17.
    Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38(1–4), 323–337 (2004)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Frederico, G.S.F., Torres, D.F.M.: Noether’s theorem for fractional optimal control problems. arXiv preprint arXiv:math/0603598 (2006)
  19. 19.
    Frederico, G.S.F., Torres, D.F.M.: Fractional conservation laws in optimal control theory. Nonlinear Dyn. 53(3), 215–222 (2008)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Frederico, G.S.F., Torres, D.F.M.: Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. arXiv preprint arXiv:0712.1844 (2007)
  21. 21.
    Jumarie, G.: Fractional Hamilton–Jacobi equation for the optimal control of nonrandom fractional dynamics with fractional cost function. J. Appl. Math. Comput. 23(1–2), 215–228 (2007)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Atanackovic, T.M., Stankovic, B.: An expansion formula for fractional derivatives and its application. Fract. Calc. Appl. Anal. 7(3), 365–378 (2004)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Jelicic, Z.D., Petrovacki, N.: Optimality conditions and a solution scheme for fractional optimal control problems. Struct. Multidiscip. Optim. 38(6), 571–581 (2009)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Tricaud, C., Chen, Y.: Solving fractional order optimal control problems in riots 95—a general purpose optimal control problem solver. In: Proceedings of the 3rd IFAC Workshop on Fractional Differentiation and Its Applications (2008)Google Scholar
  25. 25.
    Xue, D., Chen, Y.Q., Atherton, D.P.: Linear Feedback Control: Analysis and Sesign with MATLAB, vol. 14. SIAM, Philadelphia (2007)zbMATHGoogle Scholar
  26. 26.
    Pooseh, S., Almeida, R., Torres, D.F.M.: Numerical approximations of fractional derivatives with applications. Asian J. Control 15(3), 698–712 (2013)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Khader, M.M., Hendy, A.S.: An efficient numerical scheme for solving fractional optimal control problems. Int. J. Nonlinear Sci. 14(3), 287–296 (2012)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Bertsekas, D.P.: Dynamic Programming and Optimal Control, vol. 1. Athena Scientific, Belmont (2005)zbMATHGoogle Scholar
  29. 29.
    Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 47(1), 25–39 (2000)Google Scholar
  30. 30.
    Mahmoud, E.E.: Adaptive anti-lag synchronization of two identical or non-identical hyperchaotic complex nonlinear systems with uncertain parameters. J. Frankl. Inst. 349(3), 1247–1266 (2012)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Daniel Pun, W.K., Shawkat Ali, A.B.M.: Unique distance measure approach for k-means (UDMA-km) clustering algorithm. In Tencon 2007—2007 IEEE Region 10 Conference, pp. 1–4. IEEE (2007)Google Scholar
  32. 32.
    Arqub, O.A., Mohammed, A.L.-S., Momani, S., Hayat, T.: Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method. Soft Comput. 20(8), 3283–3302 (2016)zbMATHGoogle Scholar
  33. 33.
    Arqub, O.A.: Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations. Neural Comput. Appl. 28(7), 1591–1610 (2017)Google Scholar
  34. 34.
    Arqub, O.A., Al-Smadi, M., Momani, S., Hayat, T.: Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems. Soft Comput. 21(23), 7191–7206 (2017)zbMATHGoogle Scholar
  35. 35.
    Matouk, A.E., Elsadany, A.A.: Achieving synchronization between the fractional-order hyperchaotic Novel and Chen systems via a new nonlinear control technique. Appl. Math. Lett. 29, 30–35 (2014)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Aghababa, M.P.: Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique. Nonlinear Dyn. 69(1–2), 247–261 (2012)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Nick Street, W., Wolberg, W.H., Mangasarian, O.L.: Nuclear feature extraction for breast tumor diagnosis. In: Biomedical Image Processing and Biomedical Visualization, vol. 1905, pp. 861–871. International Society for Optics and Photonics, Bellingham (1993)Google Scholar
  38. 38.
    Jang, J.-S.R.: ANFIS: adaptive-network-based fuzzy inference system. IEEE Trans. Syst. Man Cybern. 23(3), 665–685 (1993)Google Scholar
  39. 39.
    Vapnik, V.: The Nature of Statistical Learning Theory. Springer, Berlin (2013)zbMATHGoogle Scholar
  40. 40.
    Nguyen, T., Khosravi, A., Creighton, D., Nahavandi, S.: Medical data classification using interval type-2 fuzzy logic system and wavelets. Appl. Soft Comput. 30, 812–822 (2015)Google Scholar
  41. 41.
    Specht, D.F.: Probabilistic neural networks. Neural Netw. 3(1), 109–118 (1990)Google Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of Electrical EngineeringUniversity of BonabBonabIran
  2. 2.Department of AutomationShanghai Jiaotong UniversityShanghaiPeople’s Republic of China
  3. 3.School of Mechatronic Engineering and AutomationShanghai UniversityShanghaiPeople’s Republic of China

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