Synchronizing Chaotic Systems with Uncertain Model and Unknown Interference Using Sliding Mode Control and Wavelet Neural Networks

  • Guo LuoEmail author
  • Zhi Yang
  • Kongming Peng


A method using sliding mode control (SMC) and wavelet neural networks (WNN) is proposed, investigated and exploited for synchronizing master and slave chaotic systems with uncertain model and unknown interference. In this paper, integral sliding surface and applying WNN for approximating uncertain model and unknown interference are further developed for designing adaptive sliding mode controller. Mexican hat wavelet function is used as activation function in WNN. The adaptive laws of network parameters are derived in the sense of Lyapunov stability analysis so that the tracking errors and convergence of the weights can be guaranteed. The error of synchronization of master–slave chaotic systems can be reached desired level in limited time by using Li function in SMC. Illustrative examples are provided and analyzed to substantiate the efficacy of proposed method for solving the problem of synchronizing master and slave chaotic systems.


Synchronizing chaotic systems Uncertain model Unknown interference Sliding mode control Wavelet neural networks 



The authors would like to thank anonymous peer reviewer.


  1. 1.
    Ott E, Grebogi C, Yorke JA (1990) Controlling chaos. Phys Rev Lett 64(11):1196–1199MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chen HK (2005) Global chaos synchronization of new chaotic systems via nonlinear control. Chaos, Solitons Fractals 23(4):1245–1251zbMATHGoogle Scholar
  3. 3.
    Cao J, Ho DWC, Yang Y (2009) Projective synchronization of a class of delayed chaotic systems via impulsive control. Phys Lett A 373(35):3128–3133MathSciNetzbMATHGoogle Scholar
  4. 4.
    Yassen MT (2006) Chaos control of chaotic dynamical systems using backstepping design. Chaos, Solitons Fractals 27(2):537–548MathSciNetzbMATHGoogle Scholar
  5. 5.
    Aghababa MP, Feizi H (2012) Design of a sliding mode controller for synchronizing chaotic systems with parameter and model uncertainties and external disturbances. Trans Inst Meas Control 34(8):990–997Google Scholar
  6. 6.
    Ouannas A, Odibat Z, Shawagfeh N et al (2017) Universal chaos synchronization control laws for general quadratic discrete systems. Appl Math Model 45:636–641MathSciNetGoogle Scholar
  7. 7.
    Yih-Yuh C (1996) Randomly synchronizing chaotic systems: condensed matter and statistical physics. Progress Theoret Phys 96(4):683–692Google Scholar
  8. 8.
    Khadra A, Liu XZ, Shen X (2005) Impulsively synchronizing chaotic systems with delay and applications to secure communication. Automatica 41(9):1491–1502MathSciNetzbMATHGoogle Scholar
  9. 9.
    Naderi B, Kheiri H, Heydari A et al (2016) Optimal synchronization of complex chaotic t-systems and its application in secure communication. J Control Autom Electr Syst 27(4):379–390Google Scholar
  10. 10.
    Carroll TL, Pecora LM (2002) Synchronizing chaotic circuits. IEEE Trans Circuits Syst 38(4):453–456zbMATHGoogle Scholar
  11. 11.
    Junwei S, Xingtong Z, Jie F et al (2018) Autonomous memristor chaotic systems of infinite chaotic attractors and circuitry realization. Nonlinear Dyn 94(4):2879–2887Google Scholar
  12. 12.
    Li WL (2008) On control and synchronization in chaotic and hyperchaotic systems via linear feedback control. Commun Nonlinear Sci Numer Simul 13(7):1246–1255MathSciNetzbMATHGoogle Scholar
  13. 13.
    Haeri M, Emadzadeh A (2007) Synchronizing different chaotic systems using active sliding mode control. Chaos, Solitons Fractals 31(1):119–129MathSciNetzbMATHGoogle Scholar
  14. 14.
    Tavazoei MS, Haeri M (2007) Determination of active sliding mode controller parameters in synchronizing different chaotic systems. Chaos, Solitons Fractals 32(2):583–591Google Scholar
  15. 15.
    Fuh CC (2009) Optimal control of chaotic systems with input saturation using an input-state linearization scheme. Commun Nonlinear Sci Numer Simul 14(8):3424–3431Google Scholar
  16. 16.
    Tan X, Zhang J, Yang Y (2003) Synchronizing chaotic systems using backstepping design. Chaos, Solitons Fractals 16(1):37–45MathSciNetzbMATHGoogle Scholar
  17. 17.
    Li GH, Zhou SP, Yang K (2006) Generalized projective synchronization between two different chaotic systems using active backstepping control. Phys Lett A 355(4):326–330Google Scholar
  18. 18.
    Chen HH, Sheu GJ, Lin YL et al (2009) Chaos synchronization between two different chaotic systems via nonlinear feedback control. Nonlinear Anal Theory Methods Appl 70(12):4393–4401MathSciNetzbMATHGoogle Scholar
  19. 19.
    Mobayen S (2018) Chaos synchronization of uncertain chaotic systems using composite nonlinear feedback based integral sliding mode control. ISA Trans 77:100–111Google Scholar
  20. 20.
    Ji DH, Jeong SC, Ju HP et al (2012) Robust adaptive backstepping synchronization for a class of uncertain chaotic systems using fuzzy disturbance observer. Nonlinear Dyn 69(3):1125–1136MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lin D, Wang X, Nian F et al (2010) Dynamic fuzzy neural networks modeling and adaptive backstepping tracking control of uncertain chaotic systems. Neurocomputing 73(16–18):2873–2881Google Scholar
  22. 22.
    Barker AE (2012) Adaptive modified function projective synchronization of general uncertain chaotic complex systems. Phys Scr 85(3):438–445Google Scholar
  23. 23.
    Mobayen S, Tchier F (2017) Synchronization of a class of uncertain chaotic systems with lipschitz nonlinearities using state-feedback control design: a matrix inequality approach. Asian J Control 20(1):71–85MathSciNetzbMATHGoogle Scholar
  24. 24.
    Mobayen S (2018) Design of novel adaptive sliding mode controller for perturbed Chameleon hidden chaotic flow. Nonlinear Dyn 92:1539–1553zbMATHGoogle Scholar
  25. 25.
    Mofid Omid, Mobayen Saleh (2018) Adaptive synchronization of fractional-order quadratic chaotic flows with nonhyperbolic equilibrium. J Vib Control 24(21):4971–4987MathSciNetGoogle Scholar
  26. 26.
    Saleh M, Jun M (2018) Robust finite-time composite nonlinear feedback control for synchronization of uncertain chaotic systems with nonlinearity and time-delay. Chaos, Solitons Fractals 114:46–54MathSciNetGoogle Scholar
  27. 27.
    Sun J, Wu Y, Cui G et al (2017) Finite-time real combination synchronization of three complex-variable chaotic systems with unknown parameters via sliding mode control. Nonlinear Dyn 88(3):1677–1690zbMATHGoogle Scholar
  28. 28.
    Sun J, Wang Y, Wang Y et al (2016) Finite-time synchronization between two complex-variable chaotic systems with unknown parameters via nonsingular terminal sliding mode control. Nonlinear Dyn 85(2):1105–1117MathSciNetzbMATHGoogle Scholar
  29. 29.
    Yan JJ, Liao TL (2017) Discrete sliding mode control for hybrid synchronization of continuous Lorenz systems with matched/unmatched disturbances. Trans Inst Meas Control 40(5):1417–1424Google Scholar
  30. 30.
    Fu YY, Wu CJ, Ko CN et al (2011) Radial basis function networks with hybrid learning for system identification with outliers. Appl Soft Comput 11(3):3083–3092Google Scholar
  31. 31.
    Fernández-Navarro F, Hervás-Martínez C, Sanchez-Monedero J et al (2011) MELM-GRBF: a modified version of the extreme learning machine for generalized radial basis function neural networks. Neurocomputing 74(16):2502–2510Google Scholar
  32. 32.
    Zhang Y, Yang Y, Tan N et al (2011) Zhang neural network solving for time-varying full-rank matrix Moore-Penrose inverse. Computing 92(2):97–121MathSciNetzbMATHGoogle Scholar
  33. 33.
    Liao B, Zhang Y (2014) From different ZFs to different ZNN models accelerated via Li activation functions to finite-time convergence for time-varying matrix pseudoinversion. Neurocomputing 133(8):512–522Google Scholar
  34. 34.
    Shuai L, Li Y, Zheng W (2013) A class of finite-time dual neural networks for solving quadratic programming problems and its k k mathContainer Loading Mathjax -winners-take-all application. Neural Netw 39(39):27–39zbMATHGoogle Scholar
  35. 35.
    Wang H, Han ZZ, Xie QY et al (2009) Finite-time chaos synchronization of unified chaotic system with uncertain parameters. Commun Nonlinear Sci Numer Simul 14(5):2239–2247Google Scholar
  36. 36.
    Cordova J, Yu W (2012) Two types of haar wavelet neural networks for nonlinear system identification. Neural Process Lett 35(3):283–300Google Scholar
  37. 37.
    Pindoriya NM, Singh SN, Singh SK (2008) An adaptive wavelet neural network-based energy price forecasting in electricity markets. IEEE Trans Power Syst 23(3):1423–1432Google Scholar
  38. 38.
    Zhang Q (1997) Using wavelet network in nonparametric estimation. IEEE Trans Neural Netw 8(2):227Google Scholar
  39. 39.
    Pai Ming-Chang (2016) RBF-based discrete sliding mode control for robust tracking of uncertain time-delay systems with input nonlinearity. Complexity 21(6):194–201MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Electrical and Computer EngineeringNanfang College of Sun Yat-sen UniversityGuangzhouChina

Personalised recommendations