Nonlinear Dynamics

, Volume 96, Issue 4, pp 2437–2447 | Cite as

Design and analysis of robust nonlinear neural dynamics for solving dynamic nonlinear equation within finite time

  • Lin XiaoEmail author
Original Paper


Dynamic nonlinear equation is a kind of important nonlinear systems, and many practical problems can be formulated as a dynamic nonlinear equation in mathematics to be solved. Inspired by the negative impact of additive noises on zeroing neural dynamics (ZND) for dynamic nonlinear equation, a robust nonlinear neural dynamics (RNND) is designed and presented to achieve noise suppression and finite-time convergence simultaneously. Compared to the existing ZND model only with finite-time convergence, the proposed RNND model inherently possesses the extra robustness property in front of additive noises, in addition to finite-time convergence. Furthermore, design process, theoretical analysis, and numerical verification of the proposed RNND model are supplied in details. Both theoretical and numerical results validate the better property of the proposed RNND model for solving such a nonlinear equation in the presence of external disturbances, as compared to the ZND model.


Nonlinear neural dynamics Dynamic nonlinear equation Convergence speedup Design formula Noise tolerance 



This work was supported in part by the National Natural Science Foundation of China under Grant 61866013, Grant 61503152, Grant 61473259, and Grant 61563017, and in part by the Natural Science Foundation of Hunan Province of China under Grant 2019JJ50478, Grant 18A289, Grant 2016JJ2101, Grant 2018TP1018, Grant 2018RS3065, Grant 15B192, and Grant 17A173. In addition, the author would like to thank the editors and anonymous reviewers for their valuable suggestions and constructive comments which have really helped the author improve very much the presentation and quality of this paper.


  1. 1.
    Xiao, L., Zhang, Y.: Solving time-varying inverse kinematics problem of wheeled mobile manipulators using Zhang neural network with exponential convergence. Nonlinear Dyn. 76, 1543–1559 (2014)CrossRefGoogle Scholar
  2. 2.
    Tlelo-Cuautle, E., Carbajal-Gomez, V.H., Obeso-Rodelo, P.J., et al.: FPGA realization of a chaotic communication system applied to image processing. Nonlinear Dyn. 82, 1879–1892 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Li, S., Zhang, Y., Jin, L.: Kinematic control of redundant manipulators using neural networks. IEEE Trans. Neural Netw. Learn. Syst. 28, 2243–2254 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Narayanan, M.D., Narayanan, S., Padmanabhan, C.: Parametric identification of nonlinear systems using multiple trials. Nonlinear Dyn. 48, 341–360 (2007)CrossRefzbMATHGoogle Scholar
  5. 5.
    Peng, J., Wang, J., Wang, W.: Neural network based robust hybrid control for robotic system: an H-\(\infty \) approach. Nonlinear Dyn. 65, 421–431 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jin, L., Zhang, Y.: Discrete-time Zhang neural network for online time-varying nonlinear optimization with application to manipulator motion generation. IEEE Trans. Neural Netw. Learn. Syst. 26, 1525–1531 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li, S., Wang, H., Rafique, M.U.: A novel recurrent neural network for manipulator control with improved noise tolerance. IEEE Trans. Neural Netw. Learn. Syst. 29, 1908–1918 (2018)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Xiao, L., Zhang, Y.: A new performance index for the repetitive motion of mobile manipulators. IEEE Trans. Cybern. 44, 280–292 (2014)CrossRefGoogle Scholar
  9. 9.
    Li, S., He, J., Li, Y., Rafique, M.U.: Distributed recurrent neural networks for cooperative control of manipulators: a game-theoretic perspective. IEEE Trans. Neural Netw. Learn. Syst. 28, 415–426 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Chun, C.: Construction of Newton-like iteration methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Abbasbandy, S.: Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method. Appl. Math. Comput. 145, 887–893 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Sharma, J.R.: A composite third order Newton–Steffensen method for solving nonlinear equations. App. Math. Comput. 169, 242–246 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ujevic, N.: A method for solving nonlinear equations. App. Math. Comput. 174, 1416–1426 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Wang, J., Chen, L., Guo, Q.: Iterative solution of the dynamic responses of locally nonlinear structures with drift. Nonlinear Dyn. 88, 1551–1564 (2017)CrossRefGoogle Scholar
  15. 15.
    Zhang, Y., Chen, D., Guo, D., Liao, B., Wang, Y.: On exponential convergence of nonlinear gradient dynamics system with application to square root finding. Nonlinear Dyn. 79, 983–1003 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Xiao, L.: A nonlinearly-activated neurodynamic model and its finite-time solution to equality-constrained quadratic optimization with nonstationary coefficients. Appl. Soft Comput. 40, 252–259 (2016)CrossRefGoogle Scholar
  17. 17.
    Zhang, Y., Xiao, L., Xiao, Z., Mao, M.: Zeroing Dynamics, Gradient Dynamics, and Newton Iterations. CRC Press, Boca Raton (2015)CrossRefzbMATHGoogle Scholar
  18. 18.
    Xiao, L.: Accelerating a recurrent neural network to finite-time convergence using a new design formula and its application to time-varying matrix square root. J. Frank. Inst. 354, 5667–5677 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Xiao, L., Zhang, Y.: Zhang neural network versus gradient neural network for solving time-varying linear inequalities. IEEE Trans. Neural Netw. 22, 1676–1684 (2011)CrossRefGoogle Scholar
  20. 20.
    Xiao, L., Zhang, Y.: Two new types of Zhang neural networks solving systems of time-varying nonlinear inequalities. IEEE Trans. Circuits Syst. I(59), 2363–2373 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Zhang, Y., Xu, P., Tan, N.: Further studies on Zhang neural-dynamics and gradient dynamics for online nonlinear equations Solving. In: Proceedings of the IEEE International Conference on Automation and Logistics, pp. 566–571 (2009)Google Scholar
  22. 22.
    Zhang, Y., Xu, P., Tan, N.: Solution of nonlinear equations by continuous-and discrete-time Zhang dynamics and more importantly their links to Newton iteration. In: Proceedings of the IEEE International Conference on Information, Communications and Signal Processing, pp. 1–5 (2009)Google Scholar
  23. 23.
    Zhang, Y., Yi, C., Guo, D., Zheng, J.: Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation. Neural Comput. Appl. 20, 1–7 (2011)CrossRefGoogle Scholar
  24. 24.
    Zhang, Y., Li, Z., Guo, D., Ke, Z., Chen, P.: Discrete-time ZD, GD and NI for solving nonlinear time-varying equations. Numer. Algorithms 64, 721–740 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Li, S., Chen, S., Liu, B.: Accelerating a recurrent neural network to finite-time convergence for solving time-varying Sylvester equation by using a sign-bi-power activation function. Neural Process. Lett. 37, 189–205 (2013)CrossRefGoogle Scholar
  26. 26.
    Xiao, L., Lu, R.: Finite-time solution to nonlinear equation using recurrent neural dynamics with a specially-constructed activation function. Neurocomputing 151, 246–251 (2015)CrossRefGoogle Scholar
  27. 27.
    Xiao, L.: A nonlinearly activated neural dynamics and its finite-time solution to time-varying nonlinear equation. Neurocomputing 173, 1983–1988 (2016)CrossRefGoogle Scholar
  28. 28.
    Xiao, L.: A finite-time recurrent neural network for solving online time-varying Sylvester matrix equation based on a new evolution formula. Nonlinear Dyn. 90, 1581–1591 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Li, S., Li, Y., Wang, Z.: A class of finite-time dual neural networks for solving quadratic programming problems and its k-winners-take-all application. Neural Netw. 39, 27–39 (2013)CrossRefzbMATHGoogle Scholar
  30. 30.
    Xiao, L., Liao, B., Li, S., Chen, K.: Nonlinear recurrent neural networks for finite-time solution of general time-varying linear matrix equations. Neural Netw. 98, 102–113 (2018)CrossRefGoogle Scholar
  31. 31.
    Xiao, L., Liao, B., Li, S., Zhang, Z., Ding, L., Jin, L.: Design and analysis of FTZNN applied to real-time solution of nonstationary Lyapunov equation and tracking control of wheeled mobile manipulator. IEEE Trans. Ind. Inf. 14, 98–105 (2018)CrossRefGoogle Scholar
  32. 32.
    Zhang, Y., Qiu, B., Liao, B., Yang, Z.: Control of pendulum tracking (including swinging up) of IPC system using zeroing-gradient method. Nonlinear Dyn. 89, 1–25 (2017)CrossRefzbMATHGoogle Scholar
  33. 33.
    Zhang, Y., Yan, X., Chen, D., Guo, D., Li, W.: QP-based refined manipulability-maximizing scheme for coordinated motion planning and control of physically constrained wheeled mobile redundant manipulators. Nonlinear Dyn. 85, 245–261 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Zhang, Y., Li, S., Guo, H.: A type of biased consensus-based distributed neural network for path plannings. Nonlinear Dyn. 89, 1803–1815 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Xiao, L., Li, S., Yang, J., Zhang, Z.: A new recurrent neural network with noise-tolerance and finite-time convergence for dynamic quadratic minimization. Neurocomputing 285, 125–132 (2018)CrossRefGoogle Scholar
  36. 36.
    Xiao, L., Zhang, Z., Zhang, Z., Li, W., Li, S.: Design, verification and robotic application of a novel recurrent neural network for computing dynamic Sylvester equation. Neural Netw. 105, 185–196 (2018)CrossRefGoogle Scholar
  37. 37.
    LaSalle, J.P.: Stability theory for ordinary differential equations. J. Differ. Equ. 4, 57–65 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Chellaboina, V., Bhat, S.P., Haddad, W.M.: An invariance principle for nonlinear hybrid and impulsive dynamical systems. Non. Anal. Theory Methods Appl. 53, 527–550 (2003)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Hunan Provincial Key Laboratory of Intelligent Computing and Language Information ProcessingHunan Normal UniversityChangshaChina

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