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

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Abstract

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.

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References

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  4. Narayanan, M.D., Narayanan, S., Padmanabhan, C.: Parametric identification of nonlinear systems using multiple trials. Nonlinear Dyn. 48, 341–360 (2007)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  8. Xiao, L., Zhang, Y.: A new performance index for the repetitive motion of mobile manipulators. IEEE Trans. Cybern. 44, 280–292 (2014)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  10. Chun, C.: Construction of Newton-like iteration methods for solving nonlinear equations. Numer. Math. 104, 297–315 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Abbasbandy, S.: Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method. Appl. Math. Comput. 145, 887–893 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Sharma, J.R.: A composite third order Newton–Steffensen method for solving nonlinear equations. App. Math. Comput. 169, 242–246 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ujevic, N.: A method for solving nonlinear equations. App. Math. Comput. 174, 1416–1426 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  17. Zhang, Y., Xiao, L., Xiao, Z., Mao, M.: Zeroing Dynamics, Gradient Dynamics, and Newton Iterations. CRC Press, Boca Raton (2015)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

  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)

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  27. Xiao, L.: A nonlinearly activated neural dynamics and its finite-time solution to time-varying nonlinear equation. Neurocomputing 173, 1983–1988 (2016)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  37. LaSalle, J.P.: Stability theory for ordinary differential equations. J. Differ. Equ. 4, 57–65 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

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.

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  1. Hunan Provincial Key Laboratory of Intelligent Computing and Language Information Processing, Hunan Normal University, Changsha, 410081, China

    Lin Xiao

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  1. Lin Xiao
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Correspondence to Lin Xiao.

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Xiao, L. Design and analysis of robust nonlinear neural dynamics for solving dynamic nonlinear equation within finite time. Nonlinear Dyn 96, 2437–2447 (2019). https://doi.org/10.1007/s11071-019-04932-8

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  • DOI: https://doi.org/10.1007/s11071-019-04932-8

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