Advertisement

Neural Processing Letters

, Volume 50, Issue 2, pp 1735–1753 | Cite as

Zhang Neural Dynamics Approximated by Backward Difference Rules in Form of Time-Delay Differential Equation

  • Yunong ZhangEmail author
  • Jinjin Guo
  • Binbin Qiu
  • Wan Li
Article
  • 89 Downloads

Abstract

Time-varying matrix inversion (TVMI) and tracking control of vehicular inverted pendulum (VIP) system are usually considered as two important benchmarking examples in mathematical science and control fields. In this paper, Zhang neural dynamics (ZND) method is used for solving these two problems. Besides, in order to investigate the effect of time delay (especially the time delay caused by the derivative approximation) on the ZND method when solving the above-mentioned two problems, backward difference rules in the form of time-delay differential equation are elegantly applied to approximating the ZND method, which is simply termed time-delay ZND method. Moreover, two theorems about time-delay ZND method are presented together with theoretical proofs. Finally, simulative tests of TVMI and tracking control of VIP system are conducted to show the effectiveness and accuracy of non-time-delay and time-delay ZND methods.

Keywords

Zhang neural dynamics (ZND) Time delay Backward difference rules Time-varying matrix inversion (TVMI) Tracking control 

Notes

Acknowledgements

The work is supported by the National Natural Science Foundation of China (with number 61473323). Kindly note that all authors of the paper are jointly of the first authorship.

References

  1. 1.
    Jin L, Li S, La HM, Luo X (2017) Manipulability optimization of redundant manipulators using dynamic neural networks. IEEE Trans Ind Electron 64:4710–4720CrossRefGoogle Scholar
  2. 2.
    Zhu WP, Ahmad MO (2002) Weighted least-square design of FIR filters using a fast iterative matrix inversion algorithm. IEEE Trans Circuits Syst I Fundam Theor Appl 49:1620–1628CrossRefGoogle Scholar
  3. 3.
    Gu C, Xue H (2009) A shift-splitting hierarchical identification method for solving Lyapunov matrix equations. Linear Algebra Appl 430:1517–1530CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Cao J, Wang X (2008) New recursive algorithm for matrix inversion. J Syst Eng Electron 19:381–384CrossRefzbMATHGoogle Scholar
  5. 5.
    Vajargah BF (2007) Different stochastic algorithms to obtain matrix inversion. Appl Math Comput 189:1841–1846MathSciNetzbMATHGoogle Scholar
  6. 6.
    Sharma G, Agarwala A, Bhattacharya B (2013) A fast parallel Gauss Jordan algorithm for matrix inversion using CUDA. Comput Struct 128:31–37CrossRefGoogle Scholar
  7. 7.
    Xiao L (2016) A new design formula exploited for accelerating Zhang neural network and its application to time-varying matrix inversion. Theor Comput Sci 647:50–58CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Stanimirovic PS, Petkovic MD, Gerontitis D (2018) Gradient neural network with nonlinear activation for computing inner inverses and the Drazin inverse. Neural Process Lett 48:109–133CrossRefGoogle Scholar
  9. 9.
    Yang C, Jiang Y, Li Z, He W, Su CY (2017) Neural control of bimanual robots with guaranteed global stability and motion precision. IEEE Trans Ind Inf 13:1162–1171CrossRefGoogle Scholar
  10. 10.
    Tang Q, Wang X (2014) Backstepping generalized synchronization for neural network with delays based on tracing control method. Neural Comput Appl 24:775–778CrossRefGoogle Scholar
  11. 11.
    Xiao L (2017) A finite-time convergent Zhang neural network and its application to real-time matrix square root finding. Neural Comput Appl.  https://doi.org/10.1007/s00521-017-3010-z
  12. 12.
    Guo Z, Yang S, Wang J (2016) Global synchronization of stochastically disturbed memristive neurodynamics via discontinuous control laws. IEEE CAA J Autom Sin 3:121–131CrossRefMathSciNetGoogle Scholar
  13. 13.
    Wang H, Chen B, Lin C (2012) Adaptive neural control for strict-feedback stochastic nonlinear systems with time-delay. Neurocomputing 77:267–274CrossRefGoogle Scholar
  14. 14.
    Wang H, Liu X, Liu K (2016) Robust adaptive neural tracking control for a class of stochastic nonlinear interconnected systems. IEEE Trans Neural Netw Learn Syst 27:510–523CrossRefMathSciNetGoogle Scholar
  15. 15.
    Chen K, Yi C (2016) Robustness analysis of a hybrid of recursive neural dynamics for online matrix inversion. Appl Math Comput 273:969–975MathSciNetzbMATHGoogle Scholar
  16. 16.
    Li S, Cui H, Li Y, Liu B, Lou Y (2013) Decentralized control of collaborative redundant manipulators with partial command coverage via locally connected recurrent neural networks. Neural Comput Appl 23:1051–1060CrossRefGoogle Scholar
  17. 17.
    Sabouri JK, Effati S, Pakdaman M (2017) A neural network approach for solving a class of fractional optimal control problems. Neural Process Lett 45:59–74CrossRefGoogle Scholar
  18. 18.
    Li S, Chen S, Liu B (2013) 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 37:189–205CrossRefGoogle Scholar
  19. 19.
    Guo J, Meng Z, Xiang Z (2018) Passivity analysis of stochastic memristor-based complex-valued recurrent neural networks with mixed time-varying delays. Neural Process Lett 47:1097–1113CrossRefGoogle Scholar
  20. 20.
    Niamsup P, Ratchagit K, Phat VN (2015) Novel criteria for finite-time stabilization and guaranteed cost control of delayed neural networks. Neurocomputing 160:281–286CrossRefGoogle Scholar
  21. 21.
    Yang S, Guo Z, Wang J (2017) Global synchronization of multiple recurrent neural networks with time delays via impulsive interactions. IEEE Trans Neural Netw Learn Syst 28:1657–1667CrossRefMathSciNetGoogle Scholar
  22. 22.
    Ratchagit K (2007) Asymptotic stability of delay-difference system of Hopfield neural networks via matrix inequalities and application. Int J Neural Syst 17:425–430CrossRefGoogle Scholar
  23. 23.
    Chen K (2016) Improved neural dynamics for online Sylvester equations solving. Inf Process Lett 116:455–459CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Saravanakumar R, Rajchakit G, Ali MS, Joo YH (2017) Extended dissipativity of generalised neural networks including time delays. Int J Syst Sci 48:2311–2320CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Zhang Y, Ma W, Cai B (2009) From Zhang neural network to Newton iteration for matrix inversion. IEEE Trans Circuits Syst Regul Pap 56:1405–1415CrossRefMathSciNetGoogle Scholar
  26. 26.
    Guo D, Zhang Y (2012) Zhang neural network, Getz-Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots’ kinematic control. Neurocomputing 97:22–32CrossRefGoogle Scholar
  27. 27.
    Guo D, Qiu B, Ke Z, Yang Z (2014) Case study of Zhang matrix inverse for different ZFs leading to different nets. In: Proceedings of international joint conference on neural networks. pp 2764–2769Google Scholar
  28. 28.
    Fella C, Abderrazak B, Abdelhak B (2018) A simplified architecture of the Zhang neural network for Toeplitz linear systems solving. Neural Process Lett 47:391–401Google Scholar
  29. 29.
    Getz N, Marsden J (1997) Dynamical methods for polar decomposition and inversion of matrices. Linear Algebra Its Appl 258:311–343CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Chen K (2013) Recurrent implicit dynamics for online matrix inversion. Appl Math Comput 219:10218–10224MathSciNetzbMATHGoogle Scholar
  31. 31.
    Zhang Y, Yan X, Liao B, Zhang Y, Ding Y (2016) Z-type control of populations for Lotka-Volterra model with exponential convergence. Math Biosci 272:15–23CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    Jin L, Zhang Y, Qiao T, Tan M, Zhang Y (2016) Tracking control of modified Lorenz nonlinear system using ZG neural dynamics with additive input or mixed inputs. Neurocomputing 196:82–94CrossRefGoogle Scholar
  33. 33.
    Zhang Y, Yi C, Guo D, Zheng J (2011) Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation. Neural Comput Appl 20:1–7CrossRefGoogle Scholar
  34. 34.
    Mao M, Li W, Qiu B, Tan H, Zhang Y (2015) ZG control for nonlinear system 2-output tracking with GD used additionally once more. In: Proceedings of the 4th international conference on computer science and network technology. pp 1541–1544Google Scholar
  35. 35.
    Zhang Y, Zhang Y, Chen D, Xiao Z, Yan X (2016) From Davidenko method to Zhang dynamics for nonlinear equation systems solving. IEEE Trans Syst Man Cybern 99:1–14Google Scholar
  36. 36.
    Yang C, Li Z, Cui R, Xu B (2014) Neural network-based motion control of an underactuated wheeled inverted pendulum model. IEEE Trans Neural Netw Learn Syst 25:2004–2016CrossRefGoogle Scholar
  37. 37.
    Gordillo F, Aracil J (2008) A new controller for the inverted pendulum on a cart. Int J Robust Nonlinear Control 18:1607–1621CrossRefMathSciNetzbMATHGoogle Scholar
  38. 38.
    Zhang Y, Qiu B, Liao B, Yang Z (2017) Control of pendulum tracking (including swinging up) of IPC system using zeroing-gradient method. Nonlinear Dyn 89:1–25CrossRefzbMATHGoogle Scholar
  39. 39.
    Zhou F, Ma C (2018) Mittag-Leffler stability and global asymptotically \(\omega \)-periodicity of fractional-order BAM neural networks with time-varying delays. Neural Process Lett 47:71–98CrossRefGoogle Scholar
  40. 40.
    Liu M, Jiang H, Hu C (2018) New results for exponential synchronization of memristive Cohen–Grossberg neural networks with time-varying delays. Neural Process Lett.  https://doi.org/10.1007/s11063-017-9728-x
  41. 41.
    Samidurai R, Rajavel S, Cao J, Alsaedi A, Ahmad B (2018) New delay-dependent stability criteria for impulsive neural networks with additive time-varying delay components and leakage term. Neural Process Lett.  https://doi.org/10.1007/s11063-018-9855-z
  42. 42.
    Li Y, Qin J, Li B (2018) Anti-periodic solutions for quaternion-valued high-order Hopfield neural networks with time-varying delays. Neural Process Lett.  https://doi.org/10.1007/s11063-018-9867-8
  43. 43.
    Tang Z, Park JH, Feng J (2018) Impulsive effects on quasi-synchronization of neural networks with parameter mismatches and time-varying delay. IEEE Trans Neural Netw Learn Syst 29:908–919CrossRefGoogle Scholar
  44. 44.
    Wei Y, Park JH, Karimi HR, Tian Y-C, Jung H (2018) Improved stability and stabilization results for stochastic synchronization of continuous-time semi-Markovian jump neural networks with time-varying delay. IEEE Trans Neural Netw Learn Syst 29:2488–2501CrossRefMathSciNetGoogle Scholar
  45. 45.
    Xiang H (2016) Oscillation of third-order nonlinear neutral differential equations with distributed time delay. Ital J Pure Appl Math 36:769–782MathSciNetzbMATHGoogle Scholar
  46. 46.
    Kumar S, Kumar M (2014) High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay. Comput Math Appl 68:1355–1367CrossRefMathSciNetzbMATHGoogle Scholar
  47. 47.
    Jin L, Zhang Y, Qiu B (2018) Neural network-based discrete-time Z-type model of high accuracy in noisy environments for solving dynamic system of linear equations. Neural Comput Appl 29:1217–1232CrossRefGoogle Scholar
  48. 48.
    Zhang Y, Qiu B, Jin L, Guo D, Yang Z (2015) Infinitely many Zhang functions resulting in various ZNN models for time-varying matrix inversion with link to Drazin inverse. Inf Process Lett 115:703–706CrossRefMathSciNetzbMATHGoogle Scholar
  49. 49.
    Hairer E, Wanner G (1991) Solving ordinary differential equations II. Springer, BerlinCrossRefzbMATHGoogle Scholar
  50. 50.
    Pearson D (1995) Calculus and ordinary differential equations. Butterworth Heinemann, OxfordGoogle Scholar
  51. 51.
    Wang J (2011) Simulation studies of inverted pendulum based on PID controllers. Simul Model Pract Theory 19:440–449CrossRefGoogle Scholar
  52. 52.
    Poorhossein A, Vahidian-Kamyad A (2010) Design and implementation of Sugeno controller for inverted pendulum on a cart system. In: Proceedings of the 8th international symposium on intelligent systems and informatics. pp 641–646Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Data and Computer ScienceSun Yat-sen UniversityGuangzhouChina
  2. 2.Key Laboratory of Machine Intelligence and Advanced ComputingMinistry of EducationGuangzhouChina

Personalised recommendations