Abstract
In this paper, a better fast convergence zeroing neural network (BFCZNN) model with a new activation function (AF) for solving dynamic nonlinear equations (DNE) and applying to control of robot manipulator is presented. The proposed BFCZNN model not only finds the solutions of DNE in fixed time, but also has better robustness than most of the previously reported studies. The numerical simulation results of the proposed BFCZNN and the previously reported robust nonlinear zeroing neural network (RNZNN) for solving third-order DNE in the same condition are presented to demonstrate the better robustness of our new BFCZNN model. Moreover, a successful kinematic control of robot manipulator of our new BFCZNN model is used to verify the realistic availability of the proposed BFCZNN model.
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References
Guo D, Zhang Y (2015) ZNN for solving online time-varying linear matrix–vector inequality via equality conversion. Appl Math Comput 259:327–338
Xiao L, Zhang Y (2014) Solving time-varying inverse kinematics problem of wheeled mobile manipulators using Zhang neural network with exponential convergence. Nonlinear Dyn 76(2):1543–1559
Li S, Zhang Y, Jin L (2017) Kinematic control of redundant manipulators using neural networks. IEEE Trans Neural Netw Learn Syst 28(10):2243–2254
Ngoc PHA, Anh TT (2019) Stability of nonlinear Volterra equations and applications. Appl Math Comput 341:1–14
Peng J, Wang J, Wang Y (2011) Neural network based robust hybrid control for robotic system: an H∞ approach. Nonlinear Dyn 65(4):421–431
Jin L, Zhang Y (2015) Discrete-time Zhang neural network for online time-varying nonlinear optimization with application to manipulator motion generation. IEEE Trans Neural Netw Learn Syst 26(7):1525–1531
Chun C (2006) Construction of Newton-like iteration methods for solving nonlinear equations. Numerische Mathematik 104(3):297–315
Abbasbandy S (2003) Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method. Appl Math Comput 145(2–3):887–893
Sharma JR (2005) A composite third order Newton-Steffensen method for solving nonlinear equations. Appl Mathe Comput 169(1):242–246
Ujevic N (2006) A method for solving nonlinear equations. Applied Mathematics and Computation, 174(2): 1416-1426
Wang J, Chen L, Guo Q (2017) Iterative solution of the dynamic responses of locally nonlinear structures with drift. Nonlinear Dyn 88(3):1551–1564
Y. Tang, L. Jiang, Y. Hou and R. Wang, (2017) "Contactless Fingerprint Image Enhancement Algorithm Based on Hessian Matrix and STFT," 2017 2nd International Conference on Multimedia and Image Processing (ICMIP), Wuhan, pp. 156-160, doi: https://doi.org/10.1109/ICMIP.2017.65
Amiri A, Cordero A, Darvishi MT et al (2019) A fast algorithm to solve systems of nonlinear equations. J Comput Appl Math 354:242–258
Dai P, Wu Q, Wu Y, Liu W (2018) Modified Newton-PSS method to solve nonlinear equations. Appl Math Lett 86:305–312
Birgin EG, Martínez JM (2019) A Newton-like method with mixed factorizations and cubic regularization for unconstrained minimization. Comput Optim Appl 73(3):707–753
Saheya B, Chen GQ, Sui YK et al (2016) A new Newton-like method for solving nonlinear equations. SpringerPlus 5(1):1269
Sharma JR, Arora H (2016) Improved Newton-like methods for solving systems of nonlinear equations. Sema J 74(2):1–17
Madhu K, Jayaraman J (2016) Some higher order Newton-like methods for solving system of nonlinear equations and its applications. Int J Appl Comput Math 1–18:2016
Ding F, Zhang H (2014) Brief Paper - Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems. IET Control Theory Appl 8(15):1588–1595
Zhang Z, Li Z, Zhang Y et al (2015) Neural-Dynamic-method-based dual-arm CMG scheme with time-varying constraints applied to humanoid robots. IEEE Trans Neural Netw Learn Syst 26(12):3251–3262
Li S, Zhang Y, Jin L (2017) Kinematic control of redundant manipulators using neural networks. IEEE Trans Neural Netw Learn Syst 28(10):2243–2254
Xiao L, Liao B, Li S et al (2018) Design and analysis of FTZNN applied to the real-time solution of a Nonstationary Lyapunov equation and tracking control of a wheeled mobile manipulator. IEEE Trans Ind Inform 14(5):98–105
Zhang Y, Ge SS (2005) Design and analysis of a general recurrent neural network model for time-varying matrix inversion. IEEE Trans. Neural Netw 16(6):1477–1490
Chen K, Yi C (2016) Robustness analysis of a hybrid of recursive neural dynamics for online matrix inversion. Appl Math Comput 273:969–975
Chen K (2013) Recurrent implicit dynamics for online matrix inversion. Appl Math Comput 219:10218–10224
Zhang Y, Ge SS (2005) Design and analysis of a general recurrent neural network model for time-varying matrix inversion. IEEE Trans Neural Netw 16:1477–1490
Li S, Chen S, Liu B (2013) Accelerating a recurrent neural network finite time convergence for solving time-varying Sylvester equation by using signbi-power activation function. Neural Proc Lett 37:189–205
Xiao L, Liao B (2016) A convergence-accelerated Zhang neural network and its solution application to Lyapunov equation. Neurocomputing 193:213–218
Yu F, Liu L, Xiao L, Li K, Cai S (2019) A robust and fixed-time zeroing neural dynamics for computing time-variant nonlinear equation using a novel nonlinear activation function. Neurocomputing 350:108–116
Jin Jie, Gong Jianqiang (2020) An interference-tolerant fast convergence zeroing neural network for dynamic matrix inversion and its application to mobile manipulator path tracking. Alex Eng J. https://doi.org/10.1016/j.aej.2020.09.059
Xiao L, Zhang Y (2014) A new performance index for the repetitive motion of mobile manipulators. IEEE Trans Cybern 44(2):280–292
Shi Y, Jin L, Li S, Qiang J (2020) Proposing, developing and verification of a novel discrete-time zeroing neural network for solving future augmented Sylvester matrix equation. J Franklin Institute 357(6):636–3655
Shi Y, Zhang Y (2020) New discrete-time models of zeroing neural network solving systems of time-variant linear and nonlinear inequalities. IEEE Trans Syst Man Cybern: Syst 50(2):565–576
Y. Shi, L. Jin, S. Li, J. Li, J. (2020) Qiang, D. K. Gerontitis, Novel Discrete-Time Recurrent Neural Networks Handling Discrete-Form Time-Variant Multi-Augmented Sylvester Matrix Problems and Manipulator Application, IEEE Transactions on Neural Networks and Learning Systems, doi: https://doi.org/10.1109/TNNLS.2020.3028136.
Xiao L, Zhang Y (2014) A new performance index for the repetitive motion of mobile manipulators. IEEE Trans Cybern 44(2):280–292
Jin J, Zhao L, Li M et al (2020) Improved zeroing neural networks for finite time solving nonlinear equations. Neural Comput Appl 32:4151–4160
Xiao L (2019) A finite-time convergent Zhang neural network and its application to real-time matrix square root finding. Neural Comput Appl 31:793–800
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Gong, J., Jin, J. A better robustness and fast convergence zeroing neural network for solving dynamic nonlinear equations. Neural Comput & Applic 35, 77–87 (2023). https://doi.org/10.1007/s00521-020-05617-9
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DOI: https://doi.org/10.1007/s00521-020-05617-9