Solving Discrete Dynamic Nonlinear Equation System Using New-Type DTG Model With Occasionally-Singular Jacobian Matrix

In this paper, a six-point discretization (6PD) formula is presented to discretize continuous-time models. Then, by using the 6PD formula and introducing the adaptivity/variability of parameter, a new-type discrete-time gradient (DTG) model is proposed to solve a discrete dynamic nonlinear equation system (DDNES) with occasionally-singular Jacobian matrix. For comparative purposes, based on the 6PD formula, a 6PD-type discrete-time zeroing (DTZ) model and an old-type (i.e., conventional) DTG model are also presented to handle the same problem. Finally, comparative numerical experiments, including an application to the discrete-time motion control of a robot manipulator, are conducted to substantiate the validity and superiority of the new-type DTG model for solving the DDNES with occasionally-singular Jacobian matrix. That is, when the Jacobian matrix of DDNES occasionally becomes singular as time evolves, the new-type DTG model can provide a feasible and effective solution to the singular Jacobian problem, whereas the other presented models fail to achieve such a solution.

[1]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[2]  Yunong Zhang,et al.  Zhang Neural Networks and Neural-Dynamic Method , 2011 .

[3]  Yang Shi,et al.  Solving future equation systems using integral-type error function and using twice ZNN formula with disturbances suppressed , 2019, J. Frankl. Inst..

[4]  Dongsheng Guo,et al.  Comparison on Zhang neural dynamics and gradient-based neural dynamics for online solution of nonlinear time-varying equation , 2011, Neural Computing and Applications.

[5]  Chen Peng,et al.  Broyden-Method Aided Discrete ZNN Solving the Systems of Time-Varying Nonlinear Equations , 2012, 2012 International Conference on Control Engineering and Communication Technology.

[6]  Bernard Bayle,et al.  Manipulability of Wheeled Mobile Manipulators: Application to Motion Generation , 2003 .

[7]  Lin Xiao,et al.  A nonlinearly activated neural dynamics and its finite-time solution to time-varying nonlinear equation , 2016, Neurocomputing.

[8]  Shuai Li,et al.  Manipulability Optimization of Redundant Manipulators Using Dynamic Neural Networks , 2017, IEEE Transactions on Industrial Electronics.

[9]  Dongsheng Guo,et al.  Theoretical analysis, numerical verification and geometrical representation of new three-step DTZD algorithm for time-varying nonlinear equations solving , 2016, Neurocomputing.

[10]  Yunong Zhang,et al.  New-Type DTZ Model for Solving Discrete Time-Dependent Nonlinear Equation System With Robotic-Arm Application , 2020, 2020 10th International Conference on Information Science and Technology (ICIST).

[11]  Xiaodong Li,et al.  New five-step DTZD algorithm for future nonlinear minimization with quartic steady-state error pattern , 2018, Numerical Algorithms.

[12]  Chen Xu,et al.  A One-Layer Recurrent Neural Network for Constrained Complex-Variable Convex Optimization , 2018, IEEE Transactions on Neural Networks and Learning Systems.

[13]  Zhen Li,et al.  Discrete-time ZD, GD and NI for solving nonlinear time-varying equations , 2012, Numerical Algorithms.

[14]  Shuai Li,et al.  A Noise-Suppressing Neural Algorithm for Solving the Time-Varying System of Linear Equations: A Control-Based Approach , 2019, IEEE Transactions on Industrial Informatics.

[15]  Yunong Zhang,et al.  Zeroing Dynamics, Gradient Dynamics, and Newton Iterations , 2015 .

[16]  W. Marsden.  I and J , 2012 .

[17]  Yunong Zhang,et al.  Convergence and stability results of Zhang neural network solving systems of time-varying nonlinear equations , 2012, 2012 8th International Conference on Natural Computation.

[18]  Ke Chen,et al.  Performance Analysis of Gradient Neural Network Exploited for Online Time-Varying Matrix Inversion , 2009, IEEE Transactions on Automatic Control.

[19]  Yunong Zhang,et al.  Link Between and Comparison and Combination of Zhang Neural Network and Quasi-Newton BFGS Method for Time-Varying Quadratic Minimization , 2013, IEEE Transactions on Cybernetics.

[20]  Aryan Mokhtari,et al.  A Class of Prediction-Correction Methods for Time-Varying Convex Optimization , 2015, IEEE Transactions on Signal Processing.

[21]  Long Jin,et al.  Neural network-based discrete-time Z-type model of high accuracy in noisy environments for solving dynamic system of linear equations , 2016, Neural Computing and Applications.

[22]  Yunong Zhang,et al.  Analysis, Verification and Comparison on Feedback‐Aided MA Equivalence and Zhang Equivalency of Minimum‐Kinetic‐Energy Type for Kinematic Control of Redundant Robot Manipulators , 2018 .

[23]  Zhi Yang,et al.  Revisit and compare Ma equivalence and Zhang equivalence of minimum velocity norm (MVN) type , 2016, Adv. Robotics.