Zhang neural network, Getz-Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots' kinematic control

In this paper, we present, develop and investigate a special kind of recurrent neural network termed Zhang neural network (ZNN) for time-varying matrix inversion. Comparing with the dynamic system proposed by Getz and Marsden (G-M) for time-varying matrix inversion, we show that such a G-M dynamic system depicted in an explicit dynamics can also be derived from the presented ZNN model depicted in an implicit dynamics. In other words, a novel result on the relationship between the ZNN model and others' model/method (i.e., the G-M dynamic system) is found for time-varying matrix inversion. In addition, we propose and investigate the discrete-time algorithms (depicted by systems of difference equations) of the aforementioned ZNN and G-M models in two situations, i.e., the time-derivative of the time-varying matrix to be inverted being known or unknown. Simulative and numerical results demonstrate the superior performance of the ZNN models for time-varying matrix inversion, as well as the efficacy of the G-M dynamic system (which has to be started with initial conditions sufficiently close to the desired initial inverse). Furthermore, the ZNN models and G-M dynamic system are applied to the kinematic control of a two-link planar manipulator via online solution of time-varying matrix inversion.

[1]  Çetin Kaya Koç,et al.  Inversion of all principal submatrices of a matrix , 1994 .

[2]  Shuzhi Sam Ge,et al.  Design and analysis of a general recurrent neural network model for time-varying matrix inversion , 2005, IEEE Transactions on Neural Networks.

[3]  Ah Chung Tsoi,et al.  Recurrent neural networks: A constructive algorithm, and its properties , 1997, Neurocomputing.

[4]  Ke Chen,et al.  Global exponential convergence and stability of gradient-based neural network for online matrix inversion , 2009, Appl. Math. Comput..

[5]  James Lam,et al.  On Smith-type iterative algorithms for the Stein matrix equation , 2009, Appl. Math. Lett..

[6]  Yunong Zhang,et al.  Simulation and verification of Zhang neural network for online time-varying matrix inversion , 2009, Simul. Model. Pract. Theory.

[7]  Binghuang Cai,et al.  From Zhang Neural Network to Newton Iteration for Matrix Inversion , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[8]  Chuanqing Gu,et al.  A shift-splitting hierarchical identification method for solving Lyapunov matrix equations , 2009 .

[9]  K. S. Yeung,et al.  Symbolic matrix inversion with application to electronic circuits , 1988 .

[10]  J. Marsden,et al.  Dynamic inversion of nonlinear maps with applications to nonlinear control and robotics , 1995 .

[11]  Hoay Beng Gooi,et al.  New ordering methods for sparse matrix inversion via diagonalization , 1997 .

[12]  Jerrold E. Marsden,et al.  Dynamical methods for polar decomposition and inversion of matrices , 1997 .

[13]  Jun Wang,et al.  A recurrent neural network for real-time matrix inversion , 1993 .

[14]  George Lindfield,et al.  Numerical Methods Using MATLAB , 1998 .

[15]  Robert H. Sturges,et al.  Analog matrix inversion [robot kinematics] , 1988, IEEE J. Robotics Autom..

[16]  Jun Wang,et al.  A recurrent neural network for solving Sylvester equation with time-varying coefficients , 2002, IEEE Trans. Neural Networks.

[17]  Jerrold E. Marsden,et al.  Joint-space tracking of workspace trajectories in continuous time , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[18]  Michael A. Fiddy,et al.  Regularized image reconstruction using SVD and a neural network method for matrix inversion , 1993, IEEE Trans. Signal Process..

[19]  Paul A. Fuhrmann,et al.  A functional approach to the Stein equation , 2010 .

[20]  Carver Mead,et al.  Analog VLSI and neural systems , 1989 .

[21]  Wen Yu,et al.  Stability Analysis of Nonlinear System Identification via Delayed Neural Networks , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

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

[23]  Gonzalo Joya,et al.  Hopfield neural networks for optimization: study of the different dynamics , 2002 .

[24]  Jun Wang,et al.  Digital hardware realization of a recurrent neural network for solving the assignment problem , 2003, Neurocomputing.

[25]  Y. K. Wong,et al.  Nonlinear system identification using optimized dynamic neural network , 2009, Neurocomputing.

[26]  Sanjit K. Mitra,et al.  Digital Signal Processing: A Computer-Based Approach , 1997 .

[27]  Ahmed El-Amawy.  A Systolic Architecture for Fast Dense Matrix Inversion , 1989, IEEE Trans. Computers.

[28]  Jerrold E. Marsden,et al.  A dynamic inverse for nonlinear maps , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[29]  Jianping Li,et al.  Notes on "Recurrent neural network model for computing largest and smallest generalized eigenvalue" , 2010, Neurocomputing.

[30]  Yunong Zhang,et al.  Improved Zhang neural network model and its solution of time-varying generalized linear matrix equations , 2010, Expert Syst. Appl..

[31]  Yu-Nong Zhang,et al.  Zhang Neural Network for Linear Time-Varying Equation Solving and its Robotic Application , 2007, 2007 International Conference on Machine Learning and Cybernetics.