ZNN for solving online time-varying linear matrix-vector inequality via equality conversion

In this paper, a special recurrent neural network termed Zhang neural network (ZNN) is proposed and investigated for solving online time-varying linear matrix-vector inequality (LMVI) via equality conversion. That is, by introducing a time-varying vector (of which each element is great than or equal to zero), such a time-varying linear inequality can be converted to a time-varying matrix-vector equation. Then, the ZNN model is developed and investigated for solving online the time-varying matrix-vector equation (as well as the time-varying LMVI) by employing the ZNN design formula. The resultant ZNN model exploits the time-derivative information of time-varying coefficients. Computer-simulation results further demonstrate the efficacy and superiority of the proposed ZNN model for solving online the time-varying LMVI (and the converted time-varying matrix-vector equation).

[1]  Xiaolin Hu,et al.  Dynamic system methods for solving mixed linear matrix inequalities and linear vector inequalities and equalities , 2010, Appl. Math. Comput..

[2]  Adi Ben-Israel,et al.  Generalized inverses: theory and applications , 1974 .

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

[4]  Chih-Chin Lai,et al.  A neural network for linear matrix inequality problems , 2000, IEEE Trans. Neural Networks Learn. Syst..

[5]  Xiaojing Yang,et al.  Lyapunov-type inequality for a class of linear differential systems , 2012, Appl. Math. Comput..

[6]  Ling Zhu,et al.  A general form of Jordan-type double inequality for the generalized and normalized Bessel functions , 2010, Appl. Math. Comput..

[7]  Yunong Zhang,et al.  A new variant of the Zhang neural network for solving online time-varying linear inequalities , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[8]  Salah M. El-Sayed,et al.  A direct method for solving circulant tridiagonal block systems of linear equations , 2005, Appl. Math. Comput..

[9]  Abhishek K Gupta,et al.  Numerical Methods using MATLAB , 2014, Apress.

[10]  Medhat Ahmed Rakha,et al.  On the Moore-Penrose generalized inverse matrix , 2004, Appl. Math. Comput..

[11]  Fan-Tien Cheng,et al.  The improved compact QP method for resolving manipulator redundancy , 1995, IEEE Trans. Syst. Man Cybern..

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

[13]  Gilles Labonté,et al.  On solving systems of linear inequalities with artificial neural networks , 1997, IEEE Trans. Neural Networks.

[14]  Yunong Zhang,et al.  Complex-valued Zhang neural network for online complex-valued time-varying matrix inversion , 2011, Appl. Math. Comput..

[15]  Jun Wang,et al.  Recurrent neural networks for solving linear inequalities and equations , 1999 .

[16]  Kuniyoshi Abe,et al.  A variant algorithm of the Orthomin(m) method for solving linear systems , 2008, Appl. Math. Comput..

[17]  John H. Mathews,et al.  Using MATLAB as a programming language for numerical analysis , 1994 .

[18]  Yunong Zhang,et al.  Zhang Neural Network Versus Gradient Neural Network for Solving Time-Varying Linear Inequalities , 2011, IEEE Transactions on Neural Networks.

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

[20]  Dongsheng Guo,et al.  Zhang neural network, Getz-Marsden dynamic system, and discrete-time algorithms for time-varying matrix inversion with application to robots' kinematic control , 2012, Neurocomputing.

[21]  Yunong Zhang,et al.  Obstacle avoidance for kinematically redundant manipulators using a dual neural network , 2004, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[22]  Fan-Tien Cheng,et al.  Resolving manipulator redundancy under inequality constraints , 1994, IEEE Trans. Robotics Autom..

[23]  K. G. Murty,et al.  New iterative methods for linear inequalities , 1992 .

[24]  Dongsheng Guo,et al.  A New Inequality-Based Obstacle-Avoidance MVN Scheme and Its Application to Redundant Robot Manipulators , 2012, IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews).

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

[26]  Andrew Klapper,et al.  Improved multicovering bounds from linear inequalities and supercodes , 2004, IEEE Transactions on Information Theory.

[27]  Bing Zheng,et al.  Generalized inverses of a normal matrix , 2008, Appl. Math. Comput..

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

[29]  Ning Tan,et al.  Zhang neural network solving for time-varying full-rank matrix Moore–Penrose inverse , 2010, Computing.

[30]  Muhammad Aslam Noor,et al.  Self-adaptive projection algorithms for general variational inequalities , 2004, Appl. Math. Comput..

[31]  A. Cichocki,et al.  Neural Networks for Solving Linear Inequality Systems , 1997, Parallel Comput..