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

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.

[1]  Bolin Liao,et al.  Control of pendulum tracking (including swinging up) of IPC system using zeroing-gradient method , 2017 .

[2]  Shuai Li,et al.  Decentralized control of collaborative redundant manipulators with partial command coverage via locally connected recurrent neural networks , 2012, Neural Computing and Applications.

[3]  Xiaoping Liu,et al.  Robust Adaptive Neural Tracking Control for a Class of Stochastic Nonlinear Interconnected Systems , 2016, IEEE Transactions on Neural Networks and Learning Systems.

[4]  Feng-Yan Zhou,et al.  Mittag–Leffler Stability and Global Asymptotically $$\omega $$ω-Periodicity of Fractional-Order BAM Neural Networks with Time-Varying Delays , 2018, Neural Processing Letters.

[5]  Bing Chen,et al.  Adaptive neural control for strict-feedback stochastic nonlinear systems with time-delay , 2012, Neurocomputing.

[6]  Bing Li,et al.  Anti-periodic Solutions for Quaternion-Valued High-Order Hopfield Neural Networks with Time-Varying Delays , 2018, Neural Processing Letters.

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

[8]  Ali Poorhossein,et al.  Design and implementation of Sugeno controller for inverted pendulum on a cart system , 2010, IEEE 8th International Symposium on Intelligent Systems and Informatics.

[9]  Behrouz Fathi Vajargah.  Different stochastic algorithms to obtain matrix inversion , 2007, Appl. Math. Comput..

[10]  Ju H. Park,et al.  Impulsive Effects on Quasi-Synchronization of Neural Networks With Parameter Mismatches and Time-Varying Delay , 2018, IEEE Transactions on Neural Networks and Learning Systems.

[11]  Long Jin,et al.  Infinitely many Zhang functions resulting in various ZNN models for time-varying matrix inversion with link to Drazin inverse , 2015, Inf. Process. Lett..

[12]  Young Hoon Joo,et al.  Extended dissipativity of generalised neural networks including time delays , 2017, Int. J. Syst. Sci..

[13]  Yunong Zhang,et al.  ZG control for nonlinear system 2-output tracking with GD used additionally once more , 2015, 2015 4th International Conference on Computer Science and Network Technology (ICCSNT).

[14]  Lin Xiao,et al.  A new design formula exploited for accelerating Zhang neural network and its application to time-varying matrix inversion , 2016, Theor. Comput. Sci..

[15]  Shuai Li,et al.  Accelerating a Recurrent Neural Network to Finite-Time Convergence for Solving Time-Varying Sylvester Equation by Using a Sign-Bi-power Activation Function , 2012, Neural Processing Letters.

[16]  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.

[17]  Haifei Xiang.  OSCILLATION OF THIRD-ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS WITH DISTRIBUTED TIME DELAY , 2016 .

[18]  Sohrab Effati,et al.  A Neural Network Approach for Solving a Class of Fractional Optimal Control Problems , 2017, Neural Processing Letters.

[19]  Dongsheng Guo,et al.  Case study of Zhang matrix inverse for different ZFs leading to different nets , 2014, 2014 International Joint Conference on Neural Networks (IJCNN).

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

[21]  Piyapong Niamsup,et al.  Novel criteria for finite-time stabilization and guaranteed cost control of delayed neural networks , 2015, Neurocomputing.

[22]  Ke Chen.  Improved neural dynamics for online Sylvester equations solving , 2016, Inf. Process. Lett..

[23]  Kreangkri Ratchagit,et al.  Asymptotic Stability of Delay-Difference System of Hopfield Neural Networks via Matrix Inequalities and Application , 2007, Int. J. Neural Syst..

[24]  Xingyuan Wang,et al.  Backstepping generalized synchronization for neural network with delays based on tracing control method , 2014, Neural Computing and Applications.

[25]  Ke Chen,et al.  Robustness analysis of a hybrid of recursive neural dynamics for online matrix inversion , 2016, Appl. Math. Comput..

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

[27]  Baidurya Bhattacharya,et al.  Technical Note: A fast parallel Gauss Jordan algorithm for matrix inversion using CUDA , 2013 .

[28]  Chun-Yi Su,et al.  Neural Control of Bimanual Robots With Guaranteed Global Stability and Motion Precision , 2017, IEEE Transactions on Industrial Informatics.

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

[30]  Zhenyuan Guo,et al.  Global synchronization of stochastically disturbed memristive neurodynamics via discontinuous control laws , 2016, IEEE/CAA Journal of Automatica Sinica.

[31]  Predrag S. Stanimirovic,et al.  Gradient Neural Network with Nonlinear Activation for Computing Inner Inverses and the Drazin Inverse , 2017, Neural Processing Letters.

[32]  Hamid Reza Karimi,et al.  Improved Stability and Stabilization Results for Stochastic Synchronization of Continuous-Time Semi-Markovian Jump Neural Networks With Time-Varying Delay , 2017, IEEE Transactions on Neural Networks and Learning Systems.

[33]  Chenguang Yang,et al.  Neural Network-Based Motion Control of an Underactuated Wheeled Inverted Pendulum Model , 2014, IEEE Transactions on Neural Networks and Learning Systems.

[34]  Zhenyuan Guo,et al.  Global Synchronization of Multiple Recurrent Neural Networks With Time Delays via Impulsive Interactions , 2017, IEEE Transactions on Neural Networks and Learning Systems.

[35]  M.N.S. Swamy,et al.  Weighted least-square design of FIR filters using a fast iterative matrix inversion algorithm , 2002 .

[36]  Long Jin,et al.  Tracking control of modified Lorenz nonlinear system using ZG neural dynamics with additive input or mixed inputs , 2016, Neurocomputing.

[37]  Wang Xuegang,et al.  New recursive algorithm for matrix inversion , 2008 .

[38]  Karline Soetaert,et al.  Solving Ordinary Differential Equations in R , 2012 .

[39]  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.

[40]  Abdelhak Bennia,et al.  A Simplified Architecture of the Zhang Neural Network for Toeplitz Linear Systems Solving , 2017, Neural Processing Letters.

[41]  Lin Xiao.  A finite-time convergent Zhang neural network and its application to real-time matrix square root finding , 2017, Neural Computing and Applications.

[42]  Yunong Zhang,et al.  From Davidenko Method to Zhang Dynamics for Nonlinear Equation Systems Solving , 2017, IEEE Transactions on Systems, Man, and Cybernetics: Systems.

[43]  Javier Aracil,et al.  A new controller for the inverted pendulum on a cart , 2008 .

[44]  Jinde Cao,et al.  New Delay-Dependent Stability Criteria for Impulsive Neural Networks with Additive Time-Varying Delay Components and Leakage Term , 2018, Neural Processing Letters.

[45]  Yunong Zhang,et al.  Z-type control of populations for Lotka-Volterra model with exponential convergence. , 2016, Mathematical biosciences.

[46]  Mei Liu,et al.  New Results for Exponential Synchronization of Memristive Cohen–Grossberg Neural Networks with Time-Varying Delays , 2018, Neural Processing Letters.

[47]  Jia-Jun Wang,et al.  Simulation studies of inverted pendulum based on PID controllers , 2011, Simul. Model. Pract. Theory.

[48]  Ke Chen.  Recurrent implicit dynamics for online matrix inversion , 2013, Appl. Math. Comput..

[49]  Jian Guo,et al.  Passivity Analysis of Stochastic Memristor-Based Complex-Valued Recurrent Neural Networks with Mixed Time-Varying Delays , 2017, Neural Processing Letters.

[50]  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.

[51]  Sunil Kumar,et al.  High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay , 2014, Comput. Math. Appl..