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GNN Model for time-varying matrix inversion With Robust Finite-time Convergence

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2024年第1期35卷 559-569页

作者： Zhang, Yinyan Li, Shuai Weng, Jian Liao, Bolin Jinan Univ Coll Cyber Secur Guangzhou 510632 Peoples R China Pazhou Lab Guangzhou 510330 Peoples R China Swansea Univ Coll Engn Swansea SA2 8PP W Glam Wales Jinan Univ Coll Informat Sci & Technol Guangzhou 510632 Peoples R China Jishou Univ Coll Informat Sci & Engn Jishou 416000 Peoples R China

As a type of recurrent neural networks (RNNs) modeled as dynamic systems, the gradient neural network (GNN) is recognized as an effective method for static matrix inversion with exponential convergence. However, when it comes to time-varying matrix inversion, most of the traditional GNNs can only track the corresponding time-varying solution with a residual error, and the performance becomes worse when there are noises. Currently, zeroing neural networks (ZNNs) take a dominant role in time-varying matrix inversion, but ZNN models are more complex than GNN models, require knowing the explicit formula of the time-derivative of the matrix, and intrinsically cannot avoid the inversion operation in its realization in digital computers. In this article, we propose a unified GNN model for handling both static matrix inversion and time-varying matrix inversion with finite-time convergence and a simpler structure. Our theoretical analysis shows that, under mild conditions, the proposed model bears finite-time convergence for time-varying matrix inversion, regardless of the existence of bounded noises. Simulation comparisons with existing GNN models and ZNN models dedicated to time-varying matrix inversion demonstrate the advantages of the proposed GNN model in terms of convergence speed and robustness to noises.

关键词： Convergence Mathematical models Computational modeling Adaptation models Biological neural networks time-varying systems Recurrent neural networks Finite-time convergence gradient neural network (GNN) robustness time-varying matrix inversion

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Novel Discrete-time Zhang Neural Network for time-varying matrix inversion

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IEEE TRANSACTIONS ON SYSTEMS MAN CYBERNETICS-SYSTEMS 2017年第8期47卷 2301-2310页

作者： Guo, Dongsheng Nie, Zhuoyun Yan, Laicheng Huaqiao Univ Coll Informat Sci & Engn Xiamen 361021 Peoples R China

In the previous work, Zhang et al. developed a special type of recurrent neural networks called Zhang neural network (ZNN) with continuous-time and discrete-time forms for time-varying matrix inversion. In this paper, a novel discretetime ZNN (DTZNN) model for time-varying matrix inversion is proposed and investigated. Specifically, a new numerical difference rule based on Taylor series expansion is established in this paper for first-order derivative approximation. Then, by exploiting this Taylor-type difference rule, the novel DTZNN model, which is a five-step iteration algorithm, is thus proposed for time-varying matrix inversion. Theoretical results are also presented for the proposed DTZNN model to show its excellent computational property. Comparative numerical results with three illustrative examples further substantiate the efficacy and superiority of the proposed DTZNN model for time-varying matrix inversion compared with previous DTZNN models.

关键词： Difference rule discrete-time Zhang neural network (DTZNN) Taylor series expansion theoretical results time-varying matrix inversion

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Simulation and verification of Zhang neural network for online time-varying matrix inversion

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SIMULATION MODELLING PRACTICE AND THEORY 2009年第10期17卷 1603-1617页

作者： Zhang, Yunong Yi, Chenfu Ma, Weimu Sun Yat Sen Univ Sch Informat Sci & Technol Guangzhou 510275 Peoples R China

Differing from gradient-based neural networks (GNN), a special kind of recurrent neural network has recently been proposed by Zhang et al. for real-time inversion of time-varying matrices. The design of such a recurrent neural network is based on a matrix-valued error function instead of a scalar-valued norm-based energy-function. In addition, it is depicted in an implicit dynamics instead of an explicit dynamics. This paper investigates the simulation and verification of such a Zhang neural network (ZNN). Four important simulation techniques are employed to simulate this system: (1) Kronecker product of matrices is introduced to transform a matrix-differential-equation (MDE) to a vector differential equation (VDE) [i.e., finally, there is a standard ordinary-differential-equation (ODE) formulation]. (2) MATLAB routine "ode45" with a mass-matrix property is introduced to Simulate the transformed initial-value implicit ODE system. (3) matrix derivatives are obtained using the routine "diff" and symbolic math toolbox. (4) Various implementation errors and different types of activation functions are investigated, further demonstrating the advantages of the ZNN model. Three illustrative computer-simulation examples substantiate the theoretical results and efficacy of the ZNN model for online time-varying matrix inversion. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Simulation and verification Recurrent neural network time-varying matrix inversion Kronecker product Symbolic math

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

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NEUROCOMPUTING 2012年 97卷 22-32页

作者： Guo, Dongsheng Zhang, Yunong Sun Yat Sen Univ Sch Informat Sci & Technol Guangzhou 510006 Guangdong Peoples R China

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. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Zhang neural network Getz-Marsden dynamic system Discrete-time algorithm time-varying matrix inversion Dynamics

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Performance Analysis of Gradient Neural Network Exploited for Online time-varying matrix inversion

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2009年第8期54卷 1940-1945页

作者： Zhang, Yunong Chen, Ke Tan, Hong-Zhou Sun Yat Sen Univ Sch Informat Sci & Technol Guangzhou 510275 Guangdong Peoples R China

This technical note presents theoretical analysis and simulation results on the performance of a classic gradient neural network (GNN), which was designed originally for constant matrix inversion but is now exploited for time-varying matrix inversion. Compared to the constant matrix-inversion case, the gradient neural network inverting a time-varying matrix could only approximately approach its time-varying theoretical inverse, instead of converging exactly. In other words, the steady-state error between the GNN solution and the theoretical/exact inverse does not vanish to zero. In this technical note, the upper bound of such an error is estimated firstly. The global exponential convergence rate is then analyzed for such a Hopfield-type neural network when approaching the bound error. Computer-simulation results finally substantiate the performance analysis of this gradient neural network exploited to invert online time-varying matrices.

关键词： Global exponential convergence rate gradient neural networks performance analysis residual error bound time-varying matrix inversion

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Design and analysis of a general recurrent neural network model for time-varying matrix inversion

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IEEE TRANSACTIONS ON NEURAL NETWORKS 2005年第6期16卷 1477-1490页

作者： Zhang, YN Ge, SS Natl Univ Singapore Dept Elect & Comp Engn Singapore 117576 Singapore

Following the idea of using first-order time derivatives, this paper presents a general recurrent neural network (RNN) model for online inversion of time-varying matrices. Different kinds of activation functions are investigated to guarantee the global exponential convergence of the neural model to the exact inverse of a given time-varying matrix. The robustness of the proposed neural model is also studied with respect to different activation functions and various implementation errors. Simulation results, including the application to kinematic control of redundant manipulators, substantiate the theoretical analysis and demonstrate the efficacy of the neural model on time-varying matrix inversion, especially when using a power-sigmoid activation function.

关键词： activation function implicit dynamics inverse kinematics recurrent neural network (RNN) time-varying matrix inversion

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A new design formula exploited for accelerating Zhang neural network and its application to time-varying matrix inversion

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THEORETICAL COMPUTER SCIENCE 2016年第0期647卷 50-58页

作者： Xiao, Lin Jishou Univ Coll Informat Sci & Engn Jishou 416000 Peoples R China

Online solution to time-varying matrix inverse is further investigated by proposing a new design formula, which can accelerate Zhang neural network (ZNN) to finite-time convergence. Compared with the existing recurrent neural networks [e.g., the gradient neural networks (GNN), and the original Zhang neural network], the proposed neural network (termed finite-time ZNN, FTZNN) makes a breakthrough in the convergence performance (i.e., from infinite time to finite time). In addition, different from the previous processing method (i.e., choosing a better nonlinear activation to accelerate convergence speed), this paper subtly proposes a new design formula to accelerate the original ZNN model and design the new FTZNN model. Besides, theoretical analyses of the design formula and the FTZNN model are given in detail. Simulative results substantiate the effectiveness and superiomess of the proposed FTZNN model for online time-varying matrix inversion, as compared with the GNN model and the original ZNN model. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Design formula time-varying matrix inversion Zhang neural network Theoretical analysis Simulation verification

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matrix ODE Tool for Solution of matrix-valued ODE with Application to Online time-varying matrix inversion 13

Matrix ODE Tool for Solution of Matrix-valued ODE with Appli...

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2016 IEEE 13th International Conference on Signal Processing（ICSP2016）

作者： Yunong Zhang Nanhao Wang Jiadi Wang Chen Peng Yingbiao Ling School of Information Science and Technology Sun Yat-sen University (SYSU) SYSU-CMU Shunde International Joint Research Institute Key Laboratory of Autonomous Systems and Networked Control Ministry of Education

ISBN: (纸本)9781509013463

In modern engineering researches, dynamics of systems are often studied by well-design computer simulations,and these dynamics are usually depicted in the form of vectoror matrix-valued ordinary differential equation(ODE) systems.A common approach for solution of matrix-valued ODE in MATLAB is to use the Kronecker product and vectorization technique. However, the vectorization procedure may be troublesome, and it can be difficult if not impossible to vectorize a complicated matrix-valued ODE(especially when it is nonlinear).Differing from the traditional approach, this paper proposes an elegant, compact and effective tool(named matrixO DE)for solving a matrix-valued ODE with arbitrary right hand side. As a complement and extension of the built-in solvers in MATLAB, the proposed matrixO DE tool allows solving a matrix-valued ODE using matrix language directly, i.e., users do not need to perform matrix-vector transformations. Application of the proposed matrixO DE tool to online time-varying matrix inversion further substantiates its effectiveness for matrix-valued ODE solving.

关键词： Special software for simulation dynamical equations scientific computation matrix-valued ODE time-varying matrix inversion

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The Application of Li-Function Activated RNN to Acceleration-Level Robots' Kinematic Control via time-varying matrix inversion 28

The Application of Li-Function Activated RNN to Acceleration...

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28th Chinese Control and Decision Conference

作者： Guo, Dongsheng Li, Kene Yan, Laicheng Nie, Zhuoyun Jin, Fujiang Huaqiao Univ Coll Informat Sci & Engn Xiamen 361021 Peoples R China Guangxi Univ Sci & Technol Sch Elect & Informat Engn Liuzhou 545006 Peoples R China

ISBN: (纸本)9781467397148

In this paper, the application of recurrent neural network (RNN) activated by Li function to acceleration-level robots' kinematic control via time-varying matrix inversion is presented and investigated. Specifically, by exploiting Li activation function and by computing the time-varying inverse of the related nonsingular matrix, the resultant RNN model is applied to kinematic control of redundant robot manipulators at the joint-acceleration level. Note that such a Li-function activated RNN (LFARNN) model can achieve the purpose of finite-time convergence, and thus is feasible to acceleration-level kinematic control of redundant robot manipulators. Simulation results based on a four-link planar robot manipulator and a PA10 robot manipulator further substantiate the effectiveness of the presented LFARNN model, as well as show the LFARNN application prospect.

关键词： Li-function activated RNN Acceleration-level kinematic control Redundant robot manipulators time-varying matrix inversion Finite-time convergence

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Five-step discrete-time noise-tolerant zeroing neural network model for time-varying matrix inversion with application to manipulator motion generation

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ENGINEERING APPLICATIONS OF ARTIFICIAL INTELLIGENCE 2021年 103卷 104306-104306页

作者： Liu, Keping Liu, Yongbai Zhang, Yun Wei, Lin Sun, Zhongbo Jin, Long Changchun Univ Technol Dept Control Engn Changchun 130012 Peoples R China Jilin Univ Minist Educ Key Lab Bion Engn Changchun 130025 Peoples R China Lanzhou Univ Sch Informat Sci & Engn Lanzhou 730000 Peoples R China

In this paper, a novel Taylor-type difference rule with O(tau(4)) pattern error is provided for the first-order derivative approximation. Then, a high accuracy noise-tolerant five-step discrete-time zeroing neural network (ZNN) (termed as FDNTZNN model) is proposed to solve the time-varying matrix inversion problem in realtime. In addition, to obtain the derivative value of time-varying variables in real-world applications, the backward-difference rule is exploited to develop the FD-NTZNN model when the derivative information is unknown (FD-NTZNN-U). Theoretical analysis demonstrates that the proposed FD-NTZNN models have the properties of 0-stability, consistency and convergence. For comparative analysis, the classical Euler-type discrete-time ZNN model (EDZNN), five-step Taylor-type discrete-time ZNN model (FDZNN) and Euler-type discrete-time noise-tolerant ZNN (NTZNN) model (ED-NTZNN) are reconsidered. Ultimately, two illustrative numerical simulations and an application example to motion generation of manipulator are simulated to substantiate the feasibility and effectiveness of the proposed FD-NTZNN model and FD-NTZNN-U model for online time-varying matrix inversion in the presence of different types of noise.

关键词： Five-step discrete-time noise-tolerant zeroing neural network (FD-NTZNN) model Noise-tolerant time-varying matrix inversion Taylor-type difference rule Theoretical results

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