Zhang Neural Network Model Research Articles

A Non-linear and Noise-Tolerant ZNN Model and Its Application to Static and Time-Varying Matrix Square Root Finding

Based on the indefinite error-monitoring function, we propose a novel Zhang neural network (ZNN) model called NNT-ZNN with two properties of nonlinear and noise-tolerant for the time-varying and static matrix square root finding in this paper. Compared to the existing models associated with the square matrix root finding, the NNT-ZNN model proposed in this study fully takes error caused by possible noise on ZNN hardware implementation into account. Under the background that the large model-implementation error, the model still has the ability to converge to the theoretical square root of the given matrix with simulative results illustrated in the paper. For the purpose of comparison, the ZNN model proposed by Zhang et al. is also introduced. Beyond that, the corresponding convergence results of the NNT-ZNN model corresponding to various activation functions, are also shown via time-varying and static positive definite matrix. In the end, the experiments are simulated with MATLAB, which further verifies the availability, effectiveness of the proposed NNT-ZNN model, and robustness against unknown noise.

A nonlinear and noise-tolerant ZNN model solving for time-varying linear matrix equation

Abstract The Zhang neural network (ZNN) has attracted a great deal of interest from a large number of researchers because of its significant advantage in solving the various time-varying problems by the monotonously increasing odd activation functions. Many related models have been proposed for time-varying matrix solutions, however, provided that the noise is zero or the preprocessing of de-noising is conducted. Therefore, many of the models previously proposed are not suitable for real-world situations. In this study, a nonlinear and noise-tolerant ZNN model, named NNT-ZNN, is proposed and discussed based on the matrix-valued error function. Theoretically, we prove that the proposed NNT-ZNN model can be globally converged to the theory solution of the considered time-varying equation, regardless of any activation function being applied. In addition, we prove that the resultant NNT-ZNN model has the superior convergence performance beside the existing ZNN models, even when noise is not zero. After that, the simulative results of the resultant NNT-ZNN model are provided by using three illustrative examples to thoroughly validate the correctness of the theoretical analysis. Moreover, the simulation comparison between the proposed NNT-ZNN model and the existing ZNN-1 model is conducted, which further show that availability and excellence of the resultant NNT-ZNN model, and robustness to noise.

Design, verification and robotic application of a novel recurrent neural network for computing dynamic Sylvester equation

To solve dynamic Sylvester equation in the presence of additive noises, a novel recurrent neural network (NRNN) with finite-time convergence and excellent robustness is proposed and analyzed in this paper. As compared with the design process of Zhang neural network (ZNN), the proposed NRNN is based on an ingenious integral design formula activated by nonlinear functions, which are able to expedite the convergence speed and suppress unknown additive noises during the solving process of dynamic Sylvester equation. In addition, the global stability, finite-time convergence and denoising property of the NRNN model are theoretically proved. The upper bound of the finite convergence time for the NRNN model is also estimated in theory. Simulative results further verify the efficiency of the NRNN model, as well as its superior robust and finite-time performance to the conventional ZNN model for dynamic Sylvester equation in front of additive noises. At last, the proposed design method for establishing the NRNN model is successfully applied to kinematical control of robotic manipulator in front of additive noises.

Three-step general discrete-time Zhang neural network design and application to time-variant matrix inversion

Abstract In recent decades, neural networks methods have widely applied to many science and engineering fields. Zhang neural network (ZNN) as a special type of recurrent neural networks was proposed by Zhang et al, which has been applied to different time-variant problems solving. ZNNs are usually used to process the continuous-time signal in time-variant systems as continuous-time ZNN (CTZNN) models. Since digit devices and computers are widely applied in science and engineering, it is necessary to develop a general method to discretize CTZNN models to discrete-time ZNN (DTZNN) models. In previous work, Euler forward difference and two types of three-step Zhang et al. discretization (ZeaD) formulas were applied to discretize CTZNN models. In this paper, a three-step general ZeaD formula based on Taylor expansion is designed to approximate the first-order derivative of the target point, and discretize CTZNN models for time-variant matrix inversion. In comparison, the two types of three-step ZeaD formulas are the special cases of the proposed general ZeaD formula. For the situation of the time derivative of objective matrix unknown, two formulas of estimating the derivative are provided, and two other corresponding three-step general DTZNN models are proposed. Theoretical analyses present the stability and convergence of the three general DTZNN models for time-variant matrix inversion. The numerical experiment results substantiate the efficacy and superiority of the three proposed general DTZNN models for time-variant matrix inversion with the theoretical steady-state residual errors, comparing with those of Newton iteration and one-step DTZNN model. In addition, by comparing the numerical results of the two general DTZNN models for the situation of the time derivative of objective matrix unknown with the one of the derivative known, the steady-state residual errors of the formers are slightly bigger than the latter.

A new recurrent neural network with noise-tolerance and finite-time convergence for dynamic quadratic minimization

Abstract To solve dynamic quadratic minimization, a nonlinearly activated integration design formula is first proposed in this paper with additive noises considered. Then, on the basis of such a design formula, a new recurrent neural network (RNN) is established to solve the dynamic quadratic minimization. Compared with the conventional Zhang neural network (ZNN) for this problem, the proposed RNN model possesses the outstanding finite-time convergence and the inherently noise-tolerant performance, and is thus called the versatile RNN (VRNN) model. In addition, the global stability, the finite-time convergence and the denoising ability of the VRNN model are proved by rigorous mathematical results in theory. The upper bound of the finite convergence time for the VRNN model is also analytically derived. Numerical simulative results are presented to validate the efficacy of the VRNN model, as well as its superior performance to the conventional ZNN model for dynamic quadratic minimization in the presence of various additive noises.

Design and Analysis of FTZNN Applied to the Real-Time Solution of a Nonstationary Lyapunov Equation and Tracking Control of a Wheeled Mobile Manipulator

The Lyapunov equation is widely employed in the engineering field to analyze stability of dynamic systems. In this paper, based on a new evolution formula, a novel finite-time recurrent neural network (termed finite-time Zhang neural network, FTZNN) is proposed and studied for solving a nonstationary Lyapunov equation. In comparison with the original Zhang neural network (ZNN) model for a nonstationary Lyapunov equation, the convergence performance has a remarkable improvement for the proposed FTZNN model and can be accelerated to finite time. Besides, by solving the differential inequality, the time upper bound of the FTZNN model is computed theoretically and analytically. Simulations are conducted and compared to validate the superiority of the FTZNN model to the original ZNN model for solving the nonstationary Lyapunov equation. At last, the FTZNN model is successfully applied to online tracking control of a wheeled mobile manipulator.

Complex ZFs for computing time-varying complex outer inverses

Abstract Our first aim is to extend the Zhang Neural Network (ZNN) algorithmic conceptual framework, developed so far in the computation of the matrix inverse, pseudoinverse, and the Drazin inverse, to the class of outer inverses with prescribed range and null space. For this purpose, two ZNN models with two types of complex activation functions, designed for calculating computing outer inverses of time-varying complex matrices are presented. In addition, an appropriate finite-time ZNN model based on the usage of Li activation function is introduced. The design of these neural networks is based on appropriate Zhang functions (ZFs) arising from known limiting representations of outer inverses. The convergence properties of the introduced ZNN models are investigated and simulative results are presented in order to demonstrate their usability, generality and effectiveness. So far known results, corresponding to the Moore-Penrose inverse, the Drazin and the group inverse, can be derived simply as particular appearances of the models defined in the present paper. Restrictions on the spectrum are avoided as well as the requirements regarding the nonsingularity of certain matrices. Moreover, a hybrid combination of the ZNN model and the GNN model for computing outer inverses with global convergence is defined for constant real matrices.

An Improved Recurrent Neural Network for Complex-Valued Systems of Linear Equation and Its Application to Robotic Motion Tracking.

To obtain the online solution of complex-valued systems of linear equation in complex domain with higher precision and higher convergence rate, a new neural network based on Zhang neural network (ZNN) is investigated in this paper. First, this new neural network for complex-valued systems of linear equation in complex domain is proposed and theoretically proved to be convergent within finite time. Then, the illustrative results show that the new neural network model has the higher precision and the higher convergence rate, as compared with the gradient neural network (GNN) model and the ZNN model. Finally, the application for controlling the robot using the proposed method for the complex-valued systems of linear equation is realized, and the simulation results verify the effectiveness and superiorness of the new neural network for the complex-valued systems of linear equation.

A finite-time recurrent neural network for solving online time-varying Sylvester matrix equation based on a new evolution formula

Sylvester equation is widely used to study the stability of a nonlinear system in the control field. In this paper, a finite-time Zhang neural network (FTZNN) is proposed and applied to online solution of time-varying Sylvester equation. Differing from the conventional accelerating method, the design of the proposed FTZNN model is based on a new evolution formula, which is presented and studied to accelerate the convergence speed of a recurrent neural network. Compared with the original Zhang neural network (ZNN) for time-varying Sylvester equation, the FTZNN model can converge to the theoretical time-varying solution within finite time, instead of converging exponentially with time. Besides, we can obtain the upper bound of the finite convergence time for the FTZNN model in theory. Simulation results show that the proposed FTZNN model achieves the better performance as compared with the original ZNN model for solving online time-varying Sylvester equation.

Novel Discrete-Time Zhang Neural Network for Time-Varying Matrix Inversion

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 discrete-time 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.

Accelerating a recurrent neural network to finite-time convergence using a new design formula and its application to time-varying matrix square root

In this paper, a new design formula is presented to accelerate the convergence speed of a recurrent neural network, and applied to time-varying matrix square root finding in real time. Then, according to such a new design formula, a finite-time Zhang neural network (FTZNN) is proposed and investigated for finding time-varying matrix square root. In comparison with the original Zhang neural network (ZNN) model, the FTZNN model makes a breakthrough in the convergence performance (i.e., from infinite time to finite time). In addition, theoretical analyses of the design formula and the FTZNN model are provided in details. Comparative results further verify the superiority of the proposed FTZNN model to the original ZNN model for finding time-varying matrix square root.

A finite-time convergent Zhang neural network and its application to real-time matrix square root finding

In this paper, a finite-time convergent Zhang neural network (ZNN) is proposed and studied for matrix square root finding. Compared to the original ZNN (OZNN) model, the finite-time convergent ZNN (FTCZNN) model fully utilizes a nonlinearly activated sign-bi-power function, and thus possesses faster convergence ability. In addition, the upper bound of convergence time for the FTCZNN model is theoretically derived and estimated by solving differential inequalities. Simulative comparisons are further conducted between the OZNN model and the FTCZNN model under the same conditions. The results validate the effectiveness and superiority of the FTCZNN model for matrix square root finding.

Two finite-time convergent Zhang neural network models for time-varying complex matrix Drazin inverse

This paper concerns the computation of the Drazin inverse of a complex time-varying matrix. Based on two Zhang functions constructed from two limit representations of the Drazin inverse, we present two complex Zhang neural network (ZNN) models with the Li activation function for computing the Drazin inverse of a complex time-varying square matrix. We prove that our ZNN models globally converge in finite time. In addition, upper bounds of the convergence time are derived analytically via the Lyapunov theory. Our simulation results verify the theoretical analysis and demonstrate the superiority of our ZNN models over the gradient-based GNN models.

ZNN models for computing matrix inverse based on hyperpower iterative methods

Our goal is to investigate and exploit an analogy between the scaled hyperpower family (SHPI family) of iterative methods for computing the matrix inverse and the discretization of Zhang Neural Network (ZNN) models. A class of ZNN models corresponding to the family of hyperpower iterative methods for computing generalized inverses is defined on the basis of the discovered analogy. The Simulink implementation in Matlab of the introduced ZNN models is described in the case of scaled hyperpower methods of the order 2 and 3. Convergence properties of the proposed ZNN models are investigated as well as their numerical behavior.

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

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 superiorness of the proposed FTZNN model for online time-varying matrix inversion, as compared with the GNN model and the original ZNN model.

Complex Neural Network Models for Time-Varying Drazin Inverse.

Two complex Zhang neural network (ZNN) models for computing the Drazin inverse of arbitrary time-varying complex square matrix are presented. The design of these neural networks is based on corresponding matrix-valued error functions arising from the limit representations of the Drazin inverse. Two types of activation functions, appropriate for handling complex matrices, are exploited to develop each of these networks. Theoretical results of convergence analysis are presented to show the desirable properties of the proposed complex-valued ZNN models. Numerical results further demonstrate the effectiveness of the proposed models.

A convergence-accelerated Zhang neural network and its solution application to Lyapunov equation

Lyapunov equation is widely encountered in scientific and engineering fields, and especially used in the control community to analyze the stability of a control system. In this paper, a convergence-accelerated Zhang neural network (CAZNN) is proposed and investigated for solving online Lyapunov equation. Different from the conventional gradient neural network (GNN) and the original Zhang neural network (ZNN), the proposed CAZNN model adopts a sign-bi-power activation function, and thus possesses the best convergence performance. Furthermore, we prove that the CAZNN model can converge to the theoretical solution of Lyapunov equation within finite time, instead of converging exponentially with time. Simulative results also verify the effectiveness and superiority of the CAZNN model for solving online Lyapunov equation, as compared with the GNN model and the ZNN model.

Performance analyses of recurrent neural network models exploited for online time-varying nonlinear optimization

In this paper, a special recurrent neural network (RNN), i.e., the Zhang neural network (ZNN), is presented and investigated for online time-varying nonlinear optimization (OTVNO). Compared with the research work done previously by others, this paper analyzes continuous-time and discrete-time ZNN models theoretically via rigorous proof. Theoretical results show that the residual errors of the continuous-time ZNN model possesses a global exponential convergence property and that the maximal steady-state residual errors of any method designed intrinsically for solving the static optimization problem and employed for the online solution of OTVNO is O(τ), where τ denotes the sampling gap. In the presence of noises, the residual errors of the continuous-time ZNN model can be arbitrarily small for constant noises and random noises. Moreover, an optimal sampling gap formula is proposed for discrete-time ZNN model in the noisy environments. Finally, computer-simulation results further substantiate the performance analyses of ZNN models exploited for online time-varying nonlinear optimization.

Taylor O(h³) Discretization of ZNN Models for Dynamic Equality-Constrained Quadratic Programming With Application to Manipulators.

In this paper, a new Taylor-type numerical differentiation formula is first presented to discretize the continuous-time Zhang neural network (ZNN), and obtain higher computational accuracy. Based on the Taylor-type formula, two Taylor-type discrete-time ZNN models (termed Taylor-type discrete-time ZNNK and Taylor-type discrete-time ZNNU models) are then proposed and discussed to perform online dynamic equality-constrained quadratic programming. For comparison, Euler-type discrete-time ZNN models (called Euler-type discrete-time ZNNK and Euler-type discrete-time ZNNU models) and Newton iteration, with interesting links being found, are also presented. It is proved herein that the steady-state residual errors of the proposed Taylor-type discrete-time ZNN models, Euler-type discrete-time ZNN models, and Newton iteration have the patterns of O(h(3)), O(h(2)), and O(h), respectively, with h denoting the sampling gap. Numerical experiments, including the application examples, are carried out, of which the results further substantiate the theoretical findings and the efficacy of Taylor-type discrete-time ZNN models. Finally, the comparisons with Taylor-type discrete-time derivative model and other Lagrange-type discrete-time ZNN models for dynamic equality-constrained quadratic programming substantiate the superiority of the proposed Taylor-type discrete-time ZNN models once again.

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).