Finite-time Zhang Neural Network Research Articles

A Novel Zeroing Neural Network for Solving Time-Varying Quadratic Matrix Equations against Linear Noises

The solving of quadratic matrix equations is a fundamental issue which essentially exists in the optimal control domain. However, noises exerted on the coefficients of quadratic matrix equations may affect the accuracy of the solutions. In order to solve the time-varying quadratic matrix equation problem under linear noise, a new error-processing design formula is proposed, and a resultant novel zeroing neural network model is developed. The new design formula incorporates a second-order error-processing manner, and the double-integration-enhanced zeroing neural network (DIEZNN) model is further proposed for solving time-varying quadratic matrix equations subject to linear noises. Compared with the original zeroing neural network (OZNN) model, finite-time zeroing neural network (FTZNN) model and integration-enhanced zeroing neural network (IEZNN) model, the DIEZNN model shows the superiority of its solution under linear noise; that is, when solving the problem of a time-varying quadratic matrix equation in the environment of linear noise, the residual error of the existing model will maintain a large level due to the influence of linear noise, which will eventually lead to the solution’s failure. The newly proposed DIEZNN model can guarantee a normal solution to the time-varying quadratic matrix equation task no matter how much linear noise there is. In addition, the theoretical analysis proves that the neural state of the DIEZNN model can converge to the theoretical solution even under linear noise. The computer simulation results further substantiate the superiority of the DIEZNN model in solving time-varying quadratic matrix equations under linear noise.

Performance Analysis and Applications of Finite-Time ZNN Models With Constant/Fuzzy Parameters for TVQPEI.

Based on extensive applications of the time-variant quadratic programming with equality and inequality constraints (TVQPEI) problem and the effectiveness of the zeroing neural network (ZNN) to address time-variant problems, this article proposes a novel finite-time ZNN (FT-ZNN) model with a combined activation function, aimed at providing a superior efficient neurodynamic method to solve the TVQPEI problem. The remarkable properties of the FT-ZNN model are faster finite-time convergence and preferable robustness, which are analyzed in detail, where in the case of the robustness discussion, two kinds of noises (i.e., bounded constant noise and bounded time-variant noise) are taken into account. Moreover, the proposed several theorems all compute the convergent time of the nondisturbed FT-ZNN model and the disturbed FT-ZNN model approaching to the upper bound of residual error. Besides, to enhance the performance of the FT-ZNN model, a fuzzy finite-time ZNN (FFT-ZNN), which possesses a fuzzy parameter, is further presented for solving the TVQPEI problem. A simulative example about the FT-ZNN and FFT-ZNN models solving the TVQPEI problem is given, and the experimental results expectably conform to the theoretical analysis. In addition, the designed FT-ZNN model is effectually applied to the repetitive motion of the three-link redundant robot and image fusion to show its potential practical value.

A parameter-changing zeroing neural network for solving linear equations with superior fixed-time convergence

Linear equations (LEs) play an essential role in mathematics. Nevertheless, the vast majority of fixed-parameter zeroing neural network (ZNN) models are not fast enough to solve LEs. So, a novel parameter-changing ZNN (PCZNN) is proposed and studied in the quest for resolving LEs more efficiently. In contrast to the conventional ZNN and finite-time ZNN (FTZNN), the tuning parameter of the presented PCZNN is time-varying, resulting in three advantages of the PCZNN model: (1) less time for parameter adjustment, (2) faster convergence rate, and (3) less conservative upper bound on the convergence time. Furthermore, the upper bound of convergence time is derived through analysis, which is independent of its initial value. Simulation comparisons conclude that the PCZNN model is more efficient than the ZNN and FTZNN models in solving LEs. Finally, the synchronization control experiments of chaotic systems by two ZNN models with time-varying parameters are offered to verify the superiority of the PCZNN model.

A family of varying-parameter finite-time zeroing neural networks for solving time-varying Sylvester equation and its application

A family of varying-parameter finite-time zeroing neural networks (VPFTZNN) is introduced for solving the time-varying Sylvester equation (TVSE). The convergence speed of the proposed VPFTZNN family is analysed and compared with the traditional zeroing neural network (ZNN) and the finite-time zeroing neural network (FTZNN). The behaviour of the proposed neural networks under various activation functions is proved theoretically and verified experimentally. In addition, the stability and noise resistance of the proposed VPFTZNN family are discussed. Further, the proposed VPFTZNN models are applied in the computation of current flows in an electrical network.

Solving the time-varying tensor square root equation by varying-parameters finite-time Zhang neural network

We study the time-varying (TV) tensor square root problem, which could be regarded as a generalization of both the time-invariant (TI) tensor square root and the TV/TI matrix square root problems. The existence and uniqueness of the solution to the TV tensor square root problem are discussed. A new general varying-parameters finite-time convergent Zhang neural network model (VPsFTZNN) is proposed. We use the original ZNN and the new VPsFTZNN model as software solutions to the TV tensor square root problem. The convergence results of ZNN and VPsFTZNN dynamical systems are given. We prove that the VPsFTZNN model converges in a finite-time which is shorter than the finite time convergence of existing finite-time ZNN models. Also, we show that a superior convergence can be achieved if we use the power-sigmoid or smooth power-sigmoid activation functions instead of linear activation function under certain conditions. For comparison, some other models are also introduced and used to compute the square root of TV tensors. Numerical examples for TI and TV tensor/matrix square root problems are elaborated to confirm the theoretical results and comparison results with various model further illustrate the reliability and superiority of the proposed VPsFTZNN dynamics. Representative example on the application of this model for solving TV linear equations is also given.

Simulation of Varying Parameter Recurrent Neural Network with application to matrix inversion

A class of adaptive recurrent neural networks (RNN) for computing the inverse of a time-varying matrix with accelerated convergence time is defined and considered. The proposed neural dynamic model involves an exponential gain time-varying term in the nonlinear activation of the finite-time Zhang neural network (FTZNN) dynamical equation. Individual models belonging to the proposed class are defined by means of corresponding error functions. It is shown theoretically and experimentally that usage of the exponential nonlinear activation accelerates the convergence rate of the error function compared to previous dynamical systems for solving the time-varying (TV) and time-invariant (TI) matrix inversion.

Improved finite‐time zeroing neural network for time‐varying division

AbstractA novel complex varying‐parameter finite‐time zeroing neural network (VPFTZNN) for finding a solution to the time‐dependent division problem is introduced. A comparative study in relation to the zeroing neural network (ZNN) and finite‐time zeroing neural network (FTZNN) is established in terms of the error function and the convergence speed. The error graphs of the VPFTZNN design show promising results and perform better than corresponding ZNN and FTZNN graphs. The proposed dynamical systems are suitable tools for overcoming the division by zero difficulty, which appears in the time‐varying division. An application of the introduced VPFTZNN model in an output tracking control time‐varying linear system is demonstrated.

A New Varying-Parameter Design Formula for Solving Time-Varying Problems

A novel finite-time convergent zeroing neural network (ZNN) based on varying gain parameter for solving time-varying (TV) problems is presented. The model is based on the improvement and generalization of the finite-time ZNN (FTZNN) dynamics by means of the varying-parameter ZNN (VPZNN) dynamics, and termed as VPFTZNN. More precisely, the proposed model replaces fixed and large values of the scaling parameter by an appropriate time-dependent gain parameter, which leads to a faster and bounded convergence of the error function in comparison to previous ZNN methods. The convergence properties of the proposed VPFTZNN dynamical evolution in its generic form is verified. Particularly, VPFTZNN for solving linear matrix equations and for computing generalized inverses are investigated theoretically and numerically. Moreover, the proposed design is applicable in solving the TV matrix inversion problem, solving the Lyapunov and Sylvester equation as well as in approximating the matrix square root. Theoretical analysis as well as simulation results validate the effectiveness of the introduced dynamical evolution. The main advantages of the proposed VPFTZNN dynamics are their generality and faster finite-time convergence with respect to FTZNN models.

Performance analysis of nonlinear activated zeroing neural networks for time-varying matrix pseudoinversion with application

By exploiting two simplified nonlinear activation functions, two zeroing neural network (ZNN) models are designed and studied to efficiently tackle the time-varying matrix pseudoinversion problem. Compared with ZNN activated by previously presented activation functions, these two simplified finite-time ZNN (SFTZNN) models (called SFTZNN1 and SFTZNN2) not only achieve faster finite-time convergence, but also possess better robustness. In addition, the SFTZNN1 and SFTZNN2 models have simpler structure compared with the widely used sign-bi-power activated ZNN model. Theoretical analysis is presented to obtain the maximum convergence time for the SFTZNN models in ideal conditions. Besides, when external perturbations are injected into the proposed SFTZNN models, upper bounds of the steady-state residual error are theoretically calculated. Comparative simulations and one engineering application case validate the feasibility and superiority of the two new SFTZNN models when solving time-varying matrix pseudoinversion.

New error function designs for finite-time ZNN models with application to dynamic matrix inversion

The zeroing neural network (ZNN), as a special kind of recurrent neural network (RNN), is often utilized to solve dynamic matrix inversion problems in the many fields recently. In this work, two finite-time ZNN (termed as ZNN-A and ZNN-B) models with the sign-bi-power (SBP) activation function are proposed by designing two novel error functions. The theoretical analysis shows the superior stability and finite-time convergence properties of the ZNN-A and ZNN-B models. Furthermore, three simulative examples show the effectiveness of the proposed ZNN-A and ZNN-B models for finding dynamic matrix inversion and the correctness of the corresponding theorems. To reveal the superior performance of the proposed ZNN-A and ZNN-B models with SBP activation function, the standard ZNN model with the linear activation function is comparatively applied in experiments.

A Finite‐Time Recurrent Neural Network for Computing Quadratic Minimization with Time‐Varying Coefficients

This paper proposes a Finite-time Zhang neural network (FTZNN) to solve time-varying quadratic minimization problems. Different from the original Zhang neural network (ZNN) that is specially designed to solve time-varying problems and possesses an exponential convergence property, the proposed neural network exploits a sign-bi-power activation function so that it can achieve the finite-time convergence. In addition, the upper bound of the finite convergence time for the FTZNN model is analytically estimated in theory. For comparative purposes, the original ZNN model is also presented to solve time-varying quadratic minimization problems. Numerical experiments are performed to evaluate and compare the performance of the original ZNN model and the FTZNN model. The results demonstrate that the FTZNN model is a more effective solution model for solving time-varying quadratic minimization problems.

A New Varying-Parameter Convergent-Differential Neural-Network for Solving Time-Varying Convex QP Problem Constrained by Linear-Equality

To solve online continuous time-varying convex quadratic-programming problems constrained by a time-varying linear-equality, a novel varying-parameter convergent-differential neural network (termed as VP-CDNN) is proposed and analyzed. Different from fixed-parameter convergent-differential neural network (FP-CDNN), such as the gradient-based recurrent neural network, the classic Zhang neural network (ZNN), and the finite-time ZNN (FT-ZNN), VP-CDNN is based on monotonically increasing time-varying design-parameters. Theoretical analysis proves that VP-CDNN has super exponential convergence and the residual errors of VP-CDNN converge to zero even under perturbation situations, which are both better than traditional FP-CDNN and FT-ZNN. Computer simulations based on different activation functions are illustrated to verify the super exponential convergence performance and strong robustness characteristics of the proposed VP-CDNN. A robot tracking example is finally presented to verify the effectiveness and availability of the proposed VP-CDNN.

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.

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.

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 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.