Zeroing Neural Network Research Articles

Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation

Complex time-dependent Lyapunov equation (CTDLE), as an important means of stability analysis of control systems, has been extensively employed in mathematics and engineering application fields. Recursive neural networks (RNNs) have been reported as an effective method for solving CTDLE. In the previous work, zeroing neural networks (ZNNs) have been established to find the accurate solution of time-dependent Lyapunov equation (TDLE) in the noise-free conditions. However, noises are inevitable in the actual implementation process. In order to suppress the interference of various noises in practical applications, in this paper, a complex noise-resistant ZNN (CNRZNN) model is proposed and employed for the CTDLE solution. Additionally, the convergence and robustness of the CNRZNN model are analyzed and proved theoretically. For verification and comparison, three experiments and the existing noise-tolerant ZNN (NTZNN) model are introduced to investigate the effectiveness, convergence and robustness of the CNRZNN model. Compared with the NTZNN model, the CNRZNN model has more generality and stronger robustness. Specifically, the NTZNN model is a special form of the CNRZNN model, and the residual error of CNRZNN can converge rapidly and stably to order 10−5 when solving CTDLE under complex linear noises, which is much lower than order 10−1 of the NTZNN model. Analogously, under complex quadratic noises, the residual error of the CNRZNN model can converge to 2∥A∥F/ζ3 quickly and stably, while the residual error of the NTZNN model is divergent.

A Multi-Joint Continuous Motion Estimation Method of Lower Limb Using Least Squares Support Vector Machine and Zeroing Neural Network based on sEMG signals

A Multi-Joint Continuous Motion Estimation Method of Lower Limb Using Least Squares Support Vector Machine and Zeroing Neural Network based on sEMG signals

On Generalized Zeroing Neural Network Under Discrete and Distributed Time Delays and Its Application to Dynamic Lyapunov Equation

Zeroing neural network (ZNN), an effective method for tracking solutions of dynamic equations, has been developed and improved by various strategies, typically the application of nonlinear activation functions (AFs) and varying parameters (VPs). Unlike VPs, AFs applied in ZNN models act directly on real-time error. The processing unit of v needs to obtain neural state in real time. In the implementation process, highly nonlinear AFs become an important cause of time delays, which eventually leads to instability and oscillation. However, most studies focus on exploring new theoretically valid AFs to improve performance of ZNNs, while ignoring the adverse effects of highly nonlinear AFs. The nonlinearity of AFs requires us fully consider time-delay tolerance of ZNNs using nonlinear AFs, so as to ensure that the model is not unstable even when disturbed by time delays. In this work, delay-perturbed generalized ZNN (DP-GZNN) is proposed to investigate time-delay tolerance of generalized ZNN (G-ZNN) in solving dynamic Lyapunov equation. Considering the nonlinearity of AFs, two delay terms are elegantly added to G-ZNN and DP-GZNN is then derived. After rigorous mathematical derivations, sufficient conditions in a linear matrix inequality (LMI) manner are presented for global convergence of DP-GZNN. Through rich numerical experiments, hyperparameters involved in the analysis process are discussed in detail. Comparative simulations are also conducted to compare the ability of different ZNN models to resist time delays. It is worth to mention that this is the first time to consider the ability of G-ZNN to resist discrete and distributed time delays.

6-Step Discrete ZNN Model for Repetitive Motion Control of Redundant Manipulator

In this article, the repetitive motion control of redundant manipulators is investigated. First, a repetitive motion control scheme is presented, and a continuous zeroing neural network (CZNN) model is obtained for solving the scheme. Meanwhile, the development of a discrete zeroing neural network (DZNN) model is desired for convenient computational processing. Based on this, this article proposes a 6-step discretization formula, which has high precision. By using the 6-step discretization formula and the 4-step backward difference formula, a 6-step DZNN (6SDZNN) model is further proposed to handle the repetitive motion control scheme. Theoretical analyses verify the efficacy of the 6SDZNN model. Additionally, some discrete forms of conventional models are developed for comparison. Computer simulations on the basis of the 4-link redundant manipulator are carried out, verifying the theoretical analyses and showing the efficacy of the 6SDZNN model. Finally, physical experiments on the basis of the Kinova Jaco <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> manipulator substantiate the practicability of the 6SDZNN model.

Solving Complex-Valued Time-Varying Linear Matrix Equations via QR Decomposition With Applications to Robotic Motion Tracking and on Angle-of-Arrival Localization.

The problem of solving linear equations is considered as one of the fundamental problems commonly encountered in science and engineering. In this article, the complex-valued time-varying linear matrix equation (CVTV-LME) problem is investigated. Then, by employing a complex-valued, time-varying QR (CVTVQR) decomposition, the zeroing neural network (ZNN) method, equivalent transformations, Kronecker product, and vectorization techniques, we propose and study a CVTVQR decomposition-based linear matrix equation (CVTVQR-LME) model. In addition to the usage of the QR decomposition, the further advantage of the CVTVQR-LME model is reflected in the fact that it can handle a linear system with square or rectangular coefficient matrix in both the matrix and vector cases. Its efficacy in solving the CVTV-LME problems have been tested in a variety of numerical simulations as well as in two applications, one in robotic motion tracking and the other in angle-of-arrival localization.

7-Instant Discrete-Time Synthesis Model Solving Future Different-Level Linear Matrix System via Equivalency of Zeroing Neural Network

Differing from the common linear matrix equation, the future different-level linear matrix system is considered, which is much more interesting and challenging. Because of its complicated structure and future-computation characteristic, traditional methods for static and same-level systems may not be effective on this occasion. For solving this difficult future different-level linear matrix system, the continuous different-level linear matrix system is first considered. On the basis of the zeroing neural network (ZNN), the physical mathematical equivalency is thus proposed, which is called ZNN equivalency (ZE), and it is compared with the traditional concept of mathematical equivalence. Then, on the basis of ZE, the continuous-time synthesis (CTS) model is further developed. To satisfy the future-computation requirement of the future different-level linear matrix system, the 7-instant discrete-time synthesis (DTS) model is further attained by utilizing the high-precision 7-instant Zhang et al. discretization (ZeaD) formula. For a comparison, three different DTS models using three conventional ZeaD formulas are also presented. Meanwhile, the efficacy of the 7-instant DTS model is testified by the theoretical analyses. Finally, experimental results verify the brilliant performance of the 7-instant DTS model in solving the future different-level linear matrix system.

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 universal noise-suppressing neural algorithm framework aided with nonconvex activation function for time-varying quadratic programming problems

Time-varying quadratic programming problems (TVQPPs) subject to equality and inequality constraints with different measurement noises often arise in the fields of scientific and economic research. Noises are always ubiquitous and unavoidable in practical problems. However, most existing methods usually assume that the computing process is free of measurement noise or the denoising has been monitored before the calculation. The zeroing neural network (ZNN), a significant neurodynamic approach, has presented potent abilities to compute a great variety of time-varying zeroing problems with odd monotonically increasing activation functions, but the existing results on ZNN cannot deal with the inequality constraints in the TVQPPs. In this article, a nonconvex function activated implicit ZNN (NIZNN) model and a noise-suppressing implicit ZNN (NNSIZNN) model, which is also known as the ZNN-based model, are proposed by the inspiration of the conventional ZNN model from a control-based framework. It guarantees an effective solution for TVQPPs in the presence of different noises. The ZNN-based model allows nonconvex sets for projection operation in activation functions and incorporates noise-tolerant techniques for handling different noises arising in TVQPPs. Besides, theoretical analyses show that the ZNN-based model globally converges to the time-varying optimization solution of TVQPPs in the presence of noises. In addition, an illustrative example is provided to substantiate the efficiency and robustness of the ZNN-based model for online solving TVQPPs with inherent tolerance to noises. Moreover, the ZNN-based model is applied to the economic model, which provides and investigates their computational efficiency and superiority. Finally, an application example to repetitive motion generation of four-wheel omnidirectional mobile robot is simulated to substantiate the feasibility and superiority of the developed NNSIZNN model for online solving TVQPPs with nonconvex constraints and measurement noise.

Double Features Zeroing Neural Network Model for Solving the Pseudoninverse of a Complex-Valued Time-Varying Matrix

The solution of a complex-valued matrix pseudoinverse is one of the key steps in various science and engineering fields. Owing to its important roles, researchers had put forward many related algorithms. With the development of research, a time-varying matrix pseudoinverse received more attention than a time-invarying one, as we know that a zeroing neural network (ZNN) is an efficient method to calculate a pseudoinverse of a complex-valued time-varying matrix. Due to the initial ZNN (IZNN) and its extensions lacking a mechanism to deal with both convergence and robustness, that is, most existing research on ZNN models only studied the convergence and robustness, respectively. In order to simultaneously improve the double features (i.e., convergence and robustness) of ZNN in solving a complex-valued time-varying pseudoinverse, this paper puts forward a double features ZNN (DFZNN) model by adopting a specially designed time-varying parameter and a novel nonlinear activation function. Moreover, two nonlinear activation types of complex number are investigated. The global convergence, predefined time convergence, and robustness are proven in theory, and the upper bound of the predefined convergence time is formulated exactly. The results of the numerical simulation verify the theoretical proof, in contrast to the existing complex-valued ZNN models, the DFZNN model has shorter predefined convergence time in a zero noise state, and enhances robustness in different noise states. Both the theoretical and the empirical results show that the DFZNN model has better ability in solving a time-varying complex-valued matrix pseudoinverse. Finally, the proposed DFZNN model is used to track the trajectory of a manipulator, which further verifies the reliability of the model.

Harmonic Noise-Tolerant ZNN for Dynamic Matrix Pseudoinversion and Its Application to Robot Manipulator.

As we know, harmonic noises widely exist in industrial fields and have a crucial impact on the computational accuracy of the zeroing neural network (ZNN) model. For tackling this issue, by combining the dynamics of harmonic signals, two harmonic noise-tolerant ZNN (HNTZNN) models are designed for the dynamic matrix pseudoinversion. In the design of HNTZNN models, an adaptive compensation term is adopted to eliminate the influence of harmonic noises, and a Li activation function is introduced to further improve the convergence rate. The convergence and robustness to harmonic noises of the proposed HNTZNN models are proved through theoretical analyses. Besides, compared with the ZNN model without adaptive compensation term, the HNTZNN models are more effective for tacking the problem of dynamic matrix pseudoinverse under harmonic noises environments. Moreover, HNTZNN models are further applied to the kinematic control of a four-link planar robot manipulator under harmonic noises. In general, the experimental results verify the effectiveness, superiority, and broad application prospect of the models.

Zeroing Neural Network Approaches Based on Direct and Indirect Methods for Solving the Yang–Baxter-like Matrix Equation

This research introduces three novel zeroing neural network (ZNN) models for addressing the time-varying Yang–Baxter-like matrix equation (TV-YBLME) with arbitrary (regular or singular) real time-varying (TV) input matrices in continuous time. One ZNN dynamic utilizes error matrices directly arising from the equation involved in the TV-YBLME. Moreover, two ZNN models are proposed using basic properties of the YBLME, such as the splitting of the YBLME and sufficient conditions for a matrix to solve the YBLME. The Tikhonov regularization principle enables addressing the TV-YBLME with an arbitrary input real TV matrix. Numerical experiments, including nonsingular and singular TV input matrices, show that the suggested models deal effectively with the TV-YBLME.

Unique non-negative definite solution of the time-varying algebraic Riccati equations with applications to stabilization of LTV systems

In the context of infinite-horizon optimal control problems, the algebraic Riccati equations (ARE) arise when the stability of linear time-varying (LTV) systems is investigated. Using the zeroing neural network (ZNN) approach to solve the time-varying eigendecomposition-based ARE (TVE-ARE) problem, the ZNN model (ZNNTVE-ARE) for solving the TVE-ARE problem is introduced as a result of this research. Since the eigendecomposition approach is employed, the ZNNTVE-ARE model is designed to produce only the unique nonnegative definite solution of the time-varying ARE (TV-ARE) problem. It is worth mentioning that this model follows the principles of the ZNN method, which converges exponentially with time to a theoretical time-varying solution. The ZNNTVE-ARE model can also produce the eigenvector solution of the continuous-time Lyapunov equation (CLE) since the Lyapunov equation is a particular case of ARE. Moreover, this paper introduces a hybrid ZNN model for stabilizing LTV systems in which the ZNNTVE-ARE model is employed to solve the continuous-time ARE (CARE) related to the optimal control law. Experiments show that the ZNNTVE-ARE and HFTZNN-LTVSS models are both effective, and that the HFTZNN-LTVSS model always provides slightly better asymptotic stability than the models from which it is derived.

An Arctan-Type Varying-Parameter ZNN for Solving Time-Varying Complex Sylvester Equations in Finite Time

Zeroing neural network (ZNN) is an effective neural solution to time-varying problems, including time-varying complex Sylvester equations. Generally, a ZNN model involves a convergence design parameter (CDP) that influences its convergence rate. In traditional fixed-parameter ZNNs (FP-ZNNs), the CDPs are set to be constant, which is not realistic since the CDPs are actually time-varying in practical hardware environments. By considering this fact, varying-parameter ZNNs (VP-ZNNs) with time-varying CDPs have been researched in the literature. Although these VP-ZNNs have been demonstrated to deliver superior convergence as compared with FP-ZNNs, they have one drawback, that is, their CDPs usually keep increasing with time, meaning that the CDPs tend to be infinity large with time progresses. Evidently, infinity large CDPs are unacceptable in practice. Moreover, computing resources will be wasted by growing the CDPs with time after the VP-ZNNs become convergent. To tackle the above issues, this article, for the first time, proposes an arctan-type VP-ZNN (ATVP-ZNN) with finite-time convergence for solving time-varying complex Sylvester equations. The ATVP-ZNN is able to adjust its CDPs that finally converge to be constant when the ATVP-ZNN becomes convergent in finite time. In theory, the finite-time convergence of the ATVP-ZNN and the upper bound of the CDPs are mathematically analyzed. Numerical studies are comparatively performed with the superior convergence of the ATVP-ZNN substantiated.

ZNN for time-variant nonlinear inequality systems: A finite-time solution

Since the solution of time-variant nonlinear inequality systems is trapped by the convergence performance of the models, this paper explores an enhanced nonlinear sign-bi-power activation function (AF) and further obtains a zeroing neural network (ZNN) model for solving time-variant nonlinear inequality systems, which is called nonlinear activated finite-time convergent zeroing neural network (NAFTCZNN). The strict theoretical analysis together with two theorems are given to demonstrate the enhanced convergence performance of the NAFTCZNN model. Furthermore, the stability and upper bound of convergent time of the NAFTCZNN model are analyzed and estimated in the theorems, which is more stable and less conservative than the ZNN models using the common sign-bi-power AFs. Numerical example results further validate the effectiveness and excellence of the NAFTCZNN model in terms of solving the time-variant nonlinear inequality systems.

Prescribed-Time Convergent Adaptive ZNN for Time-Varying Matrix Inversion under Harmonic Noise

Harmonic noises widely exist in industrial fields and always affect the computational accuracy of neural network models. The existing original adaptive zeroing neural network (OAZNN) model can effectively suppress harmonic noises. Nevertheless, the OAZNN model’s convergence rate only stays at the exponential convergence, that is, its convergence speed is usually greatly affected by the initial state. Consequently, to tackle the above issue, this work combines the dynamic characteristics of harmonic signals with prescribed-time convergence activation function, and proposes a prescribed-time convergent adaptive ZNN (PTCAZNN) for solving time-varying matrix inverse problem (TVMIP) under harmonic noises. Owing to the nonlinear activation function used having the ability to reject noises itself and the adaptive term also being able to compensate the influence of noises, the PTCAZNN model can realize double noise suppression. More importantly, the theoretical analysis of PTCAZNN model with prescribed-time convergence and robustness performance is provided. Finally, by varying a series of conditions such as the frequency of single harmonic noise, the frequency of multi-harmonic noise, and the initial value and the dimension of the matrix, the comparative simulation results further confirm the effectiveness and superiority of the PTCAZNN model.

Finite-time convergent zeroing neural network for solving time-varying algebraic Riccati equations

Various forms of the algebraic Riccati equation (ARE) have been widely used to investigate the stability of nonlinear systems in the control field. In this paper, the time-varying ARE (TV-ARE) and linear time-varying (LTV) systems stabilization problems are investigated by employing the zeroing neural networks (ZNNs). In order to solve the TV-ARE problem, two models are developed, the ZNNTV-ARE model which follows the principles of the original ZNN method, and the FTZNNTV-ARE model which follows the finite-time ZNN (FTZNN) dynamical evolution. In addition, two hybrid ZNN models are proposed for the LTV systems stabilization, which combines the ZNNTV-ARE and FTZNNTV-ARE design rules. Note that instead of the infinite exponential convergence specific to the ZNNTV-ARE design, the structure of the proposed FTZNNTV-ARE dynamic is based on a new evolution formula which is able to converge to a theoretical solution in finite time. Furthermore, we are only interested in real symmetric solutions of TV-ARE, so the ZNNTV-ARE and FTZNNTV-ARE models are designed to produce such solutions. Numerical findings, one of which includes an application to LTV systems stabilization, confirm the effectiveness of the introduced dynamical evolutions.

Concise Discrete ZNN Controllers for End-Effector Tracking and Obstacle Avoidance of Redundant Manipulators

Obstacle avoidance is usually an additional task for a redundant manipulator when the end-effector tracking task is performed, which guarantees the safety of the redundant manipulator. By formulating and combining the end-effector tracking task and the obstacle avoidance task using zeroing neural network (ZNN), in this article, a concise continuous ZNN (CZNN) controller is first proposed. To develop discrete controllers for practical control, a second-order discrete formula and a third-order discrete formula are introduced. Therefore, by utilizing two discrete formulas to discretize the CZNN controller, two concise discrete ZNN (DZNN) controllers are proposed. Detailed theoretical analyses guarantee the effectiveness of task formulations, CZNN controller, and DZNN controllers. In addition, some comparisons with existing studies are presented in details. Furthermore, three groups of simulative experiments on the basis of UR5 manipulator and two groups of physical experiments on the basis of Kinova manipulator are conducted to illustrate the effectiveness, superiority, and practicability of two DZNN controllers.

Continuous and discrete zeroing neural network for a class of multilayer dynamic system

Multilayer dynamic system is widely used in industry and other fields. Different from common systems, multilayer dynamic system has complex structure leading to challenges for research. In this paper, we study zeroing neural network(ZNN) models for a class of multilayer dynamic system(MLDS). In the case of continuous time, continuous ZNN models for continuous MLDS (MLDS-linear-ZNN, MLDS-nonlinear-ZNN and MLDS-noise-tolerant-ZNN) are proposed based on ZNN design method with theoretical analysis. In the discrete case concurrently, discrete ZNN models (MLDS-linear-mDZNN, MLDS-nonlinear-mDZNN and MLDS-noise-tolerant-mDZNN) with m-step ZeaD formula, a new Zhang et al. discretization formula presented in previous paper, are put forward and corresponding discrete algorithms are obtained. Finally, numerical experiments are carried out to verify the superiority and maneuverability of ZNN models for MLDS proposed in this paper.

Zeroing Neural Network for Pseudoinversion of an Arbitrary Time-Varying Matrix Based on Singular Value Decomposition

Many researchers have investigated the time-varying (TV) matrix pseudoinverse problem in recent years, for its importance in addressing TV problems in science and engineering. In this paper, the problem of calculating the inverse or pseudoinverse of an arbitrary TV real matrix is considered and addressed using the singular value decomposition (SVD) and the zeroing neural network (ZNN) approaches. Since SVD is frequently used to compute the inverse or pseudoinverse of a matrix, this research proposes a new ZNN model based on the SVD method as well as the technique of Tikhonov regularization, for solving the problem in continuous time. Numerical experiments, involving the pseudoinversion of square, rectangular, singular, and nonsingular input matrices, indicate that the proposed models are effective for solving the problem of the inversion or pseudoinversion of time varying matrices.

A higher-order zeroing neural network for pseudoinversion of an arbitrary time-varying matrix with applications to mobile object localization

The hyperpower family of iterative methods with arbitrary convergence order is one of the most used methods for estimating matrix inverses and generalized inverses, whereas the zeroing neural network (ZNN) is a type of neural dynamics developed to solve time-varying problems in science and engineering. Since the discretization of ZNN dynamics leads to the Newton iterative method for solving the matrix inversion and generalized inversion, this study proposes and investigates a family of ZNN dynamical models known as higher-order ZNN (HOZNN) models, which are defined on the basis of correlation with hyperpower iterations of arbitrary order. Because the HOZNN dynamical system requires error function powers, it is only applicable to square error functions. In this paper, we extend the original HOZNN dynamic flows to arbitrary time-dependent real matrices, both square and rectangular, and sign-bi-power activation is used to investigate the finite-time convergence of arbitrary order HOZNN dynamics. The proposed models are theoretically and numerically tested under three activation functions, and an application in solving the angle-of-arrival (AoA) localization problem demonstrates the effectiveness of the proposed design.