Zeroing Neural Network Research Articles

A novel adaptive parameter zeroing neural network for the synchronization of complex chaotic systems and its field programmable gate array implementation

This article proposes a novel adaptive-parameter zeroing neural network (NAP-ZNN) to control the synchronization of complex chaotic systems. Different from previous zeroing neural networks (ZNNs), the NAP-ZNN model incorporates adaptive-parameter, a unique blend of time-varying convergence factor and fuzzy control parameter, which is adjusted continuously according to the real-time synchronizing error, offering greater flexibility. The NAP-ZNN model aims to address the time-varying problem in complex-valued domain, which is scarcely explored, it rapidly impels two complex chaotic systems to synchronize within predetermined time and effectively mitigates noise interference. Moreover, the upper bound of the convergence time is theoretically calculated in noiseless environments, and the disturbance-resistant property is demonstrated under various noisy disturbances. The control performance of NAP-ZNN model has been separately compared to other general ZNNs (NAP-ZNN synchronizes at 0.097 s while three general ZNNs spend 0.43 s, 0.78 s and 0.84 s, and only NAP-ZNN complete synchronization within 0.1 s under noisy interferences while general ZNNs fail) and the traditional adaptive control method through two synchronization simulation (RMSE of NAP-ZNN under sinusoidal noise is 3.165 × 10-3 while traditional is 3.383 at t = 1 s). The numerical outcomes clearly illustrate the superior performance of the NAP-ZNN model in comparison to other synchronization strategies. The hardware implementation of the NAP-ZNN model has been successfully realized through the Field Programmable Gate Array (FPGA), and the chaos synchronization process has been vividly captured via an oscilloscope, further reinforcing the practicality and effectiveness of the NAP-ZNN model.

A Dynamic-Varying Parameter Enhanced ZNN Model for Solving Time-Varying Complex-Valued Tensor Inversion With Its Application to Image Encryption.

Time-varying complex-valued tensor inverse (TVCTI) is a public problem worthy of being studied, while numerical solutions for the TVCTI are not effective enough. This work aims to find the accurate solution to the TVCTI using zeroing neural network (ZNN), which is an effective tool in terms of solving time-varying problems and is improved in this article to solve the TVCTI problem for the first time. Based on the design idea of ZNN, an error-adaptive dynamic parameter and a new enhanced segmented signum exponential activation function (ESS-EAF) are first designed and applied to the ZNN. Then a dynamic-varying parameter-enhanced ZNN (DVPEZNN) model is proposed to solve the TVCTI problem. The convergence and robustness of the DVPEZNN model are theoretically analyzed and discussed. In order to highlight better convergence and robustness of the DVPEZNN model, it is compared with four varying-parameter ZNN models in the illustrative example. The results show that the DVPEZNN model has better convergence and robustness than the other four ZNN models in different situations. In addition, the state solution sequence generated by the DVPEZNN model in the process of solving the TVCTI cooperates with the chaotic system and deoxyribonucleic acid (DNA) coding rules to obtain the chaotic-ZNN-DNA (CZD) image encryption algorithm, which can encrypt and decrypt images with good performance.

A Dini-Derivative-Aided Zeroing Neural Network for Time-Variant Quadratic Programming Involving Multi-Type Constraints With Robotic Applications.

Time-variant quadratic programming (QP) with multi-type constraints including equality, inequality, and bound constraints is ubiquitous in practice. In the literature, there exist a few zeroing neural networks (ZNNs) that are applicable to time-variant QPs with multi-type constraints. These ZNN solvers involve continuous and differentiable elements for handling inequality and/or bound constraints, and they possess their own drawbacks such as the failure in solving problems, the approximated optimal solutions, and the boring and sometimes difficult process of tuning parameters. Differing from the existing ZNN solvers, this article aims to propose a novel ZNN solver for time-variant QPs with multi-type constraints based on a continuous but not differentiable projection operator that is deemed unsuitable for designing ZNN solvers in the community, due to the lack of the required time derivative information. To achieve the aforementioned aim, the upper right-hand Dini derivative of the projection operator with respect to its input is introduced to serve as a mode switcher, leading to a novel ZNN solver, termed Dini-derivative-aided ZNN (Dini-ZNN). In theory, the convergent optimal solution of the Dini-ZNN solver is rigorously analyzed and proved. Comparative validations are performed, verifying the effectiveness of the Dini-ZNN solver that has merits such as guaranteed capability to solve problems, high solution accuracy, and no extra hyperparameter to be tuned. To illustrate potential applications, the Dini-ZNN solver is successfully applied to kinematic control of a joint-constrained robot with simulation and experimentation conducted.

An FPGA-Implemented Antinoise Fuzzy Recurrent Neural Network for Motion Planning of Redundant Robot Manipulators.

When a robot completes end-effector tasks, internal error noises always exist. To resist internal error noises of robots, a novel fuzzy recurrent neural network (FRNN) is proposed, designed, and implemented on field-programmable gated array (FPGA). The implementation is pipeline-based, which guarantees the order of overall operations. The data processing is based on across-clock domain, which is beneficial for computing units' acceleration. Compared with traditional gradient-based neural networks (NNs) and zeroing neural networks (ZNNs), the proposed FRNN has faster convergence rate and higher correctness. Practical experiments on a 3 degree-of-freedom (DOs) planar robot manipulator show that the proposed fuzzy RNN coprocessor needs 496 lookup table random access memories (LUTRAMs), 205.5 block random access memories (BRAMs), 41384 lookup tables (LUTs), and 16743 flip-flops (FFs) of the Xilinx XCZU9EG chip.

A Fixed-Time Noise-Tolerance ZNN Model for Time-Variant Inequality-Constrained Quaternion Matrix Least-Squares Problem.

Presently, numerical algorithms for solving quaternion least-squares problems have been intensively studied and utilized in various disciplines. However, they are unsuitable for solving the corresponding time-variant problems, and thus few studies have explored the solution to the time-variant inequality-constrained quaternion matrix least-squares problem (TVIQLS). To do so, this article designs a fixed-time noise-tolerance zeroing neural network (FTNTZNN) model to determine the solution of the TVIQLS in a complex environment by exploiting the integral structure and the improved activation function (AF). The FTNTZNN model is immune to the effects of initial values and external noise, which is much superior to the conventional zeroing neural network (CZNN) models. Besides, detailed theoretical derivations about the global stability, the fixed-time (FXT) convergence, and the robustness of the FTNTZNN model are provided. Simulation results indicate that the FTNTZNN model has a shorter convergence time and superior robustness compared to other zeroing neural network (ZNN) models activated by ordinary AFs. At last, the construction method of the FTNTZNN model is successfully applied to the synchronization of Lorenz chaotic systems (LCSs), which shows the practical application value of the FTNTZNN model.

Zeroing neural network based on the equation AXA = A

According to available scientific research, there is no information that Zeroing Neural Network (ZNN) models for calculating the matrix inverse and generalized inverses have been studied on the basis of Penrose equations. Our intention is to present a new ZNN design which defined on the Penrose matrix equations and whose intention is to find the time-variant matrix inverse and pseudoinverse. We propose a novel Zeroing function (ZF) based on the first Penrose equation AXA=A. The initiated ZNN design for computing the time-varying inverse and the pseudoinverse is defined and investigated. An explicit form of the defined model is also proposed. The convergence properties of the proposed explicit dynamics are investigated in both the time-invariant and time-varying case. Illustrative simulation results are given in order to verify the obtained theoretical results.

A Survey on Biomimetic and Intelligent Algorithms with Applications.

The question "How does it work" has motivated many scientists. Through the study of natural phenomena and behaviors, many intelligence algorithms have been proposed to solve various optimization problems. This paper aims to offer an informative guide for researchers who are interested in tackling optimization problems with intelligence algorithms. First, a special neural network was comprehensively discussed, and it was called a zeroing neural network (ZNN). It is especially intended for solving time-varying optimization problems, including origin, basic principles, operation mechanism, model variants, and applications. This paper presents a new classification method based on the performance index of ZNNs. Then, two classic bio-inspired algorithms, a genetic algorithm and a particle swarm algorithm, are outlined as representatives, including their origin, design process, basic principles, and applications. Finally, to emphasize the applicability of intelligence algorithms, three practical domains are introduced, including gene feature extraction, intelligence communication, and the image process.

Noise suppression zeroing neural network for online solving the time-varying inverse kinematics problem of four-wheel mobile manipulators with external disturbances

A novel noise suppression zeroing neural network (NSZNN) is presented for the trajectory tracking problem on a four Mecanum wheeled mobile manipulator (FMWMM) by solving its time-varying inverse kinematics (TVIK) problem. The holistic kinematic model of the FMWMM is developed, which can receive synergistic control of the mobile manipulator. Different from the situation without external interference addressed in our previous work, this paper considers a variety of common time-varying interferences by studying the basic principles of various noises, and proves the NSZNN model’s of the validity and superiority, which solves the TVIK problem of the FMWMM with external disturbances through theoretical analyses. Compared with the existing gradient neural network (GNN) and the traditional zeroing neural network (ZNN), the most representative hybrid noise is selected to conduct a large number of experiments to substantiate the high efficiency and robustness of the NSZNN model. Finally, the NSZNN model is verified on the FMWMM via a robot operating system (ROS) by a successful execution of the trajectory tracking task.

A noise‐tolerant fuzzy‐type zeroing neural network for robust synchronization of chaotic systems

SummaryAs a significant research issue in control and science field, chaos synchronization has attracted wide attention in recent years. However, it is difficult for traditional control methods to realize synchronization in predefined time and resist external interference effectively. Inspired by the excellent performance of zeroing neural network (ZNN) and the wide application of fuzzy logic system (FLS), a noise‐tolerant fuzzy‐type zeroing neural network (NTFTZNN) with fuzzy time‐varying convergent parameter is proposed for the synchronization of chaotic systems in this paper. Notably the fuzzy parameter generated from FLS combined with traditional convergent parameter embedded into this NTFTZNN can adjust the convergence rate according to the synchronization errors. For the sake of emphasizing the advantages of NTFTZNN model, other three sets of contrast models (FTZNN, VPZNN, and PTZNN) are constructed for the purpose of comparison. Besides, the predefined‐time convergence and noise‐tolerant ability of NTFTZNN model are distinctly demonstrated by detailed theoretical analysis. Furthermore, synchronization simulation experiments including two chaotic systems with different dimensions are provided to verify the related mathematical theories. Finally, the schematic of NTFTZNN model for chaos synchronization is accomplished completely through Simulink, further accentuating its effectiveness and potentials in practical applications.

Predefined-Time Zeroing Neural Networks With Independent Prior Parameter for Solving Time-Varying Plural Lyapunov Tensor Equation.

As an extension of the Lyapunov equation, the time-varying plural Lyapunov tensor equation (TV-PLTE) can carry multidimensional data, which can be solved by zeroing neural network (ZNN) models effectively. However, existing ZNN models only focus on time-varying equations in field of real number. Besides, the upper bound of the settling time depends on the value of ZNN model parameters, which is a conservative estimation for existing ZNN models. Therefore, this article proposes a novel design formula for converting the upper bound of the settling time into an independent and directly modifiable prior parameter. On this basis, we design two new ZNN models called strong predefined-time convergence ZNN (SPTC-ZNN) and fast predefined (FP)-time convergence ZNN (FPTC-ZNN) models. The SPTC-ZNN model has a nonconservative upper bound of the settling time, and the FPTC-ZNN model has excellent convergence performance. The upper bound of the settling time and robustness of the SPTC-ZNN and FPTC-ZNN models are verified by theoretical analyses. Then, the effect of noise on the upper bound of settling time is discussed. The simulation results show that the SPTC-ZNN and FPTC-ZNN models have better comprehensive performance than existing ZNN models.

Predefined-time robust adaptive gradient neural network for solving linear time-varying equations and its applications

Adaptive gradient neural networks (AGNN) have been extensively employed as effective models for addressing time-varying problems. However, the existing AGNN are convergent in infinite time and only considered in noiseless environment. Consequently, the AGNN with a certain convergence time and noise tolerance is urgently required. Motivated by the advantages of predefined-time convergence, a novel predefined-time robust adaptive gradient neural network (PRAGNN) is developed for solving linear time-varying equations. It can be concluded that PRAGNN can efficiently obtain real-time solutions of linear time-varying equations within predefined time, independent of initial conditions, by properly selecting parameters. In the process of solving linear time-varying equations, PRAGNN can avoid matrix inversion, which reduces the computing cost compared with zeroing neural networks (ZNN). Furthermore, PRAGNN can completely resist both bounded vanishing noise and bounded non-vanishing noise. Numerical experiments demonstrate the effectiveness of PRAGNN. Finally, PRAGNN has been successfully utilized in 2-D dynamic positioning based on the angle of arrival (AOA) positioning algorithm, as well as in alternating currents (AC) computing.

An accelerated zeroing neural network for solving continuous coupled Lyapunov matrix equations

AbstractIn this paper, an improved zeroing neural network (ZNN) model is proposed to obtain the positive definite solutions of the continuous coupled Lyapunov matrix equations (CLMEs) associated with continuous‐time Markovian jump (CMJ) systems. To achieve this, a general ZNN model is established by constructing a matrix‐valued error function. Then, to accelerate the convergence rate of the proposed ZNN model, the latest estimation is introduced to obtain an improved ZNN model. Some convergence conditions have been derived for the presented improved ZNN model through Lyapunov theory. Comparisons among the improved ZNN model and the existing results are conducted to illustrate the advantages of the proposed improved ZNN model in numerical examples.

An Accelerated Dual-Integral Structure Zeroing Neural Network Resistant to Linear Noise for Dynamic Complex Matrix Inversion

The problem of inverting dynamic complex matrices remains a central and intricate challenge that has garnered significant attention in scientific and mathematical research. The zeroing neural network (ZNN) has been a notable approach, utilizing time derivatives for real-time solutions in noiseless settings. However, real-world disturbances pose a significant challenge to a ZNN’s convergence. We design an accelerated dual-integral structure zeroing neural network (ADISZNN), which can enhance convergence and restrict linear noise, particularly in complex domains. Based on the Lyapunov principle, theoretical analysis proves the convergence and robustness of ADISZNN. We have selectively integrated the SBPAF activation function, and through theoretical dissection and comparative experimental validation we have affirmed the efficacy and accuracy of our activation function selection strategy. After conducting numerous experiments, we discovered oscillations and improved the model accordingly, resulting in the ADISZNN-Stable model. This advanced model surpasses current models in both linear noisy and noise-free environments, delivering a more rapid and stable convergence, marking a significant leap forward in the field.

Inverse-Free DZNN Models for Solving Time-Dependent Linear System via High-Precision Linear Six-Step Method.

Time-dependent linear system (TDLS) is usually encountered in scientific research, which is the mathematical formulation of many practical applications. Different from conventional inverse-need models, by utilizing zeroing neural network (ZNN) method twice, an inverse-free continuous ZNN (CZNN) model is developed for solving TDLS. For conveniently practical use, a discrete model is naturally desired. Superior to conventional discretization methods, a general linear six-step (LSS) method with the seventh-order precision and five variable parameters is proposed for the first time. Constraints about five variable parameters are theoretically analyzed to guarantee the efficacy of the general LSS method. Within constraints, 12 specific LSS methods are further developed. Aided with the general LSS method, an inverse-free discrete ZNN (DZNN) is proposed and termed DZNN-LSS model, and its precision is greatly improved compared with conventional discrete models. For comparison, three conventional discretization methods are also utilized to generate DZNN models. Detailed theoretical analyses are provided to prove the efficacy of relevant models. In addition, a specific TDLS example is considered to show the effectiveness and superiority of the DZNN-LSS model. More than that, applications to manipulator control and sound source localization are conducted to illustrate the applicability of the DZNN-LSS model.

A Dynamic Parameter Noise-Tolerant Zeroing Neural Network for Time-Varying Quaternion Matrix Equation With Applications.

As a common and significant problem in the field of industrial information, the time-varying quaternion matrix equation (TV-QME) is considered in this article and addressed by an improved zeroing neural network (ZNN) method based on the real representation of the quaternion. In the light of an improved dynamic parameter (IDP) and an innovative activation function (IAF), a dynamic parameter noise-tolerant ZNN (DPNTZNN) model is put forward for solving the TV-QME. The presented IDP with the character of changing with the residual error and the proposed IAF with the remarkable performance can strongly enhance the convergence and robustness of the DPNTZNN model. Therefore, the DPNTZNN model possesses fast predefined-time convergence and superior robustness under different noise environments, which are theoretically analyzed in detail. Besides, the provided simulative experiments verify the advantages of the DPNTZNN model for solving the TV-QME, especially compared with other ZNN models. Finally, the DPNTZNN model is applied to image restoration, which further illustrates the practicality of the DPNTZNN model.

Hyperbolic Tangent-Type Variant-Parameter and Robust ZNN Solutions for Resolving Time-Variant Sylvester Equation in Preassigned-Time

To solve a general time-variant Sylvester equation, two novel zeroing neural networks (ZNNs) solutions are designed and analyzed. In the foregoing ZNN solutions, the design convergent parameters (CPs) before the nonlinear stimulated functions are very pivotal because CPs basically decide the convergent speeds. Nonetheless, the CPs are generally set to be constants, which is not feasible because CPs are generally time-variant in practical hardware conditions particularly when the external noises invade. So, a lot of variant-parameter ZNNs (VP-ZNNs) with time-variant CPs have been come up with. Comparing with fixed-parameter ZNNs, the foregoing VP-ZNNs have been illustrated to own better convergence, the downside is that the CPs generally increases over time, and will be probably infinite at last. Obviously, infinite large CPs would lead to be non-robustness of the ZNN schemes, which are not permitted in reality when the exterior noises inject. Moreover, even though VP-ZNNs are convergent over time, the growth of CPs will waste tremendous computing resources. Based on these factors, 2 hyperbolic tangent-type variant-parameter robust ZNNs (HTVPR-ZNNs) have been proposed in this paper. Both the convergent preassigned-time of the HTVPR-ZNN and top-time boundary of CPs are theoretically investigated. Many numerical simulations substantiated the admirable validity of the HTVPR-ZNN solutions.

New discrete-time zeroing neural network for solving time-dependent linear equation with boundary constraint

Recently, continuous- and discrete-time models of a zeroing neural network (ZNN) have been developed to provide online solutions for the time-dependent linear equation (TDLE) with boundary constraint. This paper presents a novel approach to address the bound-constrained TDLE (BCTDLE) problem by proposing a new discrete-time ZNN (DTZNN) model. The proposed DTZNN model is designed using the Taylor difference formula to discretize the previous continuous-time ZNNN (CTZNN) model. Theoretical analysis indicates the computational property of the proposed DTZNN model, and numerical results further demonstrate its validity. The applicability of the proposed DTZNN model is finally confirmed via its application to the motion planning of a PUMA560 robotic arm.

New activation functions and Zhangians in zeroing neural network and applications to time-varying matrix pseudoinversion

We are guided by the fact that zeroing neural networks (ZNN) are proven tool in online solving the time-varying (TV) matrix Moore–Penrose (M–P) inverse. This paper focuses on online computing TV full-row rank or full-column rank matrix M–P inverse using a novel ZNN model with an optimized activation function (AF) and improved error function (Zhangian). ZNN dynamical systems accelerated by the optimized class of AFs converge in a finite-time to the TV theoretical M–P inverse. The upper bounds of the estimated convergence time are obtained analytically using the Lyapunov stability theory. The simulation experiments support the theoretical analysis and demonstrate the effectiveness of the proposed ZNN dynamics.

Fixed-time solution of inequality constrained time-varying linear systems via zeroing neural networks

Zeroing neural network (ZNN) has been explored and applied in a variety of time-varying problems solving. However, ZNN models for inequality constrained time-varying linear equation (ICTVLE) solving are rarely reported. Generally, numerous practical problems can be mathematically modeled as ICTVLE problems. Motivated by the above mentioned issue, a nonlinear tunable activation function activated ZNN (NTAF-ZNN) model is constructed to effectively solve ICTVLE problems. To further improve the performances of NTAF-ZNN model, a fuzzy nonlinear tunable activation function (FNTAF) is also designed. Based on the FNTAF, a fuzzy nonlinear tunable activation function-based ZNN (FNTAF-ZNN) model is realized. Firstly, the fixed-time convergence property and disturbance rejection ability of the NTAF-ZNN and FNTAF-ZNN models are proved by theoretical analysis. Then, comparative simulation results of the two models with other existing ZNN models for solving the ICTVLE problem are provided to further verify their robustness and effectiveness. Additionally, the two models and other existing ZNN models are also employed to realize inequality constrained manipulator trajectory tracking under ideal and noisy environments for further comparisons. Finally, the superior performances of the FNTAF-ZNN model for manipulator trajectory tracking are further verified by physical experiment results.

Zeroing Neural Network With Coefficient Functions and Adjustable Parameters for Solving Time-Variant Sylvester Equation.

To solve the time-variant Sylvester equation, in 2013, Li et al. proposed the zeroing neural network with sign-bi-power function (ZNN-SBPF) model via constructing a nonlinear activation function. In this article, to further improve the convergence rate, the zeroing neural network with coefficient functions and adjustable parameters (ZNN-CFAP) model as a variation in zeroing neural network (ZNN) model is proposed. On the basis of the introduced coefficient functions, an appropriate ZNN-CFAP model can be chosen according to the error function. The high convergence rate of the ZNN-CFAP model can be achieved by choosing appropriate adjustable parameters. Moreover, the finite-time convergence property and convergence time upper bound of the ZNN-CFAP model are proved in theory. Computer simulations and numerical experiments are performed to illustrate the efficacy and validity of the ZNN-CFAP model in time-variant Sylvester equation solving. Comparative experiments among the ZNN-CFAP, ZNN-SBPF, and ZNN with linear function (ZNN-LF) models further substantiate the superiority of the ZNN-CFAP model in view of the convergence rate. Finally, the proposed ZNN-CFAP model is successfully applied to the tracking control of robot manipulator to verify its practicability.