Zeroing neural network (ZNN) can effectively solve the matrix flows inversion problem. Nevertheless, quite a few related research works focus on the improvement of the convergence and robustness performance of the ZNN models and ignore the conservatism of their predefined time. Therefore, this article adopts a polymorphous activation function (PAF) to construct a new predefined time ZNN (NPTZNN) model. The second method of Lyapunov is utilized to analyze the stability, convergence, and robustness of the NPTZNN model. The Beta function is dexterously employed in the process of calculating the predefined time of the NPTZNN model, reducing its conservatism. Furthermore, the correctness of the theoretical analyses is verified by numerous experiments. Finally, the NPTZNN model is applied to robot manipulator control and can improve the tracking speed, extending the applicability of the model.

Considering the extensive research on zeroing neurodynamic (ZN), a self-adaptive and enhanced fixed-time convergent zeroing neurodynamic (SEFC-ZN) method for addressing time-variant problems is presented in this paper based on a discrete fuzzy matrix (DFM) design parameter and a novel advanced sign-bi-power activation function (NASbpAf). Due to the distinctive design of the DFM design parameter and NASbpAf, the proposed SEFC-ZN method possesses prominent self-adaptivity and enhanced fixed-time convergence. Specifically, the DFM design parameter is actually a matrix with all elements generated from a discrete fuzzy control system, so it can self-adaptively adjust the convergence rate of every error in the SEFC-ZN method resulting in the self-adaptivity. This feature is greatly different from the conventional scalar design parameters whose values are usually fixed or increase indefinitely and different errors in the ZN method can only be adjusted by the same design parameter. By summarizing the characteristic of the activation functions designed previously according to the SbpAf, it is found that keeping two terms of the SbpAf and adding extra terms can improve the performance of the ZN method. Thereout, built on the SbpAf, the NASbpAf is presented which can make the SEFC-ZN method realize the enhanced fixed-time convergence. Three theoretical analyses and proofs, together with relative corollaries, conclude the properties of the SEFC-ZN method and the advantages of the DFM design parameter and NASbpAf. A numerical experiment about solving time-variant nonlinear equations by the SEFC-ZN method and an application to the linear-quadratic optimal control strongly verify the proposed theory and method.

Quadratic programming with equality constraint (QPEC) problems have extensive applicability in many industries as a versatile nonlinear programming modeling tool. However, noise interference is inevitable when solving QPEC problems in complex environments, so research on noise interference suppression or elimination methods is of great interest. This article proposes a modified noise-immune fuzzy neural network (MNIFNN) model and use it to solve QPEC problems. Compared with the traditional gradient recurrent neural network (TGRNN) and traditional zeroing recurrent neural network (TZRNN) models, the MNIFNN model has the advantage of inherent noise tolerance ability and stronger robustness, which is achieved by combining proportional, integral, and differential elements. Furthermore, the design parameters of the MNIFNN model adopt two disparate fuzzy parameters generated by two fuzzy logic systems (FLSs) related to the residual and residual integral term, which can improve the adaptability of the MNIFNN model. Numerical simulations demonstrate the effectiveness of the MNIFNN model in noise tolerance.

In this article, a novel distributed gradient neural network (DGNN) with predefined-time convergence (PTC) is proposed to solve consensus problems widely existing in multiagent systems (MASs). Compared with previous gradient neural networks (GNNs) for optimization and computation, the proposed DGNN model works in a nonfully connected way, in which each neuron only needs the information of neighbor neurons to converge to the equilibrium point. The convergence and asymptotic stability of the DGNN model are proved according to the Lyapunov theory. In addition, based on a relatively loose condition, three novel nonlinear activation functions are designed to speedup the DGNN model to PTC, which is proved by rigorous theory. Computer numerical results further verify the effectiveness, especially the PTC, of the proposed nonlinearly activated DGNN model to solve various consensus problems of MASs. Finally, a practical case of the directional consensus is presented to show the feasibility of the DGNN model and a corresponding connectivity-testing example is given to verify the influence on the convergence speed.

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.

Time-varying matrix inversion (TVMI) is a basic mathematical problem, which is widely involved in many scientific fields. In this article, an event-triggered control fuzzy adaptive zeroing neural network (ETC-FAZNN) model is proposed for solving the TVMI problem, where the fuzzy adaptive convergence parameter (FACP) is got by the redesigned fuzzy logic system, which makes the ETC-FAZNN model adaptive. Meanwhile, the event-triggered control is introduced to control the update of the FACP, which improves the calculation speed of the ETC-FAZNN model. Moreover, a novel activation function called segmented predefined-time activation function is put forward in this article to improve the convergence and robustness of the ETC-FAZNN model. Theoretical analysis and simulation experiments reveal that the ETC-FAZNN model can realize stability, predefined-time convergence, robustness, and adaptability performances in solving the TVMI problem.

To overcome the disadvantages of the current zeroing neural network (ZNN) in noise tolerance, this article first proposes an intensive noise-tolerant ZNN (INT-ZNN) by introducing a novel fuzzy control approach (FCA). This FCA is designed dexterously according to the variation of two errors related to the INT-ZNN. Thus, the most feature of the INT-ZNN is that the added fuzzy control can inherently restrain the various noises. Compared with the previous noise-tolerant ZNN derived by the integral design formula, the INT-ZNN with a much simpler structure can tolerate the noise in finite/fixed time. That is, the INT-ZNN activated by nonlinear functions possesses finite/fixed-time convergence while suppressing the noise, which is guaranteed by the presented theorems. Besides, it also theoretically proves that the INT-ZNN has global stability under the interference of noise. In the simulative experiment, the INT-ZNN is used to solve the time-varying Sylvester matrix equation problem and the experimental results verify the excellent noise-tolerance of the INT-ZNN. Meanwhile, the INT-ZNN is successfully applied to image processing.

In order to solve the time-varying quadratic programming (TVQP) problem more effectively, a new self-adaptive zeroing neural network (ZNN) is designed and analyzed in this article by using the Takagi-Sugeno fuzzy logic system (TSFLS) and thus called the Takagi-Sugeno (T-S) fuzzy ZNN (TSFZNN). Specifically, a multiple-input-single-output TSFLS is designed to generate a self-adaptive convergence factor to construct the TSFZNN model. In order to obtain finite- or predefined-time convergence, four novel activation functions (AFs) [namely, power-bi-sign AF (PBSAF), tanh-bi-sign AF (TBSAF), exp-bi-sign AF (EBSAF), and sinh-bi-sign AF (SBSAF)] are developed and applied in the TSFZNN model for solving the TVQP problem. Both theoretical proofs and experimental simulations show that the TSFZNN model using PBSAF or TBSAF has the property of converging in a finite time, and the TSFZNN model using EBSAF or SBSAF has the property of converging in a predefined time, which have superior convergence performance compared to the traditional ZNN model.

This article aims to studying how to solve dynamic Sylvester quaternion matrix equation (DSQME) using the neural dynamic method. In order to solve the DSQME, the complex representation method is first adopted to derive the equivalent dynamic Sylvester complex matrix equation (DSCME) from the DSQME. It is proven that the solution to the DSCME is the same as that of the DSQME in essence. Then, a state-of-the-art neural dynamic method is presented to generate a general dynamic-varying parameter zeroing neural network (DVPZNN) model with its global stability being guaranteed by the Lyapunov theory. Specifically, when the linear activation function is utilized in the DVPZNN model, the corresponding model [termed linear DVPZNN (LDVPZNN)] achieves finite-time convergence, and a time range is theoretically calculated. When the nonlinear power-sigmoid activation function is utilized in the DVPZNN model, the corresponding model [termed power-sigmoid DVPZNN (PSDVPZNN)] achieves the better convergence compared with the LDVPZNN model, which is proven in detail. Finally, three examples are presented to compare the solution performance of different neural models for the DSQME and the equivalent DSCME, and the results verify the correctness of the theories and the superiority of the proposed two DVPZNN models.

On account of the rapid progress of zeroing neural network (ZNN) and the extensive use of fuzzy logic system (FLS), this article proposes an intelligent fuzzy robustness ZNN (IFR-ZNN) model and applies it to solving the time-variant Stein matrix equation (TVSME) problem. Be different from ZNN models before, the IFR-ZNN model uses a fuzzy parameter as the design parameter and adopts a first proposed improved nonlinear piecewise activation function. Particularly, the FLS that generates the fuzzy parameter utilizes an improved membership function of nonuniform distribution which can improve the adaptability and robustness of the IFR-ZNN model. Based on the above two optimizations, the proposed IFR-ZNN model possesses three significant advantages: (1) fixed-time convergence independent of initial states; (2) superior robustness to tolerate two kinds of noises simultaneously; and (3) better adaptiveness based on computational error. Besides, the upper bounds of fixed-time convergence of the IFR-ZNN model under noisy or non-noisy situations are calculated theoretically, and the stability as well as the excellent adaptability are analyzed in detail. Finally, simulation comparison results manifest the availability and meliority of the proposed IFR-ZNN model in solving the TVSME problem.

In accordance with the advantages of zeroing neural network (ZNN) with the parallel processing character and fuzzy logic systems for calculating the uncertainties, two complex-type fuzzy ZNN (CtFZNN) models, which are mainly derived from two different limit forms of the Drazin inverse, are developed for solving the time-dependent complex-value Drazin inversion (TDCVDI) problem in this article. The most significant feature of the CtFZNN models is to use the improved fuzzy evolutionary formula, where the traditional constant or time-dependent factors are replaced by the fuzzy factors. For the non-noise or the noise disturbed CtFZNN models, the applied fuzzy factors are, respectively, generated from the single-input and single-output fuzzy logic system or the double-input and single-output fuzzy logic system. From the analytical discussions, it can conclude that the proposed CtFZNN models not only have finite-time convergence and inherent noise tolerance simultaneously, but also possess faster adaptive convergence rate even in a noisy environment. The presented theorems and the provided numerical simulations demonstrate the effectiveness of the proposed methods for addressing the TDCVDI problem, especially compared to the general ZNN model.

Aiming at the efficient online solution of the time-variant linear matrix inequality (LMI) under nonideal conditions (e.g., noise pollution), a predefined-time convergent and integral-enhanced zeroing neural network (PCIE-ZNN) model is built for the first time in this article. Compared with existing zeroing neural network (ZNN) models for settling the time-variant LMI, the PCIE-ZNN model proposed in this article is proved to have better convergence and stronger robustness even in the presence of noise interference through strict mathematical analysis and detailed numerical simulations. Specifically, the stability, predefined-time convergence, and robustness of the PCIE-ZNN model are guaranteed in theory. Then, numerical simulation cases fully compare the results of the proposed PCIE-ZNN model and the existing ZNN models for the time-variant LMI, which demonstrates the correctness of theoretical proof and the superiority of the PCIE-ZNN model in settling the time-variant LMI under various noise pollution. In addition, through comparative experiments of three sets of design parameters, the convergence speed of the PCIE-ZNN model can be further accelerated by selecting proper parameters.

Motivated from the convergence capability achieved by gradient neural network (GNN) and zeroing neural network (ZNN) for matrix inversion, in this article, a novel hybrid GNN-ZNN (H-GNN-ZNN) model is proposed by introducing a fuzzy adaptive control strategy to generate a fuzzy adaptive factor that can change its size adaptively according to the residual error. Due to its fuzzy adaptability, this novel model is called the fuzzy adaptive GNN-ZNN (FA-GNN-ZNN) model for presentation convenience. We prove that the FA-GNN-ZNN model has the better performance than the existing H-GNN-ZNN model under the same conditions. In addition, different activation functions are applied to the FA-GNN-ZNN model to improve its performance further, and the corresponding theoretical analysis is given. Finally, comparative simulation results demonstrate the validity and superiority of the FA-GNN-ZNN model for matrix inversion.

In this article, considering the effectiveness and efficiency in solving time-varying problems, a new zeroing neural network (ZNN) is proposed to solve time-varying linear equations with column full rank coefficient matrix. In addition, two novel nonlinear activation functions are developed to enhance the comprehensive performance of the ZNN model. It is demonstrated through theoretical analysis and numerical experiments that the nonlinear activated ZNN model has better noise immunity, and faster prescribed-time convergence speed. Finally, the ZNN method is successfully applied to 2-D and 3-D dynamic positioning, with lower positioning error than the traditional pseudoinverse method.

In order to improve the effect of the Exp-Sign activation function (ESAF) and the Sinh-Sign activation function (SSAF) on the convergence and robustness of the zeroing neural network (ZNN) model, two fuzzy adaptive activation functions, named FAESAF and FASSAF, are constructed by using a Mamdani fuzzy logic controller (MFLC) in this article. Thus, a novel ZNN with the FAESAF and the FASSAF is proposed to solve the time-varying linear matrix equation. Different from the ESAF and the SSAF, whose parameters are fixed, the newly constructed FAESAF and FASSAF have an adaptive property, which comes from the fact that their parameters are intelligently generated by the MFLC according to the error norm of the ZNN model. In order to highlight the superior predefined time convergence and robustness of the corresponding ZNN model with the FAESAF and the FASSAF, several theorems are provided, and the corresponding proof is given in detail. Furthermore, the ESAF and the SSAF with different values of parameters are used as a comparison in numerical experiments to verify the superior performance of the FAESAF and the FASSAF. From theoretical analysis and numerical results, we can conclude that the ZNN model with the FAESAF and the FASSAF has better predefined time convergence and robustness compared to the ZNN model with the ESAF and the SSAF under the same conditions.

Time-varying tensor inversion (TVTI) problem is a kind of general time-varying inversion problem in mathematics because scalars, vectors, and matrices can all be represented by tensors. The TVTI problem is based on a novel tensor product [termed the TensorFlow (TF) product], which is extracted from the TF. For solving such a prevalent problem, the matricization of the TF product is defined, and a novel dynamic-parameter zeroing neural-network (DP-ZNN) model is proposed by combining a ZNN design formula and a dynamic-parameter. The global convergence and the upper bound of finite-time convergence of the DP-ZNN model are analyzed theoretically. For highlighting the superior convergence performance and excellent efficiency of the DP-ZNN model in solving the TVTI problem, three comparative experiments are presented in this article. Experimental results show that the DP-ZNN model has remarkable convergent speciality.

This article, for the first time, extends the zeroing neural network (ZNN) method to address the problem of dynamic quaternion-valued matrix inversion. Due to the noncommutative property of quaternion multiplication, the complex representation method is first adopted to transform quaternion-valued matrices into the corresponding complex-valued matrices. Then, based on two kinds of ways to deal with nonlinear activation functions in the complex-valued domain, this article proposes two quaternion-valued ZNN (QVZNN) models for dynamic quaternion-valued matrix inversion. In addition, a novel nonlinear activation function is given to accelerate the convergence rate of the models to reach the predefined-time convergence. The detailed theoretical analysis, together with four theorems, are given to show the excellent properties of the QVZNN models. Furthermore, the upper bound of the convergence time is derived analytically with the residual error being zero theoretically. Finally, two numerical examples are provided to verify the theoretical results and the effectiveness of the QVZNN models for the dynamic quaternion-valued matrix inversion, and an application to mobile manipulator control is provided to indicate the practical application value of the QVZNN models.

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.

On the strength of the abundant development of zeroing neural network (ZNN) and the wide application of fuzzy logic system (FLS), this article presents a fuzzy-power ZNN (FPZNN) model for addressing the time-variant matrix Moore-Penrose inversion problem. Different from the original constant or time-variant parameters, a fuzzy power parameter is generated from the FLS, and is first embedded into the FPZNN model to adjust the convergence rate. For the purpose of highlighting the superior performance of the FPZNN model, the other three classical neural network models are developed for comparison purposes. The convergence and noise-tolerance of the FPZNN model are analyzed to guarantee its excellent performance, where the model-implementation and differential errors are taken into account in a noisy environment. Besides, simulative experiments including two kinds of examples are provided to display the advantages of the FPZNN model under three commonly used activation functions. Both the presented theorems and the simulative experiments verify the superiority of the FPZNN model.

Zeroing neural network (ZNN), as an important class of recurrent neural network, has wide applications in various computation and optimization fields. In this article, based on the traditional-type zeroing neural network (TT-ZNN) model, an adaptive fuzzy-type zeroing neural network (AFT-ZNN) model is proposed to settle time-variant quadratic programming problem via integrating an adaptive fuzzy control strategy. The most prominent feature of the AFT-ZNN model is to use an adaptive fuzzy control value to adaptively adjust its convergence rate according to the value of the computational error. Four different activation functions are injected to analyze the convergence rate of the AFT-ZNN model. In addition, different membership functions and different ranges of the fuzzy control value are discussed to study the character of the AFT-ZNN model. Theoretical analysis and numerical comparison results further show that the AFT-ZNN model has better performance than the TT-ZNN model.

This article is primarily concerned with finite-time convergence (FTC) and predefined-time convergence (PTC) design for a class of general zeroing neural network (ZNN) by constructing different activation functions (AFs). Based on the limit comparison test for improper integrals, some useful theoretical criteria are proposed to determine whether a nonlinear-activated ZNN model has FTC, PTC, or not. This novel method can avoid the valuation loss of the zoom method and the unsolvable barrier of the direct integration method that are widely used in the previous ZNN design. According to these convergence criteria, some instructive corollaries are derived to design valuable AFs to make ZNN models with FTC or PTC more easily. By taking a matrix-inversion ZNN model, some commonly used AFs are used to verify the usability of the criteria. In addition, some new AFs are constructed to further design some better ZNN models with superior FTC or PTC. Finally, convergence types of the ZNN model based on different AFs are visualized in numerical experiments.

For solving complex-valued linear matrix equations with time-varying coefficients (CV-LME-TVC) in the complex field, this article proposes a parameter-changing and complex-valued zeroing neural network (PC-CVZNN) model through integrating a new parameter-changing function. As compared to previous complex-valued zeroing neural networks (CVZNNs) with fixed parameters and existing parameter-changing functions, the PC-CVZNN model can achieve superior performance due to the accelerated role of the new parameter-changing function. In parts of theoretical analysis, we take advantage of Lyapunov methodology to prove that the proposed PC-CVZNN model can acquire the global and super-exponential convergence when the linear activation function is adopted, and even acquire super finite-time convergence when the new sign-bi-power activation function and its modified one are used. In parts of numerical comparison experiments, it is shown that the PC-CVZNN model possesses faster convergence rate than fixed-parameter CVZNN models and other analogy neural networks with parameter-changing function, when applied to finding the solution of CV-LME-TVC. Importantly, an application of the proposed method to the mobile manipulator control provides the potential practical value of the PC-CVZNN model in the industrial field.

The zeroing neural network (ZNN) activated by nonlinear activation functions plays an important role in many fields. However, conventional ZNN can only realize finite-time convergence, which greatly limits the application of ZNN in a noisy environment. Generally, finite-time convergence depends on the original state of ZNN, but the original state is often unknown in advance. In addition, when meeting with different noises, the applied nonlinear activation functions cannot tolerate external disturbances. In this article, on the strength of this idea, two prescribed-time and robust ZNN (PTR-ZNN) models activated by two nonlinear activation functions are put forward to address the time-variant Stein matrix equation. The proposed two PTR-ZNN models own two remarkable advantages simultaneously: 1) prescribed-time convergence that does not rely on original states and 2) superior noise-tolerance performance that can tolerate time-variant bounded vanishing and nonvanishing noises. Furthermore, the detailed theoretical analysis is provided to guarantee the prescribed-time convergence and noise-tolerance performance, with the convergence upper bounds of steady-state residual errors calculated. Finally, simulative comparison results indicate the effectiveness and the superiority of the proposed two PTR-ZNN models for the time-variant Stein matrix equation solving.