The nonlinear behavior with unpredicted property in engineering has received widespread attention. To explore more unpredicted nonlinear behavior, this paper focuses on the function projective lag synchronization (FPLS) of complex-valued chaotic systems. Most existing control schemes primarily address system stability, lack of consideration in convergence improvement. Zeroing neural dynamics (ZND) has significant results in the convergence performance, yet achieving a balance between model structure and performance remains a challenging task. In light of this, we propose a general time base generator (TBG), and adopt the linear ZND with the simplest structure, which has relatively weak convergence but devoid of nonlinear components, to develop a TBG based neural dynamics controller (TBGBNDC) in the FPLS of complex-valued chaotic systems. The theories illustrate that the system error under the TBGBNDC has predefined-time asymptotic convergence, which is verified in both homogeneous and heterogeneous complex-valued chaotic systems. Compared with other controllers, the TBGBNDC exhibits better performance in terms of convergence, energy and time consumption. Furthermore, the higher-quality random sequences are acquired through the FPLS between these chaotic systems, showing the significant potential application of the research topic.

Given the high coupling of state variables in second-order chaotic systems, the projective synchronization control schemes for first- or fractional-order chaotic systems are not applicable in second-order chaotic systems. Also, the current control schemes on second-order chaotic systems are difficult to tradeoff between convergence and robustness. To address the above issues, this article develops a predefined-time robust neural dynamics controller (PTRNDC). First, a predefined-time nonsingular terminal sliding mode variable (PTNTSMV) is designed to control the coupling of errors in the projective synchronization of second-order chaotic systems, ensuring the nonsingularity and convergence. Hence, a predefined-time double-integral zeroing neural dynamics (ZNDs) design formula based on a time-base generator (TBG) is devised to ensure that the sliding mode variable attains the desired sliding mode surface swiftly and robustly. The theorems about the stability, convergence, and robustness of the projective synchronization under the PTRNDC are analyzed rigorously, and the comparative simulations further verify the effectiveness of the PTRNDC. In addition, the chaotic sequences generated by the projective synchronization between the second-order chaotic systems are successfully applied in the image encryption, making the original image possess excellent visual distortion.

This paper addresses the time-varying complex-valued quadratic programming problem by designing a fuzzy adaptive complex-valued zeroing neural network (FACZNN) with fuzzy control factors. Based on a fuzzy logic system and a zeroing neural network, three types of the FACZNN model are constructed in different forms. While the first two FACZNN models utilize existing activation functions, we design a novel sinh-power-sign activation function for the third FACZNN model, enabling the FACZNN model to achieve predefined-time convergence. The model's efficiency is validated through theoretical discussion and numerical simulations in both noise-free and noisy environments. Additionally, the FACZNN model is applied to color image fusion and target detection. We processRGBimages using quaternion representation based on complex values, framing the color image fusion and target detection as a complex-valued quadratic programming problem. The results demonstrate that the model exhibits strong robustness and practical effectiveness.

Multi-view clustering (MVC) based on anchor learning has been proven to be effective in improving clustering accuracy and efficiency. Existing MVC methods are mainly based on single-granularity anchor learning, that is, the number of anchors corresponding to different views is constant and consistent, which will lead to information redundancy or insufficient mining. In addition, aggregating anchors of varying scales from all views to obtain multi-view shared clustering results remains a problem to be further explored. To address the above problems, a novel MVC method named View-adaptive Multi-granularity Anchor Learning (VMAL) is proposed in this paper, where view-adaptive anchor pruning and view-shared sample clustering are jointly optimized. On the one hand, VMAL can dynamically adjust the optimal number of anchors for each view during optimization by exploiting the reconstruction error of samples. On the other hand, an intuitive and effective mapping-aggregation message passing strategy is cleverly designed, which first maps the anchor representations of different views to the cluster space and then transfers the obtained cluster information of anchors to the sample space through an aggregation matrix. As a byproduct, VMAL can directly obtain the discrete cluster distribution of samples without additional partitioning. Finally, an iterative optimization algorithm is developed to solve the proposed VMAL method. Experimental results on multiple datasets have demonstrated the superiority of VMAL in terms of clustering results when compared with other state-of-the-art methods.

Graph Neural Networks (GNNs), due to their advanced capability in extracting high-order neighbor relationships, have become essential in multimodal recommendation tasks. However, augmenting the number of propagation layers in GNNs can result in feature redundancy, which may degrade the final recommendation performance. In addition, the existing recommendation task method directly maps the preprocessed multimodal features to the low-dimensional space, which will bring the noise unrelated to user preference, thus affecting the representation ability of the model. To tackle the aforementioned challenges, we propose Multimodal Graph Neural Network (MGNM) for Recommendation with Dynamic De-Redundancy (DDR) and Modality-Guided Feature De-Noisy, which is divided into local and global interaction. Initially, in the local interaction process, we integrate a DDR loss function which is achieved by utilizing the product of the feature coefficient matrix and the feature matrix as a penalization factor. It reduces the feature redundancy effects of multimodal and behavioral features caused by the stacking of multiple GNN layers. Subsequently, in the global interaction process, we developed modality-guided global feature purifiers for each modality to alleviate the impact of modality noise. It is a two-fold guiding mechanism eliminating modality features that are irrelevant to user preferences and captures complex relationships within the modality. Experimental results demonstrate that MGNM achieves superior performance on multimodal information denoising and removal of redundant information compared to the state-of-the-art methods.

Quadrotor unmanned aerial vehicles (UAVs) have attracted substantial attention due to their simple structure and strong adaptability, but enhancing robustness and adaptability for trajectory tracking under unbounded disturbances remains a key challenge. To address this, this article proposes a novel double integral fuzzy zeroing neural dynamics controller (DIFZNDC). The DIFZNDC integrates a double integral neural dynamics model with a novel dual-input single-output fuzzy logic system (DISOFLS), which can adaptively adjust the parameters, thereby enhancing the robustness and adaptability of the controller. In addition, the global convergence and robustness of the system under the DIFZNDC are theoretically verified. Moreover, two trajectory tracking examples demonstrate the effectiveness and superiority of the DIFZNDC for the quadrotor system. Quantitative analysis under Gaussian disturbance indicates that the DIFZNDC reduces the root-mean-square error (RMSE) by 84.61% and 48.35% compared to the modified super-twisting controller (MSTC) and the fixed-time zeroing neural dynamics controller (FTZNDC), respectively.

With the advancement of robotic-assisted minimally invasive surgery, visual servo control has become a crucial technique for improving surgical outcomes. However, traditional visual servo methods often rely on precise kinematic models and camera calibration, limiting their generalizability. Considering these, this article proposes a novel uncalibrated model-free visual servo control scheme. Specifically, we introduce a Jacobian matrix and interaction matrix estimation method based on a gradient neural network (GNN), which enables online estimation by utilizing control signals and sensor outputs. Then, the estimated results are incorporated into a visual servo control framework that considers remote center of motion (RCM) constraint, joint-drift problem, and physical constraint, formulated as a quadratic programming (QP) problem. Subsequently, focusing on the joint limits and endoscope insertion depth constraint, we develop a nonpiecewise differentiable multilevel constraint handling technique. For the formulated QP problem, a predefined-time convergent error-regulating zeroing neural network (PTCER-ZNN) solver is designed, and we can derive the optimal control signals. Detailed theoretical analyses of the developed GNN estimation method and the PTCER-ZNN solver are provided. Simulation results demonstrate the effectiveness of the proposed scheme in image feature regulation and tracking tasks, exhibiting its advantages over existing approaches.

The permanent magnet synchronous generator (PMSG) system becomes unstable when unpredicted chaos appears, and current approaches do not take how to lessen this chaos phenomenon into account. Motivated by the ability of projective synchronization (PS) to adjust the chaotic system trajectory, this research aims to use PS to reduce the chaos in PMSG system. For better control in the time estimation of PS and the robustness of systems, an adaptive predefined-time robust zeroing neural dynamic controller (APTRZNDC) for the PS between PMSG systems is proposed. In the process, an adaptive parameter determined by the system error is designed with the demand for higher convergence factor in the case of large error. In addition, a nonlinear activation function contributed to the predefined-time synchronization is created, making the upper bound of synchronization time independent of system initial states and parameters, except for a single predefined parameter. Moreover, essential theorems for the predefined-time PS and robustness under the APTRZNDC are supplied and validated. And better robustness of PMSG system with the APTRZNDC is demonstrated when compared with other controllers. Furthermore, the APTRZNDC is applied in secure communication via the PS of PMSG systems, which guarantees both the timeliness of signals and the immunity of communication.

Time-varying algebraic Riccati equations (TAREs) indeed play a crucial role in science and engineering with widespread applications. This research combines the advantages of zeroing neural network (ZNN) in handling time-varying problems with the flexibility of fuzzy logic system (FLS), proposing a ZNN-based solver for solving the TARE. One of the innovations of this article is the presentation of an adaptive input range fuzzy logic system (AFLS) with portability and adaptability, offering a novel approach for determining the input range of the FLS. The method effectively resolves the current dilemma of relying on a specific problem and model for determining the FLS input range. In addition, to enhance convergence speed and achieve predefined-time convergence of the fuzzy predefined-time robust zeroing neural network (FPRZNN) model, we introduce a novel segmental predefined-time robust activation function (SPRAF). Furthermore, three key theorems are proposed to prove the stability, convergence, and robustness of the FPRZNN model. Finally, the numerical simulations showcase the superior convergence and robustness of the FPRZNN model compared to other existing ZNN models.

Hyperspectral image (HSI) encompasses abundant spatial and spectral details, while Light Detection and Ranging (LiDAR) delivers precise elevation data. The amalgamation of HSI and LiDAR data significantly improves the precision of image classification. However, most methods focus solely on spatial features while neglecting frequency domain information, limiting the ability of deep models to characterize land cover. Furthermore, how to establish a sufficient interaction between different modalities is also an important issue. In this paper, we propose a novel multiscale cross-domain fusion network (MCFNet) for joint classification of HSI and LiDAR data. The main idea is that the wavelet transform can provide details at different resolutions simultaneously, supplementing spatial domain information and enriching feature representation. In addition, the multimodal fusion module (MFM) guided by HSI and the cross-domain fusion module (CDFM) strategy are developed to integrate features from diverse modalities and domains, respectively. Specifically, frequency domain features are extracted by discrete wavelet transform, and spatial domain features of the image are captured through a set of convolution operations. Then interactive fusion is performed by MFM and CDFM, and finally the integrated features are categorized using a classification module. Extensive experiments on three widely-used HSI and LiDAR datasets indicate that MCFNet outperforms the SOTA methods. The code will be available at https://github.com/MSFLabX/MCFNet.

Traffic Salient Object Detection (TSOD) aims to segment the objects critical to driving safety by combining semantic (e.g., collision risks) and visual saliency. Unlike SOD in natural scene images (NSI-SOD), which prioritizes visually distinctive regions, TSOD emphasizes the objects that demand immediate driver attention due to their semantic impact, even with low visual contrast. This dual criterion, i.e., bridging perception and contextual risk, re-defines saliency for autonomous and assisted driving systems. To address the lack of task-specific benchmarks, we collect the first large-scale TSOD dataset with pixel-wise saliency annotations, named TSOD10K. TSOD10K covers the diverse object categories in various real-world traffic scenes under various challenging weather/illumination variations (e.g., fog, snowstorms, low-contrast, and low-light). Methodologically, we propose a Mamba-based TSOD model, termed Tramba. Considering the challenge of distinguishing inconspicuous visual information from complex traffic backgrounds, Tramba introduces a novel Dual-Frequency Visual State Space module equipped with shifted window partitioning and dilated scanning to enhance the perception of fine details and global structure by hierarchically decomposing high/low-frequency components. To emphasize critical regions in traffic scenes, we propose a traffic-oriented Helix 2D-Selective-Scan (Helix-SS2D) mechanism that injects driving attention priors while effectively capturing global multi-direction spatial dependencies. We establish a comprehensive benchmark by evaluating Tramba and 25 existing NSI-SOD models on TSOD10K, demonstrating Tramba's superiority. Our research establishes the first foundation for safety-aware saliency analysis in intelligent transportation systems. The dataset and code will be made publicly available at https://github.com/mj129/Tramba.

In real-world scenarios, quadrotors face significant noise challenges from both internal and external sources, necessitating more robust controllers. While most existing models assume bounded noise, unbounded noise poses greater practical challenges. To address this, we propose a novel controller design method based on variant-parameter double integral multi-layer neural dynamics (VP-DIMND) for quadrotors. First, the design process of the quadrotor’s position and attitude controller based on the VP-DIMND method is presented. Second, theoretical analysis demonstrates that the VP-DIMND controller ensures rapid convergence and robust performance against various noise conditions, including bounded and unbounded noises. This is achieved through the use of variant-parameter multi-layer zeroing neural dynamics and double integral design mode. Finally, simulation experiments show that the VP-DIMND controller effectively enables the quadrotor track the given time-varying tasks in various noisy environments. Moreover, compared with similar methods, the proposed VP-DIMND method improves the convergence speed by about 40%, and the tracking results under the VP-DIMND controller outperforms the existing controllers, including convergence and robustness. These results also highlight the potential of the VP-DIMND controller for enhancing the stability and reliability of quadrotor in noisy environments.

Involving both position and orientation tracking, pose tracking control for the end-effector of a redundant manipulator is a critical problem in robotic motion control. However, existing methods often suffer from dependency on model parameters and lack joint constraints. To remedy these weaknesses, this article proposes a data-based predictive tracking control of position and orientation (DBPTCPO) for redundant manipulators with undetermined parameters. Specifically, in addition to minimizing tracking error, the DBPTCPO scheme can also minimize joint velocity and acceleration to optimize energy efficiency. Furthermore, it directly handles three-level joint constraints, effectively preventing a reduction in the feasible domain of decision variables. As for the uncertain parameters of redundant manipulators, a method based on zeroing neurodynamics (ZNs) is developed to estimate the Jacobian matrix, requiring only the sensory output and control signals. Ultimately, a ZN-based solver is designed to solve the quadratic programming (QP) problem with inequality constraints derived from the DBPTCPO scheme. Necessary theoretical analyses for the control process are provided, and the higher tracking accuracy of the proposed method is numerically validated when compared with other control schemes.

This article proposes predefined-time adaptive neural network (PTANN) and event-triggered PTANN (ET-PTANN) models to efficiently compute the time-varying tensor Moore-Penrose (MP) inverse. The PTANN model incorporates a novel adaptive parameter and activation function, enabling it to achieve strongly predefined-time convergence. Unlike traditional time-varying parameters that increase over time, the adaptive parameter is proportional to the error norm, thereby better allocating computational resources and improving efficiency. To further enhance efficiency, the ET-PTANN model combines an event trigger with the evolution formula, resulting in the adjustment of step size and reduction of computation frequency compared to the PTANN model. By conducting mathematical derivations, the article derives the upper bound of convergence time for the proposed neural network models and determines the minimum execution interval for the event trigger. A simulation example demonstrates that the PTANN and ET-PTANN models outperform other related neural network models in terms of computational efficiency and convergence rate. Finally, the practicality of the PTANN and ET-PTANN models is demonstrated through their application for mobile sound source localization.

Multimodal recommendation systems integrate diverse multimodal information into the feature representations of both items and users, thereby enabling a more comprehensive modeling of user preferences. However, existing methods are hindered by data sparsity and the inherent noise within multimodal data, which impedes the accurate capture of users' interest preferences. Additionally, discrepancies in the semantic representations of items across different modalities can adversely impact the prediction accuracy of recommendation models. To address these challenges, we introduce a novel diffusion-based contrastive learning (DiffCL) framework for multimodal recommendation. DiffCL employs a diffusion model (DM) to generate contrastive views that effectively mitigate the impact of noise during the contrastive learning phase. Furthermore, it improves semantic consistency across modalities by aligning distinct visual and textual semantic information through stable ID embeddings. Finally, the introduction of the item-item graph (I-I graph) enhances multimodal feature representations, thereby alleviating the adverse effects of data sparsity on the overall system performance. We conduct extensive experiments on three public datasets, and the results demonstrate the superiority and effectiveness of the DiffCL.

Time-varying linear equations (TVLEs) play a fundamental role in the engineering field and are of great practical value. Existing methods for the TVLE still have issues with long computation time and insufficient noise resistance. Zeroing neural network (ZNN) with parallel distribution and interference tolerance traits can mitigate these deficiencies and thus are good candidates for the TVLE. Therefore, a new predefined-time adaptive ZNN (PTAZNN) model is proposed for addressing the TVLE in this article. Unlike previous ZNN models with time-varying parameters, the PTAZNN model adopts a novel error-based adaptive parameter, which makes the convergence process more rapid and avoids unnecessary waste of computational resources caused by large parameters. Moreover, the stability, convergence, and robustness of the PTAZNN model are rigorously analyzed. Two numerical examples reflect that the PTAZNN model possesses shorter convergence time and better robustness compared with several variable-parameter ZNN models. In addition, the PTAZNN model is applied to solve the inverse kinematic solution of UR5 robot on the simulation platform CoppeliaSim, and the results further indicate the feasibility of this model intuitively.

The time-varying Lyapunov equation (TVLE) plays a crucial role in control design and system stability. However, there has been limited research conducted on the time-varying generalized Lyapunov equation in the quaternion field. To tackle the time-varying quaternion generalized Lyapunov equation, a nonlinear noise-resistant zeroing neural network (NNR-ZNN) model with a novel power activation function (NPAF) is devised. The issue of non-commutativity within quaternion is circumvented by utilizing the real representation. The theoretical analyses provide a sufficient explanation for the global stability, fixed-time convergence, and robustness of the NNR-ZNN model. Under several different kinds of noises, the exceptional robustness of the NNR-ZNN model is highlighted by comparison with other existing models. In the end, the successful applications of the NNR-ZNN model to color image fusion and color image denoising confirm the practical value of the NNR-ZNN model.

Multimodal recommendation systems improve the accuracy of recommendations by integrating information from different modalities to obtain potential representations of users and items. However, existing multimodal recommendation methods often use single user embedding to model users’ interests in different modalities, neglecting multimodal information. Furthermore, the semantics expressed by the same items in different modalities may be inconsistent, leading to suboptimal recommendation performance. To alleviate the impact of these issues, we propose a new multimodal recommendation framework called Learnable Prompts and ID-guided Contrastive Learning (LPIC). Specifically, we introduce a continuously learnable prompt embedding method, incorporating multimodal features of items to model users’ interests in specific modalities. Then, we propose an ID-guided contrastive learning component to enhance historical interaction features in textual, visual, and fused modalities, while aligning text, image, and fused modality to enhance semantic consistency between modalities. Finally, we conduct extensive experiments on three publicly available Amazon datasets to demonstrate the effectiveness of the LPIC framework.

Sliding mode control (SMC) is considered an efficacious scheme for quadrotor control. However, the control performance of the existing SMC schemes depends on initial states and multiple parameters, and the robustness needs to be improved. To address these issues, a novel predefined-time robust SMC framework based on two zeroing neural dynamics (ZND) schemes, referred to as ZND-based predefined-time robust SMC (ZNDPRSMC) framework, is developed to facilitate position and attitude tracking of a quadrotor under bounded disturbances. Initially, a nonsingular sliding mode surface (SMS) is formulated by incorporating a general ZND along with a differentiable predefined-time activation function. Following this, an approaching law is introduced by utilizing a variable-parameter noise-tolerant ZND and a novel dynamic adaptive parameter. The nonsingular SMS and the approaching law are then combined to construct a nonsingular predefined-time robust controller. The theoretical proofs provided ascertain the predefined-time convergence of the closed-loop system utilizing ZNDPRSMC and its robustness against bounded disturbances. Finally, two trajectory tracking examples of the quadrotor are presented to demonstrate the superiority of the ZNDPRSMC framework.

Dynamic equations provide mathematical frameworks to capture the evolving behavior of systems, which is essential in various fields. While the zeroing neural network (ZNN) is one of the most effective real-time solvers for dynamic equations, it is highly susceptible to noise interference, which reduces the precision and reliability of solutions. Current research struggles to address more complex noise disturbances, particularly complex-valued and random noise. To overcome this limitation, this article introduces a set of self-learning operators with real-time correction ability to counteract noise interference and obtain a new self-learning noise-resistant ZNN (SLNR-ZNN). The operators within the SLNR-ZNN model adaptively learn the physical forms of noise, utilizing the noise’s derivative properties, through continuous system oscillations to enhance noise tolerance and improve the accuracy of dynamic equation resolution. Theoretical analysis and experimental validation show that SLNR-ZNN effectively resolves linear and nonlinear dynamic equations under various types of noise, including constant, harmonic, complex spectral, and Gaussian white noise. Compared to existing ZNN models, SLNR-ZNN achieves comparable convergence rates and simultaneously maintains significantly lower steady-state errors, which are often reduced by nearly an order of magnitude under noise. Furthermore, simulation experiments demonstrate that the SLNR-ZNN-based control protocol achieves state consensus in leader-following multiagent systems and enables trajectory tracking in the UR5 robotic arm with millimeter-level accuracy, even under composite disturbances. These results highlight its practical value and robustness in robotic control applications.

Current variable-convergence-parameter zeroing neural networks (ZNNs), including the VCP-ZNN and the FCP-ZNN, are either inefficient or unintelligent. Although researchers have discussed the application of ZNN in chaos synchronization, these ZNN-based chaos synchronization controllers are rarely used in real-world applications. To the best of the authors’ knowledge, no researchers have applied the ZNN-based chaos synchronization controllers in secure voice communication. In this study, we established a novel chaos synchronization controller based on the proportional–integral-convergence-parameter ZNN (PICP-ZNN) model, which is both computationally efficient and intelligent. It was then used in secure voice communication. To demonstrate the superior features of the proposed PICP-ZNN model, we presented both theoretical analysis and numerical experiments to show its fixed-time convergence, robustness, and adaptiveness. In addition, a detailed comparison with other state-of-the-art variable-convergence-parameter ZNNs was presented to highlight our contribution further. The upper bound of the settling time is also estimated in both noisy and noise-free environments. Overall, this study offers a novel ZNN-based secure communication scheme. The PICP-ZNN models may serve as a novel source of inspiration for enhancing the variable-convergence-parameter ZNN even further.

Redundant manipulators have shown great potential in the application of robots. These manipulators possess additional degrees of freedom beyond what is essential for completing specific tasks, presenting an opportunity to optimize energy usage. However, the existence of additional degrees of freedom also brings control challenges. Due to the ability to address problems of time-varying tracking, zeroing neural network (ZNN) is gradually widely used in the control of redundant manipulators. Discrete models are often used in engineering, and the sampling gap selected during discretization is an important factor that affects the tracking precision. Large sampling gaps require less computational consumption but yield lower tracking precision, whereas small sampling gaps result in higher precision but at a greater computational cost. In this article, a minimum energy consumption scheme (MECS) for the time-varying tracking control task of redundant manipulators is presented first. By applying the ZNN design formula, the continuous ZNN (CZNN) model is established to solve the MECS. Subsequently, Euler discretization formula is utilized to transform the CZNN model into its discrete form, known as the discrete ZNN (DZNN) model. Then a dual-input–single-output fuzzy control system is designed to obtain suitable sampling gaps. The fuzzy-control-aided ZNN (FCAZNN) model enables redundant manipulators to track desired paths with the expected precision. Finally, a series of experiments are carried out in this article to demonstrate the advantages of FCAZNN model, including both computer simulations and physical experiments.

Solving nonlinear equation flows is the one critical step in statistics and control fields. Access to its real-time solutions has become a major challenge for scholars. Numerous research work has confirmed the effectiveness of zeroing neural dynamics (ZND) for time-varying problems. Accordingly, this paper constructs a novel predefined-time neural dynamics (NPTND) model for nonlinear equation flows. To address exponential convergence of the linear ZND model, the time-base generator is utilized in the design formula of NPTND model, giving the model predefined-time approximate convergence. Whereafter, numerical comparative simulations in two different dimensions are performed to assess the effectiveness of the NPTND model. Specifically, the NPTND model has predefined-time approximate convergence and a certain robustness against small noise in solving process.

This article investigates a leader-following bipartite consensus issue for uncertain nonlinear heterogeneous multiagent systems (MASs). Initially, within the framework of optimal control theory, we employ the reinforcement learning (RL) algorithm to derive an approximate solution to the Hamilton-Jacobi-Bellman equation (HJBE). Specifically, the neural networks (NNs) are utilized to construct the Actor-Critic structure with the aim of implementing control behavior and evaluating system performance, respectively. An additional network is employed to address nonlinear uncertainties existing in the system. Furthermore, we design a static threshold event-triggered mechanism (ETM) to achieve the event-triggered state feedback-based control strategy. By utilizing this event-triggered state information, we reconstruct the approximate optimal controller and update laws of neural network weights, effectively reducing the communication burden while ensuring that all signals of the MASs remain bounded. Finally, two simulation examples are carried out to demonstrate the feasibility of the proposed method.

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.

In this article, a novel adaptive robust predefined-time nonsingular sliding mode control (ARPTNSMC) scheme is investigated, which aims to achieve fast and accurate tracking control of a quadrotor subjected to external disturbance. Inspiration is drawn from a fuzzy neural network that is constructed by fuzzy logic and zeroing neural network (ZNN). Distinct from most sliding mode control approaches, two nonsingular sliding mode surfaces are formulated by employing general ZNN approaches and differentiable predefined-time activation functions. Furthermore, for the compensation of external disturbance, a dynamic adaptive parameter and a fuzzy adaptive parameter are designed in the attitude control law. The fuzzy adaptive parameter, generated by the Takagi–Sugeno fuzzy logic system, is incorporated to enhance the robustness while reducing the chattering phenomena resulting from the discontinuous sign function. Theoretical proofs are provided to demonstrate the predefined-time convergence and robustness of the closed-loop system. Finally, two trajectory tracking examples are offered to validate the convergence, robustness, and low-chattering characteristics of the closed-loop system under the developed ARPTNSMC scheme.

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.

Taking advantage of the burgeoning zeroing neural network (ZNN) and the widely used fuzzy logic system (FLS), a novel double integral noise-tolerant fuzzy ZNN (DINTFZNN) model for solving the time-varying Sylvester matrix equation (TVSME) is proposed in this article. The special feature of the DINTFZNN model lies in the adoption of a double integral design formula, which makes the DINTFZNN model has superb robustness, that is, it can effectively suppress not only linear noise but also quadratic noise. In addition, the DINTFZNN model utilizes a fuzzy parameter generated by FLS as the design parameter, which can adaptively adjust the convergence rate and enhance the robustness and adaptability of the DINTFZNN model. Theories have rigorously demonstrated the convergence and robustness of the DINTFZNN model. By the comparison experiments with the single integral noise-tolerant ZNN model, the superiority of the DINTFZNN model is further confirmed. In the end, the design method of the DINTFZNN model is applied to the synchronization of Chua's circuit chaotic systems, which epitomizes its excellent applicability.

With high wind speed, low turbulence, and high output, offshore wind power has gradually become a new area of wind power development. Nevertheless, the chaos phenomenon of offshore wind turbines manifests in some severe environments and the entire power generation system is affected. A variety of projective synchronization schemes are proposed to control this phenomenon, but the research on fixed-time projective synchronization of offshore wind turbine systems (OWTSs) is scant and has flaws in robustness. Motivated by zeroing neural dynamics (ZND) with fixed-time convergence, this article presents a fixed-time robust controller (FXTRC) based on ZND for the projective synchronization of OWTSs. In design process, a novel activation function is constructed to guarantee the projective synchronization speed and enhance the robustness. It is rigorously calculated that the upper bound of the projective synchronization time is only related to system parameters. Furthermore, the projective synchronization progress of OWTSs under the FXTRC displays better robustness compared with other controllers, which is proven by simulation results.

The coupling of multivariate repeated systems and the nonlinearity that is difficult to characterize through mechanisms, along with actuator constraints and data noise pollution, pose challenges in achieving precise tracking tasks. To address these issues, a novel data-driven robust iterative learning predictive control (ILPC) scheme is proposed. The contribution lies in its ability to achieve multivariable tracking without requiring any prior model information, all while effectively suppressing noise pollution and actively addressing actuator constraints. Specifically, a dynamic linearization data predictive model (DLDPM) is first obtained for system dynamic behavior prediction and controller synthesis. The estimation of the unknown pseudoJacobian matrix (PJM) in DLDPM was previously overlooked in terms of data noise suppression mechanisms. In this study, we utilize a noise-tolerant zeroing neural network (NT-ZNN) for its estimation. Theoretical analysis confirms that the PJM adaptive estimation law can achieve residue-free convergence and its robustness in noise suppression. Then, a constrained ILPC scheme is proposed, which transforms the multivariable tracking problem with actuator constraints into an iteration-varying quadratic programming problem with both inequality and equality constraints, which is solved using NT-ZNN. Theoretical proofs substantiate that a constrained ILPC scheme can achieve asymptotic convergence along the iterative axis. Finally, the proposed scheme is validated in a thermal management system for a proton exchange membrane fuel cell, showcasing the effectiveness in tracking tasks and handling actuator constraints in the presence of noise pollution.

A dynamic gain fixed-time (FXT) robust zeroing neural network (DFTRZNN) model is proposed to effectively solve time-variant equality constrained quaternion least squares problem (TV-EQLS). The proposed approach surmounts the shortcomings of conventional numerical algorithms which fail to address time-variant problems. The DFTRZNN model is constructed with a novel dynamic gain parameter and a novel activation function (NAF), which differs from previous zeroing neural network (ZNN) models. Moreover, the comprehensive theoretical derivation of the FXT stability and robustness of the DFTRZNN model is presented in detail. Simulation results further confirm the availability and superiority of the DFTRZNN model for solving TV-EQLS. Finally, the consensus protocols of multiagent systems are presented by utilizing the design scheme of the DFTRZNN model, which further demonstrates its practical application value.

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.

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.

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.

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.

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.

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.

In this article, a low-order zeroing neural network (LZNN), a high-order ZNN (HZNN), and a variable-parameter ZNN (VZNN) are designed and applied to the time-changing Cholesky decomposition of any positive-definite matrix, where the LZNN and HZNN models are generated based on the traditional and high-order evolutionary formulas, respectively. In addition, a new activation function (N-Acf) is applied to the LZNN, HZNN, and VZNN models to improve the convergence and robustness. Importantly, the LZNN and HZNN models activated by the N-Acf have faster predefined-time convergence velocity when solving the time-changing Cholesky decomposition problem of any positive-definite matrix, which is demonstrated via theoretical analysis and numerical experiments. Finally, in light of empirical and theoretical evidence, it can be established that the solution model of the VZNN model is able to undergo convergence to the theoretical solution of Cholesky decomposition despite the presence of interposing noise.

Image fusion can obtain the superior information and reduce the noise in the source image by designing a specific scheme. However, the noise in image fusion has been a difficult issue and hard to handle. In this article, a variable-gain fixed-time convergent and robust zeroing neural network (VFCR-ZNN) model is proposed to figure out the image fusion problem and the corresponding quadratic programming (QP) problem. In contrast to the original zeroing neural network model, the VFCR-ZNN model adopts a novel fixed-time activation function and a useful variable-gain parameter, which allows the VFCR-ZNN model to converge faster in fixed-time and realize noise immunity under external disturbance. The detailed theory is provided to support this point. Different numerical QP comparative examples are carried out to effectively corroborate the rightness of the theoretical analyses and the excellence of the VFCR-ZNN model. Additionally, the quality of fused images acquired by the VFCR-ZNN model is higher compared to existing state-of-the-art models for image fusion. Furthermore, the VFCR-ZNN model is successfully utilized in the repetitive motion of six-link robot manipulator to demonstrate its significant practical implications.

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.

Two nonlinear zeroing neural network (ZNN) models with prescribed-time convergence for time-dependent matrix LR and QR factorization are proposed in this article. To do so, two algorithms and two error functions are constructed to transform the time-dependent matrix LR and QR factorization problems into time-dependent linear equation systems, respectively. Simultaneously, a new activation function is introduced based on the initial ZNN models for the prescribed-time convergence of models. The excellent performance (robustness and convergence) of the two proposed ZNN models are analyzed theoretically. Furthermore, the prescribed-time convergence and antinoise abilities of the proposed ZNN models are well demonstrated in numerical experiments. Finally, the proposed ZNN model is applied to the moving target location problem, and the results show that the location error is at the millimeter level.

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.

As a category of the recurrent neural network (RNN), zeroing neural network (ZNN) can effectively handle time-variant optimization issues. Compared with the fixed-parameter ZNN that needs to be adjusted frequently to achieve good performance, the conventional variable-parameter ZNN (VPZNN) does not require frequent adjustment, but its variable parameter will tend to infinity as time grows. Besides, the existing noise-tolerant ZNN model is not good enough to deal with time-varying noise. Therefore, a new-type segmented VPZNN (SVPZNN) for handling the dynamic quadratic minimization issue (DQMI) is presented in this work. Unlike the previous ZNNs, the SVPZNN includes an integral term and a nonlinear activation function, in addition to two specially constructed time-varying piecewise parameters. This structure keeps the time-varying parameters stable and makes the model have strong noise tolerance capability. Besides, theoretical analysis on SVPZNN is proposed to determine the upper bound of convergence time in the absence or presence of noise interference. Numerical simulations verify that SVPZNN has shorter convergence time and better robustness than existing ZNN models when handling DQMI.

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.

A correlation between fuzzy logic systems (FLS) and zeroing neural networks (ZNN) design is investigated. It is shown that the gain parameter included in ZNN design can be dynamically adjusted over time by means of an appropriate value derived as the output of a properly defined FLS, which includes appropriately defined membership functions and fuzzy logic rules. Dynamical systems which are applicable to time-varying rank-deficient matrices are proposed. Convergence properties are investigated and illustrative simulation experiments are performed. Presented simulation experiments confirm the superiority of the FLS proposed in this article with respect to previously proposed FLS for dynamic adjustment of gain parameters. Furthermore, the superiority of the FLS-based ZNN model over the corresponding ZNN models based on the classical approach in defining the varying-gain parameter is demonstrated.

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.

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.

Matrix inversion frequently occurs in the fields of science, engineering, and related fields. Numerous matrix inversion schemes are often based on the premise that the solution procedure is ideal and noise-free. However, external interference is generally ubiquitous and unavoidable in practice. Therefore, an integrated-enhanced zeroing neural network (IEZNN) model has been proposed to handle the time-variant matrix inversion issue interfered with by noise. However, the IEZNN model can only deal with small time-variant noise interference. With slightly larger noise interference, the IEZNN model may not converge to the theoretical solution exactly. Therefore, a variable-parameter noise-tolerant zeroing neural network (VPNTZNN) model is proposed to overcome shortcomings and improve the inadequacy. Moreover, the excellent convergence and robustness of the VPNTZNN model are rigorously analyzed and proven. Finally, compared with the original zeroing neural network (OZNN) model and the IEZNN model for matrix inversion, numerical simulations and a practical application reveal that the proposed VPNTZNN model has the best robust property under the same external noise interference.

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.

For solving dynamic generalized Lyapunov equation, two robust finite-time zeroing neural network (RFTZNN) models with stationary and nonstationary parameters are generated through the usage of an improved sign-bi-power (SBP) activation function (AF). Taking differential errors and model implementation errors into account, two corresponding perturbed RFTZNN models are derived to facilitate the analyses of robustness on the two RFTZNN models. Theoretical analysis gives the quantitatively estimated upper bounds for the convergence time (UBs-CT) of the two derived models, implying a superiority of the convergence that varying parameter RFTZNN (VP-RFTZNN) possesses over the fixed parameter RFTZNN (FP-RFTZNN). When the coefficient matrices and perturbation matrices are uniformly bounded, residual error of FP-RFTZNN is bounded, whereas that of VP-RFTZNN monotonically decreases at a super-exponential rate after a finite time, and eventually converges to 0. When these matrices are bounded but not uniform, residual error of FP-RFTZNN is no longer bounded, but that of VP-RFTZNN still converges. These superiorities of VP-RFTZNN are illustrated by abundant comparative experiments, and its application value is further proved by an application to robot.

In this article, two Adams–Bashforth-type integration-enhanced discrete-time zeroing neural dynamic (ADTIZD) models are proposed to solve the time-varying complex Sylvester equation (TVCSE) problem in the first time. In ADTIZD models, Adams–Bashforth discrete formulas as novel discrete formulas are used, giving our ADTIZD models higher accuracy [truncation error being$O(\tau ^{5})$] but less time and space complexity than the ordinary multi-instant models. Enhanced by the integration part, the ADTIZD models can resist large additive noises, where even constant noises cannot decrease their precision. All convergence and robustness performance conclusions about our ADTIZD models are supported by rigorous theoretical proofs and numerical experiments. More comparisons between ADTIZD models and other discrete-time zeroing neural network models are shown in these experiments too. The efficacy of ADTIZD models is finally been validated in the simulation of adopting them in controlling a robotic manipulator.

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.

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.

A variety of realistic industrial problems can be constructed into quadratic programming (QP) problems, especially their time-varying versions. A zeroing neural network (ZNN) as a good approach for dynamic problems can solve QP problems subject to equality constraints in the past. In this article, we propose a unified predefined-time convergent and robust ZNN (PTCR-ZNN) model for solving time-varying QP problems subject to equality or inequality constraints. Compared with the normal ZNN model, the PTCR-ZNN model mainly has advantages in the following three aspects: 1) solving QP problems with or without inequality constraints in a unified model; 2) converging to the optimal solution of QP problems within a predefined time that can be determined in advance; and 3) resisting many external noises with tiny and predictable residual error. These improvements have been rigorously proved in theory. By conducting both qualitative and quantitative simulations with comparisons, the superior properties of the PTCR-ZNN model are further validated. Finally, the application of the PTCR-ZNN model to image fusion task illustrates the efficiency together with its applicability.

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.

In this article, a noise-suppression variable-parameter zeroing neural network (NSVPZNN) is proposed to handle the dynamic Sylvester equation. Differing from the previous zeroing neural networks (ZNNs), a new nonlinear activation function and an especially constructed time-variant parameter are developed to construct the novel NSVPZNN model. Therefore, the NSVPZNN model can achieve faster predefined-time convergence without noise disturbance and have stronger robust performance under multiple noises. Furthermore, the convergence upper bound of the NSVPZNN model is theoretically calculated, and a detailed proof of guaranteeing noise-tolerance performance is given. Numerical simulations verify that the NSVPZNN has better performance than the ZNN, the finite-time convergence ZNN model, the predefined-time convergence ZNN model, and the other variable-parameter ZNN when handling the dynamic Sylvester equation. Finally, the design method of the NSVPZNN is applied to the wheeled manipulator for tracking the butterfly trajectory, which further illustrates the model's reliability.

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 matrix inversion problem plays a very important role in mathematics as well as practical engineering applications. In this article, unlike the traditional fixed-parameter zeroing neural network (ZNN) model, on the basis of the original varying-parameter ZNN (VPZNN) model, an improved VPZNN (IVPZNN) model is established and researched to solve time-varying matrix inversion (TVMI). Specifically, the value of the proposed novel time-varying parameter in the IVPZNN model can grow rapidly over time, which can better meet the needs of ZNN in hardware implementation. In addition, theoretical analyses of the novel time varying parameter and the proposed IVPZNN model are given to guarantee the global superexponential convergence and finite-time convergence. Numerical calculation results verify the superior property of the established IVPZNN model for addressing the TVMI problem, as compared with the existing fixed-parameter ZNN and VPZNN models.

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.

This article focuses on the research of a general time-varying nonlinear optimization (TVNO) problem solving especially in a noise-disturbance environment. For addressing this problem more efficiently, a new noise-enduring and finite-time convergent design formula is suggested to establish a novel zeroing neural network (NZNN). In contrast to the initial zeroing neural network or the noising-enduring zeroing neural network, which either only achieves finite-time convergence or only suppresses external disturbances, the merit of the proposed NZNN model is able to find an error-free optimal solution in a finite time under various different types of external noises. In addition, the detailed mathematical analyses about finite-time convergence and noise endurance are given to prove the excellent characteristics of the NZNN model. Numerical comparative results are provided to demonstrate the accuracy, efficiency, and advantages of the NZNN model for TVNO under various types of external disturbances. Robotic tracking example further validates the applicability of the NZNN model especially in a noise-disturbance environment.

Sylvester equation is often applied to various fields, such as mathematics and control systems due to its importance. Zeroing neural network (ZNN), as a systematic design method for time-variant problems, has been proved to be effective on solving Sylvester equation in the ideal conditions. In this paper, in order to realize the predefined-time convergence of the ZNN model and modify its robustness, two new noise-tolerant ZNNs (NNTZNNs) are established by devising two novelly constructed nonlinear activation functions (AFs) to find the accurate solution of the time-variant Sylvester equation in the presence of various noises. Unlike the original ZNN models activated by known AFs, the proposed two NNTZNN models are activated by two novel AFs, therefore, possessing the excellent predefined-time convergence and strong robustness even in the presence of various noises. Besides, the detailed theoretical analyses of the predefined-time convergence and robustness ability for the NNTZNN models are given by considering different kinds of noises. Simulation comparative results further verify the excellent performance of the proposed NNTZNN models, when applied to online solution of the time-variant Sylvester equation.

The so-called zeroing neural network (ZNN) is an effective recurrent neural network for solving dynamic problems including the dynamic nonlinear equations. There exist numerous unperturbed ZNN models that can converge to the theoretical solution of solvable nonlinear equations in infinity long or finite time. However, when these ZNN models are perturbed by external disturbances, the convergence performance would be dramatically deteriorated. To overcome this issue, this paper for the first time proposes a finite-time convergent ZNN with the noise-rejection capability to endure disturbances and solve dynamic nonlinear equations in finite time. In theory, the finite-time convergence and noise-rejection properties of the finite-time convergent and noise-rejection ZNN (FTNRZNN) are rigorously proved. For potential digital hardware realization, the discrete form of the FTNRZNN model is established based on a recently developed five-step finite difference rule to guarantee a high computational accuracy. The numerical results demonstrate that the discrete-time FTNRZNN can reject constant external noises. When perturbed by dynamic bounded or unbounded linear noises, the discrete-time FTNRZNN achieves the smallest steady-state errors in comparison with those generated by other discrete-time ZNN models that have no or limited ability to handle these noises. Discrete models of the FTNRZNN and the other ZNNs are comparatively applied to redundancy resolution of a robotic arm with superior positioning accuracy of the FTNRZNN verified.

In this article, two improved finite-time convergent complex-valued zeroing neural network (IFTCVZNN) models are presented and investigated for real-time solution of time-varying reciprocal of complex matrices on account of two equivalent processing ways of complex calculations for nonlinear activation functions. Furthermore, a novel nonlinear activation function is explored to modify the comprehensive performance of such two IFTCVZNN models. Compared with existing complex-valued neural networks converging within the limited time, the proposed IFTCVZNN models with the new activation function have better finite-time convergence and less conservative upper bound. Numerical simulations verify that the maximum of convergence time estimated via Lyapunov stability is theoretically much closer to the actual convergence time.

Dynamic (or say, time-varying) problems have been a hot spot of research recently. As a general form of matrix inverse, dynamic Moore-Penrose inverse solving has received more and more attention owing to its broad applications. The approaches based on neural networks have become a popular solution to various dynamic matrix-related problems including dynamic Moore-Penrose inverse. However, existing neural models either only achieve infinitetime instead of finite-time convergence, or are sensitive to noises. Therefore, finite-time convergent neural model, which is simultaneously capable of addressing the noises, is desperately needed for dynamic Moore-Penrose inverse solving. To do that, in this paper, a novel evolution formula is designed based on the widely investigated Zhang neural network (ZNN). Accordingly, two modified ZNN models (MZNN), namely MZNN-R and MZNN-L models, are proposed and analyzed for the right and left dynamic Moore-Penrose inversion of full-rank matrices, respectively. In addition to providing detailed theoretical analyses on the desired finite-time convergence and noise-depression properties of the proposed two models, we also perform two numerical examples for further verification. Furthermore, to illustrate the potential of MZNN models in practical applications, two path-tracking control examples are also presented via a two-dimensional planar three-link and a three-dimensional Kinova Jaco2redundant robot manipulator. The feasibility, extraordinary efficacy, and superiority of the proposed MZNN models for dynamic Moore-Penrose inverse solving are corroborated by both theoretical results and simulation observations.

Complex-valued time-dependent matrix inversion (TDMI) is extensively exploited in practical industrial and engineering fields. Many current neural models are presented to find the inverse of a matrix in an ideal noise-free environment. However, the outer interferences are normally believed to be ubiquitous and avoidable in practice. If these neural models are applied to complex-valued TDMI in a noise environment, they need to take a lot of precious time to deal with outer noise disturbances in advance. Thus, a noise-suppression model is urgent to be proposed to address this problem. In this article, a complex-valued noise-tolerant zeroing neural network (CVNTZNN) on the basis of an integral-type design formula is established and investigated for finding complex-valued TDMI under a wide variety of noises. Furthermore, both convergence and robustness of the CVNTZNN model are carefully analyzed and rigorously proved. For comparison and verification purposes, the existing zeroing neural network (ZNN) and gradient neural network (GNN) have been presented to address the same problem under the same conditions. Numerical simulation consequences demonstrate the effectiveness and excellence of the proposed CVNTZNN model for complex-valued TDMI under various kinds of noises, by comparing the existing ZNN and GNN models.

It has been reported that some specially designed recurrent neural networks and their related neural dynamics are efficient for solving quadratic programming (QP) problems in the real domain. A complex-valued QP problem is generated if its variable vector is composed of the magnitude and phase information, which is often depicted in a time-dependent form. Given the important role that complex-valued problems play in cybernetics and engineering, computational models with high accuracy and strong robustness are urgently needed, especially for time-dependent problems. However, the research on the online solution of time-dependent complex-valued problems has been much less investigated compared to time-dependent real-valued problems. In this article, to solve the online time-dependent complex-valued QP problems subject to linear constraints, two new discrete-time neural dynamics models, which can achieve global convergence performance in the presence of perturbations with the provided theoretical analyses, are proposed and investigated. In addition, the second proposed model is developed to eliminate the operation of explicit matrix inversion by introducing the quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) method. Moreover, computer simulation results and applications in robotics and filters are provided to illustrate the feasibility and superiority of the proposed models in comparison with the existing solutions.

This article aims to solve the Moore-Penrose inverse of time-varying full-rank matrices in the presence of various noises in real time. For this purpose, two varying-parameter zeroing neural networks (VPZNNs) are proposed. Specifically, VPZNN-R and VPZNN-L models, which are based on a new design formula, are designed to solve the right and left Moore-Penrose inversion problems of time-varying full-rank matrices, respectively. The two VPZNN models are activated by two novel varying-parameter nonlinear activation functions. Detailed theoretical derivations are presented to show the desired finite-time convergence and outstanding robustness of the proposed VPZNN models under various kinds of noises. In addition, existing neural models, such as the original ZNN (OZNN) and the integration-enhanced ZNN (IEZNN), are compared with the VPZNN models. Simulation observations verify the advantages of the VPZNN models over the OZNN and IEZNN models in terms of convergence and robustness. The potential of the VPZNN models for robotic applications is then illustrated by an example of robot path tracking.

Zeroing neural network (ZNN) is a powerful tool to address the mathematical and optimization problems broadly arisen in the science and engineering areas. The convergence and robustness are always co-pursued in ZNN. However, there exists no related work on the ZNN for time-dependent nonlinear minimization that achieves simultaneously limited-time convergence and inherently noise suppression. In this article, for the purpose of satisfying such two requirements, a limited-time robust neural network (LTRNN) is devised and presented to solve time-dependent nonlinear minimization under various external disturbances. Different from the previous ZNN model for this problem either with limited-time convergence or with noise suppression, the proposed LTRNN model simultaneously possesses such two characteristics. Besides, rigorous theoretical analyses are given to prove the superior performance of the LTRNN model when adopted to solve time-dependent nonlinear minimization under external disturbances. Comparative results also substantiate the effectiveness and advantages of LTRNN via solving a time-dependent nonlinear minimization problem.

Computing time-varying linear systems is widely encountered in engineering practice and scientific computation. Dynamic neural networks, as a class of modeling approaches, have been intensively explored in recent decades for solving linear equations. The time-varying nature of this problem and the noisy workspace for many engineering practice require two features of practical design: 1) fast convergence in time and 2) robustness against noises and disturbance. Existing solutions usually decouple the problem into two steps by designing a fast-convergent neural controller and then topped with an additional low-pass filter to reach noise robustness. However, due to the interplay of the mentioned two dynamical parts, the overall system may lose stability if the parameters are not well tuned. In this paper, we establish the first dynamical neural model for simultaneously achieving fast-convergence, particularly finite-time convergence, and noise-robustness, with the capability to reject the unknown noise when it is constant or varies slowly. To do so, a superior design formula activated by noise-tolerant nonlinear functions is proposed to enhance the capability of zeroing neural networks (ZNNs), achieving denoising and finite-time convergence in a unified design. According to this design formula, a novel recurrent neural network (RNN) with finite-time convergence and inherently noise-suppression performance [thus termed the finite-time robust RNN (FTRRNN)] is developed and applied to robotic motion tracking illustrated via time-varying linear equation system solving. Furthermore, theoretical analyses on the global stability, the finite-time convergence and the denoising ability of the proposed design formula and the corresponding FTRRNN model are presented in details. The upper bound on the convergence time is also analytically derived. A numerical example is supplied to verify the superior property of the FTRRNN model to the ZNN model according to the results of computing time-varying linear equation system in the presence of additive noises. Finally, an application to robotic motion tracking is presented to show that the presented FTRRNN model can successfully realize the ellipse-path tracking control of a planar two-link manipulator in front of the external disturbances, while the conventional ZNN model fails under the same conditions.

Neural networks have been generally deemed as important tools to handle kinds of online computing problems in recent decades, which have plenty of applications in science and electronics fields. This paper proposes a novel recurrent neural network (RNN) to handle the perturbed time-varying underdetermined linear system with double bound limits on residual errors and state variables. Beyond that, the bound-limited underdetermined linear system is converted into a time-varying system that consists of linear and nonlinear formulas through constructing a nonnegative time-varying variable. Then, theoretical analyses are conducted to verify the superior convergence performance of the proposed RNN model. Furthermore, numerical experiment results and computer simulations demonstrate the superiority and effectiveness of the proposed RNN model for handling the time-varying underdetermined linear system with double bound limits. Finally, the proposed RNN model is applied to the physically limited PUMA560 robot to show its satisfactory applicabilities.

Zeroing neural dynamics (ZND) can be seen as an effective controller to solve various challenging scientific and engineering problems. Computing Lyapunov equation is a kind of important issue in nonlinear systems for stability analysis in control. This paper presents a systematic and constructive procedure on using ZND to design control laws based on the efficient solution of dynamic Lyapunov equation. We particularly address three important aspects in the design: 1) the global stability of ZND, to guarantee the effectiveness of the solution; 2) the robustness against additive noises, to ensure the capability of ZND for using in harsh environments; and 3) the finite-time convergence of ZND, to endow ZND for real-time solution of dynamical problems. To do so, a novel formula is first designed in a unified manner of ZND. Differing from the conventional formula appearing in ZND, the proposed formula simultaneously has finite-time convergence and noise robustness property. According to this novel formula, a novel control law (termed nonlinear neural dynamics, NND) is established to compute dynamic Lyapunov equation in the presence of various additive noises. Both theoretical and simulative results ensure the finite-time convergence and noise robustness property of the NND model for computing dynamic Lyapunov equation in front of various additive noises. As compared to the conventional ZND model for computing dynamic Lyapunov, the superior property of the NND model is further demonstrated.

In the previous work, a finite-time zeroing neural network (ZNN) has been established to find the accurate solution of Lyapunov equation in the presence of no noises. In order to further improve the convergence speed of ZNN and suppress various noises encountered in real applications, in this paper, two robust nonlinear zeroing neural networks (RNZNNs) are designed by adding two novel nonlinear activation functions (AFs) for finding the solution of the Lyapunov equation in the presence of various noises. Unlike the previous ZNN activated by known AFs (e.g., linear activation function, bipolar sigmoid activation function, and power activation function), the proposed two RNZNN models possess predefined-time convergence (instead of finite-time convergence) even in the presence of various noises. The greatest advantage of the predefined-time convergence is independent to initial states of a dynamic system, which is much superior to the finite-time convergence related to initial states, and tremendously modifies the convergence performance. In addition, the predefined-time convergence of the RNZNN models for solving the Lyapunov equation are mathematically proved in detail under various external noises. The simulation comparisons further verify the superiority of the proposed RNZNN models for finding the solution of the Lyapunov equation.

Zeroing neural network (ZNN), as a powerful calculating tool, is extensively applied in various computation and optimization fields. Convergence and noise-tolerance performance are always pursued and investigated in the ZNN field. Up to now, there are no unified ZNN models that simultaneously achieve the finite-time convergence and inherent noise tolerance for computing time-varying quadratic optimization problems, although this superior property is highly demanded in practical applications. In this paper, for computing time-varying quadratic optimization within finite-time convergence in the presence of various additive noises, a new framework for ZNN is designed to fill this gap in a unified manner. Specifically, different from the previous design formulas either possessing finite-time convergence or possessing noise-tolerance performance, a new design formula with finite-time convergence and noise tolerance is proposed in a unified framework (and thus called unified design formula). Then, on the basis of the unified design formula, a unified ZNN (UZNN) is, thus, proposed and investigated in the unified framework of ZNN for computing time-varying quadratic optimization problems in the presence of various additive noises. In addition, theoretical analyses of the unified design formula and the UZNN model are given to guarantee the finite-time convergence and inherent noise tolerance. Computer simulation results verify the superior property of the UZNN model for computing time-varying quadratic optimization problems, as compared with the previously proposed ZNN models.

Two novel nonlinearly activated recurrent neural networks (RNNs) with finite-time convergence [called finite-time RNNs (FTRNNs)] are proposed and analyzed to solve efficiently time-varying systems of nonlinear equations (SoNEs). Compared with previously presented neural networks for solving such a SoNE, the FTRNNs are activated by new nonlinear activation functions and thus possess a better finite-time convergence property. In addition, theoretical analyses about FTRNNs are presented to determine the upper bounds of convergence time under the context of using such two novel nonlinear activation functions. Computer simulations based on a numerical example validate the preponderance of the proposed FTRNNs for time-varying SoNE, as compared to the recently proposed Zhang neural network and its improved version. Finally, an engineering practical example to motion tracking of a robot manipulator demonstrates the feasibility and applicability of the FTRNNs.

Solving Sylvester equation is a common algebraic problem in mathematics and control theory. Different from the traditional fixed-parameter recurrent neural networks, such as gradient-based recurrent neural networks or Zhang neural networks, a novel varying-parameter recurrent neural network, [called varying-parameter convergent-differential neural network (VP-CDNN)] is proposed in this paper for obtaining the online solution to the time-varying Sylvester equation. With time passing by, this kind of new varying-parameter neural network can achieve super-exponential performance. Computer simulation comparisons between the fixed-parameter neural networks and the proposed VP-CDNN via using different kinds of activation functions demonstrate that the proposed VP-CDNN has better convergence and robustness properties.

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.

In this paper, a distributed scheme is proposed for the cooperative motion generation in a distributed network of multiple redundant manipulators. The proposed scheme can simultaneously achieve the specified primary task to reach global cooperation under limited communications among manipulators and optimality in terms of a specified optimization index of redundant robot manipulators. The proposed distributed scheme is reformulated as a quadratic program (QP). To inherently suppress noises originating from communication interferences or computational errors, a noise-tolerant zeroing neural network (NTZNN) is constructed to solve the QP problem online. Then, theoretical analyses show that, without noise, the proposed distributed scheme is able to execute a given task with exponentially convergent position errors. Moreover, in the presence of noise, the proposed distributed scheme with the aid of NTZNN model has a satisfactory performance. Furthermore, simulations and comparisons based on PUMA560 redundant robot manipulators substantiate the effectiveness and accuracy of the proposed distributed scheme with the aid of NTZNN model.

In this paper, online solution of complex-valued systems of linear equations is investigated in the complex domain. Different from the conventional real-valued neural network, which is only designed for real-valued linear equations solving, a fully complex-valued gradient neural network (GNN) is developed for online complex-valued systems of linear equations. The advantages of the proposed complex-valued GNN model decrease the unnecessary complexities in theoretical analysis, real-time computation and related applications. In addition, the theoretical analysis of the fully complex-valued GNN model is presented. Finally, simulative results substantiate the effectiveness of the fully complex-valued GNN model for online solution of the complex-valued systems of linear equations in the complex domain.

A mobile manipulator is a robotic device composed of a mobile platform and a stationary manipulator fixed to the platform. To achieve the repetitive motion control of mobile manipulators, the mobile platform and the manipulator have to realize the repetitive motion simultaneously. To do so, a novel quadratic performance index is, for the first time, designed and presented in this paper, of which the effectiveness is analyzed by following a neural dynamics method. Then, a repetitive motion scheme is proposed by combining the criterion, physical constraints, and integrated kinematical equations of mobile manipulators, which is further reformulated as a quadratic programming (QP) subject to equality and bound constraints. In addition, two important Bridge theorems are established to prove that such a QP can be converted equivalently into a linear variational inequality, and then equivalently into a piecewise-linear projection equation (PLPE). A real-time numerical algorithm based on PLPE is thus developed and applied for the online solution of the resultant QP. Two tracking-path tasks demonstrate the effectiveness and accuracy of the repetitive motion scheme. In addition, comparisons between the nonrepetitive and repetitive motion further validate the superiority and novelty of the proposed scheme.

By following Zhang design method, a new type of recurrent neural network [i.e., Zhang neural network (ZNN)] is presented, investigated, and analyzed for online solution of time-varying linear inequalities. Theoretical analysis is given on convergence properties of the proposed ZNN model. For comparative purposes, the conventional gradient neural network is developed and exploited for solving online time-varying linear inequalities as well. Computer simulation results further verify and demonstrate the efficacy, novelty, and superiority of such a ZNN model and its method for solving time-varying linear inequalities.