ABSTRACT This paper proposes a robust dual‐integral structure zeroing neural network (ZNN) design framework, effectively overcoming the limitations of existing single‐integral enhanced ZNN models in completely suppressing linear noise. Based on this design framework, a complex‐type dual‐integral structure ZNN (DISZNN) model with inherent linear noise suppression capability is constructed for computing dynamic complex matrix inversion (DCMI) online. The stability, convergence, and robustness of the proposed DISZNN model are ensured via rigorous theoretical analyses. In three distinct experiments involving DCMI (including cases with only imaginary parts, both real and imaginary parts, and high‐dimensional scenarios), the state trajectories of the DISZNN model are well and quickly fitted to the dynamic trajectories of the theoretical solutions with very low residual errors in various linear noise environments. More specifically, the residual errors of the DISZNN model for online computation of DCMI under linear noise environments are consistently below the order of , representing one‐thousandth of the residual errors in existing noise‐tolerant ZNN models. Finally, the DISZNN design framework is applied to construct a controlled chaotic system of a permanent magnet synchronous motor (PMSM) with uncertainties and external disturbances based on real‐world modeling. Experimental results demonstrate that the three state errors of the controlled PMSM chaotic system converge to zero quickly and stably under various conditions (system parameters, external disturbances, and uncertainties), further highlighting the superiority and generalizability of the DISZNN design framework.

Real-time online optimisation plays a crucial role in high-frequency trading (HFT) strategies. The Markowitz model, as a Nobel Prize-winning framework, is widely used for portfolio management optimisation by framing the problem as a constrained quadratic programming task. While conventional analytical methods are typically effective for solving quadratic programming problems with linear constraints, the introduction of both linear equality and inequality constraints in the Markowitz model necessitates the use of numerical methods. The complexity of these numerical solutions presents technical challenges for real-time online optimisation, especially in HFT environments where computational speed and efficiency are critical. To address this challenge, we propose a simplified model that decomposes the problem into analytically solvable and unsolvable components, alongside an innovative dynamic neural network designed to quickly solve the unsolvable components. Overall, this method helps reduce computational load and is well-suited for real-time online computations in HFT settings. Furthermore, we conducted a theoretical analysis and proof of the optimality and global convergence of the solutions obtained using this method. Finally, based on a large set of real stock data, we performed three numerical experiments to validate its effectiveness. Notably, in an experiment using Dow Jones Industrial Average (DJIA) stock data, our approach reduced total costs by 5.54% compared to the commonly used MATLAB quadprog() solver, demonstrating the potential of this method as an efficient tool for portfolio management in HFT scenarios.

With the rapid development of the multi-robot systems, formation control has become a fundamental challenge. Traditional approaches focus mainly on the design of control algorithms to realize specific formation patterns, while neglecting how to determine the desired formation. In this paper, the optimal formation problem based on shape theory is reformulated as a convex optimization problem. A predetermined time convergent zeroing neural dynamics (PDTZND) approach, derived from zeroing neural networks (ZNN), is proposed to efficiently solve this problem. The PDTZND approach ensures that the system error converges in a strict and predetermined time, which provides an efficient, accurate solution for optimal formation. In addition, the convergence of the proposed approach is rigorously analyzed by means of Lyapunov theory, and its validity and superiority are verified by numerical simulations and physical experiments.

Abstract Zeroing neural networks (ZNNs), a specialized class of recurrent neural networks, have demonstrated remarkable effectiveness in matrix computation and dynamic optimization problems due to their inherent parallel computing capabilities. In this paper, a predefined-time and noise reduction ZNN (PTNRZNN) model is proposed for solving convex quadratic programming problems with equality and inequality constraints. Additionally, a new activation function is proposed, demonstrating enhanced accelerated convergence and noise reduction performance compared to previous models. The convergence and robustness of the PTNRZNN model are effectively proven through theoretical assessment. Furthermore, the performance of the PTNRZNN model is further validated through simulation experiments. Finally, the PTNRZNN model is applied to the binary assignment problem in logistics (BAPL), yielding optimized results with an error margin as low as $$10^{-2}$$ 10 - 2 compared to theoretical values. The strong robustness of the method makes it an excellent performer in solving BAPL under noise interference.

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.

The robot industry is developing rapidly, and how to control the redundant manipulators precisely and effectively has become a new hot topic in industry’s development. In recent years, many scholars in the industry have also proposed various control methods. However, most of these methods are proposed assuming that the Jacobian matrix is known. Actually, in practical applications, the detailed information of Jacobian matrix is often not precisely known. Therefore, this article develops a novel data-driven recurrent neural network (RNN) model that can update the Jacobian matrix and joint angles. By defining two dynamic error functions, two RNN designed formulas are used to obtain a continuous RNN (CRNN) model. Subsequently, the CRNN model is discretized by using Euler forward formula, and a discrete RNN (DRNN) model is generated. Then, a classic sliding mode control (SMC) algorithm is introduced, and DRNN-SMC model is further proposed. Moreover, the corresponding rigorous mathematical derivation and proof are carried out. In addition, simulation tests are carried out by using the Kinova Gen2 manipulator, comparing the DRNN model and PD controller, as well as the DRNN-SMC model and DRNN model, validating the precision of the DRNN-SMC model. Additionally, practical experiments using the Kinova Gen3 manipulator are performed to showcase the applicability and versatility of the DRNN-SMC model.

Cloud-assisted Internet of Things (IoT) has become an increasingly popular paradigm to greatly improve the performance of IoT applications by delegating the cloud to manage the massive IoT data. How to achieve secure and real-time traceable data sharing (STDS) is crucial in this paradigm, especially, a large amount of sensitive data produced by IoT devices needs to be stored or accessed to/from the clouds. This article proposes an STDS scheme, which leverages the acrlong DIFC model to allow data owners to not only securely and efficiently share their data produced by IoT devices with data users but also have the capability of tracking the data users’ identity with nonrepudiation based on the hash chain technique. Subsequently, the acrlong HLPN, acrlong SMT-Lib, and Z3 solver are used to formally analyze and verify STDS based on acrlong BMC technique to prove the correctness and security STDS. The formal analysis results show that STDS fulfills its intended security goals. Finally, the performance evaluation results have demonstrated the efficiency of STDS.

As a type of recurrent neural networks (RNNs) modeled as dynamic systems, the gradient neural network (GNN) is recognized as an effective method for static matrix inversion with exponential convergence. However, when it comes to time-varying matrix inversion, most of the traditional GNNs can only track the corresponding time-varying solution with a residual error, and the performance becomes worse when there are noises. Currently, zeroing neural networks (ZNNs) take a dominant role in time-varying matrix inversion, but ZNN models are more complex than GNN models, require knowing the explicit formula of the time-derivative of the matrix, and intrinsically cannot avoid the inversion operation in its realization in digital computers. In this article, we propose a unified GNN model for handling both static matrix inversion and time-varying matrix inversion with finite-time convergence and a simpler structure. Our theoretical analysis shows that, under mild conditions, the proposed model bears finite-time convergence for time-varying matrix inversion, regardless of the existence of bounded noises. Simulation comparisons with existing GNN models and ZNN models dedicated to time-varying matrix inversion demonstrate the advantages of the proposed GNN model in terms of convergence speed and robustness to noises.

Dynamic neural networks are considered as an effective method in the field of scientific computing, among which gradient neural networks (GNNs) are an efficient method for solving static problems. However, when solving dynamic problems, the current GNNs are often subject to lagging errors. In this article, we develop a finite-time convergent GNN (FTCGNN) model for solving static and time-varying systems of linear equations. Different from zeroing neural networks (ZNNs) dedicated to time-varying problem solving, the FTCGNN model has finite-time convergence regardless of the existence of time-varying noises. Simulation results show that the FTCGNN model is effective, among which the comparisons with existing GNNs and ZNNs validate the advantages of the FTCGNN model.

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.

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.

Repetitive motion planning (RMP) is a crucial issue encountered in studies on redundant robot manipulators. Numerous RMP schemes have been established in previous studies wherein simulations are assumed to be free of noise. However, noise is ubiquitous and can severely affect RMP schemes to the point of causing failure. This paper attempts address the limitations imposed by noise by providing the first RMP scheme with inherent noise-suppression capability. The new RMP scheme for redundant robot manipulators in a noisy environment is proposed on the basis of an equality criterion that is robust against additive noise. The equality criterion is established by incorporating the proportional and integral information of the desired end-effector path. The proposed scheme is reformulated as a quadratic program and is calculated by using a recurrent neural network. Comparative simulation results obtained with PA10 and four-link robot manipulators illustrate the effectiveness and superiority of the proposed RMP scheme over the traditional RMP scheme.

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.

Varying-parameter recurrent neural network, being a special kind of neural-dynamic methodology, has revealed powerful abilities to handle various time-varying problems, such as quadratic minimization (QM) and quadratic programming (QP) problems. In this paper, a novel power-type varying-parameter recurrent neural network (PT-VP-RNN) is proposed to solve the perturbed time-varying QM and QP problems. First, based on the generalization of time-varying QM and QP problems, the design process of the PT-VP-RNN is presented in detail. Second, the robustness performance of the proposed PT-VP-RNN is theoretically analyzed and proved. What is more, two numerical examples are simulated to illustrate the robustness convergence performance of PT-VP-RNN even in a large disturbance condition. Finally, two practical application examples (i.e., a robot tracking example and a venture investment example) further verify the effectiveness, accuracy, and widespread applicability of the proposed PT-VP-RNN.

Input disturbances and physical constraints are important issues in the kinematic control of redundant manipulators. In this paper, we propose a novel recurrent neural network to simultaneously address the periodic input disturbance, joint angle constraint, and joint velocity constraint, and optimize a general quadratic performance index. The proposed recurrent neural network applies to both regulation and tracking tasks. Theoretical analysis shows that, with the proposed neural network, the end-effector tracking and regulation errors asymptotically converge to zero in the presence of both input disturbance and the two constraints. Simulation examples and comparisons with an existing controller are also presented to validate the effectiveness and superiority of the proposed controller.

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.

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.

Using neural networks to handle intractability problems and solve complex computation equations is becoming common practices in academia and industry. It has been shown that, although complicated, these problems can be formulated as a set of equations and the key is to find the zeros of them. Zeroing neural networks (ZNN), as a class of neural networks particularly dedicated to find zeros of equations, have played an indispensable role in the online solution of time-varying problem in the past years and many fruitful research outcomes have been reported in the literatures. The aim of this paper is to provide a comprehensive survey of the research on ZNNs, including continuous-time and discrete-time ZNN models for various problems solving as well as their applications in motion planning and control of redundant manipulators, tracking control of chaotic systems, or even populations control in mathematical biosciences. By considering the fact that real-time performance is highly demanded for time-varying problems in practice, stability and convergence analyses of different continuous-time ZNN models are reviewed in detail in a unified way. For the case of discrete-time problems solving, the procedures on how to discretize a continuous-time ZNN model and the techniques on how to obtain an accuracy solution are summarized. Concluding remarks and future directions of ZNN are pointed out and discussed.

In this paper, a new Taylor-type numerical differentiation formula is first presented to discretize the continuous-time Zhang neural network (ZNN), and obtain higher computational accuracy. Based on the Taylor-type formula, two Taylor-type discrete-time ZNN models (termed Taylor-type discrete-time ZNNK and Taylor-type discrete-time ZNNU models) are then proposed and discussed to perform online dynamic equality-constrained quadratic programming. For comparison, Euler-type discrete-time ZNN models (called Euler-type discrete-time ZNNK and Euler-type discrete-time ZNNU models) and Newton iteration, with interesting links being found, are also presented. It is proved herein that the steady-state residual errors of the proposed Taylor-type discrete-time ZNN models, Euler-type discrete-time ZNN models, and Newton iteration have the patterns of O(h(3)), O(h(2)), and O(h), respectively, with h denoting the sampling gap. Numerical experiments, including the application examples, are carried out, of which the results further substantiate the theoretical findings and the efficacy of Taylor-type discrete-time ZNN models. Finally, the comparisons with Taylor-type discrete-time derivative model and other Lagrange-type discrete-time ZNN models for dynamic equality-constrained quadratic programming substantiate the superiority of the proposed Taylor-type discrete-time ZNN models once again.

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.

As a special class of recurrent neural network, Zhang neural network (ZNN) has been recently proposed since 2001 for solving various time-varying problems, and has shown high efficiency and excellent performance for solving the problems in the real domain. In this paper, to solve online the time-varying complex generalized inverse (in most cases, the pseudoinverse) problem in the complex domain, a new type of complex-valued ZNN is further proposed and investigated. The design of such a complex ZNN is based on a complex Zhang function (ZF) which is indefinite and quite different from the usual error function (specially, the scalar-valued energy function) in the studies of conventional algorithms. By introducing five different complex ZFs, five different complex ZNN models (termed complex ZNN-I, ZNN-II, ZNN-III, ZNN-IV, and ZNN-V models) are proposed, developed, and investigated for the online solution of the time-varying complex generalized inverse matrices. Theoretical results of convergence analysis are presented to show the desirable properties of complex ZNN models. In addition, we discover the link between the proposed complex ZNN models and the Getz-Marsden dynamic system in the complex domain. Computer-simulation results further demonstrate the effectiveness of complex ZNN models based on different complex ZFs for the time-varying complex generalized inverse matrices.