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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.