A novel finite-time complex-valued zeoring neural network for solving time-varying complex-valued Sylvester equation

A novel finite-time complex-valued zeroing neural network (FTCVZNN) for solving time-varying Sylvester equation is proposed and investigated. Asymptotic stability analysis of this network is examined with any general activation function satisfying a condition or with an odd monotonically increasing activation function. So far, finite-time model studies have been investigated for the upper bound time of convergence using a linear activation function with design formula for the derivative of the error or with variations of sign-bi-power activation functions to zeroing neural networks. A function adaptive coefficient for sign-bi-power activation function (FA-CSBP) is introduced and examined for faster convergence. An upper bound on convergence time is derived with the two components in the function adaptive coefficients of sign-bi-power activation function. Numerical simulation results demonstrate that the FTCVZNN with function adaptive coefficient for sign-bi-power activation function is faster than applying a sign-bi-power activation function to the zeroing neural network (ZNN) and the other finite-time complex-valued models for the selected example problems.     

Solving future equation systems using integral-type error function and using twice ZNN formula with disturbances suppressed

In this paper, for solving future equation systems, two novel discrete-time advanced zeroing neural network models are proposed, analyzed and investigated. First of all, by using integral-type error function and twice zeroing neural network (or termed, Zhang neural network) formula, as the preliminaries and bases of future problems solving, two continuous-time advanced zeroing neural network models are presented for solving continuous time-variant equation systems. Secondly, a one-step-ahead numerical differentiation rule termed 5-instant discretization formula is presented for the first-order derivative approximation with higher computational precision. By exploiting the presented 5-instant discretization formula to discretize the continuous-time advanced zeroing neural network models, two novel discrete-time advanced zeroing neural network models are proposed. Theoretical analyses on the convergence and precision of the discrete-time advanced zeroing neural network models are proposed. In addition, in the presence of disturbance, the proposed discrete-time advanced zeroing neural network models still possess excellent performance. Comparative numerical experimental results further substantiate the efficacy and superiority of the proposed discrete-time advanced zeroing neural network models for solving the future equation systems.     

Fixed-time convergence integral-enhanced ZNN for calculating complex-valued flow matrix Drazin inverse

The complex-valued flow matrix Drazin inverse has recently attracted considerable interest from researchers due to its great academic value. In this paper, a fixed-time convergence integral-enhanced zeroing neural network (FTCIEZNN) model is proposed and investigated for calculating the Drazin inverse of complex-valued flow matrix. Since the FTCIEZNN model possesses fixed-time convergence, its upper limit of convergence time is irrelevant to initial conditions and can be adjusted by specified system parameters. Meanwhile, by adopting the newly designed reformed nonlinear activation function (RNAF) and variable parameters, the FTCIEZNN model converges rapidly in a relatively fast fixed-time and its robustness is dramatically strengthened. In addition, the upper limit of the convergence time in the absence of noise and the upper limit of the steady-state error in the presence of time-varying bounded noise are given by a scrupulous mathematical logic calculation. Furthermore, the outcomes of the numerical simulations demonstrate that the FTCIEZNN model outshines existing zeroing neural network models in calculating complex-valued flow matrix Drazin inverse. Finally, an application based on the FTCIEZNN model in image encryption fully illustrates the practical value of the FCIEZNN model.     

Prescribed-time robust ZNN models for solving equality and inequality systems

At present, there are few studies on solving time-variant linear equality and inequality systems (TVLEIS) under noise interference, and the numerical algorithm has limitations in solving the TVLEIS problems. Therefore, to determine the online solution of the TVLEIS in a complex environment, two prescribed-time robust zeroing neural network (PTRZNN) models are proposed, investigated, and verified in this paper. The PTRZNN models have a faster convergence rate and superior robustness compared with other zeroing neural network models activated by common activation functions. In addition, the detailed theoretical derivation of the prescribed-time convergence and robustness of the PTRZNN models is provided. The effectiveness and superiority of the PTRZNN models for determining the TVLEIS are further demonstrated by simulation results. It is worth mentioning that the design idea of the PTRZNN models is applied to the multi-agent system, which shows the practical value of the PTRZNN models.     

A modified noise-tolerant ZNN model for solving time-varying Sylvester equation with its application to robot manipulator

Recently, Xiao et al. (2021) proposed an efficient noise-tolerant zeroing neural network (NTZNN) model with fixed-time convergence for solving the time-varying Sylvester equation. In this paper, we propose a modified version of their NTZNN model, named the modified noise-tolerant zeroing neural network (MNTZNN) model. It extends the NTZNN model to a more general form and then we prove that, with appropriate parameter selection, our new MNTZNN model can significantly accelerate the convergence of the NTZNN model. Numerical experiments confirm that the MNTZNN model not only maintains fixed-time convergence and noise-tolerance but also has a faster convergence rate than the NTZNN model under certain conditions. In addition, the design strategy of the MNTZNN is also successfully applied to the path tracking of a 6-link planar robot manipulator under noise disturbance, which demonstrates its applicability and practicality.     

A robust fast convergence zeroing neural network and its applications to dynamic Sylvester equation solving and robot trajectory tracking

In order to find the theoretical solution of a dynamic Sylvester equation (DSE) in noisy environment, a robust fast convergence zeroing neural network (RFCZNN) is proposed in this paper. Unlike the original zeroing neural network (ZNN) model with existing activation functions (AF), by introducing a new AF, the proposed RFCZNN model guarantees fixed-time convergence to theoretical solution of DSE and robustness against noise simultaneously. The effectiveness and robustness of the proposed RFCZNN model are investigated in theory and demonstrated through simulation results. In addition, its effectiveness and robustness are further verified by a successful robotic trajectory tracking application in noisy environment.     

Accelerating a recurrent neural network to finite-time convergence using a new design formula and its application to time-varying matrix square root

In this paper, a new design formula is presented to accelerate the convergence speed of a recurrent neural network, and applied to time-varying matrix square root finding in real time. Then, according to such a new design formula, a finite-time Zhang neural network (FTZNN) is proposed and investigated for finding time-varying matrix square root. In comparison with the original Zhang neural network (ZNN) model, the FTZNN model makes a breakthrough in the convergence performance (i.e., from infinite time to finite time). In addition, theoretical analyses of the design formula and the FTZNN model are provided in details. Comparative results further verify the superiority of the proposed FTZNN model to the original ZNN model for finding time-varying matrix square root.     

Finite-time adaptive neural prescribed tracking control of stochastic nonlinear systems with multiple power terms and unknown time-varying powers

This research addresses the problem of finite-time tracking error constrained control for a class of non-strict stochastic nonlinear systems with unknown time-varying powers and multiple power terms. Based on the conversion from constrained tracking error to an unconstrained signal with the same effect, by adopting the backstepping technique together with adaptive neural network control, a controller with upper and lower time-varying power bounds is designed to meet the prescribed performance control scheme in finite-time. Finally, two simulation examples are shown to verify the effectiveness of the commendatory control method.     

Finite-time convergent zeroing neural network for solving time-varying algebraic Riccati equations

Various forms of the algebraic Riccati equation (ARE) have been widely used to investigate the stability of nonlinear systems in the control field. In this paper, the time-varying ARE (TV-ARE) and linear time-varying (LTV) systems stabilization problems are investigated by employing the zeroing neural networks (ZNNs). In order to solve the TV-ARE problem, two models are developed, the ZNNTV-ARE model which follows the principles of the original ZNN method, and the FTZNNTV-ARE model which follows the finite-time ZNN (FTZNN) dynamical evolution. In addition, two hybrid ZNN models are proposed for the LTV systems stabilization, which combines the ZNNTV-ARE and FTZNNTV-ARE design rules. Note that instead of the infinite exponential convergence specific to the ZNNTV-ARE design, the structure of the proposed FTZNNTV-ARE dynamic is based on a new evolution formula which is able to converge to a theoretical solution in finite time. Furthermore, we are only interested in real symmetric solutions of TV-ARE, so the ZNNTV-ARE and FTZNNTV-ARE models are designed to produce such solutions. Numerical findings, one of which includes an application to LTV systems stabilization, confirm the effectiveness of the introduced dynamical evolutions.     

Fixed-time outer synchronization of hybrid-coupled delayed complex networks via periodically semi-intermittent control

In this paper, the fixed-time synchronization between two delayed complex networks with hybrid couplings is investigated. The internal delay, transmission coupling delay and self-feedback coupling delay are all included in the network model. By introducing and proving a new and important differential equality, and utilizing periodically semi-intermittent control, some fixed-time synchronization criteria are derived in which the settling time function is bounded for any initial values. It is shown that the control rate, network size and node dimension heavily influence the estimating for the upper bound of the convergence time of synchronization state. Finally, numerical simulations are performed to show the feasibility and effectiveness of the control methodology by comparing with the corresponding finite-time synchronization problem.     

Finite-time synchronization of delayed dynamical networks via aperiodically intermittent control

In this paper, we concern the finite-time synchronization problem for delayed dynamical networks via aperiodically intermittent control. Compared with some correspondingly previous results, the intermittent control can be aperiodic which is more general. Moreover, by establishing a new differential inequality and constructing Lyapunov function, several useful criteria are derived analytically to realize finite-time synchronization for delay complex networks. Additionally, as a special case, some sufficient conditions ensuring the finite-time synchronization for a class of coupled neural network are obtained. It is worth noting that the convergence time is carefully discussed and does not depend on control widths or rest widths for the proposed aperiodically intermittent control. Finally, a numerical example is given to demonstrate the validness of the proposed scheme.     

Delay-dependent robust exponential stability of Markovian jumping reaction-diffusion Cohen–Grossberg neural networks with mixed delays

This paper is devoted to investigating the robust stochastic exponential stability for reaction-diffusion Cohen–Grossberg neural networks (RDCGNNs) with Markovian jumping parameters and mixed delays. The parameter uncertainties are assumed to be norm bounded. The delays are assumed to be time-varying and belong to a given interval, which means that the lower and upper bounds of interval time-varying delays are available. Some criteria for delay-dependent robust exponential stability of RDCGNNs with Markovian jumping parameters are established in terms of linear matrix inequalities (LMIs), which can be easily checked by utilizing Matlab LMI toolbox. Numerical examples are provided to demonstrate the efficiency of the proposed results.     

Synchronization of fractional-order spatiotemporal complex-valued neural networks in finite-time interval and its application

This paper focuses on the synchronization of fractional-order complex-valued neural networks (FOCVNNs) with reaction–diffusion terms in finite-time interval. Different from the existing complex-valued neural networks (CVNNs), the reaction–diffusion phenomena and fractional derivative are first considered into the system, meanwhile, the parameter switching (the system parameters will switch with the state) is considered, which makes the presented model more comprehensive. By choosing an appropriate Lyapunov function, the driver and response systems achieve Mittag-Leffler synchronization under a suitable controller. In addition, based on the fractional calculus theorem and the basic inequality methods, a criterion of synchronization for the error system in finite-time interval is derived and the upper bound of the corresponding finite synchronization time can be obtained. Finally, two examples are provided, one is a numerical example to explain the effectiveness of the main results, and the other shows that the results of this paper can be applied to image encryption for any size with high-security coefficient.     

Finite-time congestion tracking control for TCP/AWM network systems employing event-triggered mechanism

This paper addresses the tracking control problem of TCP/AWM network systems in presence of nonresponsive data flows of category user datagram protocol (UDP) flows. Firstly, a modified network system model is established by a certain suitable variable transformation, and then a fuzzy logic system (FLS) emulator is used to approximate the nonlinear terms in the network dynamics representation system. Secondly, inspired by the idea of the prescribed performance control (PPC), a novel finite-time performance function (NFTPF) is proposed. In turn, an adaptive finite-time congestion control strategy is designed by compatible usage as appropriate of a barrier Lyapunov function (BLF), the backstepping control synthesis, and an event-triggered mechanism. The proposed control strategy can not only make the tracking error to satisfy the pre-assigned transient and steady-state performance, but also ensure that all the closed-loop signals remain semi-globally uniformly ultimately bounded (SGUUB). In addition, the designed congestion control strategy eliminates potential occurrence of Zeno behavior. A set of simulation results are presented to clarify the feasibility and effectiveness of proposed methodological approach and the designed congestion controller.     

Fast fixed-time synchronization control analysis for a class of coupled delayed Cohen-Grossberg neural networks

In a fixed-time control system, the convergence rate and the fixed settling time are two important performance indexes. In this paper, a novel fixed-time control law is proposed and designed to control a class of coupled delayed Cohen-Grossberg neural networks to achieve synchronization with fast convergence rate within a fixed settling time. It should be emphasized that the derived settling time approach can provide a tighter settling time to more effectively reflect the performance for fast convergence rate of the considered controlled system. Moreover, to show the advantages of the proposed fixed-time control law and the derived fixed settling time approach, the existing related control laws and fixed settling time approaches are further discussed. In addition, the obtained fixed-time synchronization control theory is applied to a secure communication scenario, which further shows the feasibility and innovation of the addressed theoretical results.     

Finite-time event-triggered containment control of multiple Euler–Lagrange systems with unknown control coefficients

This paper researches the finite-time event-triggered containment control problem of multiple Euler–Lagrange systems (ELSs) with unknown control coefficients. To realize an accurate convergence time, the settling-time performance function is employed to ensures the steady-state and dynamic properties of the containment errors in the resulting system. Meanwhile, to handle unknown control coefficients, adaptive neural networks (ANNs) with an additional saturated term are designed, which removes the requirement of full rank control coefficients in traditional control methods. By establishing an event-triggered mechanism, a novel finite-time event-triggered containment control law is designed, which yields the semi-global practical finite-time stable (SGPFS) of the resulting closed-loop system without Zeno phenomenon according to the finite-time stability criterion. The effectiveness of the designed methodology is verified by simulation.     

Finite-time stabilization of nonlinear systems via impulsive control with state-dependent delay

This paper focuses on the issue of finite-time stability for a general form of nonlinear systems subject to state-dependent delayed impulsive controller. Based on the Lyapunov theory and the impulsive control theory, sufficient conditions for finite-time stability (FTS) and finite-time contractive stability (FTCS) are obtained. Additionally, we apply theoretical results to finite-time synchronization of chaotic systems and design the effective state-dependent delayed impulsive controllers in terms of techniques of linear matrix inequality (LMI). Finally, we present two numerical examples of finite-time synchronization of cellular neural networks and Chua’s circuit to verify the effectiveness of our results.     

On novel trajectory tracking control of quadrotor UAV: A finite-time guaranteed performance approach

In this work, aiming at the trajectory tracking control of the quadrotor UAV subject to external disturbances and model uncertainties, a finite-time approach with preassigned performance guaranteed is proposed. First, the control system is decoupled into translational and rotational subsystems. Then, in both two subsystems, the performance bounds constructed by the newly established appointed-time performance functions are devised for guaranteeing the tracking performance, and the controllers are designed via applying the dynamic surface control technique with integral barrier Lyapunov functions involved. Moreover, finite-time tracking differentiators and finite-time multivariable disturbance observers are exploited to estimate the target signals and the lumped disturbances, respectively. Finally, two examples of simulation are carried out to validate the effectiveness and superiority of the proposed control method.     

Dynamic output feedback sliding mode control for spacecraft hovering without velocity measurements

In consideration of target angular velocity uncertainty and external disturbance, a modified dynamic output feedback sliding mode control (DOFSMC) method is proposed for spacecraft autonomous hovering system without velocity measurements. As a stepping-stone, an additional dynamic compensator is introduced into the design of sliding surface, then an augmented system is reconstructed with the system uncertainty and external disturbance. Based on the linear matrix inequality (LMI), a sufficient condition is given, which guarantees the disturbance attenuation performance of sliding mode dynamics. By introducing an auxiliary variable, a modified version of adaptive sliding mode control (ASMC) law is designed, and the finite-time stability of sliding variable is established by the Lyapunov stability theory. Compared with other results, the proposed method is less conservative and can decrease the generated control input force significantly. Finally, two simulation examples are performed to validate the effectiveness of the proposed method.     

Fixed-time synchronization of delayed BAM neural networks via new fixed-time stability results and non-chattering quantized controls

In this paper, in order to obtain a smaller estimation of settling time, reduce chattering caused by sign function and improve network communication efficiency, the fixed-time (FXT) synchronization of delayed BAM neural networks is analyzed based on some new FXT stability results and non-chattering quantized controllers. Firstly, by comprehensively discussing the conditions of power laws in differential inequalities, a new FXT stability lemma is presented and a smaller upper bound of settling time is estimated. Then, unlike previous controllers with sign functions, a non-chattering quantized state feedback control and a non-chattering quantized pinning control are designed, and some sufficient conditions are derived to ensure FXT synchronization of the established system. Finally, two numerical simulations are given to verify the effectiveness of the theoretical results. The results show that compared with the previous researches, this paper provides a smaller upper bound. However, the convergence time of the uncontrolled nodes is indirectly affected by the coupling of the controlled nodes and is much longer than the estimated upper bound.