This article focuses on dynamic integral sliding mode control (SMC) of uncertain Takagi–Sugeno fuzzy delayed systems on time scales. SMC approaches for both continuous-and discrete-time fuzzy delayed systems are designed in a unified framework. First, we design a dynamic controller to guarantee global asymptotic stabilization (GAS) withH∞performance of the addressed systems. Second, to better adapt to the uncertainty characteristics of fuzzy models, an integral sliding mode surface (SMS) considering states, inputs, and uncertainties is proposed, which is an important contribution of this article. By utilizing the Lyapunov function and timescale calculus, it is shown that all states of the addressed control system can be driven close enough to the SMS and global asymptotic convergence (GAC) of the sliding motion can be ensured under matrix inequality criteria. In addition, the chattering phenomenon near the origin of discrete-time SMC system can be avoided in this article. Finally, three simulation examples are offered to illustrate the feasibility of the proposed control schemes.

This article proposes a novel distributed neuro-adaptive 3-D formation tracking control framework of multiple autonomous underwater vehicles (multi-AUVs) subject to marine environmental disturbances. On the one hand, we assume that all AUVs can obtain the real-time states. By introducing a series of variable transformations, the multi-AUV system is transformed into an underactuated nonlinear system with virtual control input. Radial basis function neural networks (RBFNNs), whose weights are updated online, are utilized to approximate nonlinear functions. Considering environmental disturbances, a virtual controller is designed such that all AUVs track the leader while maintaining the desired formation geometry. Then, the actual controller is given as an adaptive form according to the virtual control signals. On the other hand, we assume that all AUVs can only obtain the sampling states of themselves and their neighbors under the predefined event-triggered conditions. Multi-AUV system is transformed into a second-order system with complex nonlinear dynamics, then their states are reconstructed via a neuro-adaptive state observer using sampling states, and a virtual controller is proposed such that all AUVs track the leader while maintaining the desired formation geometry under local communication with no Zeno behavior. Finally, numerical simulations are carried out to demonstrate the effectiveness of the proposed control design.

This article studies a time-segmented design method to achieve prescribed-time formation control (PTFC) while ensuring collision avoidance (CA) by introducing the virtual reference signal for multiagent systems (MASs). Simultaneously achieving these control objectives is challenging, especially in situations where there is a conflict between PTFC and CA. Three different Lyapunov functions (LFs) in the steady-state and transient-state stages play an important role in analyzing the prescribed-time convergence and collision-free characteristics, where the prescribed time can also be set in advance and is not affected by the initial values and parameters of MASs. In addition, the mismatching terms in transient-state stages are dealt with by the designed event-triggered mechanism (ETM), which makes this control strategy more intelligent and determines whether MASs should perform the formation task or CA according to the relationship between the real-time distances of the agents. The conclusion is extended to a class ofn-order MASs, where they can compensate for the mismatching terms and adjust the transient behaviors by the backstepping method and high-order filter, rather than designing distinct controller forms for each of these tasks. Finally, the validity of the proposed control strategy is verified through simulation examples.

Most of the existing event-triggered mechanisms (ETMs) were designed according to the difference between the quadratic form of measurement errors and the quadratic form of sampling states (or real-time states). In order to reduce the amount of data transmission and develop ETMs for continuous-time and discrete-time delayed nonlinear systems (NSs) simultaneously, this article investigates quasi-synchronization (QS) of NSs on time scales based on a novel ETM, which is designed according to the convergence rate instead of measurement errors of the addressed systems. First, a novel ETM is designed under known nonlinear dynamics, and it is demonstrated that QS with given convergence rate and error level can be achieved under matrix inequality criteria. Second, if the nonlinear functions are unknown, we adapt our ETM to handle this special case. Not only QS but also complete synchronization with given convergence rate can be achieved under the ETMs. If the constructed Lyapunov functions passes through 0, the designed ETM will keep it at the origin. In this case, finite-time synchronization is achieved. Third, under the designed ETMs, it is proved that Zeno behavior can be excluded. At last, four numerical simulations are presented to demonstrate the feasibility and the advantage of the designed ETMs in this article.

In recent years, adaptive drive-response synchronization (DRS) of two continuous-time delayed neural networks (NNs) has been investigated extensively. For two timescale-type NNs (TNNs), how to develop adaptive synchronization control schemes and demonstrate rigorously is still an open problem. This article concentrates on adaptive control design for synchronization of TNNs with unbounded time-varying delays. First, timescale-type Barbalat lemma and novel timescale-type inequality techniques are first proposed, which provides us practical methods to investigate timescale-type nonlinear systems. Second, using timescale-type calculus, novel timescale-type inequality, and timescale-type Barbalat lemma, we demonstrate that global asymptotic synchronization can be achieved via adaptive control under algebraic and matrix inequality criteria even if the time-varying delays are unbounded and nondifferentiable. Adaptive DRS is discussed for TNNs, which implies our control schemes are suitable for continuous-time NNs, their discrete-time counterparts, and any combination of them. Finally, numerical examples on TNNs and timescale-type chaotic Ikeda-like oscillator with unbounded time-varying delays are carried out to verify the adaptive control schemes.

The vast majority of published event-triggered mechanisms (ETMs) are constructed based on measurement errors, which introduces a problem naturally that they are updated when the measurement errors exceed the thresholds although the current obtained sampling states can make systems converge well. With this problem in mind, we redesign ETMs for quasi-synchronization of T-S fuzzy neural networks (FNNs) with time delays on time scales. First, a novel ETM is designed for continuous-time FNNs with time-varying delays to achieve quasi-synchronization, with which synchronization errors is suppressed to globally exponentially converge to a ball. Second, we introduce the ETM for continuous-time FNNs to discrete-time FNNs, owing to the existence of discrete-time states, the Lypunov function of synchronization errors run over the exponentially decay curve, but it can be suppressed to evolve under another exponentially decay curve. Third, for FNNs on time scales with constant and time-varying delays, we estimate the forward jump operator of the Lyapunov functions and design ETMs to guarantee that the Lypunov functions evolve under the exponentially decay curves, so quasi-synchronization can be achieved. Last but not least, we prove that Zeno behavior will not happen and four numerical examples are introduced to verify the validity and the superiority of the proposed ETMs in reducing information transmission.

This article attempts to design the prescribed-time time-varying deployment schemes for first-order and second-order nonlinear multiagent systems (MASs). We assume that all agents can obtain the information of their current and final relative positions with their neighbors, and the final absolute velocities (as well as their current and final relative velocities, the final absolute accelerations for the second-order MASs) through a communication network, whereas two boundary agents are able to obtain their current and final absolute positions (as well as their current and final absolute velocities for the second-order MASs). The neighbor relationship of all agents is described by a spatial variable and two static-feedback controllers are introduced, which can be expressed as a second-order space difference of the spatial variable. Then, the deployment of MASs can be transformed into the stabilization of discrete-space partial differential equation (PDE) systems. Three virtual agents are introduced to constitute the Dirchlet and Neumann boundary conditions. Several algebraic inequality criteria are derived to guarantee that the prescribed-time time-varying deployment can be achieved within a prescribed time under the Dirchlet and mixed boundary conditions. Unlike the published results, our results are derived based on the discrete-space PDE systems instead of continuous-space PDE systems, which is consistent with the discrete spatial distribution of agents. Finally, two numerical examples are given to illustrate the effectiveness of our results.

This article attempts to study observer-based output-feedback tracking control problems of multiple-input multiple-output strict-feedback nonlinear systems (MIMOSFNSs) on time scales. If all nonlinear dynamics are known, the corresponding virtual and actual controllers are designed based on system output and observer states such that system output closely tracks the desired signal. If all nonlinear dynamics are unknown, they are approximated by introducing several Takagi–Sugeno fuzzy systems, then a fuzzy observer, the corresponding virtual and actual controllers, and novel parameter adaptive laws are proposed to achieve output-feedback tracking control for MIMOSFNSs on time scales. It is worth emphasizing that our control schemes are applicable to high-order high-dimensional MIMOSFNSs without using complex nonlinear transformations, and the contradiction of causality in the recursive control design can be avoided. Last but not least, since MIMOSFNSs have been investigated on time scales, our theoretical results can be appropriate for continuous-time MIMOSFNSs, their discrete-time counterparts and arbitrary combination of them. Finally, simulation studies are conducted to verify the validity of the proposed output-feedback tracking control schemes.

This article addresses distributed adaptive fuzzy 3-D formation tracking control of multiple autonomous underwater vehicles (AUVs) subject to marine environmental disturbances. First, by constructing a coordinate transformation, AUV model is transformed into a simple second-order systems with nonlinear dynamics. Second, fuzzy logic systems are utilized to approximate the complex nonlinear dynamics and a distributed adaptive control scheme is carried out under the assumption that all AUVs have access to the real-time information of themselves and their neighbors. Third, we assume that only sampling states of their neighbors under the event-triggered conditions are available. System states are reconstructed via the fuzzy state observers, and then the other distributed adaptive control scheme is proposed to ensure that all AUVs track the leader with the desired formation configuration under local communication. The coordinate transformation and fuzzy logic approximation method not only reduce computation but also simplify control design. Finally, a numerical simulation is carried out to demonstrate the effectiveness of the proposed control design.

Global exponential stability (GES) for impulsive timescale-type nonautonomous neural networks (ITNNNs) with mixed delays is investigated in this article. Discrete time-varying and infinite distributed delays (DTVIDDs) are taken into consideration. First, an improved timescale-type Halanay inequality is proven by timescale theory. Second, several algebraic inequality criteria are demonstrated by constructing impulse-dependent functions and utilizing timescale analytical techniques. Different from the published works, the theoretical results can be applied to GES for ITNNNs and impulsive stabilization design of timescale-type nonautonomous neural networks (TNNNs) with mixed delays. The improved timescale-type Halanay inequality considers time-varying coefficients and DTVIDDs, which improves and extends some existing ones. GES criteria for ITNNNs cover the stability conditions of discrete-time nonautonomous neural networks (NNs) and continuous-time ones, and these theoretical results hold for NNs with discrete-continuous dynamics. The effectiveness of our new theoretical results is verified by two numerical examples in the end.

Almost all event-triggered control (ETC) strategies were designed for discrete-time or continuous-time systems. In order to unify these existing theoretical results of ETC and develop ETC strategies for nonlinear systems, whose state variables evolve steadily at one time and change intermittently at another time, this article investigates quasisynchronization of delayed neural networks (NNs) on time scales with discontinuous activation functions via ETC approaches. First, the existence of the Filippov solutions is proved for discontinuous NNs with finite discontinuities. Second, two static event-triggered conditions and two dynamic event-triggered conditions are established to avoid continuous communication between the master-slave systems under algebraic/matrix inequality criteria. Third, under static/dynamic event-triggered conditions, a positive lower bound of event-triggered intervals is demonstrated to be greater than a positive number for each event-based controller, which shows that the Zeno behavior will not occur. Finally, two numerical simulations are carried out to show the effectiveness of the presented theoretical results in this article.

This article reconsiders synchronization problem of linear complex networks with time-varying delay on time scales. For different types of time scales, aperiodically intermittent control scheme is established by using a matrix-based convex combination method, which has great potential in reducing control consumption and saving communication bandwidth. By employing a common Lyapunov function, aperiodically intermittent controllers are utilized successfully to achieve synchronization of linear delayed complex networks on special time scales onto an isolated node. Next, by constructing a special Lyapunov function with time-varying coefficients, sufficient criteria that consist of two linear matrix inequalities are demonstrated to make linear delayed complex networks on general time scales synchronized onto an isolated system with an exponential convergence rate given in advance. Due to delayed complex networks in this article defined on time scales, the proposed control schemes are applicable to continuous-time networks, their discrete-time forms, and any combination of them. Four numerical examples are offered to highlight the effectiveness and superiority of the proposed aperiodically intermittent control schemes at last.

In this article, we focus on adaptive tracking control design for strict-feedback nonlinear systems with full state constraints. Unlike the barrier Lyapunov function method, all theoretical results are derived by direct method instead of introducing nonlinear state-dependent transformations, which reduces the calculated amount and the high sensitivity of control signals, and provides a new perspective to solve state-constrained problems. Then, by using the backstepping technique, the proposed adaptive controller maintains that the output closely tracks the desired trajectory, all signals in the closed-loop systems are bounded and the time-varying state constraints are not violated. Finally, two numerical simulations are provided to exhibit the effectiveness and superiority of the proposed adaptive tracking controller.

This article focuses on the impulsive stabilization of nonautonomous timescale-type neural networks (NTNNs) with constant and unbounded time-varying delays. By choosing two discontinuous piecewise linear functions and employing the convex combination method, several algebraic criteria are demonstrated to achieve globally asymptotic stabilization of NTNNs with constant and unbounded time-varying delays. First, the asymptotic stability theorem for timescale-type systems is constructed by utilizing the timescale theory. Based on this asymptotic stability theorem, an impulsive control scheme is designed to guarantee global asymptotic stabilization of NTNNs with constant delay. Second, we propose several impulsive control schemes to achieve globally asymptotic stabilization of NTNNs with unbounded delay by virtue of the comparison strategy. Globally asymptotic stabilization criteria for NTNNs include the theoretical results of continuous-time neural networks (NNs), their discrete-time forms and NNs on continuous-discrete hybrid time scales. Especially, impulsive control for globally asymptotic stabilization of NTNNs with proportional delay is demonstrated without any variable transformation. Finally, the effectiveness of our proposed theoretical results is verified by four numerical examples.

Most synchronization criteria are scale-free on time evolution, whose main research objects are discrete-time/continuous-time systems. Unlike these theoretical results, in order to develop impulsive control schemes for discrete-time and continuous-time neural networks (NNs) in a unified framework. This article investigates impulsive control design of timescale-type NNs (TNNs) with parameter mismatches and time-varying delays (TVDs). First, several timescale impulsive differential inequalities are demonstrated by the timescale theory, which offer new inequality techniques for the investigation of timescale-type impulsive systems. Next, some criteria are proved for discrete-time NNs and TNNs by utilizing impulsive control theory, timescale inequality techniques, and the average impulsive interval method. Unlike the published works, this article gives some impulsive control schemes to ensure quasi-synchronization (QS) even if there exist TVDs in TNNs. In the end, four simulation examples are offered to demonstrate the validness of the obtained theoretical results.

The existing results of Lagrange stability for neural networks with distributed time delays are scale-free, which introduces conservativeness naturally. A class of Takagi–Sugeno fuzzy memristive neural networks (FMNNs) on time scales with discrete time-varying and infinite distributed delays is brought in this article. First, a new scale-limited Halanay inequality is demonstrated by timescale theory. Next, on the basis of inequality techniques on time scales, some new scale-limited algebraic criteria and linear matrix inequality criteria of Lagrange stability are obtained by comparison strategy and generalized Halanay inequality. All scale-limited sufficient criteria of Lagrange stability for FMNNs not only apply to continuous-time FMNNs and their discrete-time analogs, but also could deal with the arbitrary combination of them. Finally, two numerical simulations are given to verify the validity of the obtained theoretical results.

The published stability criteria for impulsive neural networks are scale-free on time line, which is only appropriate for discrete or continuous ones. The issue of global exponential stability for impulsive delayed neural networks on time scales is analyzed by employing the convex combination method in this article. Several algebraic and linear matrix inequality conditions are proved by constructing impulse-dependent Lyapunov functionals and using timescale inequality techniques. Unlike the published works, impulsive control strategies can be designed by utilizing our theoretical results to stabilize delayed neural networks on time scales if they are unstable before introducing impulses. Sufficient criteria for global exponential stability in this article are derived based on the timescale theory, and they are applicable to discrete-time impulsive neural networks, their continuous-time analogues, and neural networks whose states are discrete at one time and continuous at another time. Four numerical examples are offered to demonstrate the effectiveness and superiority of our new theoretical results in the end.

The extensively studied fractional-order dynamical networks only contain a uniform fractional derivative, which restricts the scope of fractional-order investigation. One type of Takagi–Sugeno fuzzy second-fractional-order linear networks is established in this article. First, by defining a novel Lyapunov functional, a sufficient criterion is offered to ensure global asymptotic stability of the addressed networks with time delay. Second, a fuzzy state-feedback control scheme is designed for the Takagi–Sugeno fuzzy second-fractional-order delayed linear networks to achieve global stabilization. Third, considering that only a few scholars have worked on adaptive control of fractional-order networks, an adaptive control method is also designed to realize the global stabilization of the Takagi–Sugeno fuzzy second-fractional-order linear networks, which does not contain the sign function and chattering problem can be averted. All results, given as algebraic criteria, are proved directly from the Takagi–Sugeno fuzzy second-fractional-order networks without utilizing the reduced-order approach. Three simulation examples are conducted to show the validity of the theoretical results at last.

This article investigates the problem of multistability of almost-periodic solutions of Takagi-Sugeno fuzzy neural networks with nonmonotonic discontinuous activation functions and time-varying delays. Based on the geometrical properties of nonmonotonic activation functions, by using the Ascoli-Arzela theorem and the inequality techniques, it is demonstrated that under some reasonable conditions, the addressed networks have a locally exponentially stable almost-periodic solution in some hyperrectangular regions. We also estimate the attraction basins of the locally stable almost-periodic solutions, which indicates that the attraction basins of the locally exponentially stable almost-periodic solution can be larger than original hyperrectangular regions. These results, which include boundedness, globally attractivity, multiple stability, and attraction basins, generalize and improve the earlier publications, and can be extended to monostability and multistability of Takagi-Sugeno fuzzy neural networks with nonmonotonic discontinuous activation functions. Finally, several numerical examples are given to show the feasibility, the effectiveness, and the merits of the theoretical results.

This article is dedicated to designing a novel periodic impulsive control strategy for producing globally exponentially stable periodic solutions for switched neural networks with discrete and finite distributed time-varying delays. First, tunable parameters and cubic convex combination approach are proposed to study the globally exponential convergence of switched neural networks. Second, a sufficient criterion for the existence, uniqueness, and globally exponential stability of a periodic solution is demonstrated by using contraction mapping theorem and the impulse-delay-dependent Lyapunov-Krasovskii functional method. It is worth emphasizing that the addressed Lyapunov-Krasovskii functional covers both triple integral terms and novel quadruple integral terms, which makes the conservatism of the above criteria decrease. Even if the original neural network models are unstable or the impulsive effects are strong, the addressed neural network model can produce a globally exponentially stable periodic solution. These results here, which include boundedness, globally uniformly exponential convergence, and globally exponentially stability of the periodic solution, generalize and improve the earlier publications. Finally, two numerical examples and their computer simulations are given to show the effectiveness of theoretical results.

Since the unsaturating activation function is unbounded, more complex dynamics may exist in neural networks with this kind of activation function. In this article, monostability and multistability results of almost-periodic solutions are developed for fractional-order neural networks with unsaturating piecewise linear activation functions. Some globally Mittag-Leffler attractive sets are given, and the existence of globally Mittag-Leffler stable almost-periodic solution is demonstrated by using Ascoli-Arzela theorem. In particular, some sufficient conditions are provided to ascertain the multistability of almost-periodic solutions based on locally positively invariant set. It shows that there exists an almost-periodic solution in each positively invariant set, and all trajectories converge to this periodic trajectory in that rectangular area. Two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria.