New results on adaptive fixed-time control for convex-delayed neural networks

This paper studies the adaptive fixed-time synchronization issue for convex-delayed neural networks. First, the convex delay is introduced to address the state delay of neural networks in order to reflect the impacts of multiple delay components such as input transition time and switching communication. Then, a new fixed-time control method is presented to adaptively determine multi-control gains with a unified update law. Afterward, some sufficient criteria are figured out by using Lyapunov stability theorem to ensure that the delayed neural networks are fixed-timely stable. Finally, simulated examples are adopted to validate our theoretical results.

l₂-l∞ State Estimation for Persistent Dwell-Time Switched Coupled Networks Subject to Round-Robin Protocol

This article is concerned with the issue of l₂ - l_∞ state estimation for nonlinear coupled networks, where the variation of coupling mode is governed by a set of switching signals satisfying a persistent dwell-time property. To solve the problem of data collisions in a constrained communication network, the round-robin protocol, as an important scheduling strategy for orchestrating the transmission order of sensor nodes, is introduced. Redundant channels with signal quantization are used to improve the reliability of data transmission. The main purpose is to determine an estimator that can guarantee the exponential stability in mean square sense and an l₂ - l_∞ performance level of the estimation error system. Based on the Lyapunov method, sufficient conditions for the addressed problem are established. The desired estimator gains can be obtained by addressing a convex optimization case. The correctness and availability of the developed approach are finally explained via two illustrative examples.

Synchronization of Nonidentical Neural Networks With Unknown Parameters and Diffusion Effects via Robust Adaptive Control Techniques

This paper considers the self-synchronization and tracking synchronization issues for a class of nonidentically coupled neural networks model with unknown parameters and diffusion effects. Using the special structure of neural networks with global Lipschitz activation function, nonidentical terms are treated as external disturbances, which can then be compensated via robust adaptive control techniques. For the case where no common reference trajectory is given in advance, a distributed adaptive controller is proposed to drive the synchronization error to an adjustable bounded area. For the case where a reference trajectory is predesigned, two distributed adaptive controllers are proposed, respectively, to address the tracking synchronization problem with bounded and unbounded reference trajectories, different decomposition methods are given to extract the heterogeneous characteristics. To avoid the appearance of global information, such as the spectrum of the coupling matrix, corresponding adaptive designs on coupling strengths are also provided for both cases. Moreover, the upper bounds of the final synchronization errors can be gradually adjusted according to the parameters of the adaptive designs. Finally, numerical examples are given to test the effectiveness of the control algorithms.

Synchronization of the Networked System With Continuous and Impulsive Hybrid Communications

Many networked systems display some kind of dynamics behaving in a style with both continuous and impulsive communications. The cooperation behaviors of these networked systems with continuous connected or impulsive connected or both connected topologies of communications are important to understand. This paper is devoted to the synchronization of the networked system with continuous and impulsive hybrid communications, where each topology of communication mode is not connected in every moment. Two kind of structures, i.e., fixed structure and switching structures, are taken into consideration. A general concept of directed spanning tree (DST) is proposed to describe the connectivity of the networked system with hybrid communication modes. The suitable Lyapunov functions are constructed to analyze the synchronization stability. It is showed that for fixed topology having a jointly DST, the networked system with continuous and impulsive hybrid communication modes will achieve asymptotic synchronization if the feedback gain matrix and the average impulsive interval are properly selected. The results are then extended to the switching case where the graph has a frequently jointly DST. Some simple examples are then given to illustrate the derived synchronization criteria.

Adaptive Synchronization of Reaction-Diffusion Neural Networks and Its Application to Secure Communication

This paper is mainly concerned with the synchronization problem of reaction-diffusion neural networks (RDNNs) with delays and its direct application in image secure communications. An adaptive control is designed without a sign function in which the controller gain matrix is a function of time. The synchronization criteria are established for an error model derived from master-slave models through solving the set of linear matrix inequalities derived by constructing the suitable novel Lyapunov-Krasovskii functional candidate, Green's formula, and Wirtinger's inequality. If the proposed sufficient conditions are satisfied, then the global asymptotic synchronization of the error model is guaranteed. The numerical illustrations are provided to demonstrate the validity of the derived synchronization criteria. In addition, the role of system parameters is picturized through the chaotic nature of RDNNs and those unprecedented solutions is utilized to promote better security of image transactions. As is evident, the enhancement of image encryption algorithm is designed with two levels, namely, image watermarking and diffusion process. The contributions of this paper are discussed as concluding remarks.

Exponential Synchronization and L2 -Gain Analysis of Delayed Chaotic Neural Networks Via Intermittent Control With Actuator Saturation

By using an intermittent control approach, this paper is concerned with the exponential synchronization and L₂ -gain analysis for a class of delayed master-slave chaotic neural networks subject to actuator saturation. Based on a switching strategy, the synchronization error system is modeled as a switched synchronization error system consisting of two subsystems, and each subsystem of the switched system satisfies a dwell time constraint due to the characteristics of intermittent control. A piecewise Lyapunov-Krasovskii functional depending on the control rate and control period is then introduced, under which sufficient conditions for the exponential stability of the constructed switched synchronization error system are developed. In addition, the influence of the exogenous perturbations on synchronization performance is constrained at a prescribed level. In the meantime, the intermittent linear state feedback controller can be derived by solving a set of linear matrix inequalities. More incisively, the proposed method is also proved to be valid in the case of aperiodically intermittent control. Finally, two simulation examples are employed to demonstrate the effectiveness and potential of the obtained results.

Synchronization of Multiple Reaction-Diffusion Neural Networks With Heterogeneous and Unbounded Time-Varying Delays

The synchronization problem of multiple/coupled reaction-diffusion neural networks with time-varying delays is investigated. Differing from the existing considerations, state delays among distinct neurons and coupling delays among different subnetworks are included in the proposed model, the assumptions posed on the arisen delays are very weak, time-varying, heterogeneous, even unbounded delays are permitted. To overcome the difficulties from this kind of delay as well as diffusion effects, a comparison-based approach is applied to this model and a series of algebraic criteria are successfully obtained to verify the global asymptotical synchronization. By specifying the existing delays, some M -matrix-based criteria are derived to justify the power-rate synchronization and exponential synchronization. In addition, new criterion on synchronization of general connected neural networks without diffusion effects is also given. Finally, two simulation examples are given to verify the effectiveness of the obtained theoretical results and provide a comparison with the existing criterion.

Novel aperiodically intermittent stability criteria for Markovian switching stochastic delayed coupled systems

This paper concerns p th moment exponential stability of stochastic coupled systems with multiple time-varying delays, and Markovian switching topologies via intermittent control. Compared with previous research results, the mathematical model of this kind of stochastic coupled systems with multiple time-varying delays and Markovian switching topologies is studied for the first time. The intermittent control designed in this paper is aperiodical, which is more general in practice. Moreover, the restriction between control width and time delays is removed. By constructing a new differential inequality on delayed dynamical systems with Markovian switching topologies and combining the graph-theoretic approach with M-matrix theory, two sufficient criteria are derived to guarantee p th moment exponential stability of systems. Moreover, the exponential convergence rate has a close relationship with the maximum ratio of the rest width to the aperiodical time span (the sum of the control width and the rest width). Finally, we employ the theoretical results to study the exponential stability of stochastic coupled oscillators with multiple time-varying delays and Markovian switching topologies. Meanwhile, a numerical example is presented to illustrate the effectiveness and feasibility of the proposed results.

Event-Triggered Cluster Consensus of Leader-Following Linear Multi-Agent Systems

This paper is concerned with cluster consensus of linear multi-agent systems via a distributed event-triggered control scheme. Assume that agents can be split into several clusters and a leader is associated with each cluster. Sufficient conditions are derived to guarantee the realization of cluster consensus by a feasible event-triggered controller if the network topology of each cluster has a directed spanning tree and the couplings within each cluster are sufficiently strong. Further, positive inner-event time intervals are ensured for the proposed event-triggered strategy to avoid Zeno behaviors. Finally, a numerical example is given to illustrate the effectiveness of the theoretical results.

Synchronization of Uncertain Euler-Lagrange Systems With Uncertain Time-Varying Communication Delays

A decentralized controller is designed for leader-based synchronization of communication-delayed networked agents. The agents have heterogeneous dynamics modeled by uncertain, nonlinear Euler-Lagrange equations of motion affected by heterogeneous, unknown, exogenous disturbances. The developed controller requires only one-hop (delayed) communication from network neighbors and the communication delays are assumed to be heterogeneous, uncertain, and time-varying. Each agent uses an estimate of communication delay to provide feedback of estimated recent tracking error. Simulation results are provided to demonstrate the improved performance of the developed controller over other popular control designs.

State Estimation for Discrete-Time Dynamical Networks With Time-Varying Delays and Stochastic Disturbances Under the Round-Robin Protocol

This paper is concerned with the state estimation problem for a class of nonlinear dynamical networks with time-varying delays subject to the round-robin protocol. The communication between the state estimator and the nodes of the dynamical networks is implemented through a shared constrained network, in which only one node is allowed to send data at each time instant. The round-robin protocol is utilized to orchestrate the transmission order of nodes. By using a switch-based approach, the dynamics of the estimation error is modeled by a periodic parameter-switching system with time-varying delays. The purpose of the problem addressed is to design an estimator, such that the estimation error is exponentially ultimately bounded with a certain asymptotic upper bound in mean square subject to the process noise and exogenous disturbance. Furthermore, such a bound is subsequently minimized by the designed estimator parameters. A novel Lyapunov-like functional is employed to deal with the dynamics analysis issue of the estimation error. Sufficient conditions are established to guarantee the ultimate boundedness of the estimation error in mean square by applying the stochastic analysis approach. Then, the desired estimator gains are characterized by solving a convex problem. Finally, a numerical example is given to illustrate the effectiveness of the estimator design scheme.

Graph Theory-Based Pinning Synchronization of Stochastic Complex Dynamical Networks

This paper is concerned with the adaptive pinning synchronization problem of stochastic complex dynamical networks (CDNs). Based on algebraic graph theory and Lyapunov theory, pinning controller design conditions are derived, and the rigorous convergence analysis of synchronization errors in the probability sense is also conducted. Compared with the existing results, the topology structures of stochastic CDN are allowed to be unknown due to the use of graph theory. In particular, it is shown that the selection of nodes for pinning depends on the unknown lower bounds of coupling strengths. Finally, an example on a Chua's circuit network is given to validate the effectiveness of the theoretical results.

Leader-Following Consensus of Nonlinear Multiagent Systems With Stochastic Sampling

This paper is concerned with sampled-data leader-following consensus of a group of agents with nonlinear characteristic. A distributed consensus protocol with probabilistic sampling in two sampling periods is proposed. First, a general consensus criterion is derived for multiagent systems under a directed graph. A number of results in several special cases without transmittal delays or with the deterministic sampling are obtained. Second, a dimension-reduced condition is obtained for multiagent systems under an undirected graph. It is shown that the leader-following consensus problem with stochastic sampling can be transferred into a master-slave synchronization problem with only one master system and two slave systems. The problem solving is independent of the number of agents, which greatly facilitates its application to large-scale networked agents. Third, the network design issue is further addressed, demonstrating the positive and active roles of the network structure in reaching consensus. Finally, two examples are given to verify the theoretical results.

Event-Triggered Distributed Average Consensus Over Directed Digital Networks With Limited Communication Bandwidth

In this paper, we consider the event-triggered distributed average-consensus of discrete-time first-order multiagent systems with limited communication data rate and general directed network topology. In the framework of digital communication network, each agent has a real-valued state but can only exchange finite-bit binary symbolic data sequence with its neighborhood agents at each time step due to the digital communication channels with energy constraints. Novel event-triggered dynamic encoder and decoder for each agent are designed, based on which a distributed control algorithm is proposed. A scheme that selects the number of channel quantization level (number of bits) at each time step is developed, under which all the quantizers in the network are never saturated. The convergence rate of consensus is explicitly characterized, which is related to the scale of network, the maximum degree of nodes, the network structure, the scaling function, the quantization interval, the initial states of agents, the control gain and the event gain. It is also found that under the designed event-triggered protocol, by selecting suitable parameters, for any directed digital network containing a spanning tree, the distributed average consensus can be always achieved with an exponential convergence rate based on merely one bit information exchange between each pair of adjacent agents at each time step. Two simulation examples are provided to illustrate the feasibility of presented protocol and the correctness of the theoretical results.

FLS-Based Adaptive Synchronization Control of Complex Dynamical Networks With Nonlinear Couplings and State-Dependent Uncertainties

This paper is concerned with the problem of synchronization control of complex dynamical networks (CDN) subject to nonlinear couplings and uncertainties. An fuzzy logical system-based adaptive distributed controller is designed to achieve the synchronization. The asymptotic convergence of synchronization errors is analyzed by combining algebraic graph theory and Lyapunov theory. In contrast to the existing results, the proposed synchronization control method is applicable for the CDN with system uncertainties and unknown topology. Especially, the considered uncertainties are allowed to occur in the node local dynamics as well as in the interconnections of different nodes. In addition, it is shown that a unified controller design framework is derived for the CDN with or without coupling delays. Finally, simulations on a Chua's circuit network are provided to validate the effectiveness of the theoretical results.

Exponential Synchronization of Coupled Stochastic Memristor-Based Neural Networks With Time-Varying Probabilistic Delay Coupling and Impulsive Delay

This paper deals with the exponential synchronization of coupled stochastic memristor-based neural networks with probabilistic time-varying delay coupling and time-varying impulsive delay. There is one probabilistic transmittal delay in the delayed coupling that is translated by a Bernoulli stochastic variable satisfying a conditional probability distribution. The disturbance is described by a Wiener process. Based on Lyapunov functions, Halanay inequality, and linear matrix inequalities, sufficient conditions that depend on the probability distribution of the delay coupling and the impulsive delay were obtained. Numerical simulations are used to show the effectiveness of the theoretical results.

On Extended Dissipativity of Discrete-Time Neural Networks With Time Delay

In this brief, the problem of extended dissipativity analysis for discrete-time neural networks with time-varying delay is investigated. The definition of extended dissipativity of discrete-time neural networks is proposed, which unifies several performance measures, such as the H∞ performance, passivity, l2 - l∞ performance, and dissipativity. By introducing a triple-summable term in Lyapunov function, the reciprocally convex approach is utilized to bound the forward difference of the triple-summable term and then the extended dissipativity criterion for discrete-time neural networks with time-varying delay is established. The derived condition guarantees not only the extended dissipativity but also the stability of the neural networks. Two numerical examples are given to demonstrate the reduced conservatism and effectiveness of the obtained results.

Global synchronization of complex dynamical networks through digital communication with limited data rate

This paper studies the global synchronization of complex dynamical network (CDN) under digital communication with limited bandwidth. To realize the digital communication, the so-called uniform-quantizer-sets are introduced to quantize the states of nodes, which are then encoded and decoded by newly designed encoders and decoders. To meet the requirement of the bandwidth constraint, a scaling function is utilized to guarantee the quantizers having bounded inputs and thus achieving bounded real-time quantization levels. Moreover, a new type of vector norm is introduced to simplify the expression of the bandwidth limit. Through mathematical induction, a sufficient condition is derived to ensure global synchronization of the CDNs. The lower bound on the sum of the real-time quantization levels is analyzed for different cases. Optimization method is employed to relax the requirements on the network topology and to determine the minimum of such lower bound for each case, respectively. Simulation examples are also presented to illustrate the established results.

Hierarchical cooperative control for multiagent systems with switching directed topologies

The hierarchical cooperative control problem is concerned for a two-layer networked multiagent system under switching directed topologies. The group cooperative objective is to achieve finite-time formation control for the upper layer of leaders and containment control for the lower layer of followers. Two kinds of cooperative strategies, including centralized-distributed control and distributed-distributed control, are proposed for two types of switching laws: 1) random switching law with the dwell time and 2) Markov switching law with stationary distribution. Utilizing the state transition matrix methods and matrix measure techniques, some sufficient conditions are derived for asymptotical containment control and exponential almost sure containment control, respectively. Finally, some numerical examples are provided to demonstrate the effectiveness of the proposed control schemes.

Graph Theory-Based Approach for Stability Analysis of Stochastic Coupled Systems With Lévy Noise on Networks

In this paper, a novel class of stochastic coupled systems with Lévy noise on networks (SCSLNNs) is presented. Both white noise and Lévy noise are considered in the networks. By exploiting graph theory and Lyapunov stability theory, criteria ensuring p th moment exponential stability and stability in probability of these SCSLNNs are established, respectively. These principles are closely related to the topology of the network and the perturbation intensity of white noise and Lévy noise. Moreover, to verify the theoretical results, stochastic coupled oscillators with Lévy noise on a network and stochastic Volterra predator-prey system with Lévy noise are performed. Finally, a numerical example about oscillators' network is provided to illustrate the feasibility of our analytical results.

Synchronization of stochastic dynamical networks under impulsive control with time delays

In this paper, the stochastic synchronization problem is studied for a class of delayed dynamical networks under delayed impulsive control. Different from the existing results on the synchronization of dynamical networks under impulsive control, impulsive input delays are considered in our model. By assuming that the impulsive intervals belong to a certain interval and using the mathematical induction method, several conditions are derived to guarantee that complex networks are exponentially synchronized in mean square. The derived conditions reveal that the frequency of impulsive occurrence, impulsive input delays, and stochastic perturbations can heavily affect the synchronization performance. A control algorithm is then presented for synchronizing stochastic dynamical networks with delayed synchronizing impulses. Finally, two examples are given to demonstrate the effectiveness of the proposed approach.

Pinning synchronization of hybrid-coupled directed delayed dynamical network via intermittent control

This paper concerns the problem of exponential synchronization for a class of general delayed dynamical networks with hybrid coupling via pinning periodically intermittent control. Both the internal delay and coupling delay are taken into account in the network model. Meanwhile, the transmission delay and self-feedback delay are involved in the delayed coupling term. By establishing a new differential inequality, several simple and useful exponential synchronization criteria are derived analytically. It is shown that the controlled synchronization state can vary in comparison with the conventional synchronized solution, and the degree of the node and the inner delayed coupling matrix play important roles in the controlled synchronization state. By choosing different inner delayed coupling matrices and the degrees of the node, different controlled synchronization states can be obtained. Furthermore, the detail pinning schemes deciding what nodes should be chosen as pinned candidates and how many nodes are needed to be pinned for a fixed coupling strength are provided. The simple procedures illuminating how to design suitable intermittent controllers in real application are also given. Numerical simulations, including an undirected scale-free network and a directed small-world network, are finally presented to demonstrate the effectiveness of the theoretical results.

Exponential synchronization of coupled switched neural networks with mode-dependent impulsive effects

This paper investigates the synchronization problem of coupled switched neural networks (SNNs) with mode-dependent impulsive effects and time delays. The main feature of mode-dependent impulsive effects is that impulsive effects can exist not only at the instants coinciding with mode switching but also at the instants when there is no system switching. The impulses considered here include those that suppress synchronization or enhance synchronization. Based on switching analysis techniques and the comparison principle, the exponential synchronization criteria are derived for coupled delayed SNNs with mode-dependent impulsive effects. Finally, simulations are provided to illustrate the effectiveness of the results.

Distributed synchronization of coupled neural networks via randomly occurring control

In this paper, we study the distributed synchronization and pinning distributed synchronization of stochastic coupled neural networks via randomly occurring control. Two Bernoulli stochastic variables are used to describe the occurrences of distributed adaptive control and updating law according to certain probabilities. Both distributed adaptive control and updating law for each vertex in a network depend on state information on each vertex's neighborhood. By constructing appropriate Lyapunov functions and employing stochastic analysis techniques, we prove that the distributed synchronization and the distributed pinning synchronization of stochastic complex networks can be achieved in mean square. Additionally, randomly occurring distributed control is compared with periodically intermittent control. It is revealed that, although randomly occurring control is an intermediate method among the three types of control in terms of control costs and convergence rates, it has fewer restrictions to implement and can be more easily applied in practice than periodically intermittent control.

Synchronization of Coupled Neutral-Type Neural Networks With Jumping-Mode-Dependent Discrete and Unbounded Distributed Delays

In this paper, the synchronization problem is studied for an array of N identical delayed neutral-type neural networks with Markovian jumping parameters. The coupled networks involve both the mode-dependent discrete-time delays and the mode-dependent unbounded distributed time delays. All the network parameters including the coupling matrix are also dependent on the Markovian jumping mode. By introducing novel Lyapunov-Krasovskii functionals and using some analytical techniques, sufficient conditions are derived to guarantee that the coupled networks are asymptotically synchronized in mean square. The derived sufficient conditions are closely related with the discrete-time delays, the distributed time delays, the mode transition probability, and the coupling structure of the networks. The obtained criteria are given in terms of matrix inequalities that can be efficiently solved by employing the semidefinite program method. Numerical simulations are presented to further demonstrate the effectiveness of the proposed approach.

Inphase and antiphase synchronization in a delay-coupled system with applications to a delay-coupled FitzHugh-Nagumo system

A time delay is inevitable in the coupled system and is an essential property of the coupling, which cannot be neglected in many realistic coupled systems. In this paper, we first study the existence of a Hopf bifurcation induced by coupling time delay and then investigate the influence of coupling time delay on the patterns of Hopf-bifurcating periodic oscillations. How the coupling time delay leads to complex scenarios of synchronized inphase or antiphase oscillations is analytically investigated. As an example, we study the delay-coupled FitzHugh-Nagumo system. We find conditional stability, absolute stability, and stability switches of the steady state provoked by the coupling time delay. Then we investigate the inphase and antiphase synchronized periodic solutions induced by delay, and determine the direction and stability of these bifurcating periodic orbits by employing the center manifold reduction and normal form theory. We find that in the region where stability switches occur, there exist synchronization transitions, i.e., synchronized dynamics can be switched from inphase (antiphase) to antiphase (inphase) and back to inphase (antiphase) and so on just by progressive increase of the coupling time delay.

Synchronization error estimation and controller design for delayed Lur'e systems with parameter mismatches

This paper investigates the problem of master-slave synchronization of two delayed Lur'e systems in the presence of parameter mismatches. First, by analyzing the corresponding synchronization error system, synchronization with an error level, which is referred to as quasi-synchronization, is established. Some delay-dependent quasi-synchronization criteria are derived. An estimation of the synchronization error bound is given, and an explicit expression of error levels is obtained. Second, sufficient conditions on the existence of feedback controllers under a predetermined error level are provided. The controller gains are obtained by solving a set of linear matrix inequalities. Finally, a delayed Chua's circuit is chosen to illustrate the effectiveness of the derived results.

H∞ state estimation for discrete-time complex networks with randomly occurring sensor saturations and randomly varying sensor delays

In this paper, the state estimation problem is investigated for a class of discrete time-delay nonlinear complex networks with randomly occurring phenomena from sensor measurements. The randomly occurring phenomena include randomly occurring sensor saturations (ROSSs) and randomly varying sensor delays (RVSDs) that result typically from networked environments. A novel sensor model is proposed to describe the ROSSs and the RVSDs within a unified framework via two sets of Bernoulli-distributed white sequences with known conditional probabilities. Rather than employing the commonly used Lipschitz-type function, a more general sector-like nonlinear function is used to describe the nonlinearities existing in the network. The purpose of the addressed problem is to design a state estimator to estimate the network states through available output measurements such that, for all probabilistic sensor saturations and sensor delays, the dynamics of the estimation error is guaranteed to be exponentially mean-square stable and the effect from the exogenous disturbances to the estimation accuracy is attenuated at a given level by means of an H∞-norm. In terms of a novel Lyapunov-Krasovskii functional and the Kronecker product, sufficient conditions are established under which the addressed state estimation problem is recast as solving a convex optimization problem via the semidefinite programming method. A simulation example is provided to show the usefulness of the proposed state estimation conditions.

Impulsive synchronization of coupled dynamical networks with nonidentical Duffing oscillators and coupling delays

This paper aims to investigate the synchronization problem of coupled dynamical networks with nonidentical Duffing-type oscillators without or with coupling delays. Different from cluster synchronization of nonidentical dynamical networks in the previous literature, this paper focuses on the problem of complete synchronization, which is more challenging than cluster synchronization. By applying an impulsive controller, some sufficient criteria are obtained for complete synchronization of the coupled dynamical networks of nonidentical oscillators. Furthermore, numerical simulations are given to verify the theoretical results.

Exponential synchronization of complex networks with finite distributed delays coupling

In this paper, the exponential synchronization for a class of complex networks with finite distributed delays coupling is studied via periodically intermittent control. Some novel and useful criteria are derived by utilizing a different technique compared with some correspondingly previous results. As a special case, some sufficient conditions ensuring the exponential synchronization for a class of coupled neural networks with distributed delays are obtained. Furthermore, a feasible region of the control parameters is derived for the realization of exponential synchronization. It is worth noting that the synchronized state in this paper is not an isolated node but a non-decoupled state, in which the inner coupling matrix and the degree of the nodes play a central role. Additionally, the traditional assumptions on control width, non-control width, and discrete delays are removed in our results. Finally, some numerical simulations are given to demonstrate the effectiveness of the proposed control method.