Synchronization of chaotic Lur'e systems with delayed feedback control using deadzone nonlinearity

In this paper we present a synchronization method for chaotic Lur'e systems by constructing a new piecewise Lyapunov function. Using a delayed feedback control scheme, a delay-dependent stability criterion is derived for the synchronization of chaotic systems that are represented by Lur'e systems with deadzone nonlinearity. Based on the Lyapunov--Krasovskii functional and by using some properties of the nonlinearity, a new delay-dependent stabilization condition for synchronization is obtained via linear matrix inequality (LMI) formulation. The criterion is less conservative than existing ones, and it will be verified through a numerical example.     

Pinning control of complex Lur'e networks

This paper addresses the control problem of a class of complex dynamical networks with each node being a Lur'e system whose nonlinearity satisfies a sector condition, by applying local feedback injections to a small fraction of the nodes. The pinning control problem is reformulated in the framework of the absolute stability theory. It is shown that the global stability of the controlled network can be reduced to the test of a set of linear matrix inequalities, which in turn guarantee the absolute stability of the corresponding Lur'e systems whose dimensions are the same as that of a single node. A circle-type criterion in the frequency domain is further presented for checking the stability of the controlled network graphically. Finally, a network of Chua's oscillators is provided as a simulation example to illustrate the effectiveness of the theoretical results.     

Global synchronization in general complex delayed dynamical networks and its applications

The main objective of the present paper is further to investigate global synchronization of a general model of complex delayed dynamical networks. Based on stability theory on delayed dynamical systems, some simple yet less conservative criteria for both delay-independent and delay-dependent global synchronization of the networks are derived analytically. It is shown that under some conditions, if the uncoupled dynamical node is stable itself, then the network can be globally synchronized for any coupling delays as long as the coupling strength is small enough. On the other hand, if each dynamical node of the network is chaotic, then global synchronization of the networks is heavily dependent on the effects of coupling delays in addition to the connection configuration. Furthermore, the results are applied to some typical small-world (SW) and scale-free (SF) complex networks composing of coupled dynamical nodes such as the cellular neural networks (CNNs) and the chaotic FHN neuron oscillators, and numerical simulations are given to verify and also visualize the theoretical results.     

Cluster synchronization in fractional-order complex dynamical networks

Cluster synchronization of complex dynamical networks with fractional-order dynamical nodes is discussed in the Letter. By using the stability theory of fractional-order differential system and linear pinning control, a sufficient condition for the stability of the synchronization behavior in complex networks with fractional order dynamics is derived. Only the nodes in one community which have direct connections to the nodes in other communities are needed to be controlled, resulting in reduced control cost. A numerical example is presented to demonstrate the validity and feasibility of the obtained result. Numerical simulations illustrate that cluster synchronization performance for fractional-order complex dynamical networks is influenced by inner-coupling matrix, control gain, coupling strength and topological structures of the networks.     

Adaptive input-to-state stable synchronization for uncertain time-delay Lur'e systems

This paper is dedicated to the study of adaptive input-to-state stable synchronization of uncertain time-delay Lur'e systems with exterior interference. With the help of the Lyapunov function approach, a sufficient condition for the input-to-state stability of the synchronization-error system is derived, which is theoretically less conservative than a previously reported criterion in the absence of parameter uncertainties. On the basis of the present condition, a co-design of the feedback gain and estimates of the uncertain parameters is given to determine the desired adaptive synchronization controller. Finally, an example with simulations is provided to demonstrate the applicability and superiority of the analysis and design strategies.     

Generalized projective synchronization between two different general complex dynamical networks with delayed coupling

In this Letter, generalized projective synchronization (GPS) between two different complex dynamical networks with delayed coupling is investigated. Two complex networks are distinct if they have diverse node dynamics, or different number of nodes, or different topological structures. By using the adaptive control scheme, a sufficient synchronization criterion for this GPS is derived based on the LaSalle invariance principle. Three corollaries are also obtained. It is noticed that the synchronization speed sensitively depends on the adjustable positive constants μi. Furthermore, the coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix need not be symmetric. In addition, the node dynamic need not satisfy the very strong and conservative uniformly Lipschitz condition. Numerical simulations further demonstrate the feasibility and effectiveness of the theoretical results.     

Synchronization in complex delayed dynamical networks with impulsive effects

The present paper is mainly concerned with the issues of synchronization dynamics of complex delayed dynamical networks with impulsive effects. A general model of complex delayed dynamical networks with impulsive effects is formulated, which can well describe practical architectures of more realistic complex networks related to impulsive effects. Based on impulsive stability theory on delayed dynamical systems, some simple but less conservative criterion are derived for global synchronization of such dynamical network. It is shown that synchronization of the networks is heavily dependent on impulsive effects of connecting configuration in the networks. Furthermore, the theoretical results are applied to a typical SF network composing of impulsive coupled chaotic delayed Hopfield neural network nodes, and are also illustrated by numerical simulations.     

Pinning sampled-data synchronization for complex networks with probabilistic coupling delay

We deal with the problem of pinning sampled-data synchronization for a complex network with probabilistic time-varying coupling delay. The sampling period considered here is assumed to be less than a given bound. Without using the Kronecker product, a new synchronization error system is constructed by using the property of the random variable and input delay approach. Based on the Lyapunov theory, a delay-dependent pinning sampled-data synchronization criterion is derived in terms of linear matrix inequalities （LMIs） that can be solved effectively by using MATLAB LMI toolbox. Numerical examples are provided to demonstrate the effectiveness of the proposed scheme.     

Robust H_∞ observer-based control for synchronization of a class of complex dynamical networks

This paper is concerned with the robust H∞ synchronization problem for a class of complex dynamical networks by applying the observer-based control. The proposed feedback control scheme is developed to ensure the asymptotic stability of the augmented system, to reconstruct the non-measurable state variables of each node and to improve the H∞ performance related to the synchronization error and observation error despite the external disturbance. Based on the Lyapunov stability theory, a synchronization criterion is obtained under which the controlled network can be robustly stabilized onto a desired state with a guaranteed H∞ performance. The controller and the observer gains can be given by the feasible solutions of a set of linear matrix inequalities (LMIs). The effectiveness of the proposed control scheme is demonstrated by a numerical example through simulation.     

Synchronization criteria for a generalized complex delayed dynamical network model

It has been demonstrated that most complex networks display synchronization phenomena, and this problem has attracted great attention of various fields including science and engineering. In this paper, a generalized complex dynamical network model with time-varying delays was presented. Kronecker product was adopted to investigate this model. Moreover, several synchronization criteria were derived for both delay-independent and delay-dependent asymptotical stability. Especially, it has been shown that synchronization of such dynamical network is determined by the linear matrix inequality consisting of coupling configuration matrices, inner-coupling matrices and isolated cells. At last, illustrative examples were given to validate the above-acquired.     

Synchronization criteria for complex dynamical networks with neutral-type coupling delay

A generalized complex dynamical networks model with neutral-type coupling delay is proposed, which is an extension for the systems without time delay and with the retarded delay. By some transformation, the synchronization problem of the complex networks is transferred equally into the asymptotical stability problem of a group of uncorrelated neutral delay functional differential equations. Furthermore, the less conservative sufficient conditions for both delay-independent and delay-dependent asymptotical synchronization stability criteria are derived in the form of linear matrix inequalities based on the free weighting matrix strategy. Numerical examples are given to illustrate the theoretical results.     

Synchronization in complex delayed dynamical networks with nonsymmetric coupling

A new general complex delayed dynamical network model with nonsymmetric coupling is introduced, and then we investigate its synchronization phenomena. Several synchronization criteria for delay-independent and delay-dependent synchronization are provided which generalize some previous results. The matrix Jordan canonical formalization method is used instead of the matrix diagonalization method, so in our synchronization criteria, the coupling configuration matrix of the network does not required to be diagonalizable and may have complex eigenvalues. Especially, we show clearly that the synchronizability of a delayed dynamical network is not always characterized by the second-largest eigenvalue even though all the eigenvalues of the coupling configuration matrix are real. Furthermore, the effects of time-delay on synchronizability of networks with unidirectional coupling are studied under some typical network structures. The results are illustrated by delayed networks in which each node is a two-dimensional limit cycle oscillator system consisting of a two-cell cellular neural network, numerical simulations show that these networks can realize synchronization with smaller time-delay, and will lose synchronization when the time-delay increase larger than a threshold.     

Impulsive synchronization of chaotic Lur'e systems via partial states

In this Letter, the problem of impulsive synchronization of two identical Lur'e systems via partial states is studied. The problem arises in the situation when only partial states of the driven systems are available. By using the method of the variation of parameters for linear impulsive systems and some analysis technique, a sufficient condition for the existence of the impulsive control law for synchronization via partial states is derived. The sufficient condition is given in terms of linear matrix inequalities concerning the interconnection matrices. By using this result, we propose a new impulsive synchronization scheme for a class of Lur'e systems. The new impulsive synchronization scheme only exerts the impulsive input on partial states of the driven system and is characterized by a set of conditions related to the impulsive interval bound, the impulsive magnitude and a coupling condition between them. The proposed impulsive synchronization method is illustrated through Chua's circuit.     

节点含时滞的不确定复杂网络的自适应同步研究

研究了节点带有时滞,网络结构已知或者完全未知时的不确定动态网络模型的同步问题.基于李雅普诺夫稳定性理论,并按照参数的已知和未知情况分别设计了复杂网络同步控制器和复杂网络同步自适应控制器,给出了网络同步的充分条件,保证了动态网络渐进同步于任意指定的网络中的单独节点的状态.最后,数值结果表明了方法的有效性. 关键词： 自适应同步 不确定复杂网络 Lyapunov稳定理论     

Master-slave synchronization of Lur'e systems with sector and slope restricted nonlinearities

This Letter presents a synchronization method for Lur'e systems with sector and slope restricted nonlinearities. A static error feedback controller based on the Lyapunov stability theory is proposed for asymptotic synchronization. The Lyapunov function candidate is chosen as a quadratic form of the error states and nonlinear functions of the systems. The nonlinearities are expressed as convex combinations of sector and slope bounds by using convex properties of the nonlinear function so that equality constraints are converted into inequality constraints. Then, the feedback gain matrix is derived through a linear matrix inequality (LMI) formulation. Finally, a numerical example shows the effectiveness of the proposed method.     

The impulsive control synchronization of the drive-response complex system

This Letter investigates projective synchronization between the drive system and response complex dynamical system. An impulsive control scheme is adapted to synchronize the drive-response dynamical system to a desired scalar factor. By using the stability theory of the impulsive differential equation, the criteria for the projective synchronization are derived. The feasibility of the impulsive control of the projective synchronization is demonstrated in the drive-response dynamical system.     

Pinning control of weighted general complex dynamical networks with time delay

Time delays commonly exist in the real world. In the present work we consider weighted general complex dynamical networks with time delay, which are undirected and connected. Control of such networks, by applying local feedback injections to a fraction of network nodes, is investigated for both continuous-time and discrete-time cases. Both delay-independent and delay-dependent asymptotical stability criteria for network stabilization are derived. It is also shown that the whole network can be stabilized by controlling only one node. The efficiency of the derived results was illustrated by numerical examples.     

Adaptive cluster synchronization in complex dynamical networks

Cluster synchronization is investigated in different complex dynamical networks. In this Letter, a novel adaptive strategy is proposed to make a complex dynamical network achieve cluster synchronization, where the adaptive strategy of one edge is adjusted only according to its local information. A sufficient condition about the global stability arbitrarily grouped of cluster synchronization is derived. Several numerical simulations show the effectiveness of the adaptive strategy.     

Robust adaptive synchronization of general dynamical networks with multiple delays and uncertainties

In this article, a general complex dynamical network which contains multiple delays and uncertainties is introduced, which contains time-varying coupling delays, time-varying node delay, and uncertainties of both the inner- and outer-coupling matrices. A robust adaptive synchronization scheme for these general complex networks with multiple delays and uncertainties is established and raised by employing the robust adaptive control principle and the Lyapunov stability theory. We choose some suitable adaptive synchronization controllers to ensure the robust synchronization of this dynamical network. The numerical simulations of the time-delay Lorenz chaotic system as local dynamical node are provided to observe and verify the viability and productivity of the theoretical research in this paper. Compared to the achievement of previous research, the research in this paper seems quite comprehensive and universal.     

Non-fragile synchronization of genetic regulatory networks with randomly occurring controller gain fluctuation

This study examines the non-fragile synchronization of genetic regulatory networks (GRNs) with time-varying delays. Genetic regulatory network is formulated and sufficient conditions are derived to guarantee its synchronization based on master-slave system approach. The non-fragile observer based feedback controller gains are assumed to have the random fluctuations, two different types of uncertainties which perturb the gains are taken into account. By constructing a suitable Lyapunov-Krasovskii stability theory together with linear matrix inequality (LMI) approach we derived the delay-dependent criteria to ensure the asymptotic stability of the error system, which guarantees the master system synchronize with the slave system. The expressions for the non-fragile controller can be obtained by solving a set of LMIs using standard softwares. Finally, some numerical examples are included to show that the proposed method is less conservative than existing ones.