Proportional‐delay adaptive control for global synchronization of complex networks with time‐delay and switching outer‐coupling matrices

In this paper, the global synchronization problem for complex dynamical networks with time‐delay and switching outer‐coupling matrices is studied using the method of adaptive control. The mixed time‐delays in the considered system comprise both discrete time‐varying delay and distributed delay. The outer‐coupling matrices in the complex networks are assumed to vary with time and are described by a Markov process with finite state values. Based on the Lyapunov stability theory, the complete‐square method, and the Kronecker product analysis technique, a sufficient condition, which guarantees the global synchronization of all of nodes is obtained. The updated laws of controller do not only depend on the lag states of the complex dynamical networks but also depend on the Markov process. Finally, a numerical example is provided to show the effectiveness of the proposed method. Copyright © 2011 John Wiley & Sons, Ltd.     

Synchronization of Complex Networks with Coupled and Self‐Feedback Delays Via Aperiodically Intermittent Strategy

In this paper, we concern the exponential synchronization problem for hybrid‐coupled delayed dynamical networks via aperiodically intermittent control. Different from previous works, the delayed coupling term considered here contains the transmission delay and self‐feedback delay, and the intermittent control can be aperiodic. By utilizing a different technique compared with some previous results, several useful criteria are derived analytically to realise exponential synchronization for a class of coupled complex network. As a special case, some sufficient conditions ensuring the exponential synchronization for a class of coupled neural network are obtained. Finally, a numerical example is given to demonstrate the validness of the proposed scheme.     

Crossed Synchronization of Multiple Subnets Complex Network System with Time‐Varying Delay

A dual‐time varying delay complex network system is formed by a plurality of sub‐networks. This paper discusses the crossed synchronization stability of such systems on the basis of crossed synchronization definition between the sub‐nets. By the Lyapunov stability theory with the Zero Theorem, we obtained a sufficient condition that a synchronization exponentially stable controller exists in the complex network system, with characteristic variable delay. The relationship between the complex network nodes are discussed, and the two nodes must be connected and can interact with each other so it has the actual coupling strength. Finally, combining the given conditions, a numerical simulation illustrates its effectiveness.     

Improved stability conditions for discrete‐time systems under dynamic network protocols

This paper deals with the stability of discrete‐time networked systems with multiple sensor nodes under dynamic scheduling protocols. Access to the communication medium is orchestrated by a weighted try‐once‐discard or by an independent and identically‐distributed stochastic protocol that determines which sensor node can access the network at each sampling instant and transmit its corresponding data. Through a time‐delay approach, a unified discrete‐time hybrid system with time‐varying delays in the dynamics and in the reset conditions is formulated under both scheduling protocols. Then, a new stability criterion for discrete‐time systems with time‐varying delays is proposed by the discrete counterpart of the second‐order Bessel‐Legendre integral inequality. The developed approach is applied to guarantee the stability of the resulting discrete‐time hybrid system model with respect to the full state under try‐once‐discard or independent and identically‐distributed scheduling protocol. The communication delays can be larger than the sampling intervals. Finally, the efficiency of the presented approach is illustrated by a cart‐pendulum system.     

Synchronization in a network of identical continuous‐ or discrete‐time agents with unknown nonuniform constant input delay

This paper studies the synchronization problem for multiagent systems with identical continuous‐ or discrete‐time agents with unknown nonuniform constant input delays. The agents are connected via full‐ or partial‐state coupling. The agents are assumed to be asymptotically null controllable, ie, all eigenvalues are in the closed left‐half complex plane for continuous‐time agents or in the closed unit disc for discrete‐time agents. We derive an upper bound for the input delay tolerance, which explicitly depends on the agent dynamics. Moreover, for any unknown delay satisfying this upper bound, a low‐gain–based protocol design methodology is proposed without relying on exact knowledge of the network topology such that synchronization is achieved among agents for any network graph in a given set.     

On pinning synchronization of complex dynamical networks

There exist some fundamental and yet challenging problems in pinning control of complex networks: (1) What types of pinning schemes may be chosen for a given complex network to realize synchronization? (2) What kinds of controllers may be designed to ensure the network synchronization? (3) How large should the coupling strength be used in a given complex network to achieve synchronization? This paper addresses these technique questions. Surprisingly, it is found that a network under a typical framework can realize synchronization subject to any linear feedback pinning scheme by using adaptive tuning of the coupling strength. In addition, it is found that the nodes with low degrees should be pinned first when the coupling strength is small, which is contrary to the common view that the most-highly-connected nodes should be pinned first. Furthermore, it is interesting to find that the derived pinning condition with controllers given in a high-dimensional setting can be reduced to a low-dimensional condition without the pinning controllers involved. Finally, simulation examples of scale-free networks are given to verify the theoretical results.     

Robust synchronization in an array of fuzzy delayed cellular neural networks with stochastically hybrid coupling

In this paper, the robust synchronization control problem of an array of fuzzy cellular neural networks with uncertain stochastically coupling is investigated, which involves constant coupling, discrete time-varying delay coupling and distributed time-varying delay coupling. By using adaptive feedback control scheme and exploiting some stochastic analysis techniques, several sufficient conditions are developed to ensure the synchronization of stochastically hybrid coupled fuzzy neural networks with all admissible uncertainties in mean square. Finally, a numerical example illustrated by scale-free complex networks is provided to show the effectiveness and the applicability of the proposed method.     

Synchronization of Uncertain Complex Networks with Time‐Varying Node Delay and Multiple Time‐Varying Coupling Delays

This paper investigates the synchronization problem of a class of complex dynamical networks via an adaptive control method. It differs from existing works in considering intrinsic delay and multiple different time‐varying coupling delays, and uncertain couplings. A simple approach is used to linearize the uncertainties with the norm‐bounded condition. Simple but suitable adaptive controllers are designed to drive all nodes of the complex network locally and globally synchronize to a desired state. In addition, several synchronization protocols are deduced in detail by virtue of Lyapunov stability theory and a Cauchy matrix inequality. Finally, a simulation example is presented, in which the dynamics of each node are time‐varying delayed Chua chaotic systems, to demonstrate the effectiveness of the proposed adaptive method.     

Synchronization in a network of identical discrete‐time agents with uniform constant communication delay

This paper studies the synchronization problem for a network of identical discrete‐time agents with unknown uniform constant communication delay. When the agents are non‐introspective, the problem is solvable via a decentralized low‐gain‐based synchronization controller if the delay satisfies the proposed upper bound. When the agents are introspective, the synchronization problem can be solved with arbitrary bounded communication delay. Copyright © 2013 John Wiley & Sons, Ltd.     

Pinning adaptive synchronization of a general complex dynamical network

There are two challenging fundamental questions in pinning control of complex networks: (i) How many nodes should a network with fixed network structure and coupling strength be pinned to reach network synchronization? (ii) How much coupling strength should a network with fixed network structure and pinning nodes be applied to realize network synchronization? To fix these two questions, we propose a general complex dynamical network model and then further investigate its pinning adaptive synchronization. Based on this model, we attain several novel adaptive synchronization criteria which indeed give the positive answers to these two questions. That is, we provide a simply approximate formula for estimating the detailed number of pinning nodes and the magnitude of the coupling strength for a given general complex dynamical network. Here, the coupling-configuration matrix and the inner-coupling matrix are not necessarily symmetric. Moreover, our pinning adaptive controllers are rather simple compared with some traditional controllers. A Barabási-Albert network example is finally given to show the effectiveness of the proposed synchronization criteria.     

Solvability conditions and design for synchronization of discrete‐time multiagent systems

This paper provides solvability conditions for state synchronization with homogeneous discrete‐time multiagent systems with a directed and weighted communication network under partial‐ or full‐state coupling. Our solvability conditions reveal that the synchronization problem is solvable for all possible, a priori given, set of graphs associated with a communication network only under the condition that the agents are at most weakly unstable (ie, agents have all eigenvalues in the closed unit disc). However, if an upper bound on the eigenvalues inside the unit disc of the row stochastic matrices associated with any graph in a given set of graphs is known, then one can achieve synchronization for a class of unstable agents. We provide protocol design for at most weakly unstable agents based on a direct eigenstructure assignment method and a standard H₂ discrete‐time algebraic Riccati equation. We also provide protocol design for strictly unstable agents (ie, agents have at least one eigenvalue outside the unit disc) based on the standard H₂ discrete‐time algebraic Riccati equation.     

Adaptive aperiodically intermittent control for pinning synchronization of directed dynamical networks

In this paper, we investigate the pinning synchronization of directed complex network with time‐varying delay under the adaptive aperiodically intermittent control. First, the model of a directed complex network is established, the interaction graph of which is not required to contain a directed spanning tree, and for which an adaptive aperiodically intermittent feedback controller is well designed. In addition, based on the Lyapunov function method and linear matrix inequality technique, several asymptotical synchronization criteria are derived via an aperiodically intermittent strategy, which is superior to the periodical scheme. Moreover, the adaptive strategy providing the exponential convergence rate and the pinning algorithm demonstrating how to select pinned nodes are presented. Finally, numerical examples are provided to illustrate the effectiveness of the derived theoretical results.     

Synchronization Criteria for Complex Dynamical Networks with State and Coupling Time‐Delays

This paper is concerned with the problem of synchronization of complex dynamical networks with state and coupling time‐delays. Therefore, larger class and more complicated complex dynamical networks could be considered for the synchronization problem. Based on the Lyapunov‐Krasovskii functional, some delay‐independent and delay‐dependent criteria are obtained and formulated in the form of linear matrix inequalities (LMIs) to ascertain the synchronization between each node of the complex dynamical network. The effectiveness of the proposed method is illustrated using some numerical simulations.     

Global Synchronization for Discrete-Time Stochastic Complex Networks With Randomly Occurred Nonlinearities and Mixed Time Delays

In this paper, the problem of stochastic synchronization analysis is investigated for a new array of coupled discrete-time stochastic complex networks with randomly occurred nonlinearities (RONs) and time delays. The discrete-time complex networks under consideration are subject to: 1) stochastic nonlinearities that occur according to the Bernoulli distributed white noise sequences; 2) stochastic disturbances that enter the coupling term, the delayed coupling term as well as the overall network; and 3) time delays that include both the discrete and distributed ones. Note that the newly introduced RONs and the multiple stochastic disturbances can better reflect the dynamical behaviors of coupled complex networks whose information transmission process is affected by a noisy environment (e.g., internet-based control systems). By constructing a novel Lyapunov-like matrix functional, the idea of delay fractioning is applied to deal with the addressed synchronization analysis problem. By employing a combination of the linear matrix inequality (LMI) techniques, the free-weighting matrix method and stochastic analysis theories, several delay-dependent sufficient conditions are obtained which ensure the asymptotic synchronization in the mean square sense for the discrete-time stochastic complex networks with time delays. The criteria derived are characterized in terms of LMIs whose solution can be solved by utilizing the standard numerical software. A simulation example is presented to show the effectiveness and applicability of the proposed results.      

Synchronization of uncertain complex dynamical networks via adaptive control

In this paper, synchronization of an uncertain dynamical network with time‐varying delay is investigated by means of adaptive control schemes. Time delays and uncertainties exist universally in real‐world complex networks. Especially, parameters of nodes in these complex networks are usually partially or completely uncertain. Considering the networks with unknown or partially known nodes, we design adaptive controllers for the corresponding complex dynamical networks, respectively. Several criteria guaranteeing synchronization of such systems are established by employing the Lyapunov stability theorem. Analytical and numerical results show that the proposed controllers have high robustness against parameter variations including network topologies, coupling structures, and strength. Copyright © 2008 John Wiley & Sons, Ltd.     

Non‐Fragile Synchronization Control For Markovian Jumping Complex Dynamical Networks With Probabilistic Time‐Varying Coupling Delays

This paper studies the problem of non‐fragile synchronization control for Markovian jumping complex dynamical networks with probabilistic time‐varying coupling delays. By constructing a new Lyapunov–Krasovskii functional (LKF) and combining the reciprocal convex technique, sufficient conditions for the complex dynamical networks to be globally asymptotically synchronized in the mean square sense are derived. The probability distribution of the delays have been proposed and delay probability‐distribution‐dependent conditions are derived in the form of linear matrix inequalities (LMIs). The derived conditions depend not only on the size of the delay but also on the probability of the delay taking values in some intervals. Further, a non‐fragile synchronization controller is proposed. Finally, a numerical example is given to demonstrate the effectiveness of the proposed methods.     

Synchronization of coupled neural networks with random coupling strengths and mixed probabilistic time‐varying delays

This paper investigates the global asymptotic synchronization in an array of coupled neural networks with random coupling strengths, probabilistic interval time‐varying coupling delays as well as unbounded distributed delays (mixed delays). Two important integral inequalities that include the Jensen's inequality as a special case are developed. On the basis of the developed inequalities, the properties of random variables and Lyapunov functional method, several delay‐dependent sufficient synchronization criteria are derived for the considered model. The derived synchronization criteria are formulated by linear matrix inequalities (LMIs) and can be easily verified by using MATLAB LMI Toolbox. Some existing results are improved and extended by taking different values of parameters of the obtained results. Numerical simulations are finally given to demonstrate the effectiveness of the theoretical results. Copyright © 2012 John Wiley & Sons, Ltd.     

Synchronization in heterogeneous networks of discrete‐time introspective right‐invertible agents

This paper studies output synchronization problem, formation problem, and regulated synchronization problem for a heterogenous network of discrete‐time introspective right‐invertible agents. We first propose a decentralized control scheme to solve the output synchronization problem for a set of communication topologies. Moreover, if the synchronization trajectories are assumed to be bounded, a universal controller can be constructed for all communication topologies, which contain a directed spanning tree. The design can be applied to solve the formation problem with arbitrary formation vectors. In the regulated synchronization problem, we assume only the root receives information from exosystem. We then design a decentralized controller to solve the problem for a set of communication topologies. Copyright © 2013 John Wiley & Sons, Ltd.     

L2‐Gain Analysis for Synchronization of Stochastic Complex Network with Switched Topology and Time‐Varying Delay

We investigate the exponential stability and L₂‐gain analysis for the synchronization of stochastic complex networks under average dwell time switched topology with consideration of external disturbance, internal noise and fast time‐varying delay in the synchronized process. Based on the proposed stochastic network, a new L₂‐gain synchronization is proposed to solve the mean‐square exponential stable under switched topology with an H_∞ performance from the extrinsic disturbances to the synchronization error. The obtained results are applicable for the fast time‐varying case with larger‐than‐1 delay derivative. Finally, numerical simulations are performed to demonstrate the effectiveness of our strategies.     

Discretized LKF method for stability of coupled differential‐difference equations with multiple discrete and distributed delays

Time‐delay systems described by coupled differential‐functional equations include as special cases many types of time‐delay systems and coupled differential‐difference systems with time delays. This article discusses the discretized Lyapunov–Krasovskii functional (LKF) method for the stability problem of coupled differential‐difference equations with multiple discrete and distributed delays. Through independently dividing every delay region that the plane regions consists in two delays to discretize LKF, the exponential stability conditions for coupled systems with multiple discrete and distributed delays are established based on a linear matrix inequality (LMI). The numerical examples show that the analysis limit of delay bound in which the systems are stable may be approached by our result. Copyright © 2011 John Wiley & Sons, Ltd.