Target layer state estimation in multi-layer complex dynamical networks considering nonlinear node dynamics

In many engineering networks, only a part of target state variables are required to be estimated.On the other hand,multi-layer complex network exists widely in practical situations.In this paper, the state estimation of target state variables in multi-layer complex dynamical networks with nonlinear node dynamics is studied.A suitable functional state observer is constructed with the limited measurement.The parameters of the designed functional observer are obtained from the algebraic method and the stability of the functional observer is proven by the Lyapunov theorem.Some necessary conditions that need to be satisfied for the design of the functional state observer are obtained.Different from previous studies, in the multi-layer complex dynamical network with nonlinear node dynamics, the proposed method can estimate the state of target variables on some layers directly instead of estimating all the individual states.Thus, it can greatly reduce the placement of observers and computational cost.Numerical simulations with the three-layer complex dynamical network composed of three-dimensional nonlinear dynamical nodes are developed to verify the effectiveness of the method.

Topology identification for a class of complex dynamical networks using output variables

A problem of topology identification for complex dynamical networks is investigated in this paper. An adaptive observer is proposed to identify the topology of a complex dynamical networks based on the Lyapunov stability theory. Here the output of the network and the states of the observer are used to construct the updating law of the topology such that the communication resources from the network to its observer are saved. Some convergent criteria of the adaptive observer are derived in the form of linear inequality matrices. Several numerical examples are shown to demonstrate the effectiveness of the proposed observer.

SYNCHRONIZATION OF MASTER-SLAVE MARKOVIAN SWITCHING COMPLEX DYNAMICAL NETWORKS WITH TIME-VARYING DELAYS IN NONLINEAR FUNCTION VIA SLIDING MODE CONTROL

In this article, a synchronization problem for master-slave Markovian switching complex dynamical networks with time-varying delays in nonlinear function via sliding mode control is investigated. On the basis of the appropriate Lyapunov-Krasovskii functional, introducing some free weighting matrices, new synchronization criteria are derived in terms of linear matrix inequalities （LMIs）. Then, an integral sliding surface is designed to guarantee synchronization of master-slave Markovian switching complex dynamical networks, and the suitable controller is synthesized to ensure that the trajectory of the closed-loop error system can be driven onto the prescribed sliding mode surface. By using Dynkin＇s formula, we established the stochastic stablity of master-slave system. Finally, numerical example is provided to demonstrate the effectiveness of the obtained theoretical results.

Synchronization of complex switched delay dynamical networks with simultaneously diagonalizable coupling matrices

This paper studies local exponential synchronization of complex delayed networks with switching topology via switched system stability theory. First, by a common unitary matrix, the problem of synchronization is transformed into the stability analysis of some linear switched delay systems. Then, when all subnetworks are synchronizable, a delay-dependent sufficient condition is given in terms of linear matrix inequalities （LMIs） which guarantees the solvability of the synchronization problem under an average dwell time scheme. We extend this result to the case that not all subnetworks are synchronizable. It is shown that in addition to average dwell time, if the ratio of the total activation time of synchronizable and non-synchronizable subnetworks satisfy an extra condition, then the problem is also solvable. Two numerical examples of delayed dynamical networks with switching topology are given, which demonstrate the effectiveness of obtained results.

Disturbance rejection and H_∞ pinning control of linear complex dynamical networks

This paper concerns the disturbance rejection problem of a linear complex dynamical network subject to external disturbances. A dynamical network is said to be robust to disturbance, if the H∞ norm of its transfer function matrix from the disturbance to the performance variable is satisfactorily small. It is shown that the disturbance rejection problem of a dynamical network can be solved by analysing the H∞ control problem of a set of independent systems whose dimensions are equal to that of a single node. A counter-intuitive result is that the disturbance rejection level of the whole network with a diffusive coupling will never be better than that of an isolated node. To improve this, local feedback injections are applied to a small fraction of the nodes in the network. Some criteria for possible performance improvement are derived in terms of linear matrix inequalities. It is further demonstrated via a simulation example that one can indeed improve the disturbance rejection level of the network by pinning the nodes with higher degrees than pinning those with lower degrees.

Fault diagnosis of time-delay complex dynamical networks using output signals

This paper proposes a novel approach for fault diagnosis of a time-delay complex dynamical network. Unlike the other methods, assuming that the dynamics of the network can be described by a linear stochastic model, or using the state variables of nodes in the network to design an adaptive observer, it only uses the output variable of the nodes to design an observer and an adaptive law of topology matrix in the observer of a complex network, leading to simple design of the observer and easy realisation of topology monitoring for the complex networks in real engineering. The proposed scheme can monitor any changes of the topology structure of a time-delay complex network. The effectiveness of this method is successfully demonstrated by virtue of a complex networks with Lorenz model.

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

This paper is concerned with the robust Hoo 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.

Robust adaptive synchronization of uncertain and delayed dynamical complex networks with faulty network

This paper presents a new robust adaptive synchronization method for a class of uncertain dynamical complex networks with network failures and coupling time-varying delays. Adaptive schemes are proposed to adjust controller parameters for the faulty network compensations, as well as to estimate the upper and lower bounds of delayed state errors and perturbations to compensate the effects of delay and perturbation on-line without assuming symmetry or irreducibility of networks. It is shown that, through Lyapunov stability theory, distributed adaptive controllers con- structed by the adaptive schemes are successful in ensuring the achievement of asymptotic synchronization of networks in the present of faulty and delayed networks, and perturbation inputs. A Chua＇s circuit network example is finally given to show the effectiveness of the proposed synchronization criteria.

Fault Tolerant Synchronization for a General Complex DynamicalNetwork with Random Delay

A fault tolerant synchronization strategy is proposed to synchronize a complex network with random time delays and sensor faults. Random time delays over the network transmission are described by using Markov chains. Based on the Lyapunov stability theory and stochastic analysis, several passive fault tolerant synchronization criteria are derived,which can be described in the form of linear matrix inequalities. Finally,a numerical simulation example is carried out and the results show the validity of the proposed fault tolerant synchronization controller.

A novel robust fixed time synchronization of complex network subject to input nonlinearity in the presence of uncertainties and external disturbances

This paper studies the global fixed time synchronization of complex dynamical network,including non-identical nodes with disturbances and uncertainties as well as input nonlinearity.First,a novel fixed time sliding manifold is constructed to achieve the fixed time synchronization of complex dynamical network with disturbances and uncertainties.Second,a novel sliding mode controller is proposed to realize the global fixed time reachability of sliding surfaces.The outstanding feature of the designed control is that the fixed convergence time of both reaching and sliding modes can be adjusted to the desired values in advance by choosing the explicit parameters in the controller,which does not rest upon the initial conditions and the topology of the network.Finally,the effectiveness and validity of the obtained results are demonstrated by corresponding numerical simulations.

Synchronization and Bifurcation of General Complex Dynamical Networks

In the present paper, synchronization and bifurcation of general complex dynamical networks are investigated. We mainly focus on networks with a somewhat general coupling matrix, i.e., the sum of each row equals a nonzero constant u. We derive a result that the networks can reach a new synchronous state, which is not the asymptotic limit set determined by the node equation. At the synchronous state, the networks appear bifurcation if we regard the constant u as a bifurcation parameter. Numerical examples are given to illustrate our derived conclusions.

Hash function construction using weighted complex dynamical networks

A novel scheme to construct a hash function based on a weighted complex dynamical network （WCDN） generated from an original message is proposed in this paper. First, the original message is divided into blocks. Then, each block is divided into components, and the nodes and weighted edges are well defined from these components and their relations. Namely, the WCDN closely related to the original message is established. Furthermore, the node dynamics of the WCDN are chosen as a chaotic map. After chaotic iterations, quantization and exclusive-or operations, the fixed-length hash value is obtained. This scheme has the property that any tiny change in message can be diffused rapidly through the WCDN, leading to very different hash values. Analysis and simulation show that the scheme possesses good statistical properties, excellent confusion and diffusion, strong collision resistance and high efficiency.

Impulsive synchronization of two coupled complex networks with time-delayed dynamical nodes

In this paper, we investigate the impulsive synchronization between two coupled complex networks with time- delayed dynamical nodes. Based on the Lyapunov stability, the linear feedback control and the impulsive control theories, the linear feedback and the impulsive controllers are designed separately. By using the generalized Barbalat＇s lemma, the global asymptotic impulsive synchronization of the drive-response complex networks is derived and some corresponding sufficient conditions are also obtained. Numerical examples are presented to verify the effectiveness and the correctness of the synchronization criteria.

Exponential Synchronization of Uncertain Complex Dynamical Networks with Delay Coupling

This paper studies the global exponential synchronization of uncertain complex delayed dynamical networks. The network model considered is general dynamical delay networks with unknown network structure and unknown coupling functions but bounded. Novel delay-dependent linear controllers are designed via the Lyapunov stability theory. Especially, it is shown that the controlled networks are globally exponentially synchronized with a given convergence rate. An example of typical dynamical network of this class, having the Lorenz system at each node, has been used to demonstrate and verify the novel design proposed. And, the numerical simulation results show the effectiveness of proposed synchronization approaches.

Decentralized adaptive synchronization of an uncertain complex delayed dynamical network

In this paper, we investigate the locally and globally adaptive synchronization problem for an uncertain complex dynamical network with time-varying coupling delays based on the decentralized control. The coupling terms here are bounded by high-order polynomials with known gains that are ubiquitous in a large class of complex dynamical networks. We generalize the usual technology of searching for an appropriate coordinates transformation to change the network dynamics into a series of decoupled lower-dimensional systems. Several adaptive synchronization criteria are derived by constructing the Lyapunov-Krasovskii functional and Barbalat lemma, and the proposed criteria are simple in form and convenient for the practical engineering design. Numerical simulations illustrated by a nearest-neighbor coupling network verify the effectiveness of the proposed synchronization scheme.

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.

Comparison between pinning control of different chaotic complex dynamical networks

The stabilization properties of various typical complex dynamical networks composed of chaotic nodes are theoretically investigated and numerically simulated in detail. Some local stability properties of such pinned networks are derived and the valid stability regions are estimated based on eigenvalue analysis. Numerical simulations of such networks are given to explain why significantly less local controllers are needed by the specifically pinning scheme, which pins the most highly connected nodes in scale-free networks, than that required by the randomly pinning scheme. Also, it is explained why there is no significant difference between the two schemes for controlling random-graph networks and small-world networks.

IMPULSIVE EXPONENTIAL SYNCHRONIZATION OF FRACTIONAL-ORDER COMPLEX DYNAMICAL NETWORKS WITH DERIVATIVE COUPLINGS VIA FEEDBACK CONTROL BASED ON DISCRETE TIME STATE OBSERVATIONS

This article aims to address the global exponential synchronization problem for fractional-order complex dynamical networks(FCDNs)with derivative couplings and impulse effects via designing an appropriate feedback control based on discrete time state observations.In contrast to the existing works on integer-order derivative couplings,fractional derivative couplings are introduced into FCDNs.First,a useful lemma with respect to the relationship between the discrete time observations term and a continuous term is developed.Second,by utilizing an inequality technique and auxiliary functions,the rigorous global exponential synchronization analysis is given and synchronization criterions are achieved in terms of linear matrix inequalities(LMIs).Finally,two examples are provided to illustrate the correctness of the obtained results.

Secure Synchronization Control for a Class of Cyber-Physical Systems With Unknown Dynamics

This paper investigates the secure synchronization control problem for a class of cyber-physical systems(CPSs)with unknown system matrices and intermittent denial-of-service(DoS)attacks.For the attack free case,an optimal control law consisting of a feedback control and a compensated feedforward control is proposed to achieve the synchronization,and the feedback control gain matrix is learned by iteratively solving an algebraic Riccati equation(ARE).For considering the attack cases,it is difficult to perform the stability analysis of the synchronization errors by using the existing Lyapunov function method due to the presence of unknown system matrices.In order to overcome this difficulty,a matrix polynomial replacement method is given and it is shown that,the proposed optimal control law can still guarantee the asymptotical convergence of synchronization errors if two inequality conditions related with the DoS attacks hold.Finally,two examples are given to illustrate the effectiveness of the proposed approaches.

Cost and effect of pinning control for network synchronization

The problem of pinning control for the synchronization of complex dynamical networks is discussed in this paper. A cost function of the controlled network is defined by the feedback gain and the coupling strength of the network. An interesting result is that a lower cost is achieved by using the control scheme of pinning nodes with smaller degrees. Some strict mathematical analyses are presented for achieving a lower cost in the synchronization of different star-shaped networks. Numerical simulations on some non-regular complex networks generated by the Barabasi-Albert model and various star-shaped networks are performed for verification and illustration.