Discrete-time Complex Dynamical Networks Research Articles

Aperiodic intermittent event-triggered synchronization control for discrete-time complex dynamical networks

In this paper, the synchronization control of discrete-time complex dynamical networks (CDNs) is studied under the framework of intermittent networked communication. To reduce transmission frequency, a novel aperiodic intermittent event-triggered mechanism (ETM) with discontinuous characteristics is proposed. Under this mechanism, the controller only works within predetermined working intervals, and whether the control signal is updated depends on a preset triggering condition. In addition, an exponential term is designed in this mechanism to achieve integration and correlation between the intermittent mechanism and the ETM, which depends on the initial instants of the working intervals. Compared with the existing ETMs, this mechanism saves communication and computing resources effectively. To improve flexibility in the design of working/rest intervals, the concepts of average working time ratio (AWTR) and average working period (AWP) are introduced to describe intermittent aperiodic performance. A sufficient criterion for the synchronization of discrete-time CDNs is obtained, and the feasible domain of AWTR is calculated. Finally, a simulation example is given to verify the effectiveness of the proposed method.

Cluster Synchronization Control for Discrete-Time Complex Dynamical Networks: When Data Transmission Meets Constrained Bit Rate.

In this article, the cluster synchronization control problem is studied for discrete-time complex dynamical networks when the data transmission is subject to constrained bit rate. A bit-rate model is presented to quantify the limited network bandwidth, and the effects from the constrained bit rate onto the control performance of the cluster synchronization are evaluated. A sufficient condition is first proposed to guarantee the ultimate boundedness of the error dynamics of the cluster synchronization, and then, a bit-rate condition is established to reveal the fundamental relationship between the bit rate and the certain performance index of the cluster synchronization. Subsequently, two optimization problems are formulated to design the desired synchronization controllers with aim to achieve two distinct synchronization performance indices. The codesign issue for the bit-rate allocation protocol and the controller gains is further discussed to reduce the conservatism by locally minimizing a certain asymptotic upper bound of the synchronization error dynamics. Finally, three illustrative simulation examples are utilized to validate the feasibility and effectiveness of the developed synchronization control scheme.

Distributed periodic event-triggered synchronization for discrete-time complex dynamical networks with time-varying delay

In this paper, we study the synchronization control problem for a class of discrete-time complex dynamical networks (CDNs) under an event-triggered mechanism (ETM). First, a discrete-time distributed periodic ETM is developed based on communication measurement. Under this mechanism, the signal is sampled in a periodic manner, but whether the control signal is updated or not depends on the pre-designed triggering condition. The sampling signal is designed as the communication measurement calculated by both state feedback and inner communication among nodes. Compared with the existing discrete-time ETMs, this method considers the topology connection among network nodes and avoids the continuous measurement of signals. Therefore, this mechanism effectively saves communication resources and reduces the waste of computing resources. Second, in order to analyze the serrated constraint caused by the periodic sampling error, a piecewise Lyapunov function is constructed. Third, under the distributed periodic ETM, sufficient conditions are derived for bounded synchronization of discrete-time CDNs with time-varying delay. Finally, a simulation example verifies the effectiveness and accuracy of the conclusion.

Impulsive Pinning Control of Discrete-Time Complex Networks with Time-Varying Connections

Complex dynamical networks with time-varying connections have characteristics that allow a better representation of real-world complex systems, especially interest in their not static behavior and topology. Their applications reach areas such as communication systems, electrical systems, medicine, robotic, and more. Both continuous and discrete-time complex dynamical networks and the pinning control technique have been studied. However, even with interest in the research on complex networks combining characteristics of discrete-time, time-varying connections, pinning control, and impulsive control, there are few studies reported in the literature. There are some previous studies dealing with impulsively pin-controlling a discrete-time complex network. Nevertheless, they neglect to deal with time-varying connections; they deal with these systems by experimentally using continuous-time methods or linearizing the node dynamics. In this manner, this paper presents a control scheme that not only deals with pin control on discrete-time complex networks but also includes time-varying connections. This paper proposes an impulsive pin control to a zero state using passivity degrees considering a discrete-time complex network with undirected, linear, and diffusive couplings. Additionally, a corresponding mathematical analysis, which allows the representation of the dynamics as a set of symmetric matrices, is presented. With this, certain kinds of time-varying connections can be integrated into the analysis. Moreover, a particular criterion for selecting nodes to pin is also presented. The behavior of the controller for the non-varying and time-varying coupling cases is shown via numeric simulations.

Periodic Event-Triggered Synchronization for Discrete-Time Complex Dynamical Networks.

In this article, we investigate the periodic event-triggered synchronization of discrete-time complex dynamical networks (CDNs). First, a discrete-time version of periodic event-triggered mechanism (ETM) is proposed, under which the sensors sample the signals in a periodic manner. But whether the sampling signals are transmitted to controllers or not is determined by a predefined periodic ETM. Compared with the common ETMs in the field of discrete-time systems, the proposed method avoids monitoring the measurements point-to-point and enlarges the lower bound of the inter-event intervals. As a result, it is beneficial to save both the energy and communication resources. Second, the "discontinuous" Lyapunov functionals are constructed to deal with the sawtooth constraint of sampling signals. The functionals can be viewed as the discrete-time extension for those discontinuous ones in continuous-time fields. Third, sufficient conditions for the ultimately bounded synchronization are derived for the discrete-time CDNs with or without considering communication delays, respectively. A calculation method for simultaneously designing the triggering parameter and control gains is developed such that the estimation of error level is accurate as much as possible. Finally, the simulation examples are presented to show the effectiveness and improvements of the proposed method.

Matrix projective synchronization for a class of discrete-time complex networks with commonality via controlling the crucial node

This article proposes two control strategies to realize matrix projective synchronization (MPS) for a class of discrete-time complex dynamical networks (DTCDNs). It is worth pointing out that our network model is composed of nodes with different state dimensions and its outer coupling configuration matrix can be asymmetric. In addition, since a small percentage of nodes that play more important role than others usually exist in a network, thus, one important node, which is named as crucial node in this paper, is taken into account in our network model. Besides, the commonality between each node and the crucial node is also precisely expressed. Further, based on the crucial node and the commonality and associated with the Lyapunov stability theory, two control strategies are addressed to realize the exponential matrix projective synchronization (EMPS) and asymptotic matrix projective synchronization (AMPS), respectively. It should be stressed that only the crucial node is controlled and only the state of the crucial node is applied to design the EMPS and AMPS controllers. This is promising to reduce the control cost as much as possible and to improve the feasibility of our MPS strategies. The effectiveness and feasibility of our MPS schemes are rigorously proved in theory as well as verified by two numerical examples.

Pinning Synchronization of Complex Dynamical Networks on Time Scales

The purpose of this paper is to investigate the synchronization problem of complex dynamical networks on time scales, which includes the synchronization problem of continuous-time and discrete-time complex dynamical networks as special cases. A pinning control strategy is designed to achieve synchronization of complex dynamical networks on time scales. Based on the theory of calculus on time scales and the Lyapunov method, pinning synchronization criteria for complex dynamical networks on time scales are established. Moreover, a numerical example is given to verify the effectiveness of theoretical results.

Synchronization control for discrete-time complex dynamical networks with dynamic links subsystem

In this paper, the synchronization of discrete-time complex dynamical network with dynamic links is investigated. From the angle of large-scale system, if the links and nodes are time-varying, the complex dynamical network may be regarded to be composed of nodes subsystem (NS) and links subsystem (LS). The weighted value of links between nodes can be regarded as the state variables of LS, the above two subsystems are mutually coupled. The two subsystems are modeled mathematically by the state difference equations, especially, the dynamics of LS is modeled as Riccati matrix difference equation without control input. Different from the previous researches, this work concerns simultaneously the dynamics of LS and NS, by which the synchronization of NS is investigated. Associated with the given dynamic reference target for LS, the nodes controller is synthesized to ensure the state synchronization of NS to be achieved with the assistance of the LS. Finally, the numerical simulation is given to illustrate the effectiveness of the proposed theoretical results in this paper.

Pinning Synchronization Control for a Class of Dynamical Networks with Coupled Time-Varying Delays: An Interval-Observer-Based Approach

This paper is concerned with the pinning synchronization control issue for a class of discrete-time complex dynamical networks. By resorting to the interval observer approach, a novel pinning synchronization control strategy is adopted to control a small fraction of the network nodes with hope to reduce the implementation cost. By using the Lyapunov stability theorem, some synchronization criteria have been derived to ensure the desired synchronization performance. Finally, a numerical simulation is presented to illustrate the effectiveness and usefulness of the theoretical results.

Structural balance for discrete-time complex dynamical network associated with the controlled nodes

In this paper, the discrete-time complex dynamical networks with dynamic weighted value of connection relationships are regarded to be composed of the node and link subsystems, and the state variables of the two subsystems are mutually coupled. Different from most of the existing researches on synchronization or stabilization of nodes, the emphasis of this paper is on the links instead of nodes. This paper mainly focuses on the generation mechanism of structural balance in the link subsystem, the nodes only play an auxiliary role. Associated with the dynamic coupling term in the link subsystem, the suitable controller is proposed for node subsystem such that the structural balance of link subsystem without control input be achieved indirectly. Finally, a numerical simulation is given to show the effectiveness of the method in this paper.

Tracking control for the dynamic links of discrete-time complex dynamical network via state observer

In this paper, a class of discrete-time complex dynamical networks with time-varying links is regarded to be composed of the nodes subsystem and links subsystem which are mutually coupled. In the existing literatures on complex dynamical networks, the emphasizes is synchronization or stabilization of nodes, the dynamic behaviors of the links between nodes are always ignored. It is noted that the state of links is very important to the networks in practical, such as, the tension value of webs in web-winding system, the relationships between individuals in social network, therefore, discussing the tracking control problem for links subsystem is also of practical significance. Considering the state of links is difficult to be measured accurately in practice applications, this makes it difficult to design the controller. In view of this, a state observer is proposed for the links subsystem modeled mathematically by the Riccati matrix difference equation in this paper. And then the control scheme is proposed for the links subsystem by employing the state observer such that the links subsystem asymptotically converging to the given target. Finally, the simulations are used to show the validity of the method in this paper.

A New Model for Complex Dynamical Networks Considering Random Data Loss.

Model construction is a very fundamental and important issue in the field of complex dynamical networks. With the state-coupling complex dynamical network model proposed, many kinds of complex dynamical network models were introduced by considering various practical situations. In this paper, aiming at the data loss which may take place in the communication between any pair of directly connected nodes in a complex dynamical network, we propose a new discrete-time complex dynamical network model by constructing an auxiliary observer and choosing the observer states to compensate for the lost states in the coupling term. By employing Lyapunov stability theory and stochastic analysis, a sufficient condition is derived to guarantee the compensation values finally equal to the lost values, namely, the influence of data loss is finally eliminated in the proposed model. Moreover, we generalize the modeling method to output-coupling complex dynamical networks. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed model.

Tracking Control for the Connection Relationships of Discrete-time Complex Dynamical Network Associated with the Controlled Nodes

In this paper, a general discrete-time complex dynamical network is regarded to be composed of the nodes subsystem and the links subsystem which are mutually coupled. Different from the existing researches on the stabilization and synchronization of nodes group, this paper mainly focuses on the dynamical behavior of connection relationships between the nodes, and the network nodes only play a helping and secondary role in the dynamics of connection relationships. Due to the state of the links subsystem is difficult to be measured accurately in practice appliances, the Riccati difference equation without any control input is employed as the dynamic model of the links subsystem, in which the dynamic coupling term is the first order polynomial about the state of nodes. With the helping of controlled nodes, the tracking problem is discussed for the links subsystem. By using the coupling algebraic relation between the reference tracking targets of the nodes subsystem and the given tracking target of the links subsystem, the tracking control scheme is proposed for the nodes subsystem to force the links subsystem converges asymptotically to the given target. Finally, the simulations are used to show the validity of the method in this paper.

Synchronization for discrete-time complex networks with probabilistic time delays

This paper investigates the problem of adaptive synchronization for discrete-time complex dynamical networks with random delays. In such complex systems, there occurs two kinds of nonidentical probability delays, i.e. self-feedback delays and coupling delays, and these delays are assumed to take values in a given finite sets with probability distributions. By means of the Lyapunov theory, discrete-time Jensen inequality and reciprocal convex combination approach, several delay-probability-distribution-dependent conditions are derived in the linear matrix inequality (LMI) format such that the discrete-time complex dynamical networks with random delays are globally synchronization in mean square. In addition, we use Watts–Strogatz (WS) random complex networks and regular Rulkov networks as two numerical examples to illustrate the effectiveness of our theoretical analysis.

Extended dissipativity state estimation for switched discrete-time complex dynamical networks with multiple communication channels: A sojourn probability dependent approach

In this paper the problem of extended dissipative state estimation for discrete-time switched complex dynamical networks (CDNs) with mixed time delays is investigated. The switching approach is based on sojourn probabilities and it is assumed that these probabilities are known aprior. One primary channel and multiredundant channels which constitute multiple communication channels are considered to coexist for the state estimation of underlying switched CDNs. To solve for the H∞, l2−l∞, passive and dissipative state estimation, the concept of the extended dissipativity is used by adjusting the weighting matrices in a new performance index. Suitable Lyapunov–Krasovskii functional is constructed in terms of Kronecker product and based on the Lyapunov stability theory, new delay-dependent sufficient stability conditions are derived in terms of linear matrix inequalities (LMIs). The effectiveness of the developed theoretical results is demonstrated via a numerical example.

Synchronization of discrete-time Markovian jump complex dynamical networks with random delays via non-fragile control

In this proposed article, a framework is presented for the analysis of the problem of synchronization of Markovian jumping discrete-time complex dynamical networks (CDNs) with probabilistic interval time-varying delay in the dynamical node and in the network coupling. The networks are expressed in terms of Kronecker product technique. The delay in time is taken to be unexpected and the probability distribution is known a prior. The synchronization is achieved by introducing a non-fragile procedure. This controller is subject to randomly occurring perturbation and is assumed to belong to the Binomial sequence. A suitable Lyapunov–Krasovskii functional (LKF) with triple summation terms is considered. By utilizing the reciprocal convex combination approach and Finsler׳s Lemma, conditions for the synchronization of networks are established in terms of linear matrix inequalities (LMIs). The effectiveness of the results obtained theoretically are illustrated through two numerical examples.

Synchronization of discrete-time complex dynamical networks with interval time-varying delays via non-fragile controller with randomly occurring perturbation

This paper addresses synchronization problem for discrete-time complex dynamical networks with interval time-varying delays. In order to achieve the synchronization, a feedback controller subjected to randomly occurring perturbations will be considered. The randomly occurring perturbations are assumed to belong to the Binomial sequence. By constructing a suitable Lyapunov–Krasovskii functional, and utilizing reciprocally convex approach and Finsler׳s lemma, the synchronization criteria for the networks are established in terms of linear matrix inequalities (LMIs) which can be easily solved by various effective optimization algorithms. The networks are represented by the use of Kronecker product technique. The effectiveness of the proposed methods will be verified via numerical examples.

Synchronization stability of delayed discrete-time complex dynamical networks with randomly changing coupling strength

Abstract This paper addresses a delay-dependent synchronization stability problem for discrete-time complex dynamical networks with interval time-varying delays and randomly changing coupling strength. The randomly changing coupling strength is considered with the concept of binomial distribution. By constructing a suitable Lyapunov-Krasovskii functional and utilizing reciprocally convex approach and Finsler’s lemma, the proposed synchronization stability criteria for the networks are established in terms of linear matrix inequalities which can be easily solved by various effective optimization algorithms. The networks are represented by use of the Kronecker product technique. The effectiveness of the proposed methods will be verified via numerical examples.

On synchronized regions of discrete-time complex dynamical networks

In this paper, the local synchronization of discrete-time complex networks is studied. First, it is shown that for any natural number n, there exists a discrete-time network which has at least disconnected synchronized regions for local synchronization, which implies the possibility of intermittent synchronization behaviors. Different from the continuous-time networks, the existence of an unbounded synchronized region is impossible for discrete-time networks. The convexity of the synchronized regions is also characterized based on the stability of a class of matrix pencils, which is useful for enlarging the stability region so as to improve the network synchronizability.

Synchronization of Discrete-time CDNs via Delayed Impulsive Control

This paper investigates the delayed impulsive control for synchronization of discretetime complex dynamical networks (CDNs). The original dynamical network is complex with respect to the unstable or chaotic nodes, coupling time-delays, and time-varying, and thus it does not have the synchronization property. A synchronization scheme of impulsive control with delayed signal inputs is proposed for the discrete-time CDNs via hybrid discrete-time. By utilizing methods such as matrix spectrum and eigenvalue theory and dwell time, global uniform exponential stability (GUES) criteria are derived for the error system, under which exponential synchronization is achieved for the CDN. Moreover, by only delayed impulsive control signals, the exponential synchronization is realized for the CDN with coupling time-delays. Finally, two examples with numerical simulations are worked out for illustration.