Adaptive lag synchronization for uncertain complex dynamical network with delayed coupling
Introduction
A complex dynamical network (CDN) is a set of coupled nodes interconnected by edges, in which each node represents a dynamical system. The structure of many real systems in nature can be described by the CDNs, such as social relationship networks, metabolic networks, food chain, disease transmission networks, Internet, the World-Wide-Web, power grids, and so on [1], [2], [3]. This has led to much interest to the studies of the CDNs. In particular, synchronization of the network has been one of the main topics due to its various applications. In the literature, a number of researchers have proposed many synchronization methods including linear state feedback control [4], pinning control [5], [6], state observer based control [7], control of CDN with impulsive effect [8], [9], and adaptive control methods [10], [11], [12], [13], [14]. It should be noted that these studies dealt with complete synchronization scheme. However, several different types of synchronization phenomena have been reported, such as generalized [15], [16], lag [17], phase [18], projective [19], anticipating synchronization [20], and so on. Among them, lag synchronization can be a reasonable scheme from the viewpoint of engineering applications and characteristics of channel. This is why the time delay is inevitable when signals between systems are transferred. Therefore, lag synchronization has become a hot topic and attracted much attention from authors in many fields [17], [21], [22], [23]. Unfortunately, there exist few results of lag synchronization method for CDNs [24]. In [24], a control method was proposed to lag synchronize the network with an identical node. Although the approach achieved the lag synchronization for CDN, there are still some problems which should be studied. They include: (1) coupling delay, (2) parameter uncertainty and external disturbance, and (3) synchronization with non-identical node. (1) Coupling delay between nodes is an inevitable factor in the network. Because the speed of signal travel between nodes is limited and the network nodes may be required to have non-local interconnections such as telecommunications [25], [26]. (2) It is well-known that parameter uncertainty and external disturbance are unavoidable factors in many practical situations. Moreover, they can destroy the system stability or can make control of dynamic systems more difficult due to their effects. Therefore, some approaches such as updating law for unknown parameters or robust controller have been developed to deal with the uncertainty and disturbance [12], [13], [27]. (3) It is not realistic to assume that all nodes of the network are synchronized with an identical reference node. In real life applications such as laser array and biological systems, it is recognized that the network synchronization with non-identical node can be demanded [28], [29]. Therefore, it is worth proposing a lag synchronization method in which the problems mentioned above are considered.
In this paper, a lag synchronization method between uncertain complex network with delayed coupling and a non-identical reference node has been proposed. Both the network nodes and reference one have parameter uncertainties and bounded external disturbances. All of the unknown parameters are estimated by adaptive laws derived from Lyapunov stability theory, which are used in the proposed synchronization method. By use of the updating laws, a robust controller is designed to synchronize the network despite the disturbances bounded by unknown constants. In the end, the network is globally asymptotically synchronized with the proposed method. Results of numerical example show the effectiveness of the proposed approach.
The notation throughout the paper is quite standard. denotes n-dimensional Euclidean space, and is the set of all n × m real matrices. The notation X > 0(⩾0) means that X is real symmetric and positive definite (semi-definite). diag (⋯) denotes the block diagonal matrix. The superscript ‘T’ denotes the transpose of the matrix. Sometimes, the arguments of a function or a matrix will be omitted in the analysis when no confusion can arise.
Section snippets
Problem statement
Consider a controlled complex dynamic network consisting of N linearly and diffusively non-delayed and delayed coupled nodes with both parameter uncertainty and disturbance. The ith node can be described as follows:where i = 1, 2, … , N, is the state vector of node i, is input vector, and are the known continuous nonlinear function matrices, is the
Controller design for lag synchronization
In this section, we propose an adaptive lag synchronization method for the uncertain complex dynamical network (1) with delayed coupling.
From (1), (2), the error dynamics for lag synchronization is obtained as
The following theorem provides the control input and adaptive laws design method to make the errors ei(t) for i = 1, … , N globally asymptotically stabilized. Theorem 1 Consider the
Numerical simulation
Let us consider an example to demonstrate the effectiveness of the proposed lag synchronization method. The reference node is described as Chua’s circuit (Fig. 1(a)) with unknown parameters and disturbancewhere with η1 = −1.4325 and η2 = −0.7831, and the parameter vector and disturbance signal are chosen as
Conclusion
An adaptive lag synchronization method was presented for uncertain CDNs with delayed coupling. Both the network and a non-identical reference node are affected by parameter uncertainties and disturbances. The unknown parameters were estimated by the adaptive laws obtained from Lyapunov stability theory. Even if there exist unknown bounded disturbances, the proposed controller with the estimated parameters achieved the lag synchronization of the network. Numerical results showed the
Acknowledgements
This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0009373). This research was also supported by the Yeungnam University Research Grants in 2010.
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2021, NeurocomputingCitation Excerpt :For instance, many experts and scholars have asserted that the mammalian brain can storage associative memories and modulate oscillatory neuronal synchronization by selective perceive attention. Recently, some synchronization criteria of CDNs have been proposed based on Lyapunov stability theory and many kinds of control methods, such as pinning control [11,12], intermittent control [13,14], impulsive control [15,16], adaptive control [17,18], and sliding mode control (SMC) [19,20]. SMC as an effective nonlinear control method is formed by reaching mode and sliding mode.