Synchronization criterion of complex networks with time-delay under mixed topologies

doi:10.1016/j.neucom.2018.01.019

Neurocomputing

Volume 295, 21 June 2018, Pages 8-16

https://doi.org/10.1016/j.neucom.2018.01.019 Get rights and content

Abstract

Two kinds of synchronization issues are studied in this paper. First, the synchronization problems of complex network with time-delay are investigated using pinning control. A type of original mixed topologies with time-delay is proposed, which contains directed networks and undirected networks. The pinning controllers are constructed to make the networks synchronization. Second, the synchronization problems between two different complex dynamical networks with time-delay are investigated using Taylor expansion. The two different complex networks have various topological structures. Finally, simulation examples illustrate that the theoretical methods are effective.

Introduction

Complex dynamical networks are broadly used to simulate lots of systems, like the electricity grid, communication network, social networks, that attracted many people’s attention [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. Synchronization, as a very important phenomenon, has often appeared in papers [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [25], [28], [29], [30]. The consensus of multiagent systems is a similar topic to synchronization. Thus, some ideas from MASs could be meaningful in investigating networks synchronization. There are many kinds of synchronization, such as, synchronization of multiple networks, self-synchronization of networks, outer synchronization and so on. Time-delay, always be considered when scholars study the synchronization problem of complex networks [20], [21], [22], [23], [26], [27], [29], [31], [32], [33], [34], [35], [36], [37], [38], [39]. In [1], Fan studied heterogeneous complex networks, which has disconnected topology. In [20], Zeng investigated the stability analysis of systems with time-delay, and proposed a new method to study the stability analysis. This new method reduces the conservativeness which is generated by the Lyapunov–Krasovskii functional. However, only the general topology is handled.

Pinning control is broadly discussed [4], [5], [6], [15], [16], [17], [24], [25], [26], [27], [28], [37]. Feng investigated complex networks with periodically intermittent pinning control [13]. Lu studied pinning control synchronization problem of complex networks [28]. Some synchronization criteria were proposed for the networks. But, the topology of the networks was undirected graph. Besides that, literature [28] did not consider the effects of time-delay.

In this paper, two kinds of synchronization problems are studied using pinning control method and Taylor expansion, respectively. In the first synchronization issue, the whole networks contain leaders, controlled nodes, uncontrolled nodes, time-delay and various topologies. In the second issue, the synchronization problem between two complex dynamical networks with different topological structures has been studied.

The major contributions are as follows:

(1)
An original mixed topological structure with time-delay is studied in the first synchronization problem, including directed networks and undirected networks. The networks are under hybrid topologies, which is rarely arisen in other papers.
(2)
In most literatures, the nonlinear functional of system is under restrictions, for example, Lipschitz condition or some other restricted conditions. In the second synchronization problem, restriction is released. As far as the authors know, Taylor expansion is used to solve related function problems for the first time.
(3)
The synchronization problems for two complex time-delay networks with various topological structures have been studied for the first time. One are directed networks and the other are undirected networks.
(4)
Because the consensus of multi-agent and the synchronization of complex network have some similarity, some graphs theories are used.

This paper is structured as hereunder mentioned. The studied complex dynamical network model and some propaedeutics are stated in Section 2. Synchronization criterion of complex dynamical network is studied in Section 3. In Section 4, the synchronization problem between two different topological structures complex networks has been studied. In Section 5, numerical examples show our methods are effective and feasible. Some conclusions are presented in Section 6.

Throughout this paper, P > 0 means the matrix P is a symmetric and positive definite matrix; Symmetric terms in a matrix are denote by ′⋆′; I is a dimensioned identity matrix; $S y m [X] = X + X^{T}$ .

Section snippets

Preliminaries

Let G is a weighted digraph of order N, and the associated adjacency matrix is $A = [a_{i j}] \in R^{N \times N}$ . a_ij > 0 means a directed edge from node j to i. Assume that $a_{i i} = 0 (i = 1, . . ., N) .$ $L = [l_{i j}]$ is the Laplacian matrix of G, $l_{i j} = - a_{i j} (i \neq j),$ and $l_{i i} = \sum_{j = 1, j \neq i}^{N} a_{i j},$ $L 1_{N} = 0$ .

Suppose there are two time-delay networks, including N nodes and n₁ leaders, respectively. Consider a time-delay complex networks : $\begin{matrix} \begin{matrix} {\dot{s}}_{i} (t) & = f (s_{i} (t)) + c \sum_{j = 1, j \neq i}^{n_{1}} G_{i j}^{1} Γ (s_{j} (t - τ (t)) - s_{i} (t - τ (t))) \\ = f (s_{i} (t)) - c \sum_{j = 1}^{n_{1}} L_{i j}^{1} Γ s_{j} (t - τ (t)), i = 1, 2, \dots n_{1}, \end{matrix} \end{matrix}$ where $s_{i} (t) = (s_{i 1} (t$

The pinning synchronization of complex dynamical network

For the first synchronization problem, we regard the convex combination of all the leaders’ states as the reference state. The goal is to make all the states reach this reference state. Let $\bar{s} (t) = \sum_{i = 1}^{n_{1}} β_{1}^{i} s_{i} (t)$ is the reference state, $e_{i} (t) = s_{i} (t) - \bar{s} (t)$ and $e_{i}^{*} (t) = x_{i} (t) - \bar{s} (t)$ represent the relative states from the leaders and the followers to the reference state.

The derivative of $\bar{s} (t)$ are $\begin{matrix} \dot{\bar{s}} (t) & = & \sum_{i = 1}^{n_{1}} β_{1}^{i} {\dot{s}}_{i} (t) \\ = & \sum_{i = 1}^{n_{1}} β_{1}^{i} (f (s_{i} (t)) - c \sum_{j = 1}^{n_{1}} L_{i j}^{1} Γ s_{j} ((t - τ (t)))) \\ = & \sum_{i = 1}^{n_{1}} β_{1}^{i} f (s_{i} (t)) - c \sum_{j = 1}^{n_{1}} \sum_{i = 1}^{n_{1}} β_{1}^{i} L_{i j}^{1} Γ s \end{matrix}$

Global synchronization criterion between two different complex networks with time-delay

This section, we will consider the global synchronization criterion between two different complex networks with time-delay. One network is described by Eq. (1), and another network is given below: ${\dot{x}}_{i} (t) = f (x_{i} (t)) - c \sum_{j = 1}^{N} L_{i j} Γ x_{j} (t - τ (t)), i = 1, 2, \dots, N .$

Obviously, the two teams of network have different topological structures. One is directed and the other is undirected. We rewrite Eq. (7) in the following compact form: $\dot{e} (t) = [(I_{n_{1}} - 1_{n_{1}} β_{1}^{T}) \otimes I_{n}] F (s (t)) - c (L^{1} \otimes Γ) e (t - τ (t)),$ where $\begin{matrix} e (t) & = & {(e_{1}^{T} (t), e_{2}^{T} (t), \dots, e_{n_{1}}^{T} (t))}^{T}, \end{matrix}$

Simulation

In this section, three examples are simulated to verify the theoretical analysis in this paper. There are three leaders in the digraph network, which is strongly connected and three nodes in the undirect network. it’s Laplacian matrix is defined as follow: $L = (\begin{matrix} 1 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 1 \end{matrix}) .$ Γ is an identity matrix.

Example 1

The function f is described: $f (x_{i} (t)) = A x_{i} (t) = - 5 x_{i} (t) .$ Accordingly, applying LMI method, we can get the following symmetric positive definite matrixes: $\begin{matrix} P & = & (\begin{matrix} 1.2144 & 0.5066 & 0.5066 \\ 0.5066 & 1.2144 & 0.5066 \\ 0.5066 \end{matrix} \end{matrix}$

Conclusion

In this paper, the synchronization problems of complex networks with time-delay are investigated using pinning control method and Taylor expansion, respectively. There are two teams complex networks with time-delay. One’s topology is a strongly connected digraph, while the other is an undirected graph. This paper studies two kinds of synchronization issues. Some ideas from the consensus analysis for MASs, such as graphical viewpoints, have been used in this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (61433004, 61627809, 61621004), and IAPI Fundamental Research Funds 2013ZCX14.

Rui Yu received the B.S. degree in information and computing science from Northeastern University, Shenyang, China, in 2011, and the M.S. degree in computational mathematics form Northeastern University, Shenyang, China, in 2013. She is currently working toward the Ph.D. degree in control theory and control engineering, Northeastern University. Her research interests include complex networks, neural networks and network control.

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Huaguang Zhang (M’03, SM’04, F’14) received the B.S. degree and the M.S. degree in control engineering from Northeast Dianli University of China, Jilin City, China, in 1982 and 1985, respectively. He received the Ph.D. degree in thermal power engineering and automation from Southeast University, Nanjing, China, in 1991.

He joined the Department of Automatic Control, Northeastern University, Shenyang, China, in 1992, as a Postdoctoral Fellow for two years. Since 1994, he has been a Professor and Head of the Institute of Electric Automation, School of Information Science and Engineering, Northeastern University, Shenyang, China. His main research interests are fuzzy control, stochastic system control, neural networks based control, nonlinear control, and their applications. He has authored and coauthored over 280 journal and conference papers, six monographs and co-invented 90 patents.

Dr. Zhang is the fellow of IEEE, the E-letter Chair of IEEE CIS Society, the former Chair of the Adaptive Dynamic Programming & Reinforcement Learning Technical Committee on IEEE Computational Intelligence Society. He is an Associate Editor of AUTOMATICA , IEEE Transactions on neural networks, ieee transactions on cybernetics, and Neurocomputing, respectively. He was an Associate Editor of IEEE Transactions on fuzzy systems (2008–2013). He was awarded the Outstanding Youth Science Foundation Award from the National Natural Science Foundation Committee of China in 2003. He was named the Cheung Kong Scholar by the Education Ministry of China in 2005. He is a recipient of the IEEE Transactions on Neural Networks 2012 Outstanding Paper Award.

Zhiliang Wang received the B.S. degree in applied mechanics and the M.S. degree in computation almechanics from Jilin University, Changchun, China,in 1997 and 2000, respectively, and the Ph.D. degree in control theory and control engineering from Northeastern University, Shenyang, China, in 2004. He is currently as an Associate Professor with the School of Information Science and Engineering,Northeastern University. His research interests include fractional-order systems, nonlinear and adaptive control and chaos theory and its applications.

Yang Liu received the B.S. degree in automation control, M.S. degree in control theory and control engineering from Liaoning Technical University, Fuxin, China, in 2008 and 2011, respectively. He is currently pursuing the Ph.D. degree in control theory and control engineering with Northeastern University, Shenyang, China.His current research interests include reinforcement learning, optimal control,adaptive dynamic programming, adaptive control, and their industrial applications.

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Synchronization criterion of complex networks with time-delay under mixed topologies

Abstract

Introduction

Section snippets

Preliminaries

The pinning synchronization of complex dynamical network

Global synchronization criterion between two different complex networks with time-delay

Simulation

Conclusion

Acknowledgment

Neurocomputing

Neurocomputing

Neurocomputing

Phys. Rep.

BioSystems

Chaos, Solitons Fract.

Appl. Math. Comput.

Optik

Neural Netw.

Neurocomputing

Appl. Math. Comput.

Neurocomputing

ISA Trans.

Chaos, Solitons Fract.

Automatica

J. Frankl. Inst.

Optik

Appl. Math. Comput.

Neurocomputing