Elsevier

Neurocomputing

Volume 157, 1 June 2015, Pages 199-207
Neurocomputing

Pinning exponential synchronization of complex networks via event-triggered communication with combinational measurements

https://doi.org/10.1016/j.neucom.2015.01.018Get rights and content

Highlights

  • Two novel event-triggered control strategies are proposed for pinning synchronization of complex networks.

  • The convergence rates for complex networks under such two event-triggered control strategies are estimated.

  • The proposed event-triggered control strategies are proved to be Zeno-free.

Abstract

In this paper, we consider the pinning exponential synchronization of complex networks via event-triggered communication. By using the combinational measurements, two classes of event triggers are designed, one depends on continuous communications between the agents, the other avoids the continuous communications. The controller updates when triggering function reaches certain threshold. For such classes of two event triggers, the exponential synchronization as well as the convergence rate of the controlled complex networks are studied, respectively, by employing the M-matrix theory, algebraic graph theory and the Lyapunov method. Two simulation examples are provided to illustrate the effectiveness of the proposed two classes of event triggering strategies. It is noteworthy that the event trigger with combinational measurements avoids decoupling the actual state of the nodes, which is more effective than the error-based event trigger.

Introduction

Complex networks that consist of vast interconnected nodes are ubiquitous in the real world. What we often encounter such as social networks, biological networks, electrical power grids, the Internet and World Wide Web all can be modeled and analyzed by complex networks [1], [2]. The past decade has witnessed the dramatic progresses in studying synchronization problems of complex networks, including global synchronization [3], local synchronization [4], exponential synchronization [5], [6], anticipation synchronization [7], [8], cluster synchronization [9], to name just a few.

In practice, autonomous agent such as mobile robots are often equipped with digital microprocessors which coordinate the data acquisition, communication with other agents, and control actuation. Thus, it is necessary to implement controllers on a discrete time instants. There are mainly two classes of discrete time updated controllers: the time-triggered controller; the event-triggered controller. The updates of the former are determined by the preset triggering time instants, and the updates of the latter are determined by the preset triggering event instants. The impulsive controller [10], [11], [12] and the sampled data controller [13], [14], [15] are two typical types of the time-triggered controllers. However, the time-triggered controller may lead control wastes, a typical example is that the time-triggered controller will keep updating even when the control goal is achieved. To overcome this conservativeness of the time-triggered controller, the event-triggered controller has been proposed [16], [17], where controller updates are determined by certain events that are triggered depending on the agents’ behavior. The event-based controllers have been adopted for control engineering applications [18], [19], [20], [22].

Zeno behavior often occurs in the event-based control systems. It describes the phenomenon that the event-based controller undergoes an unbounded number of updates in a finite and bounded length of time. This can happen, for example, when a controller unsuccessfully attempts to satisfy an invariance specification by updating the measurement error and the measurement threshold faster and faster [21]. It should be pointed out that the Zeno behavior is harmful to the event-based control systems: when Zeno behavior occurs, the event-based controller updates continuously, that is, the event-based controller degenerates to the continuous-time controller, which leads to the control wastes. Hence, Zeno behavior should be avoided when designing the event-based controllers.

Tabuada [22] seminally presented a triggering condition based on norms of the state and the state error. Zhu et al. [23] extended the work of Tabuada. They studied the event-based consensus problem for general LTI systems, where the triggering function depends on the measurement errors, the states of the neighboring agents and arbitrary small constants. However, this error-dependent nature of the event triggers makes the network executions decoupled from the actual state of the agents. Fan et al. [24] proposed the combinational measurement-based triggering function, which is a combination of its own state and its neighbors’ states. This type of triggering controllers avoids the decoupling of the actual state of the agents. They also provided a new iterative event-triggered algorithm that can avoid the continuous measurement between the agents.

For large-scale networks, to reduce the control consumption, one often employ the pinning control strategy. Namely, controllers only apply control actions to a very few fraction of nodes [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. However, most of the existing works are based on continuous-time communications between nodes that have a obvious insurmountable deficiency: the designed control law requires real-time updates, which promotes network nodes to be equipped with high performance processors and high-speed communication channels [36], [37], [38], [39]. Gao et al. considered the pinning controllability of complex networks with a distributed event-triggered mechanism [40]. They proposed an error-based event trigger, and proved that the pinning controlled complex network can achieve synchronization under certain sufficient conditions. However, they did not prove that the Zeno behavior can be excluded for the designed event-trigger.

In this paper, we consider the pinning exponential synchronization problem for N identical nodes via event-triggered communication with combinational measurements. The controller updates of each agent are driven by properly defined events, which depends only on the combinational measurements. The primary contributions are stated as follows. Firstly, novel event-triggered pinning control approaches using combinational measurements are applied to the exponential synchronization problem. Here, the Laplacian matrix of the communication graph is no longer assumed to be symmetric. Second, two classes of event triggers for pinning exponential synchronization are designed. One requires continuous communications, the other avoids the continuous communications. The exponential synchronization as well as the convergence rate of the controlled complex networks are studied. Last, for such two event triggers, we prove that the Zeno behavior can be excluded, respectively.

Notations: Let x and A be the Euclidean norm of a vector x and a matrix A, respectively. Let In be the n×n identity matrix. Let 1n be the vector whose elements are 1. Let λmin(A) and λmax(A) be the smallest and largest eigenvalue of symmetric matrix A. Let ρ(A) be the spectral radius of matrix A. For a matrix A, A>0(0) means all elements in A is positive (nonpositive). For a symmetric matrix A, A0(0) means A is positive definite (negative definite). diag{} stands for a block-diagonal matrix. The superscript “T” represents the vector and matrix transpose. Let ⊗ be the Kronecker product.

Section snippets

Algebraic graph theory

Several definitions and notations in the graph theory are introduced in the following which will be used in the later analysis. The interested readers refer to some textbooks of graph theory [41] for more details.

Let G=(V,E,A) be a weighted digraph of order N with the set of nodes V={1,2,,N}, set of directed edges E=V×V, and a weighted adjacency matrix A=(aij)N×N. An edge in network G is denoted by (i,j) where i and j are called the terminal and initial nodes, respectively, which means that

Event trigger design

Under Assumption 1, Assumption 2: Using combination measurement, let qi(t)=j=1Nl˜ijεj(t) and gi(t)=qi(tki)qi(t), where {tki} is the triggering time to be determined. Then, qi(tki)=gi(t)+qi(t). Let ε(t)=(ε1T(t),ε2T(t),,εNT(t))T, F(t,x(t))=(fT(t,x1(t)),fT(t,x2(t)),,fT(t,xN(t)))Tq(t)=(q1T(t),q2T(t),,qNT(t))T and g(t)=(g1T(t),g2T(t),,gNT(t))T. Then consider the vector form of (1), we haveε̇(t)=F(t,x(t))1Nf(t,s(t))c(INΓ)(g(t)+q(t)).Consider the following Lyapunov function: V(t)=εT(t)(ΞIn)ε

Simulation example

In this section, we will provide two simulation examples to illustrate the proposed approaches. Consider the information interactive network with communication graph G given in Fig. 1 and the corresponding Laplacian matrix L and the pinning adjacency matrix D is given by L=[2200035200011002024000033],D=[2000000000000000000000000].Thus, L˜=[4200035200011002024000033].

Each network node׳s kinematic under study is modeled by a Lorenz system which is described by {ẋi1(t)=10xi1(t)+10xi

Conclusions

Based on algebraic graph theory, matrix theory, and Lyapunov method, the pinning synchronization of complex networks via event-triggered communication has been studied. Two classes of piecewise constant feedback controller with combinational measurements have been designed. One depends on continuous communications between agents and the second one can avoid continuous communication between agents. With different classes of triggering functions, two sufficient conditions have been presented to

Acknowledgment

This work was supported in part by the National Natural Science Foundation of China under Grants 61170249, 61273021, in part by the Natural Science Foundation Project of CQ cstc2013jjB40008, in part by the Research Fund of Preferential Development Domain for the Doctoral Program of Ministry of Education of China under Grant 201101911130005, in part by the Program for Changjiang Scholars and in part by the fundamental research funds for the central universities under Grant XDJK2014D029. This

Bo Zhou received the B.Sc. degree in applied mathematics and the M.Sc. degree in computer application, both from Chongqing Jiaotong University, Chongqing, China, in 2010 and in 2013, respectively. Currently, he is working towards the Ph.D. degree at the College of Electronic and Information Engineering, Southwest University, Chongqing, China. From November 2014 to January 2015, he was a program aid at Texas A&M University at Qatar, Doha, Qatar. His research interest involves neural networks,

References (45)

  • W.L. Lu et al.

    Global stabilization of complex networks with digraph topologies via a local pinning algorithm

    Automatica

    (2010)
  • S.P. Wen et al.

    Circuit design and exponential stabilization of memristive neural networks

    Neural Netw.

    (2015)
  • J. Hu et al.

    Leader-following coordination of multi-agent systems with coupling time-delay

    Physica A

    (2007)
  • D.J. Watts et al.

    Collective dynamics of ‘small-world’ networks

    Nature

    (1998)
  • A.L. Barabasi et al.

    Emergence of scaling in random networks

    Science

    (1999)
  • W.W. Yu et al.

    Local synchronization of a complex network model

    IEEE Trans. Syst. Man Cybern. B

    (2009)
  • Q. Han et al.

    Anticipating synchronization of chaotic systems with time delay and parameter mismatch

    Chaos

    (2009)
  • Q. Han et al.

    Anticipating synchronization of a class of chaotic systems

    Chaos

    (2009)
  • W.L. Lu et al.

    Cluster synchronization in networks of coupled nonidentical dynamical systems

    Chaos

    (2010)
  • T. Yang

    Impulsive Control Theory

    (2001)
  • T. Chen et al.

    Optimal Sampled-Data Control Systems

    (1995)
  • W.W. Yu et al.

    Consensus in multi-agent systems with second-order dynamics and sampled data

    IEEE Trans. Ind. Inform.

    (2013)
  • Cited by (57)

    • Fully distributed event-triggered pinning group consensus control for heterogeneous multi-agent systems with cooperative-competitive interaction strength

      2021, Neurocomputing
      Citation Excerpt :

      It is not dependent on the global information of the system which is different from the work [19–25], in where they need either the total number of the agents [19–21], or the system’s Laplace matrix [22–25]. In these circumstances, there will be less communication traffic for the heterogeneous MASs. (3) Different from the existing work employed hybrid pinning and event-triggered control mechanism [6,20,31,39,40], the pinning control strategies are presented. It shows that the zero in-degree nodes must be pinned.

    View all citing articles on Scopus

    Bo Zhou received the B.Sc. degree in applied mathematics and the M.Sc. degree in computer application, both from Chongqing Jiaotong University, Chongqing, China, in 2010 and in 2013, respectively. Currently, he is working towards the Ph.D. degree at the College of Electronic and Information Engineering, Southwest University, Chongqing, China. From November 2014 to January 2015, he was a program aid at Texas A&M University at Qatar, Doha, Qatar. His research interest involves neural networks, multi-agent networks, complex networks and their applications.

    Xiaofeng Liao received the B.S. and M.S. degrees in mathematics from Sichuan University, Chengdu, China, in 1986 and 1992, respectively, and the Ph.D. degree in circuits and systems from the University of Electronic Science and Technology of China in 1997. From 1999 to 2012, he was a professor at Chongqing University. At present, he is a professor at Southwest University and the Dean of School of Electronic and Information Engineering. He is also a Yangtze River Scholar of the Ministry of Education of China. From November 1997 to April 1998, he was a research associate at the Chinese University of Hong Kong. From October 1999 to October 2000, he was a research associate at the City University of Hong Kong. From March 2001 to June 2001 and March 2002 to June 2002, he was a senior research associate at the City University of Hong Kong. From March 2006 to April 2007, he was a research fellow at the City University of Hong Kong.

    Professor Liao holds 4 patents, and published 4 books and over 300 international journal and conference papers. His current research interests include neural networks, nonlinear dynamical systems, bifurcation and chaos, and cryptography.

    Tingwen Huang received his B.S. degree in mathematics from Southwest Normal University, Chongqing, China, in 1990, M.S. degree in applied mathematics from Sichuan University, Chengdu, China, in 1993 and Ph.D. degree in mathematics from Texas A&M University, College Station, Texas, USA, in 2002.

    He was a lecturer at Jiangsu University, China, from 1994 to 1998, and Visiting Assistant Professor at Texas A&M University, College Station, USA, from January of 2003 to July of 2003, and from August of 2003 to June of 2009, he was an Assistant Professor, from July of 2009 to June of 2013, he was an Associate Professor at Texas A&M University at Qatar, Doha, Qatar, from July of 2013 to now, he is a professor at Texas A&M University at Qatar, Doha, Qatar.

    His research areas are including neural networks, complex networks, chaos and dynamics of systems and operator semi-groups and their applications.

    Guo Chen received the Ph.D. degree from the University of Queensland, Brisbane, Australia, in 2010. He is currently a research fellow at the School of Electrical and Information Engineering, the University of Sydney, Australia. He previously held research position at the Australian National University, Canberra, and the University of Newcastle, Australia. His research interests include complex networks and complex systems, optimization and control, intelligent algorithms and their applications in smart grid.

    View full text