通过具有非周期性采样数据的事件触发控制，二阶多主体系统中的领导者遵循共识

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 62, NO. 10, OCTOBER 2015 1007

Leader-Following Consensus in Second-Order

Multiagent Systems via Event-Triggered

Control With Nonperiodic Sampled Data

Nankun Mu, Xiaofeng Liao, and Tingwen Huang

Abstract—The problem of leader-following consensus in second-

order multiagent systems is investigated in this brief, where the

data are sampled randomly within a certain known bound and

the data transmission is driven by an event-triggered control pro-

tocol. A distributed event-triggered control protocol is designed,

in which the Zeno behavior is naturally excluded by the strictly

positive sampling intervals and the data transmission is largely

reduced. Under the proposed protocol, the sufﬁcient condition

is derived for assuring the consensus, which declares that the

consensus can be achieved if the control gains and the sampling

intervals are reasonable. Some numerical examples are provided

to demonstrate the effectiveness of the proposed protocol.

Index Terms—Consensus, event-triggered control, nonperiodic

sampling, second-order dynamics.

I. INTRODUCTION

N the past decade, the collective behaviors in autonomous

agents have received increasing attention due to their broad

applications in many areas. One of the attractive collective

behavior is consensus, which refers to the problem that an

agreement state can be reached as a result of local data transmis-

sion among multiple autonomous agents, and the reader may

refer to [1]–[11] for the speciﬁc studies.

Recently, the event-triggered control protocols have attracted

a lot of research interests due to its advantage of reducing data

transmission. The so-called event-triggered control protocol is

that the data transmission only occurs when a measurement

error increases to a certain threshold; thus, the same control

goal can be achieved with less data transmission. There have

been considerable research efforts on event-triggered control

Manuscript received April 4, 2015; revised April 29, 2015; accepted May 19,

2015. Date of publication July 17, 2015; date of current version September 25,

2015. This work was supported in part by the National Natural Science

Foundation of China under Grant 61170249 and Grant 61472331, by the

Qatar National Research Fund (a member of the Qatar Foundation) through

the National Priorities Research Program under Grant 4-1162-1-181, by the

Graduate Student Research Innovation Project of Chongqing, and by the Talents

of Science and Technology Promote Plan, Chongqing Science Technology

Commission. This brief was recommended by Associate Editor J. Lu.

N. Mu and X. Liao are with the College of Electronic and Informa-

tion Engineering, Southwest University, Chongqing 400715, China (e-mail:

nankun.mu@qq.com; xﬂiao@swu.edu.cn).

T. Huang is with Texas A&M University at Qatar, Doha 23874, Qatar

(e-mail: tingwen.huang@qatar.tamu.edu).

Color versions of one or more of the ﬁgures in this brief are available online

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TCSII.2015.2458036

protocol, including the problem of packet loss [12], time-

dependent threshold [13], periodic sampled data [14], [15],

combinational measurement errors [16]–[20], potential Zeno

behavior [21], time delay [20], pinning control [22], [23], and

second-order dynamics [19].

In practice, it is possible that the sampling interval cannot

be constant [24], [25]; thus, it is natural that the nonperiodic

sampling systems are more favorable than the periodic sam-

pling systems. Additionally, compared with most of the existing

works, the second-order dynamics are more complicated; thus,

the problem of the event-based consensus in such systems is

more challenging. Although the event-triggered control proto-

cols for second-order multiagent systems have been proposed

[13], [19], the effectiveness of the event-triggered control pro-

tocols can be further improved. These discussions motivates

this brief.

In this brief, we consider the problem of leader-following

consensus in second-order multiagent systems via event-

triggered control. The distributed event-triggered control

protocol based on nonperiodic sampled data and original mea-

surement errors and time-dependent threshold is designed.

Then, the analysis on consensus is given, and the sufﬁcient con-

sensus conditions are derived. Finally, the numerical examples

also demonstrate the effectiveness of the proposed protocol.

II. P

RELIMINARIES

The communication topology in a multiagent system, which

consists of a leader and N follower agents, can be modeled as

a directed graph

G = {0}∩G,where{0} denotes the leader,

and G = {V, E, W} is a weighted directed graph of the N

agents, with the set of nodes V = {v

,...,v

},thesetof

directed edges E⊆V×V, and the weighted adjacency matrix

W =(w

) ∈ R

N×N

. If the directed edge (j, i) ∈E, then agent

j is called a neighbor of agent i with w

> 0, and agent

i can receive information from agent j;otherwise,w

=0.

The neighbor index set of agent i is denoted by N

= {j ∈

V|(j, i) ∈E}. The Laplacian matrix L =(l

) ∈ R

N×N

associated with the adjacency matrix W. The matrix B =

diag{b

,...,b

} is the leader adjacency matrix associated

with the graph

G,whereb

> 0 if agent i is a neighbor of the

leader, and b

=0otherwise. A graph is said to have a spanning

tree if and only if there is a node that can reach all the other

nodes following the edge directions.

Lemma 1: If and only if

G has a directed spanning tree with

the leader at the root, all eigenvalues of (L + B) have positive

real parts [26].

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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