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ARTICLE IN PRESS

JID: INS [m3Gsc;December 21, 2015;18:11]

Information Sciences 000 (2015) 1–13

Contents lists available at ScienceDirect

Information Sciences

journal homepage: www.elsevier.com/locate/ins

∞

consensus of event-based multi-agent systems with

switching topology

Hao Zhang

a,∗

, Ruohan Yang

, Huaicheng Yan

b,c

, Fuwen Yang

Department of Control Science and Engineering, Tongji University, Shanghai 200092, China

Key Laboratory of Advanced Control and Optimization for Chemical Process of Ministry of Education, East China University of Science and

Technology, Shanghai 200237, China

School of Information Science and Engineering, East China University of Science and Technology, Shanghai 200237, China

School of Engineering, Griﬃth University, Gold Coast QLD 4222, Australia

article info

Article history:

Received 20 January 2015

Revised 14 September 2015

Accepted 12 November 2015

Available online xxx

Keywords:

Multi-agent systems

Consensus

Event-triggered control

Markov switching

abstract

This paper is concerned with the consensus problem for multi-agent systems with switching

topology. A novel distributed control strategy is proposed to reduce the frequency of con-

troller update and save network resources. Based on communication uncertainty of practical

networks, partial information exchange and switching topology which subjects to a heteroge-

neous Markov chain are considered in controller design. An H

∞

consensus criterion is derived

by using linear matrix inequality and Lyapunov methods. According to this consensus crite-

rion, a suﬃcient condition on designing H

∞

state-feedback controller is presented. Finally, a

simulation example is given to illustrate the effectiveness of the proposed event-based control

strategy.

1. Introduction

In recent years, consensus problem of multi-agent systems has been widely studied due to its broad applications in multi-

vehicle formation control, wireless sensor networks [5,22] and so on. There are several literatures [4,9–11,13,14,18,24] that have

focused on consensus problem. Nowadays, two dominant methods, continuous control and periodic sampling control, have been

used to solve consensus problem. However, continuous control strategy needs continuous controller update, and in periodic sam-

pling control strategy, the actuation is more frequency than necessary. Therefore, many researchers have suggested that event-

triggered control [1,6,12,15,16,19,20,26,28,29] could be a promising technique to reduce controller’s update and computation cost

because control signals are kept constant until triggering condition is violated. Owing to the above advantages, event-triggered

control strategy has been ﬁrstly proposed in [2] that could be applied to solve consensus problem of ﬁrst-order multi-agent

systems. Following this thought, an event-based control scheme presented in [23] has released the threshold of event triggering

condition to be state-independent. Thus continuous monitoring between neighbor agents could be avoided. Ref. [27] solves the

consensus problem of both ﬁrst-order and second-order heterogeneous multi-agent systems with a distributed event-triggered

consensus protocol. In [33], two event-based control strategies under continuous and discrete communication among neighbor

agents are considered, by which the consensus of general linear multi-agent systems can be achieved. Different from [33],the

consensus problem of multi-agent systems is studied in [28] by using observer-based output feedback event-triggered control

∗

Corresponding author. Tel.: +86 21 69589495.

E-mail addresses: zhang_hao@mail.tongji.edu.cn (H. Zhang), xh_yrh@163.com (R. Yang), hcyan@ecust.edu.cn (H. Yan), fuwen.yang@griﬃth.edu.au (F. Yang).

http://dx.doi.org/10.1016/j.ins.2015.11.019

Please cite this article as: H. Zhang et al., H

∞

consensus of event-based multi-agent systems with switching topology, Informa-

tion Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.019

2 H. Zhang et al. / Information Sciences 000 (2015) 1–13

ARTICLE IN PRESS

JID: INS [m3Gsc;December 21, 2015;18:11]

strategy, where states of agents are unavailable and they have been observed by output signals. In [8], a novel distributed event-

triggered sampled-data transmission strategy is proposed to achieve consensus of multi-agent systems. By using this method,

the consensus of multi-agent systems can be transformed into the stability of a time-delay systems.

In aforementioned work, the communication topology of multi-agent systems is assumed to be ﬁxed. However, in prac-

tice, it is diﬃcult to avoid network uncertainty, such as the changes of system components. Thus, the research on consensus of

multi-agent systems with switching topology is of great practical signiﬁcance, and it has been extensively studied in literatures

[3,17,21,25,30–32].In[21], consensus problem is studied for multi-agent systems where agents communicate via a stochastic

information network, and the communication topology is governed by a periodic switching graph. Both ﬁxed and switching net-

work topologies are considered in [17], and average consensus of multi-agent systems can be achieved. The consensus problem

of ﬁrst-order multi-agent systems is investigated in [32] with ﬁxed and switching topologies, where the switching topologies

are described by a ﬁnite-time Markov chain. In [31], the mean square consentability problem is presented for second-order

multi-agent systems with Markov switching topology. The formation problem for Lipschitz nonlinear multi-agent systems with

random jumping process is considered in [25], where the random jumping process is modeled as a Markov chain. The sampled-

data leader-following consensus problem is studied in [3] for Lipschitz nonlinear systems, where random switching network

topologies are modeled by Markov process and communication delay is solved by proposed variable periodic sampling control

protocol. In most of the articles referenced above, the dynamics of the agents are limited to be ﬁrst-order/second-order systems,

control strategies are restricted to be continuous control strategies and the Markov switching chain is assumed to be homoge-

neous. This motivates us to investigate the H

∞

consensus problem for general linear multi-agent systems with heterogeneous

Markov switching topology via event-driven controller.

The main contributions of this paper are summarized as follows:

1. A novel event-based control strategy has been proposed. By using the proposed control strategy, a global H

∞

consensus of

discrete-time multi-agent systems can be achieved. Moreover, computation cost and actuation update of agents can be reduced.

2. Considering the uncertainty of practical networks, a switching topology which subjects to heterogeneous Markov chain has

been presented to describe the communication among agents.

3. Partial information exchange has been considered among neighbor agents, which is more reasonable compared with full

information exchange in practical networks.

Notations. R

n×n

means the set of n × n real matrices. N denotes the set of positive integers. B

†

denotes generalized inverse of

B. A ⊗ B denotes the Kronecker product of matrices A and B. I

represents the identity matrix of dimension N. The superscript 

stands for matrix transposition. E

{·} stands for the mathematical expectation. Symbol  ·  represents Euclidean norm.

2. Problem statement

The communication topology can be described by a directed graph G = (V, E, A), where the set of nodes is V = {1, 2,...,N},

the set of edges is E ⊆ V × V. The set of neighbors of agent i is denoted by N

= { j : ( j, i) ∈ E}. The adjacent matrix A =

] ∈ R

N×N

with nonnegative adjacency elements a

.If j ∈ N

, a

= 1, otherwise, a

= 0.Thedegreematrixisdescribedby

D = diag

, D

,...,D

} with D



j∈N

. The matrix L = D − A is Laplacian matrix [7].

The multi-agent systems including N agents are described by the following discrete-time linear system:

(k + 1) = Ax

(k) + Bu

(k) + Cν

(k), (1)

where x

(k) ∈ R

represents the state value of agent i, u

(k) ∈ R

denotes the control input, ν

(k) is the disturbance of the system,

A ∈ R

n×n

, B ∈ R

n×m

and C ∈ R

n×1

are constant matrices.

The transmission signal received at node i from node j can be described by

(k) = E

(k) + G

(k), (2)

where ϑ

(k) is channel noise, and satisﬁes E{ϑ

(k)} = 0, E

∈ R

n×n

and G

∈ R

n×n

are known constant matrices with appropriate

dimensions. The sequence of event-triggered instants of agent i is denoted by k

, k

,....Eachk

is deﬁned when the triggering

condition is satisﬁed, for i ∈ V, l = 0, 1, 2,.... Deﬁne

= x



(k)

), k ∈ [k

, k

l+1

), where l



(k) = arg max

∈N

{|k ≥ k



Remark 1. The information transmission between neighbor agents can be given by Eq. (2), which considers partial information

exchange among agents and the channel noise which is produced by signal transmission. If matrix E

is unﬁlled rank, there is only

partial information of agent j’s state has been transmitted to agent j, otherwise, full information exchange between neighbors

will occur. Specially, if there is full information exchange and no channel noise, i.e., E

is non-singular matrix and ϑ

(k) = 0, the

information transfer from agent j to agent i can be simpliﬁed as y

(k) =

(k) [2,23].

The event-triggered transmission scheme is designed as in [19]



(k)(r(k))e

(k) ≤ ε(r(k))ω

r(k)

(k)(r(k))ω

r(k)

(k), (3)

where

r(k)

(k) =



j∈N

r(k)

(k) − x

(k)), e

(k) =

(k) − x

(k) is the state measurement error,

(k) = x

), for k ∈ [k

, k

l+1

ε(r(k)) > 0 denotes the threshold, (r(k)) ∈ R

n×n

> 0 represents the weighting matrix.

Remark 2. If e



(k)(r(k))e

(k) ≤ ε(r(k))ω

r(k)

(k)(r(k))ω

r(k)

(k) is fulﬁlled, no event is triggered, then the next event trigger

is generated when e



(k)(r(k))e

(k) > ε(r(k))ω

r(k)

(k)(r(k))ω

r(k)

(k) is satisﬁed. The threshold ε and the weighting matrix

Please cite this article as: H. Zhang et al., H

∞

consensus of event-based multi-agent systems with switching topology, Informa-

tion Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.11.019

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