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2412 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 5, MAY 2014

Coordination for Linear Multiagent Systems With

Dynamic Interaction Topology in the

Leader-Following Framework

Jiahu Qin, Member, IEEE, Changbin Yu, Senior Member, IEEE, and Huijun Gao, Senior Member, IEEE

Abstract—This paper considers mainly the leader-following

consensus for multiple agents with general linear system dynamics

under switching topologies. Three different settings are systemat-

ically considered. We ﬁrst consider the setting that the underly-

ing interaction topologies switch arbitrarily among the possible

weakly connected digraphs and then extend it to a more general

setting that the weak connectivity of the interaction topologies is

kept for some disconnected time intervals with short length due

to the communication constraints among agents. Exponentially,

consensus control is proved to be achieved, and the convergence

rate can be speciﬁed as well for both settings in spite of the relaxed

conditions on the system dynamics of each individual agent which

even allow that each agent has exponentially unstable mode, while

for the last case where the weak connectivity is only maintained on

the joint of the interaction topologies, consensus control is proved

to be achieved when the system matrix of each individual agent

satisﬁes certain stability conditions.

Index Terms—Consensus control, general linear agents, leader-

following consensus, switching topologies.

I. INTRODUCTION

ESIGNING a feasible and simple distributed control law

for each agent such that a team of agents as a whole

can perform complex tasks has been the focus of signiﬁcant

research subjects in systems and control community. This is

partly motivated by its wide applications in such areas as for-

mation control [1]–[3], swarming/ﬂocking [4], [5], consensus/

Manuscript received January 22, 2013; revised April 4, 2013; accepted

May 7, 2013. Date of publication July 16, 2013; date of current version

October 18, 2013. This work was supported in part by the Australian Research

Council through Discovery Project DP130103610, by a Queen Elizabeth II

Fellowship under Grant DP110100538, by the Overseas Expert Program of

Shandong Province, by the National Natural Science Foundation of China

under Grant 61333012, by a Grant from the Shandong Academy of Science

Development Fund for Science and Technology, and by the Pilot Project for

Science and Technology in Shandong Academy of Science.

J. Qin was with the Shandong Provincial Key Laboratory of Computer

Network, Shandong Computer Science Center, Jinan 250014, China, and

also with the Australian National University, Canberra, ACT 0200, Australia

(e-mail: jiahu.qin@anu.edu.au).

C. Yu is with the Australian National University, Canberra, ACT 0200,

Australia. He is also with the National ICT (Information & Communica-

tions Technology) Australia Ltd., Canberra, ACT 2601, Australia, and also

with the Shandong Computer Science Center, Jinan 250014, China (e-mail:

brad.yu@anu.edu.au).

H. Gao is with the Research Institute of Intelligent Control and Systems,

Harbin Institute of Technology, Harbin 150001, China, and also with King

Abdulaziz University (KAU), Jeddah, Saudi Arabia (e-mail: huijungao@

gmail.com; hjgao@hit.edu.cn).

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

at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TIE.2013.2273480

synchronization [6]–[13], congestion control in communication

networks, networked control systems [14]–[18], and sensor

networks [19]–[23].

Information consensus, as one of the central problems aris-

ing in multiagent coordination with the collective objective

of reaching agreement about some variables of interests such

as attitude, position, velocity, voltage, etc., has been studied

over the past two decades from various different perspectives;

see, e.g., [24]–[33] and the references therein for an overview

and recent developments on the topic. In these works, the

agent usually has no dynamics in the absence of information

exchange between agents, and it is the exchange of information

only that determines the evolutions of the states of agents. In

addition to these works, research has recently begun to address

the consensus (synchronization) for multiagent systems of a

more general model, i.e., each agent is assumed to take general

linear system dynamics. Different from the integrator agents,

here, the individual dynamics of the agent and the exchange of

information between agents combined will affect the consensus

behavior. Some efforts have been made toward this line; see,

e.g., [34]–[40].

In the discrete-time setting, Tuna [38] studies the leaderless

consensus problem via the distributed state feedback controller,

and You and Xie [40] provide the necessary and sufﬁcient

conditions for the consensusability of multiagent system via

the static state feedback controller, while in the continuous-time

setting, Ma and Zhang [35] consider the consensusability using

the static output feedback controller, and Li et al. [34] consider

the consensus via the use of the distributed observer-type

consensus protocol based on relative output measurements. All

of these works are conducted under a ﬁxed interaction topology.

The consensus of general linear multiagent systems under the

switching interaction topology is investigated in [37] and [39]

via the dynamic feedback controller rather than the static one.

However, as also indicated in [37], conditions on the individual

agent dynamics as well as the interaction topology become very

restrictive when using the static feedback controller. Using the

static feedback controller, Qin et al. [41] study the leaderless

consensus under arbitrarily switching topologies provided that

such topologies are being kept strongly connected and bal-

anced, and Ni and Cheng [36] consider the leader-following

consensus under the undirected interaction topology where the

system matrix of each individual agent needs to satisfy a certain

restrictive assumption and the weighting factors associated

with the interactions among agents are assumed to vary in a

piecewise constant mode.

QIN et al.: COORDINATION FOR LINEAR MULTIAGENT SYSTEMS WITH DYNAMIC INTERACTION TOPOLOGY 2413

In this paper, we consider the convergence of leader-

following consensus algorithms for general linear agents un-

der the switching topology by expanding on our preliminary

work reported in [41]. More speciﬁcally, with the balanced

underlying i nteraction topology condition among the follower

agents, the leader-following consensus is studied regarding

the following three aspects: 1) the case that the underlying

interaction topology among the leader and the follower agents,

for example,

σ(t)

, i s being kept weakly connected; 2) the case

that the weak connectedness of

σ(t)

can only be guaranteed at

some disconnected time intervals, where the length of such time

intervals may be a very small number due to the communication

constraints among agents; and 3) the case that

σ(t)

is repeat-

edly jointly weakly connected. For cases 1) and 2), we will

show that consensus can be achieved exponentially fast even if

each individual agent may have exponentially unstable mode,

while consensus control for case 3) requires somewhat restric-

tive stability conditions on each individual agents. It is worth

mentioning that, for all such cases, this paper takes account

of the general case that the weighting factors modeling the

communications among agents are assumed to be dynamically

changing in an inﬁnite set as opposed to the ﬁnite case in most

of the existing literature such as [36], [39], and [28]. Indeed,

in most practical settings, the weight factors may be time

varying in a piecewise continuous mode rather than piecewise

constant [8].

The rest of this paper is structured as follows. Section II

describes the existing graph theory notions applied as well as

the problem statement in this paper while Section III aims

to analyze the leader-following consensus in various different

settings, including the setting that the weak connectivity can be

guaranteed for

σ(t)

, the setting that such a weak connectivity

can be kept for a short time across each time interval, and the

setting that

σ(t)

is repeatedly jointly weakly connected. Some

concluding remarks are made in Section IV.

Basic Notation and Notions: The f ollowing notations will be

used throughout this paper. Denote by M>0(M<0) that M

is symmetric positive (negative) deﬁnite and by M ≥ 0(M ≤

0) that M is symmetric and positive (negative) semideﬁnite. If

all the eigenvalues of M are real, then denote by λ

max

(M)

and λ

min

(M) the maximum and minimum eigenvalues of

M, respectively. Denote by diag{A

,...,A

} the block

diagonal matrix with its ith main diagonal matrix being a

square matrix A

,i=1,...,n. A matrix is called nonnegative

(positive) whenever all its elements are nonnegative (positive).

II. P

RELIMINARIES AND PROBLEM STATEMENT

Let G =(V,ε,A) be a weighted digraph of order N with

a ﬁnite nonempty set of nodes V = {1, 2,...,N}, a set of

edges ε ⊂V×V, and a weighted adjacency matrix A =

] ∈ R

N×N

in which a

> 0 ⇔ (j, i) ∈ ε. Moreover, we as-

sume a

=0,i=1,...,N. Denote by L(G)=L =[

] the

Laplacian matrix associated with G, where 

= −a

,i= j,

and l



k=1,k=i

. Con versely , any given N × N matrix

L =[

] with nonpositive off-diagonal elements and satisfying

=0, which shall be called graph Laplacian in the sequel

for convenience, can be deemed as a Laplacian matrix of a

weighted digraph, for example, G(L), which is deﬁned as

follows: G(L) is a digraph with node set {1, 2,...,N}, and

there is an edge in G(L) from j to i (j = i) if and only if 

0. Digraph G is called balanced if and only if 1

L(G)=0[28].

Given a digraph G, denote by G the mirror graph [28] of G.IfG

is a balanced digraph, then L(G)+L

(G)/2 is the Laplacian

matrix of G, i.e., L(G )=L(G)+L

(G)/2 (see [28, Th. 7]).

A digraph is called strongly connected if any two distinct

nodes of the graph can be connected by a directed path, while

it is called weakly connected if replacing all of its directed

edges with undirected edges produces a connected graph. For

an undirected graph G with the Laplacian matrix being L,

denote by λ

(L(G)) the algebraic connectivity of G [42] which

is deﬁned as

(L(G)) = min

x=0,1

x=0

L(G)x

. (1)

Note that λ

(L(G)) > 0 if and only if undirected graph G is

connected [42].

Consider a multiagent system consisting of N agents, in-

dexed as agents 1, 2,...,N, and a leader with index 0 which

evolves according to the following dynamics:

˙x

= Ax

. (2)

Each agent is regarded as a node in a digraph G, the dynamics

of which is described by

˙x

= Ax

+ Bu

,i=1,...,N (3)

where x

∈ R

is the ith agent’s state and u

∈ R

is the

control input for agent i which uses only the state information

from its neighboring agents.

In this paper, we will consider the following control law for

agent i, i =1, 2,...,N:

(t)=K



j∈N

(t)

(t)(x

(t) − x

(t))

+ Kd

(t)(x

(t) − x

(t)) (4)

where N

(t) is the set of neighbors of agent i at time t,

K ∈ R

m×n

is a feedback matrix to be designed, and d

(t) ∈

{0}∪[d

d],

d>d> 0. Agent i can receive information from

the leader at time t if and only if d

(t) ≥ d. It is further assumed

that all the nonzero and hence positive weighting factors are

chosen from [α

, ¯α], i.e., a

(t) ∈ [α, ¯α], where 0 <α< ¯α,if

and only if j ∈ N

(t).

Let G be the set of all possible balanced interaction topolo-

gies among the N agents, i.e.,

G = {G|G (V,,A =[a

])

is balanced and a

∈{0}∪[α, ¯α]}

and σ(t):[0, ∞) →I

be a switching signal whose image at

time t is the index associated with a graph in G.

Remark 1: Different from most of the existing literature (see,

e.g., [28], [36], and [39]) in which the index set I

is ﬁnite,

2414 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 5, MAY 2014

G here is an inﬁnite set as the weighting factors are allowed

to be dynamically changing in an inﬁnite set {0}∪[α

, ¯α], and

thus, I

is also inﬁnite.

Deﬁnition 1: The leader-following consensus of system (2)

and (3) is said to be reached by employing control law (4) if,

for any initial state x

(0), i =0, 1,...,N, there holds

lim

t→∞

x

(t) − x

(t) =0,i=1,...,N.

III. C

ONSENSUS ANALYSIS UNDER SWITCHING TOPOLOGY

The following is a standard assumption made on matrix pair

(A, B) when considering the consensus for linear multiagent

systems; see, e.g., [36]–[38].

Assumption 1: The pair (A, B) is stabilizable.

Remark 2: Assumption 1 allows that A has exponentially

unstable mode, i.e., A may have eigenvalues with positive real

parts.

A. Consensus Under Weakly Connected Topologies

Denote the state error between agent i and the

leader as e

(t)=x

(t) − x

(t), i =1, 2,...,N.Let

e(t)=[e

(t),e

(t),...,e

(t)]

, and then, from (3) and

(4), one obtains the following compact form of the error

dynamics:

˙e(t)=



⊗ A −



σ(t)

+ D

σ(t)



⊗ BK



e(t) (5)

where D

σ(t)

=diag{d

(t),d

(t),...,d

(t)} and L

σ(t)

is the

Laplacian matrix of G

σ(t)

Deﬁnition 2: Given any digraph G with node set

V = {1,...,N} and nonnegative diagonal matrix

D =diag{d

,...,d

}, denote by

G the induced digraph

from G and D. More speciﬁcally,

G consists of digraph G,

node 0, and the directed edges from the node 0 to the nodes

in G, where the existence of such edges follows the following

rule: There is a directed edge from node 0 to node, for example,

i,inG if and only if d

> 0, i =1,...,N.

To proceed further, we also need the following results con-

cerning the induced graph

Lemma 1 [43, Lemma 2]: Given any digraph G,

G is weakly

connected if and only if there exists at least a directed edge

from node 0 to each of the weakly connected component of G.

Furthermore, if G is balanced, then L(G)+L

(G)/2+D>

0 if and only if

G is weakly connected.

Lemma 2: Let

Π={L + D =[

]+diag{d

,...,d

}

|L is a graph Laplacian, 1

L =0;

− 

∈{0}∪[α, ¯α],d

∈{0}∪[d,

d],

i, j =1,...,N, i = j; and

the digraph induced by G(L) and D,

is weakly connected} .

Then, there exists a μ>0 such that λ

min

(

L +

/2) ≥ μ,

∀

L ∈ Π.

Proof: Denote by Ω the set of all N × N nonnegative

matrices with zero diagonal elements. It is clear that there are

only ﬁnite different types of matrices (two nonnegative matrices

and P

are said to be of the same type [44], P

∼ P

,ifthey

have zero elements and positive elements in the same places)

in Ω and all the digraphs associated with the matrices having

the same type in Ω are also with the same topological structure.

Note that Ω can be partitioned by the equivalence relation ∼.

Let [A]:={B ∈ Ω|B ∼ A} denote the equivalence class to

which A (A ∈ Ω) belongs. Without loss of generality, denote

by [A

], [A

],...,[A

] all the equivalence classes in Ω, and



L =[

]|L is a graph Laplacian, 1

L =0;

− 

∈{0}∪[α, ¯α],i,j =1,...,N,i= j;

− L +diag{

,...,

}∼A

} .

Note that the set of all N × N matrices can be viewed as the

metric space R

. In what follows, we ﬁrst prove that Π

k =1, 2,...,M, is compact in R

. We only prove that Π

is a compact set, the compactness of Π

, k =2,...,M,fol-

low exactly the same way. In fact, each L =[

] in Π

can

be viewed as a vector [

1,1

,...,

1,N

,

2,1

,...,

2,N

,

N,1

,...,



N,N

] in R

. To prove that Π

is compact in the Euclidean

space R

, it is equivalent to prove that it is a closed and

bounded set. Let Υ

={[

]|−

∈{0}∪[α, ¯α],i,j=1,...,

N,i = j; 

∈ [0,N¯α],i=1,...,N}, Υ

={[



j=1



0,i=1,...,N}, and Υ

={[



i=1



=0,j=1,...,N}.

It can be easily derived from the deﬁnition of the Laplacian

matrix that Υ

∩ Υ

is the set consisting of all graph

Laplacian L satisfying 1

L =0and the off-diagonal elements

of L are chosen from the set 0 ∪ [α

, ¯α]. We ﬁrst prove that

∩ Υ

is closed and bounded in R

. In fact, Υ

closed and bounded, and the sets Υ

and Υ

are closed but

unbounded. The former argument holds since it is the product

space of N

closed and bounded sets in R

. For the latter

argument,we only prove in the following that the set Υ

closed, and a similar proof can be derived for that of Υ

.Let

={[

i,1

,...,

i,N

]|[

i,1

,...,

i,N

] is the vector taken from

the i−th row of [

] ∈ Υ

} ,i=1,...,N.

Then, Υ

= S

× S

×···×S

. It is clear that Υ

is a closed

set in R

if each S

,i=1, 2,...,N is closed in R

.In

order to prove that S

is closed, we introduce the following

continuous multivariate function:

f : R

→ R

,f(x):=



i=1

, ∀x =[x

,...,x

]∈ R

Since f is continuous and {1} is a closed set in R

, f

−1

({1})

is closed in R

, i.e., each set S

,i=1,...,N is closed in R

Therefore, Υ

∩ Υ

is closed and bounded.

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