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Systems & Control Letters 59 (2010) 340–348

Contents lists available at ScienceDirect

Systems & Control Letters

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

Group consensus in multi-agent systems with switching topologies and

communication delays

Junyan Yu

∗

, Long Wang

Institute of Intelligent Engineering, Center for Systems and Control, College of Engineering, State Key Laboratory for Turbulence and Complex Systems, Peking University, Beijing,

100871, PR China

a r t i c l e i n f o

Article history:

Received 24 August 2009

Received in revised form

16 March 2010

Accepted 23 March 2010

Available online 22 April 2010

Keywords:

Multi-agent systems

Group consensus

Switched topologies

Communication delays

a b s t r a c t

Group consensus problems in networks of dynamic agents are addressed for two cases: (i) communication

topologies are switching and the switching occurs among finite topologies arbitrarily; (ii) communication

topologies are switching and the switching occurs among finite topologies arbitrarily, and there exist

communication delays. We introduce double-tree-form transformations under which dynamic equations

of agents are transformed into reduced-order systems. Based on the reduced-order systems, we obtain

some analysis results for the two cases. In addition, we further investigate multi-group consensus as an

extension of the group consensus, and present some analysis results by similar techniques. Simulation

results are presented to demonstrate the effectiveness of the theoretical results.

1. Introduction

As one type of critical problems for cooperative control of

multiple agents, consensus problems have been found to possess

broad applications in many areas such as computer science, vehicle

systems and unmanned air vehicles. This has resulted in tremen-

dous amount of interest in consensus problems of multi-agent sys-

tems [1–6] in recent years. Generally speaking, the main objective

in the consensus problems is to design appropriate protocols and

algorithms such that a group of agents converges to some consis-

tent value under exchanged information between each other. The

consistent value might represent physical quantities such as atti-

tude, position, temperature, voltage and so on.

Vicsek et al. proposed a discrete-time model of autonomous

agents [4]. It was assumed that agents move in the plane with the

same speed but with different headings. The concept of neighbors

of agents was introduced and some simulation results to demon-

strate the nearest neighbor rule were presented. Consequently,

Jadbabaie et al. provided a theoretical explanation for the observed

behavior of the Vicsek model in [7]. It was shown that consen-

sus can be achieved if the union of the interaction graphs for

the team are connected frequently enough as the system evolves.

Olfati-Saber and Murray established a systematical framework of a

consensus problem in continuous-time multi-agent systems with

∗

Corresponding author.

E-mail addresses: junyanyu@pku.edu.cn (J. Yu), longwang@pku.edu.cn

(L. Wang).

fixed/switching topology and communication time-delays in [8].

Two consensus protocols have been introduced for networks with

and without time delays to solve an average-consensus problem

with directed graphs, and sufficient and/or necessary algebraic cri-

terions were established based on algebraic graph theories.

In recent years, much interest has been primarily originated

from the aforementioned papers. More specifically, in [9], Ren

and Beard studied more comprehensive discrete-time consensus

scheme which includes the results in [7] as special cases. For

discrete-time models, Xiao and Wang discussed a consensus prob-

lem in the existence of time delays when agents exchange in-

formation between each other in [10]. Second-order dynamics of

continuous-time multi-agent systems were addressed in [11,12].

Moreau introduced a set-valued Lyapunov function to deal with

consensus problems for multiple agents with undirected commu-

nication links in [13]. A linear matrix inequality (LMI) approach

was adopted to study consensus problems in [14], where it was

proved that all the agents in a network achieve average consensus

asymptotically for appropriate communication delays if the net-

work topology is connected.

Moreover, there has been an increasing attention in extended

consensus problems: asynchronous consensus problems were

studied for continuous-time multi-agent systems with discontin-

uous information transmission in [15] and discrete-time multi-

agent systems in [6]; finite-time state consensus problems were

investigated for continuous-time multi-agent systems by using a

finite-time Lyapunov function method [5]; stochastic consensus

problems were studied in [16,17]. For details, please refer to survey

papers [18,19] and the references therein.

doi:10.1016/j.sysconle.2010.03.009

J. Yu, L. Wang / Systems & Control Letters 59 (2010) 340–348 341

Note that all the aforementioned results were concerned with

such consensus that all the agents in a network share a common

value. In cooperative control, for cooperative control strategies to

be effective, agents need to reach consensus on a shared data.

A group of agents must be able to respond to unanticipated

situations or any changes when a cooperative task is carried out.

This might result in that the agreements are different with the

changes of environments, situations, cooperative tasks or even

time. Motivated by the analysis above, we study a group consensus

problem. It aims to design appropriate protocols and algorithms

such that agents in a network reach more than one consistent

state, i.e., to find some appropriate control inputs such that some

agents in a network reach a consistent state, while others reach

other consistent states. In contrast to the networks studied in the

aforementioned results, the group consensus problem concerns

a network which is divided into multiple sub-networks, and

information exchange exists not only two agents in a group but

also in different groups. This is more suitable for complex practical

applications, since many applicable networks tend to be complex.

The motivation of this work is to extend the consensus in the

existing results to more general consensus–group consensus. The

work mainly builds on [20,21]. As a comparison, [20] studied a

consensus problem for continuous-time multi-agent systems in

undirected networks with switching topologies and time-varying

delays. Some conditions guaranteeing all the agents reaching a

consistent state were established under the assumption that each

topology graph has a spanning tree. In addition, [21] solved a

group average-consensus problem with undirected graphs, and

it required the interaction between any two sub-graphs to be

balanced. [21] only discussed a group average-consensus prob-

lem for networks with fixed topologies, and provided sufficient

and/or necessary conditions for agents achieving group average

consensus. In practical applications, the interaction topology be-

tween agents may change dynamically, which has resulted in a

tremendous amount of interest in switching topologies such as

[9,8,20] and so on. In this paper, we study a group consensus

problem for a multi-agent network with time-varying topologies.

To solve the group consensus problem, we introduce a double-

tree-form transformation under which the dynamic equation of

agents is transformed into a reduced-order system. Some nec-

essary and/or sufficient conditions are presented for the agents

achieving the group consensus.

The paper is organized as follows. Section 2 contains the prob-

lem formulation as well as some definitions. Section 3 is the main

results. Simulation results are presented in Section 4. The conclu-

sion is given in Section 5.

Notations. Throughout this paper, the following notations are

used: the superscripts ‘‘T’’ and ‘‘−1’’ stand for matrix transposi-

tion and matrix inverse, respectively; 1 = [1 . . . 1]

with proper

dimension and 1

= [1 . . . 1]

∈ R

; 0 represents any zero

matrix with an appropriate dimension; diag{M

, M

, . . . , M

} de-

notes a block diagonal matrix whose diagonal blocks are given by

, M

, . . . , M

; in symmetric block matrices, ‘‘∗’’ represents an

ellipsis for the term introduced by symmetry.

2. Problem formulation

Let G = (V, E , A) be a weighted directed graph of order

n (n ≥ 2) with the set of nodes V = {v

, . . . , v

}, set of edges

E ⊆ V × V , and the nonsymmetric weighted adjacency matrix

A = [a

] with real adjacency elements a

. The node indexes

belong to a finite index set I = {1, 2, . . . , n}. An edge of G is

denoted by e

= (v

, v

). The adjacency elements associated with

the edges of the graph are nonzero, i.e., e

∈ E if and only if a

6= 0.

Moreover, we assume a

= 0 for all i ∈ I. The set of neighbors of

node v

is denoted by N

= {v

∈ V : (v

, v

) ∈ E }. A directed path

is a sequence of ordered edges of the form (v

, v

), (v

, v

), . . .,

where (v

, v

j+1

) ∈ E . If a directed graph has the property that

, v

) belongs to E for any (v

, v

) ∈ E , the directed graph is

called undirected. A directed graph is called strongly connected

(connected for undirected graph) if any two distinct nodes of the

graph can be connected via a directed path (path for undirected

graph) that follows the edges of the graph. A directed tree is a

directed graph, where every node, except one special node without

any parent, which is called the root, has exactly one parent, and

the root can be connected to any other nodes through paths. A

spanning tree of a digraph is a directed tree formed by graph edges

that connect all the nodes of the graph.

Given a network consisting of n agents, let x

∈ R be the state of

the ith agent v

. We refer to (G, x) as a network with state x ∈ R

and topology graph G. Suppose that each agent is regarded as a

node in a directed graph, and available information linking from

agent j to agent i corresponds to each edge e

∈ E . Each agent

updates its current state based upon the information received from

its neighbors.

Suppose that each agent has the dynamics as follows.

(t) = u

(t), ∀i ∈ I. (1)

A state feedback

= k

, . . . , x

) (2)

is said to be a protocol with topology G if the cluster J

, . . . , v

} of nodes with {j

, . . . , j

} ⊆ I satisfies the property

⊆ {v

} ∪ N

Consider a complex network (G, x) consisting of n + m (n, m >

1) agents. We assume that topology graph G = (V , E , A) and state

x = (x

, x

, . . . , x

n+m

)

. Denote I

= {1, 2, . . . , n}, I

= {n +

1, n + 2, . . . , n + m}, V

= {v

, . . . , v

}, V

= {v

n+1

, . . . , v

n+m

V = V

∪ V

and I = I

∪ I

. Furthermore, let N

= {v

∈ V

, v

) ∈ E }, N

= {v

∈ V

: (v

, v

) ∈ E } and N

= N

∪ N

Now, for convenience we restate two definitions in [21] as

follows.

Definition 1 (Group Consensus). Protocol (2) is said to solve a

group consensus problem asymptotically if the states of agents

satisfy (i) lim

t→∞

(t) − x

(t)k = 0, ∀i, j ∈ I

and (ii) lim

t→∞

(t) − x

(t)k = 0, ∀i, j ∈ I

Definition 2 (Sub-graph). A network with topology G

= (V

, E

) is said to be a sub-network of a network with topology G =

(V, E , A) if (i) V

⊆ V , (ii) E

⊆ E and (iii) the weighted adjacency

matrix A

inherits A. Correspondingly, we call G

a sub-graph of G.

Furthermore, if the inclusion relations in (i) and (ii) are strict, and

= {(v

, v

) : i, j ∈ V

, (v

, v

) ∈ E }, we say that the first network

is a proper sub-network of the second one. Correspondingly, we

call G

a proper sub-graph of G.

Let x

(0) be the initial state of agent v

. Then, protocol (2) is said

to solve a group average-consensus problem asymptotically if the

states of agents satisfy (i) lim

t→∞

(t) =

j=1

(0), ∀i ∈ I

and (ii) lim

t→∞

(t) =

n+m

j=n+1

(0), ∀i ∈ I

. To solve the

group average-consensus problem, [21] introduced the following

protocol

(t) =











∈N

(t) − x

(t)) +

∈N

(t), ∀i ∈ I

∈N

(t) +

∈N

(t) − x

(t)), ∀i ∈ I

(3)

where a

≥ 0, ∀i, j ∈ I

, a

≥ 0, ∀i, j ∈ I

and a

∈ R,

∀(i, j) ∈ Φ = {(i, j) : i ∈ I

, j ∈ I

} ∪ {(i, j) : j ∈ I

, i ∈ I

In [21], it was required that (i)

n+m

j=n+1

n+m

j=n+1

= 0

for all i ∈ I

and (ii)

j=1

= 0 for all i ∈ I

. In

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