Int. J. Communications, Network and System Sciences, 2010, 3, 893-898
doi:10.4236/ijcns.2010.311121 Published Online November 2010 (http://www.SciRP.org/journal/ijcns)
Consensus of a kind of Dynamical Agents in Network with
Time Delays
Hongwang Yu, Baoshan Zhang
School of Mathematics and Statistic, Nanjing Audit University, Nanjing, China
Received August 4, 2010; revised September 18, 2010; accepted October 23, 2010
Abstract
This paper investigates the collective behavior of a class of dynamic agents with time delay in transmission
networks. It is assumed that the agents are Lyapunov stable distributed on a plane and their location coordi-
nates are measured by some remote sensors with certain error and transmitted to their neighbors. The control
protocol is designed on the transmitted information by a linear decentralized law. The coordination of dy-
namical agents is shown under the condition that the error is small enough. Numerical simulations demon-
strate that our theoretical results are valid.
Keywords: Multi-Agent System, Time Delay, Consensus Protocol
1. Introduction
Distributed coordination of network of dynamic agents
has attracted a great attention in recent years. Modeling
and exploring these coordinated dynamic agents have
become an important issue in physics, biophysics, sys-
tems biology, applied mathematics, mechanics, computer
science and control theory [1-10]. How and when coor-
dinated dynamic agents achieve aggregation is one of the
interesting topics in the research area. Such problem may
also be described as a consensus control problem.
To describe the collective behavior of agents in a large
scale network, the agent in the network usually is mod-
eled by a very simple mathematical model, which is an
approximation of real objects. Saber and Murray [3,4]
proposed a systematical framework of consensus prob-
lems in networks of dynamic agents. In their work the
dynamics of the agent is modelled by a simple scalar
continuous-time integrator
x
u
, the convergence
analysis is provided in different types of the network
topologies. Following the work of [2,3], G. Xie and L.
Wang [8] study the case where the dynamics of each
agent is second order. In their work, they show that by
means of a simple linear control protocol based on the
structure of the graph, the dynamical agents will eventu-
ally achieve aggregation, i.e., all agents will gradually
move into a fixed position of the line, meanwhile their
velocities converge to zero.
In networks of the dynamic agents, time delays and
sensor error may arise naturally, e.g., because of the
moving of the agents, the congestion of the communica-
tion channels and the finite transmission speed due to the
physical characteristics of the medium transmitting the
information. The different protocols with time delays
have been investigated [2,4]. And the sensor error or
communicated errors are also to be considered in the
consensus protocol [9]. It is shown that the collective
behavior of dynamical agent will depend on the commu-
nicated error and the algebraic characterization of the
communicated network topology.
In this paper, we study the consensus of multiagent
systems where the dynamics of each agent is second or-
der. The agents may represent the vehicles or mobile
robots spread over a wild area and they communicate by
means of some remote sensors with certain error and
time delay. When the agents are moving in a plane, the
consensus conditions will depend on the time delay,
communicated error and the algebraic characterization of
network topology, as well as the dynamical behavior of
agents.
This paper is organized as follows. In Section 2, we
recall some properties on graph theory and give the
problem formulation. In Section 3 the main results of this
paper are given and some simulation results are pre-
sented in Section 4. Final section is a conclusion.

894 H. W. YU ET AL.
2. Preliminaries
By , we denote an undirected graph with an
weighted adjacency matrix
(, , )GVEA
[]
ij
A
a
EVV
, where
is the set of nodes, is the set of edges. The
node indexes belong to a finite index set
1
{, ,}
M
Vp p


1, ,
M
M .
An edge of G is denoted by . The adjacency
elements are defined in following way:
(,
ji )p
ij
eE
ij
ep
ij
a

. Moreover, we assume for all 0
ij
a 0
ii
aiM

.
The set of neighbors of node is denoted by
i
p


|(
ij
NpV ,
i
p)
j
pE
Ddiagd 
.
A diagonal matrix is a degree
matrix of G with . Then the Laplacian of the

,,d


1M
1
M
iij
j
da


weighted graph G is defined as . A graph is
called connected if there exists a path between any two
distinct vertices of the graph.
LDA
An important fact of L is that all the row sums are zero,
therefore, is an eigenvector of L associ-
ated with zero eigenvalue. Moreover, the graph G is
connected if and only if its Laplacian L is satisfied
rank(L) = M − 1 and all eigenvalues of L are of positive
real numbers except that only one eigenvalue is zero [2].

11,,1
T
M
Consider a network of dynamical agents defined by a
graph . The node set V consists of dy-
namical agents
(, , )GVEA
,
i
Pi M. Let 2
12
(, )
T
iii
x
xx R
P be the coordinate of dynamical agent i, then the dynamics
of are identical and described as follows.
i
p
ii
iii i
i
ii
x
v
mvKv u
x
yF
v






 (1)
where i
x
is indicates the location vector of agent i
in the plane, represents its velocity vector
p
12
(, )
T
iii
vvv
of the ith agent, is its mass and is
i
m11 12
21 22
kk
Kkk



a dynamical feedback matrix of the agent. F is an obser-
vation matrix of the agent by some remote sensor. In
what follows we simply assume that and ii 1
i
mpx

.
Let which means that the location informa-
tion of the ith agent is only measured by some remote
sensor and is transmitted to its neighbors through the
network. The matrix C is assumed to be the form
[0]FC
1
1
i
i
C








. The parameter i

indicates that the
network communicated error or the coordinates used for
sensor could be different from that of the agents.
For the dynamic agent (1) in network we have follow-
ing assumption.
Assumption 1. The dynamics (1) is Lyapunov stable
when it disconnected with its neighbors, meaning that the
dynamical agent as an autonomous will gradually stop by
moving a finite distance for any non-zero initial velocity.
We shall give the conditions, under which the network
of dynamical agents (1) achieve asymptotical consensus
meaning that there exists a fixed position (equilibrium)
such that for
2
RxiM

21
lim( )
lim( )0
i
t
i
t
x
tx
vt





 (2)
Due to time-delay in communicated network, the con-
trol protocol of the dynamical agent i is a neighbor-
based linear control law in the form that
p
(( ())( ()))
ji
iijjijiij
pN
uayttytt 



(3)
where i is the set of neighbors of agent i and
ij are adjacency elements of A. The
N p
a() 0
ij t

, denoting
the communication transmission time-delay from agent
j
p to agent i. In this paper, we discuss the consensus
of dynamical agent under the condition of the communi-
cated error and the time delay. We focus on the simplest
possible case where the time-delays in all channels are
equal to
p

and the communicated errors are equal to

.
Remark 1. If we choose 0

 and 11 22 kkk

,
1221 0kk

, then the two-dimension agent systems (1)
with the control protocol (3) can be decoupled into two
identical linear systems and it was discussed by some
literatures with or without time delays( such as [8,9]).
3. Consensus of Dynamic Agents
We denote the initial locations and the initial velocities
of the agents by
1
(0) ( (0)(0))
TT
M
xx xT
1
(0) ( (0)(0))
TT
M
vv v ,
T
respectively. Under control protocol (3) with ()
ij t



,
the dynamical equation of agent is written by
i
p
()()( ()())
ji
iiijj i
pN
tAtB att



  


where , () ( (),())
TT
iii
txtvt  T iM
22 22
22
0
0
I
A
K








, . (4)
22 22
22 22
00
0
BI







T
Furthermore, let 1, then the
dynamic network is of the following form
() ((),,())
TT
M
tt t   
12
()()( )ttt




 (5)
where
12
,
M
I
ALB

  (6)

H. W. YU ET AL.
895
And L is the Laplacian associated with the connected
graph G. Because (5) is a standard linear time-delay dy-
namical systems, its stability analysis is equivalent to
analyzing the eigenvalues and eigenvectors of matrix
.
12
Lemma 1. The matrix 12 has two eigen-
vectors associated with zero eigenvalue. Let
e

 e

 

,

be
the left and right eigenvectors (denoted by matrices) of
matrix associated with zero eigenvalue, respectively.
Then

22
11
T
TM
K
I
M







,
1
22
110
M
K
M








(7)
and 22
I



, where

11,,1
T
M
MR.
Proof It is well known that (refer to [2,11]) the graph
is connected if and only if its Laplacian
satisfies that rank(L) = M − 1. Moreover,
is an eigenvector of L associated with zero eigenvalue.
Then, there is only one zero eigenvalue of L, all the other
ones are positive and real. By the definition of (6) and
, one has
(, ,)GVEA
1
e

 

11,,1
T
M
2
 


22 12
22 22
22
22
22 22
22
22 22
42
11
0
110
00
110
0
M
M
M
T
TM
T
TM
KI e
M
I
IKI K
M
eLeLKII
M












  




















Thus,
22
11
T
TM
K
I M








represents the two left
eigenvectors of .

Similarly, it is easy to check 42 0M


 , which im-
plies that
1



22
110
M
K
M




 represents two left-
eigenvectors of . And it is holds that 22
I



 .
Theorem 1 If the dynamical feedback matrix K in
dynamical agent (1) satisfies Assumption 1, i



in
the C of (1) and the time delays ()
ij t

  in (3) satis-
fies
min max

  (8)
with
min 22
2( )
abc d
eab



, max 22
2( )
abc d
eab



 (9)
where , ,,
and
21 12
ak k
22
()4(e a

  11 22
bkk 
22
)bb c11 2212 21
ckkkk
d abc
M

  denotes
the biggest eigenvalue of matrix L, then all of the eigen-
values of 12
e


 
0
defined in (6) are of negative
real parts except for only two zero eigenvalues.
Proof As the graph G is connected, one denotes the
eigenvalues of L by 12M



 . There
exists an orthogonal matrix W such that
12
{, ,,}
M
iag
T
WLW d

 
44
44
44
44
()
()
00
M
T
T
WI W
WII
AeB
.
One can verify the following formulae.
44
44
144 44
2 44
44
()
( )(
00
00
M
I
AeLBWI
AeB
)
A
eB











 


























Then the dynamical behavior of the network (1) is char-
acterized by the eigenvalues of ,
iiMAeB

  .
First we discuss the block with 10

. By Assump-
tion 1, one has
1122 0,kk12211122
kk kk

 (10)
and 1
()Ae B

 rank 2


12
0, (ss
, its four eigenvalues satisfy
3 4
) 0,() 0Res Res


For 0
i

, one has 22 2
0
ii
I
AB C eek









. As
()rank C2

, then . Therefore,
12
()
i
AeB  rank


4
e

  has only two zero eigenvalues. Consider
the characteristic polynomial of i
A
Be

  for iM

.
11 12
21 22
43 2
1234
()( ())
010
00
i
AB i
ii
ii
sdetsIA B
s
s1
s
kk
ksk
e
ee
e
s
e
asasas a







 





 
where
11122211 221221
321121122
22 2
4
, 2
()()],
(
,
[
1)
i
i
i
e
e
akkakkkk
akk
e
kk
a






 


(11)
Construct the Routh array of ()
i
AB
s



4
24
3
13
2
12
1
1
0
1
1
0
0
0
aa
s
aa sbb
s
c
s
d
s

896 H. W. YU ET AL.
with 22 2
12 3
1214
1
,
i
aa a
bbdae
a

(1),




22
131212331 4
1
111
.
ba abaaa aaa
cbab


By the criterion, for stability it is
necessary that 111 1 . Therefore, the
dynamical network is stable if and only if the following
inequalities hold
Routh Hurwith
0, 0,ab0, 0 cd
1
4
12 3
22
12331 4
0
0
0
0
a
a
aa a
aaaaa a






 

(12)
By (10) and (11), it is obvious that the first and second
inequalities in (12) hold, and one may easily verify that
the third inequality in (12) holds only if the fourth ine-
qualities holds. Then
22
12331 4
112211 2212212112
222 22
1122112221 12
{()() [()
()]} [()(
ii
aaaaa a
kk kkkkkk
kk eekkkk





 )]
2
So the fourth inequality in (12) can be rewritten as the
following form.
222
(0)
ii i
bc abceab

  
(13)
where 2112 , ,.
From it, one may obtain
ak k 11 22 bkk 11 2212 21 ckkkk
22 22
2( )2(
ii
i
ii
abc dabc d
eab eab





)
2
where . Under the case,
the matrix i
222
()4 ()
ii
dabceabb c

  
A
Be

  is
H
urwith. Then it is obvious
that the matriies i
A
Be

 are
H
urwith

for all
under the condition of (8) and (9). There-
fore, all of the eigenvalues of the matrix 12
defined in (6) are of negative real parts except for only
two zero eigenvalues.
2,i,M
e

 
Theorem 2 Under conditions of Theorem 1, the con-
trol protocol (3) globally and asymptotically achieves the
collective behavior of the dynamic agents.
Proof Under conditions of Theorem 1, all of the ei-
genvalues of the matrix 12 defined in (6)
are of negative real parts except for only two zero eigen-
values. By
e

 
4 14
, ,,)44
12
:(,
M
M
R 

 
MM
 

 one
denotes right-eigenvectors of

associated with eigen-
values 12 2
,,,
M

 
 

respectively.
It holds that , 22 1,
1


0,,,
M
diagJ J

J
denotes the Jordan form of two order associated with
the eigenvalues 1

and 2

. i
J
denotes the Jordan
form of four order associated with the eigenvalues
43 4241
,,
iii




and 4i

for all . 2, ,iM
414
, )
Let 14
:(, ,
TT 4
12
,
M
M
R

MM
TT T
 

 



,
where ;4
ii
 
M are 4
M
row left-eigenvectors of

correspondingly. Then it is holds that . 

41
(Re R

As 34 )( 4 ) )(
MM
Re e0(Re)




2(42)
42)(42)
M
M

 
  
)2(
0
00
MM

,
one has 22
(42
I

limJt
te








2(4 2)
(42)(42
0
00
M
MM

 
and
22
42)2 )M

(
I
lim( )
texp t 



 





Let 12
()



and
2
1








, one has li
tm()exp t


.



Due to the fact that 1
44
··
M
M
I 


 ,

and

satisfy the property 22
I



.


1
22
2
22
(
0
exp
k
2
(0 )
·(
·(0)
kI

1
22
lim()lim)(0)
10)
1
00
tt
MM
MM
t t
M
Ik
M










 





11
11










Therefore, 1 (0) (
ij
x kv


, 2,,} M
1
1
lim ({[0)]}
M
j
tj
xt M
 


)
0,
and
it is obvious that
lim( ){1
i
tvt i



This implies the protocol (3) globally asymptotically
achieves the collective behavior of the dynamic agents.
Corollary 1 The dynamical feedback matrix K is
chosen to be 1221 1 kkk


and 11222 , then the
control protocol (3) globally and asymptotically achieves
the collective behavior of the dynamic agents if the time
kkk

satisfy
22
21
2
delays ln kk 



.
Proof It is easy to verify according to the proof of
Theorem 1 and Theorem 2. Under Assumption 1, one has
122111 22
kk kk

. Thus, one finds that and it further
implies that min
0c
0


and max 0

 in (9) for bounded
time delay. Thus we have the following result.
Corollary 2 The dynamical agents achieve collective
behavior if the network communicated error

is suf-
ficient small for the bounded time delays.
4. Simulations
Numerical simulations will be given to illustrate the
theoretical results obtained in the previous section. Con-

H. W. YU ET AL.
897
sider five dynamic agents under network described in
Figure 1.
We can obtain the Laplacian matrix L of the graph G
of Figure 1 and its eigenvalues are ,
10
 23




1(713)
2, 45
1(7 13)
2
  , 64

.
We consider that the dynamic agent (1) in the network
has ,
10.1
0.2 1
k




0.1

 in the observation matrix C
and the time delay 0.1

 in (3). Thus, it is
stable and satisfies Assumption 1. One can get
Lyapunov
0.1a

,
, ,20.98cb12.132 4and the 0.1

 belongs
to the range of parameters, i.e.,
min max
0.2125 0.2764

 
When a control protocol (3) is applied into the agents
in network, the collective behavior of dynamic agents
takes place according to our result. Figures 2-3 give
simulation results of the collective behavior of the agents
while their velocities.
5. Conclusion
In this paper, we discuss the collective behavior of dy-
namical agents in network which associated with a graph
Figure 1. An undirected graph G with M = 6 nodes.
Figure 2. State trajectories of the agents in G.
Figure 3. Velocity trajectories of the agents in G.
G. It is assumed that the agents are Lyapunov stable dis-
tributed on a plane and their location coordinates are
measured by some remote sensor with certain error and
transmitted to its neighbors. Based on the transmitted
information with time-delay, the control protocol is de-
signed as a linear decentralized law. The coordination of
dynamical agents is shown under the condition that the
error is small enough. There is a tradeoff between ro-
bustness of a protocol to time-delays and the sensor er-
ror.
6. Acknowledgement
This work is supported by the Nanjing Audit University
Fund (No. 20720035).
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