Wireless Sensor Network, 2010, 2, 141-147
doi:10.4236/wsn.2010.22019 Published Online February 2010 (http://www.SciRP.org/journal/wsn/).
Copyright © 2010 SciRes. WSN
Coordination for Networks of Dynamic Agents with
Time-Varying Delays
Hongwang Yu1, Baoshan Zhang1, Yufan Zheng2
1School of Mathematics and Statistic, Nanjing Audit University, Nanjing, China
2Department of Mathematics, Shanghai University, Shangha, China
E-mail: {yuhongwang, zbs}@nau.edu.cn, {yfzheng, yuhw05820107}@shu.edu.cn
Received November 21, 2009; revised December 5, 2009; accepted December 7, 2009
Abstract
This paper is devoted to the study of the coordinate stability in undirected networks of dynamical agents with
time-varying transmission delay. Neighbor-based rules are adopted to realize local control strategies for
these continuous-time autonomous agents. Sufficient and necessary conditions in terms of linear matrix ine-
qualities (LMIs) are given to guarantee the coordination of dynamical agents. Numerical simulations are
given and demonstrate that our theoretical results are effective.
Keywords: Multi-Agent System, Time-Varying Delay, Consensus Protocol, Laplacian
1. Introduction
The coordinate stability of multi-agent systems is an im-
portant research topic in engineering applications, in-
cluding moving in formation for fleets of unmanned ae-
rial vehicles (UAVs), satellite clusters and mobile sensor
networks. In the last few years, it has attracted more at-
tention in diverse fields in physics, biophysics, systems
biology, applied mathematics, mechanics, computer sci-
ence and control theory.
In the multi-agent systems, agents are usually coupled
and interconnected with some simple rules including a
proposed first-order/two-order dynamical model and
nearest-neighbor rules. Using graph theory, Jadbabaie et
al. [1] provided a theoretical explanation for the consen-
sus behavior of dynamical multi-agents. The extended
results under some more relaxable conditions are studied
in [2]. Saber et al. investigated a systematical framework
of consensus problem under a variety of assumptions on
the network topology (fixed or switching), presence or
lack of communication delays, and directed or undirected
network information flow [3,4].
In networks of the dynamic agents, time-varying de-
lays may arise naturally, e.g., because of the moving of
the agents, the congestion of the communication chan-
nels, the asymmetry of interactions, and the finite trans-
mission speed due to the physical characteristics of the
medium transmitting the information. The different con-
sensus protocols have been investigated in [46], where
the communication delay is a fixed constant. The aver-
age-consensus problem of agents under continuous-time
networks with both switching topology and time-delay is
studied in [7,8], where the dynamics order of each agent
is one. A leader-following consensus problem for multi-
ple agent with communication transmission time delays
is discussed in [9], where the dynamics of each agent is
second order.
Motivated by [7] and [9], we study the coordinated
stability of multi-agent systems where the dynamics of
each agent is second order in this paper. The communi-
cation transmission time delays of multi-agent systems
are varying and the interconnection graph of the agents is
undirected. The method used in this paper is partly mo-
tivated by the work of [10,11].
This paper is organized as follows. In Section 2, we
recall some properties of graph and give the problem
formulation. Coordinated stability analysis of the agents
under network is given in Section 3. Section 4 gives a
simulation example. Section 5 is a conclusion.
2. Preliminaries
By
(,,)
GVEA
=
, we denote an undirected graph with
an weighted adjacency matrix
[]
Aa
ij
=, where
12
{,,,}
M
Vppp
=
L
is the set of nodes,
⊆×
is the
set of edges. The node indexes belong to a finite index
set
{
}
1,2,,
MM
@L
. An edge of
G
is denoted by
(,)
ijji
epp
=for some ,
ijM
. The adjacency elements
ij
a
are defined in following way: ij
eE
0
ij
a
⇔>
H. W. YU ET AL.
Copyright © 2010 SciRes. WSN
142
and ij
eE
0
ij
a
⇔=
. Moreover, we assume
0
ii
a
=
for all
iM
. The set of neighbors of node
i
p
is de-
noted by
{
}
|(,)
ijij
NpVppE
=∈∈.
A diagonal matrix
{
}
1,,
MM
M
DddR
×
=∈
L
is a de-
gree matrix of
G
, whose diagonal elements
1
M
iij
j
da
=
=
for
iM
. Then the Laplacian of the weighted graph
G
is defined as
=−
. A graph is called connected if
there exists a path between any two distinct vertices of
the graph.
Lemma 1 The graph
G
with the Laplacian L is con-
nected if and only if rank(L)=M-1 and all eigenvalues
of L are of positive real numbers except that only one
eigenvalue is zero with eigenvector
( )
11,,1
T
M=L.
In this paper, we consider a network of dynamical
agents defined by a connected graph
(,,)
GVEA
=
. The
node set
V
consists of dynamical agents ,
i
piM
.
The dynamics of ,
i
piM
are identical and described
as follows.
ii
iiii
i
i
i
xv
mvkvu
x
yF
v
=
=+

=


&
& (1)
where
n
i
xR
is the location vector of agent
i
p
,
n
i
vR
represents its velocity vector of the i-th agent,
n
i
uR
is its coupling inputs and
n
i
mR
is its mass.
The control gain
k
is designed later. The output map
indicates that the state information of dynamical agents
is measured by, for example, some remote sensor and
transmitted to other agents in network [10].
Due to time-delay in communicated network, the con-
trol protocol of the dynamical agent
i
p
is a neighbor-
based linear control law in the form that
((())(()))
ji
iijjijiij
pN
uayttytt
ττ
=−−
(2)
where
i
N
is the set of neighbors of agent
i
p
and
ij
a
are adjacency elements of
A
. The
()0
ij tτ
, denot-
ing the communication transmission time-delay from
agent
j
p
to agent
i
p
. In the following, we assume that
time-varying delays in (2) satisfy
0(),()
ijij
tdth
ττ
≤≤
&
(3)
or 0()
ij
td
τ
≤≤
(4)
For
0
t
. That is to say, nothing has been known
about the derivative of
()
ij
t
τ
, where
d
and
h
are posi-
tive constant numbers.
To focus our study in a main stream, we simply as-
sume that
1
i
m
=
, the observation matrix
F
=
[
]
0
nnnn
I××
and
()()
ij
tt
ττ
= for all ,
ijM
in this pa-
per. We shall give the conditions, under which the net-
work of dynamical agents (\ref{dyn0}) achieve asymp-
totical consensus stability meaning that there exists a
fixed position (equilibrium)
xR
such that for
iM
1
lim()1
lim()0
in
t
in
t
xtx
vt
→∞
×
→∞
=⊗
= (5)
3. Coordination of Dynamic Agents with
Time-Varying Delay
We study the collective behavior of dynamical agents
under a class of communicated networks. The collective
behavior of dynamical agents in network can be de-
scribed by 1
()((),,()),
TTTMn
M
xtxtxtR=∈
L
()
vt
=
1
((),,())
TTT
M
vtvt
L
and its communication topology is
characterized by a connected graph
G
. By
1
(0)((0),,(0)),
TTT
M
xxx=
L
1
(0)((0),,(0))
TTT
M
vvv=
L
, we
denote the initial locations and the initial velocities of the
agents, respectively.
3.1. Description of Dynamic Systems
Under control protocol (2) with
()()
ij
tt
ττ
=for all
,
ijM
, the dynamical equations of each agent of
multi-agent systems are written by
()()((())(()))
ji
iiijji
pN
tAtBatttt
ξξξτξτ
=+−−
&
(6)
where
()((),())
TTT
iii
txtvt
ξ=,
iM
0
0
nnnn
nnnn
I
AkI
××
××

=


, 00
0
nnnn
nnnn
BI
××
××

=


.
Furthermore, let 1
()((),,())
TTT
M
ttt
ξξξ=
L
, then the
dynamic network is of the following form
()()()()(())
M
tIAtLBtt
ξξξτ=⊗−
&
(7)
where L is the Laplacian associated with the connected
graph G. Moreover, we have the following result, which
is similar to the dynamic systems without time- delay
[10].
Lemma 2 The dynamics of System 7 is stabilized if
and only if
M
systems
()()(())
iiii
tAtBtt
ηηλητ=−−
&
(8)
are globally asymptotical stable, where ,
i
iM
λare the
nonnegative eigenvalues of L.
H. W. YU ET AL.
Copyright © 2010 SciRes. WSN
143
Proof Since the Laplacian
L
of undirected graph
G
is real symmetric matrix, there exists an orthogonal ma-
trix
W
such that
1
2
00
00
00
T
M
WLW
λ
λ
λ



=Λ=



L
L
MMOM
L
where ,
i
iM
λare the nonnegative real eigenvalues of
Laplacian
L
. By the transform 22
()
nn
WI
ηξ
×
=⊗ , we
may obtain
22
()[()()()(())]
()()(())
nnM
M
WIIAtLBtt
IAttt
ηξξτ
ηητ
×
=⊗−
=Λ−
&
which implies that the dynamics of System 7 is stabilized
if and only if
M
Systems 8 are globally asymptotical
stable.
3.2. Main Results
First, by means of linear matrix inequality (LMI), we
study consensus stability of dynamic Systems 8 with
certain communication transmission time-varying delay
()
t
τ
.
Theorem 1 The dynamic equations of (8) that the ei-
genvalues of Laplacian
L
are zero, i.e.,
0
s
λ
=
for
some
sM
, achieves globally asymptotical stable if
the control gain
0
k
<
.
Moreover, let 1
2
()
()1
()
s
sn
s
t
tt
η
ηη

=⊗


, then
lim()1
0
sn
t
x
tη
→∞

=⊗


, (9)
where 12
1
(0)(0)
ss
xk
ηη
=− .
Proof Consider the dynamic equations of (8) with
0
s
λ
=
for some
sM
, it is easy to obtain their ex-
pressions as the following
()()
ss
tAt
ηη=
&
.
Denoting 1
2
()
()1
()
s
sn
s
t
tt
η
ηη

=⊗


, we have
1222
()(), ()()
ssss
tttkt
ηηηη==
&&
.
Then
22
()(0)
kt
ss
teηη=, 221
1
()(0)(0)
kt
sss
e
t
k
ηηη
=+.
Since
0
k
<
, one gets
112
1
lim()(0)(0)
sss
t
tk
ηηη
→∞ =− , 2
lim()0
s
ttη
→∞
=
.
which leads to the result of Theorem 1.
In order to prove our main result relevant to the dy-
namic Systems 8 with communication transmission
time-varying delay, we recite the following lemma [7].
Lemma 3 For any real differentiable vector function
()
n
ztR
and any
nn
×
symmetric positive definite
matrix
Γ
, one has the following inequality
()
[()(())][()(())]
()()
T
tT
tt
ztzttztztt
dzszsds
τ
ττ
Γ−−
≤Γ
&&
where
()
t
τ
satisfies 0()
td
τ
≤≤
.
Theorem 2 Assume that the control gain
0
k
<
and
the communication transmission time-varying delay sat-
isfies (3). If there exist symmetric positive definite ma-
trices
22
,,
nn
iii
PQRR
×
such that the following condi-
tions hold:
12
23
0
ii
T
ii
ΦΦ

<

ΦΦ

, 3
0
i
Φ>
(10)
where
1
2
2
3
()(),
(1)(),
1(1),
[()()].
T
iiiiii
T
iiiiiii
T
iiiii
T
iiiii
hQdABRAB
PBhQdABRB
RhQdBRB
d
ABPPAB
λλ
λλλ
λ
λλ
Φ=++−−
Φ=++−
Φ=+−−
=+−
(11)
with properly choosing
0
d
,
0
h
. Then the origin
of the
i
-th dynamic System 8 is asymptotical stable
equilibrium point if and only if the communication net-
worked topology
G
is connected.
Proof (Sufficiency) Since the undirected communica-
tion networked topology
G
is connected, the eigenval-
ues
i
λ
,
2,,
iM
=
L
of Laplacian
L
are positive num-
bers in addition to 1
0
λ
=
from Lemma 1. Consider the
characteristic polynomial of
i
AB
λ
,
2,,
iM
=
L
2
()det(())()
i
n
ABii
ssIABsks
λ
πλλ
==−+ .
Since
0
k
<
and
0
i
λ
>
, it is easy to be verify that
i
AB
λ
is Hurwitz. Then there exists a symmetric posi-
tive definite matrix
i
P
such that
[()()]
T
iiiii
ABPPAB
λλ=+− is positive definite
matrix. So (10) is always feasible for appropriate positive
scalars
h
and
d
.
Take a Lyapunov function for the
i
-th dynamic Sys-
tem 8 as follows:
()
0
()()()()()
()()()
t
TT
iiiiiii
tt
T
iii
d
VttPtsQsds
sdtsRtsds
τ
ηηηη
ηη
=+
++++
&&
Rewrite the
i
-th dynamic System 8 as the following
equivalent form
H. W. YU ET AL.
Copyright © 2010 SciRes. WSN
144
()()()()
iiiii
tABtBt
ηληλζ=−+
&
(12)
where
()()(())
iii
tttt
ζηητ
=−−
. Along the trajectory of
the solution of System 12, we have
0
()()[()()]()
[()()()()]
()()(1())(())(())
()()()().
TT
iiiiiii
TTT
iiiiiii
TT
iiiiii
TT
iiiiii
d
VttABPPABt
tPBttBPt
tQttttQtt
dtRttsRtsds
ηλλη
ληζζη
ηητητητ
ηηηη
=+−
++
+−−
+++
&
&
&&&&
With the condition (3) and Lemma 2, we have
21
()()[()()
()()]()()[
(1)()]()()
[(1)()]()
()[(1)]()
TT
iiiiiii
TT
iiiiiii
TT
iiiiii
TT
iiiiiii
TT
iiiiii
VttABPPABhQ
dABRABttPB
hQdABRBtt
BPhQdBRABt
thQdBRBdRt
ηλλ
λληηλ
λλζζ
λλλη
ζλζ
+−+
+−+
++−+
++−⋅
−+
&
( )
12
23
()
()()
()
iii
TT
ii T
iii
t
tt
t
η
ηζ ζ
ΦΦ

=


Φ−Φ

where
,1,2,3
ij
j
Φ=
are defined in (10). Therefore,
there exists a positive constant
i
β
such that
()
()()
()
i
iiii
i
t
Vtt
t
η
ββη
ζ
≤−
&.
This implies that the
i
-th dynamic System of 8
achieve asymptotical stable for 0()
td
τ
≤≤
and
0()
th
τ
≤≤
&
.
(Necessary) Since the origin of the dynamic Systems 8
is asymptotical stable equilibrium point, the eigenvalue
of
i
AB
λ
have negative real-parts except that at most
n
eigenvalues are zero. Considering the Laplacian
L
and the characteristic polynomial of
i
AB
λ
, one
may get 1
0
λ
=
and the eigenvalues
i
λ
,
2,,
iM
=
L
of
Laplacian
L
are positive numbers. By Lemma 1, we
may get the communication networked topology is con-
nected.
Due to the reversible orthogonal transform, the
M
dynamic Systems 8 are equivalent to the System 7. So
we get the same result of stability for the System 7.
Theorem 3 Assume that the graph
G
is connected,
the control gain
0
k
<
and the communication trans-
mission time-varying delay satisfies (3). If there exist
symmetric positive definite matrices
22
,,
nn
iii
PQRR
×
such that the following linear matrix inequalities hold
21222222
22232222
222212
222223
00
00
0
00
00
nnnn
Tnnnn
nnnnMM
T
nnnnMM
××
××
××
××
ΦΦ


Φ−Φ


<

ΦΦ


Φ−Φ

L
L
MMOMM
L
L
,
3
0
i
Φ>
(13)
with properly choosing positive scalars
h
and
d
,
where
ij
Φ
(
{
}
2,,
iM
L
and
1,2,3
j
=
) are defined in
(11). Then the dynamic System 7 achieves globally
asymptotical consensus stability if and only if the com-
munication networked topology
G
is connected.
Moreover,
lim()11
0
Mn
ttξ
ξ
→∞

=⊗⊗

 , (14)
where
1
11
[(0)(0)]1
M
iin
i
xv
Mk
ξ
=
=−⊗
.
Proof By Lemma 1, the networked topology
G
is
connected if and only if the real eigenvalues of Laplacian
L
with
123
0
M
λλλλ
=<≤≤
L
. Under the given
conditions, the
M
dynamic Systems 8 achieve globally
asymptotical stable if and only if the communication
networked topology
G
is connected from Theorem 1
and Theorem 2.
Since the transform 22
()
nn
WI
ηξ
×
=⊗ is reversible
orthogonal, the
M
dynamic Systems 8 are equivalent to
System 7. Hence, for appropriate positive scalars
h
and
d
, one can conclude that the System 7 achieves globally
asymptotically consensus stability if and only if the
communication networked topology
G
is connected.
And the allowable
h
and
d
can be obtained by the
feasible linear matrix Inequality 13.
As the origin of the dynamic Systems of 8 for
{
}
2,,
iM
L
is asymptotically stable equilibrium, we
have
(
)
212(1)
()()0
TT
MnM
ttηη
×−
L.
Due to the fact that one eigenvalue of Laplacian is
zero with eigenvector
( )
11,,1
T
M=L, one may get
1
11
11
1
1
T
M
M
WW
M

=


with
111
11
MM
W
−−
=− ,
11111
11
TT
MMM
WWMI
−−
=− .
By Lemma \ref{lem1} and Lemma 2, one gets
( )
( )
21221
1222
1
,,(1)
1
(),,
T
TT
MMnn
T
TTT
nnM
I
M
WI
M
ηηξ
ξξ
−⊗
=⊗
+⊗
L
L
Then, we can obtain that
1
1M
im
m
M
ξξξ
=
==
,
{
}
2,,
iM
L
.
One can get 11
1M
i
i
M
ηξ
=
=
by Lemma 1. Then
from Theorem 1, it is hold that 1
1
lim()
0
n
ttxη
→∞

=


, with
H. W. YU ET AL.
Copyright © 2010 SciRes. WSN
145
1112 1
111
(0)(0)[(0)(0)]
M
ii
i
xxv
kk
M
ηη
=
==−
.
Therefore, we can obtain
1
11
lim()[(0)(0)]1
M
iiin
ti
xtxv
Mk
→∞ =
=−⊗
,
1
lim()0
in
tvt
×
→∞ =
The proof of the Theorem is completed.
Remark For any
0
h
, the maximal allowable
d
guaranteeing average consensus in Theorem 2 and/or
Theorem 3 can be obtained from the following optimiza-
tion problem:
Maximize
d
s.t.
01
h
≤<
,
0,0,0
iii
PQR
>>>
and (13).
This optimization problem can be solved by using the
GEVP solver in Matlabs Control Systems Toolbox [5].
Considering the matrices in the linear matrix Inequal-
ity 13 are continuous for
0
i
λ
>
,
{
}
2,,
iM
L
, we may
obtain the following corollary for estimation of conser-
vative upper bound
h
and
d
.
Corollary 1. Assume that the control gain
0
k
<
and
the communication time-varying delay satisfies (3). If
there exist symmetric positive definite matrices
22
,,
nn
iii
PQRR
×
such that the following linear matrix
inequalities hold
21222222
22232222
222212
222223
00
00
0
00
00
nnnn
Tnnnn
nnnnMM
T
nnnnMM
××
××
××
××
ΦΦ


Φ−Φ

<

ΦΦ

Φ−Φ

,
23
0
Φ>
3
0
M
Φ>
(15)
with properly choosing
0
d
and
0
h
, where
2
j
Φ
,
Mj
Φ
(
1,2,3
j
=
) are defined in (13). Then the dynamic
System 7 achieves globally asymptotical consensus sta-
bility if and only if the communication networked topol-
ogy
G
is connected.
When time-varying delays satisfy (4), that is to say,
nothing has been known about the derivative of
()
t
τ
.
For Systems 8, one may construct the following Lyapu-
nov function as
0
()()()()()
t
TT
iiiiiii
dt
WttStsRsdsd
θ
ηηηηθ
−+
=+
∫∫ && .
Similar to the proof of Theorem 2 and Theorem 3, it is
easy to get the following results and we may omit their
proof here.
Theorem 4 Assume that the control gain
0
k
<
and
the communication transmission time-varying delay sat-
isfies (4). If there exist symmetric positive definite ma-
trices
22
,
nn
ii
STR
×
such that the following conditions
hold:
12
23
0
ii
T
ii
ΨΨ

<

ΨΨ

, 3
0
i
Ψ>
(16)
where
1
2
()(),
(),
T
iiiii
T
iiiiii
dABTAB
SBdABTB
λλ
λλλ
Ψ=Γ+−−
Ψ=+−
2
3
1
,
T
iiii
TdBTB
dλΨ=−
[()()].
T
iiiii
ABSSAB
λλΓ=+− (17)
with properly choosing
0
d
. Then the origin of the
i
-th dynamic System of 8 is asymptotical stable equilib-
rium point if and only if the communication networked
topology
G
is connected.
Theorem 5 Assume that the graph
G
is connected,
the control gain
0
k
<
and the communication trans-
mission time-varying delay satisfies (4). If there exist
symmetric positive definite matrices
22
,
nn
ii
STR
×
such
that the following linear matrix inequalities hold
21222222
22232222
222212
222223
00
00
0
00
00
nnnn
Tnnnn
nnnnMM
T
nnnnMM
××
××
××
××
ΨΨ


Ψ−Ψ


<

ΨΨ


Ψ−Ψ

L
L
MMOMM
L
L
,
3
0
i
Ψ>
(18)
with properly choosing positive scalars
d
, where
ij
Ψ
(
{
}
2,,
iM
L
and
1,2,3
j
=
) are defined in (17).
Then the dynamic System 7 achieves globally asymp-
totical consensus stability if and only if the communica-
tion networked topology
G
is connected.
Moreover,
lim()11
0
Mn
ttξ
ξ
→∞

=⊗⊗

 ,
where
1
11
[(0)(0)]1
M
iin
i
xv
Mk
ξ
=
=−⊗
.
Figure 1 Undirected connected graph
G
with five nodes.
H. W. YU ET AL.
Copyright © 2010 SciRes. WSN
146
Figure 2. State trajectories of the agents in
G
.
Figure 3. Velocity trajectories of the agents in
G
.
4. Simulations
Numerical simulations will be given to illustrate the
theoretical results obtained in the previous section. Con-
sider five dynamic agents under network described in
Figure 1.
Here we consider the dynamical equations (\ref{dyn0})
with
2
n
=
. By employing the LMI Toolbox in Matlab,
one gets that the maximum time-delay bound is
2.1152
d= when
0
h
=
, i.e. the value of time-delay is
fixed. When
0.5
h=, the maximum delay bound is
1.2799
d=. And we may get the corresponding feasible
solutions in the following.
22
1
22
1.49780.2519
0.25190.9299
II
P
II

=


22
2
22
1.39690.2175
0.21750.6666
II
P
II

=


22
3
22
1.23850.0985
0.09850.2196
II
P
II

=


22
4
22
1.68960.5056
0.20560.2557
II
P
II

=


22
1
22
0.17840.0334
0.03340.6915
II
Q
II

=


22
2
22
0.37660.0488
0.04880.6925
II
Q
II

=


22
3
22
2.06320.1171
0.11710.0099
II
R
II

=


22
4
22
2.41740.1864
0.18640.0471
II
R
II

=


22
322
0.24630.0197
0.01970.3858
II
Q
II

=


22
422
0.85080.1547
0.15470.0829
II
Q
II

=


22
122
1.46550.0197
0.01970.1397
II
R
II

=


22
222
1.98950.0281
0.02810.0714
II
R
II

=


The agents have initial conditions
( )
1
()212
T
xθ=− ,
( )
2
()810
T
xθ=−− ,
( )
3
()154
T
xθ=− ,
( )
4
()122
T
xθ=,
( )
5
()2525
T
xθ=,
( )
1
()1213
T
vθ=− ,
( )
2
()125
T
vθ=− ,
( )
3
()718
T
vθ=,
( )
4
()1525
T
vθ=−,
( )
5
()2015
T
vθ=− for
[1,0]
θ
∈−
. The eigenvalues of the Laplacian matrix are
1
0
λ
=
, 2
0.8299
λ=, 3
2.6889
λ=, 4
4
λ
=
,
5
4.4812
λ=. Figure 2 and Figure 3 show the state and
velocity trajectories of the multi-agent systems with
time-varying delay
()0.2sin
tt
τ=.
5. Conclusions
In this paper, we discuss the coordinate stability of
multi-agent systems where the agent is described by dou-
ble-integrator with time-varying transmission delay in
their communicated network. Two different time-varying
delays are considered for dynamical systems. We firstly
decompose the multi-agent systems into $M$ dynamical
systems by certain transformation of state space under the
condition of undirected connected communication net-
work. By the methods of linear matrix inequality (LMI),
we study each dynamical system with time-varying delay
and show that the agents of multi-agent systems can
achieve globally asymptotical consensus stability. Mean-
while, the upper bound parameters of time-varying delay
can be estimated by checking solutions of LMI. Numerical
simulation results are provided and demonstrate the effec-
tiveness of our theoretical results.
6. Acknowledgment
This work is supported by National Nature Science
Foundation of China under Grant 60674046, the Nanjing
Audit University Scientific Research Start-up Fund for
High-level Talents and the Theory and Application of
Differential Equations Foundation of the Nanjing Audit
University.
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