Commun. Theor. Phys. (Beijing, China) 51 (2009) pp. 101–109
c Chinese Physical Society and IOP Publishing Ltd
Vol. 51, No. 1, January 15, 2009
Second-Order Consensus of Multiple Agents with Coupling Delay
SU Hou-Sheng∗ and ZHANG Wei
Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
(Received April 11, 2008)
Abstract In this paper, we investigate two kinds of second-order consensus algorithms for multiple agents with
coupling delay under general fixed directed information topology. Stability analysis is performed based on Lyapunov–
Krasovskii functional method. Delay-dependent asymptotical stability condition in terms of linear matrix inequalities
(LMIs) is derived for the second-order consensus algorithm of delayed dynamical networks. Both delay-independent and
delay-dependent asymptotical stability conditions in terms of LMIs are derived for the second-order consensus algorithm
with information feedback.
PACS numbers: 05.45.-a
Key words: consensus, time delay, linear matrix inequality, multi-agent systems
1 Introduction
Recently, consensus problems have attracted much attention among researchers studying biology, physics, computer science, and control engineering.[1] This is partly
due to broad applications of multi-agent systems in many
areas including cooperative control of mobile robots, unmanned air vehicles (UAVs), autonomous underwater vehicles (AUVs), automated highway systems, and so on.
First-order consensus problems for networked dynamic systems have been extensively studied by many
researchers.[2] In reality, however, a broad class of agents
should be described by second-order dynamic models, such
as torque motor and gas jet, which are adjusted for their
desired motion directly by their acceleration rather than
by their speeds. Hence, the consensus problem of the
multi-agent system modeled by double integrators is more
challenging. Ren et al. proposed a set of second-order
consensus algorithms,[3] which include fundamental consensus algorithm and consensus algorithm with information feedback under directed networks. Lee and Spong
proposed a second-order consensus algorithm of multiple
inertial agents on balanced graphs.[4]
Due to the finite speeds of transmission and spreading as well as traffic congestions, there usually are time
delays in spreading and communication in reality. Therefore, time delay should be considered in designing the
consensus algorithms. First-order consensus algorithms
with time delays have been broadly investigated.[5−8] In
Ref. [5], time-varying delays of first-order consensus are
considered for directed networks. In Ref. [6], average consensus in directed networks of agents with coupling timedelay is investigated. In Ref. [7], uniformly constant delay
is considered in directed networks. In Ref. [8], time delay problems have been studied for the discrete consensus
algorithm.
However, less attention has been paid to the delayrelated consensus problem for multiple agents with double
∗ Corresponding
author, E-mail: [email protected]
integrator dynamics. In this paper, we study the secondorder consensus algorithms with a coupling delay under
general directed networks. Simple second-order consensus
algorithms analogous to that in Refs. [3] and [4] are applied to solving the consensus problem of the multi-agent
system with a coupling delay. By using the passivity decomposition methods in Ref. [4], we decompose the system
into two subsystems: locked system dynamics and shape
system dynamics. Based on the shape system dynamics and Lyapunov–Krasovskii functional method,[9−11] a
convergence condition for delay-dependent asymptotical
stability in terms of LMIs is derived for fundamental
second-order consensus in time delayed networks. Moreover, based on Lyapunov–Krasovskii functional method,
convergence conditions for both delay-independent and
delay-dependent asymptotical stabilities in terms of LMIs
are derived for second-order consensus algorithm with information feedback in time delayed networks. Numerical
simulations are worked out to illustrate our theoretical
results.
An outline of this paper is as follows. In Sec. 2, we
formulate the problem to be investigated. Consensus conditions for both delay-independent and delay-dependent
asymptotical stabilities in terms of linear matrix inequalities are introduced in Sec. 3. Numerical examples are
given in Sec. 4 to illustrate the theoretical results. Concluding remarks are stated in Sec. 5.
2 Problem Formulations
The interaction topology of a network of agents is represented by a directed graph G = {V, E, W } with the
set of nodes V = {n1 , . . . , nN }, edges E ⊆ V × V and
W : E → R+ is a map assigning a positive weight to each
edges. In this paper, we exclude the self-joining edges
from E.
102
SU Hou-Sheng and ZHANG Wei
The second-order agent dynamics is modeled by
ẋi = vi ,
n
v̇i = ui ,
i = 1, . . . , N ,
n
(1)
n
where xi ∈ R , vi ∈ R , and ui ∈ R , and a second-order
consensus protocol is proposed as follows,[3,4]
N
X
ui = −
gij wij [(xi − xj ) + γ(vi − vj )], i = 1, . . . , N, (2)
j=1
where gij = 1 if information flows from agent i to agent j
and 0 otherwise, ∀i 6= j, and weight wij > 0 and scaling
factor γ > 0 are uniformly bounded. The adjacency matrix A of the information exchange topology is defined accordingly as aii = 0 and aij = gij wij , ∀i 6= j. The goal of
consensus protocol (2) is to guarantee that kxi − xj k → 0
and kvi − vj k → 0 as t → ∞.
In order to achieve a desired value, a second-order consensus protocol with information feedback is proposed as
follows,[3]
ui = v̇d − β[(xi − xd ) + γ(vi − vd )]
−
N
X
gij wij [(xi − xj )+γ(vi −vj )] ,
i = 1, . . . , N , (3)
j=1
where β > 0 are uniformly bounded, xd ∈ Rn and vd ∈ Rn
are desired value for xi and vi respectively and satisfy
ẋd = vd . The goal of consensus protocol (3) is to guarantee that x1 = · · · = xN = xd and v1 = · · · = vN = vd as
t → ∞.
Time delays commonly exist in the real world. In
this paper, according to protocols (2) and (3), we consider second-order consensus algorithms with a coupling
delay for weighted general directed networks, which can
be written as
N
X
ui = −
gij wij [(xi (t − τ ) − xj (t − τ ))
j=1
+ γ(vi (t − τ ) − vj (t − τ ))] ,
(4)
and
ui = v̇d − β[(xi − xd ) + γ(vi − vd )]
−
N
X
gij wij [(xi (t − τ ) − xj (t − τ ))
j=1
+ γ(vi (t − τ ) − vj (t − τ ))] ,
(5)
Equations (4) and (5) can be written in matrix form as
u = v̇ = −(L ⊗ In )x(t − τ ) − γ(L ⊗ In )v(t − τ ) ,
and
u = v̇d ⊗ 1N − β(x − xd ⊗ 1N ) − γβ(v − vd ⊗ 1N )
− (L ⊗ In )x(t − τ ) − γ(L ⊗ In )v(t − τ ) ,
where x = [x1 , . . . , xN ]T , v = [v1 , . . . , vN ]T , L = [lij ] is
P
given as lii = j6=i gij wij and lij = −gij wij , ∀i 6= j, and
τ is the time delay (we assume that all the delays are the
same in the network). Hereafter, consensus protocol (4)
(or consensus protocol (5)) is said to achieve asymptotic
consensus if kxi − xj k → 0 and kvi − vj k → 0 as t → ∞
Vol. 51
(or if x1 = · · · = xN = xd and v1 = · · · = vN = vd as
t → ∞).
Before stating the main results of this paper, we need
the following preliminaries:
Lemma 1[10] Suppose that a symmetric matrix is partitioned as
E1 E2
E=
,
E2T E3
where E1 and E3 are square. E is positive definite if and
only if both E1 and E3 − E2T E1−1 E2 are positive definite.
Lemma 2[11] Suppose that a and b are vectors, then for
any positive-definite matrix E, the following inequality
holds:
−2aT b ≤ inf {aT Ea + bT E −1 b} .
E>0
3 Main Results
Using the Lyapunov–Krasovskii functional approach,
some sufficient conditions for ensuring the stability of the
consensus states of protocols (4) and (5) are derived in
this section.
3.1 Fundamental Second-Order Consensus of
Delayed Dynamical Networks
Following the passive decomposition,[4] we define the
following coordinate transformation:
 1
1
1
1 
...
N
N
N
N


 1 −1 0 . . . 0 


z = (S ⊗ In )x; S = 
1 −1 . . . 0 
 0
 , (6)

 ·
..
..
..
·

 ·
.
.
.
·
·
·
0 . . . . . . 1 −1
T
where z = [z1 , z2 , . . . , zn ] ∈ RnN is the transformed coordinate, and S ∈ RN ×N is the (full-rank) transformation
matrix. Let us define ze = [z2 , z3 , . . . , zN ]T ∈ Rn(N −1) so
that z = [z1 , zeT ]T . Then, from Eq. (6), we can show that
ze describes the internal group shape as it is given by
ze = [x1 − x2 , x2 − x3 , . . . , xn−1 − xn ]T ,
(7)
and z1 abstracts the overall group maneuver, as
N
1 X
z1 =
xi .
N i=1
(8)
Using Eq. (6), we can rewrite the protocol (4) with
respect to z such that
((S −T S −1 ) ⊗ In )z̈ + γ((S −T LS −1 ) ⊗ In )ż(t − τ )
+ ((S −T LS −1 ) ⊗ In )z(t − τ ) = 0 .
And we can get that
0
−T −1
−T
−1
S S = diag [N, S̄] , S LS =
0(N −1)×1
(9)
D̄T
,
L̄
where S̄ ∈ R(N −1)×(N −1) is a symmetric and positivedefinite matrix, the j-th component of D̄ ∈ RN −1 is given
by
N
X
D̄j = −
(L1k + L2k + · · · + Lnk ) ,
k=j+1
No. 1
Second-Order Consensus of Multiple Agents with Coupling Delay
j ∈ {1, 2, . . . , N − 1}, and the ij-th component of the matrix L̄ ∈ R(N −1)×(N −1) is given by
N
N
N − i
X
X
L̄ij =
D̄j +
Lpq ,
N
p=i+1 q=j+1
103
(i) For arbitrary initial conditions and scaling factor
γ, the weighted average value x̄˙ (t) is invariant, i.e.
x̄˙ (t) =
N
X
ξi ẋi (t) = x̄˙ (0) ,
∀t ≥ 0 .
(12)
i=1
i, j ∈ {1, 2, . . . , N − 1}, and protocol (4) can be decomposed as the locked system dynamics
N z̈1 (t)+γ(D̄T ⊗In )że (t−τ )+(D̄T ⊗In )ze (t−τ ) = 0 , (10)
and the shape system dynamics
(S̄ ⊗ In )z̈e (t) + γ(L̄ ⊗ In )że (t − τ )
(ii) If the following linear time-varying delayed differential equation is asymptotically stable about its zero solution:
+ (L̄ ⊗ In )ze (t − τ ) = 0 .
(11)
Denote the weighted average value of the differential
vector ẋ as
N
X
x̄˙ (t) =
ξi ẋi (t) ,
where
i=1
where ξ = [ξ1 , ξ2 , . . . , ξn ]T ∈ Rn is the left eigenvector of
L
Pncorresponding to eigenvalue of zero. And we assume
i=1 ξi = 1.
Theorem 1 Consider a system of N agents with dynamics (1), each steered by protocol (4).
ẇ(t) = Cw(t) + Bw(t − τ ) ,
(13)
w(t) = [zeT , żeT ]T ,
0
0
B=
−(S̄ ⊗ In )−1 (L̄ ⊗ In ) −γ(S̄ ⊗ In )−1 (L̄ ⊗ In )
0 I(N −1)×(N −1) ⊗ In
C=
,
0
0
then the consensus states (kxi − xj k → 0 and kvi − vj k →
0 as t → ∞ for all i, j ∈ {1, 2, . . . , N }) are asymptotically
stable.
Proof
(i) From Eq. (4), we have
(ξ ⊗ In )T u = (ξ ⊗ In )T ẍ = −(ξ ⊗ In )T (L ⊗ In )x(t − τ ) − γ(ξ ⊗ In )T (L ⊗ In )ẋ(t − τ ) = 0 .
Then, the weighted average value x̄˙ (t) is invariant.
(ii) From the shape system dynamics (11), we can get that
z̈e (t) = (S̄ ⊗ In )−1 γ(L̄ ⊗ In )że (t − τ ) + (S̄ ⊗ In )−1 (L̄ ⊗ In )ze (t − τ ) ,
(14)
and then we can get Eq. (13). From Eq. (7), if the linear time-varying delayed differential equation (13) is asymptotically
stable about its zero solution, then the protocol (4) is said to achieve (asymptotical) consensus.
Based on the Lyapunov–Krasovskii functional method, a convergence condition for delay-dependent asymptotical
stability in terms of linear matrix inequality is derived for consensus protocol (4).
Theorem 2 Suppose that the time-invariant delay τ ∈ [0, h] for some h < ∞, if there exist some symmetric matrices
P > 0, Q > 0, such that
"
#
P (C + B) + (C + B)T P + 2h(C T QC + B T QB) P B
< 0,
(15)
1
BTP
− Q
h
then the consensus states of consensus protocol (4) are
asymptotically stable for any time-delay τ ∈ [0, h].
Proof Choose a Lyapunov–Krasovskii functional,
V = V1 + V2 + V3 ,
where
V1 = wT (t)P w(t) ,
Z τ
Z t
dβ
ẇT (α)Qẇ(α)dα ,
V2 =
0
V3 = 2
t−β
Z
t
τ wT (α)B T QBw(α)dα .
t−τ
(16)
The equation in system (13) can be written as
Z t
ẇ(t) = (C + B)w(t) − B
ẇ(α)dα ,
t−τ
and thus, the derivate of V1 satisfies
V̇1 = wT (t)((C + B)T P + P (C + B))w(t)
Z t
−
[ẇT (α)B T P w(t) + wT (t)P B ẇ(α)]dα ,
t−τ
T
V̇2 = τ ẇ (t)Qẇ(t) −
Z
t
ẇT (α)Qẇ(α)dα .
t−τ
Furthermore
− ẇT (α)B T P w(t) − wT (t)P B ẇ(α) − ẇT (α)Qẇ(α)
= −[B T P w(t) + Qẇ(α)]T Q−1 [B T P w(t) + Qẇ(α)] + wT (t)P BQ−1 B T P w(t) .
But
τ ẇT (t)Qẇ(t) + V̇3 = τ [wT (t)C T QCw(t) + wT (t − τ )B T QBw(t − τ ) + wT (t)C T QBw(t − τ )
104
SU Hou-Sheng and ZHANG Wei
Vol. 51
+ wT (t − τ )B T QCw(t)] + 2τ wT (t)B T QBw(t) − 2τ wT (t − τ )B T QBw(t − τ ) .
Note that
[wT (t)C T QCw(t) + wT (t − τ )B T QBw(t − τ ) + wT (t)C T QBw(t − τ )
+ wT (t − τ )B T QCw(t)] − 2wT (t − τ )B T QBw(t − τ ) − 2wT (t)B T QBw(t)
= −wT (t)C T QCw(t) − wT (t − τ )B T QBw(t − τ ) + wT (t)C T QBw(t − τ ) + wT (t − τ )B T QCw(t)
= −(Cw(t) − Bw(t − τ ))T Q(Cw(t) − Bw(t − τ )) .
As a result, we obtain
T
T
T
−1
V̇ = w (t)((C + B) P + P (C + B) + 2τ C QC + 2τ B QB + τ P BQ
T
T
B P )w(t) −
Z
t
[B T P w(t) + Qẇ(α)]T Q−1
t−τ
× [B T P w(t) + Qẇ(α)]d α − τ (Cw(t) − Bw(t − τ ))T Q(Cw(t) − Bw(t − τ )) .
From the Shur complements (Lemma 1), this derivative is negative under condition (15). This completes the
proof of Theorem 2.
Remark 1 From Eq. (15), the convergence condition for
fundamental second-order consensus of delayed dynamical networks is determined by L̄ and γ. Hence, the stabilization of protocol (4) is determined by the topological
structure, coupling weight and scaling factor.
3.2 Second-Order Consensus with Information
Feedback of Delayed Dynamical Networks
Denote the difference vectors between agent i and the
reference as x̃i = xi − xd and ṽi = vi − vd , then
x̃˙ i = ṽi , ũi = ṽ˙ i = v̇i − v̇d ,
(18)
and the protocol (5) can be written as
ũ = − β[x̃ + γṽ] − (L ⊗ In )x̃(t − τ )
(17)
terms of linear matrix inequality are derived for consensus
protocol (5).
Theorem 3 If there exist two symmetric matrices P > 0,
Q > 0, such that
P M + M TP + Q P N
< 0,
(22)
N TP
−Q
then the consensus states of consensus protocol (5) are
asymptotically stable for any time-delay τ > 0.
Proof Choose a Lyapunov–Krasovskii functional
Z t
T
V = J (t)P J(t) +
J T (α)QJ(α)dα .
(23)
t−τ
Clearly, V is positive-definite. The derivative of V along
the trajectories of function (21) is
˙
V̇ = J˙T (t)P J(t) + J T (t)P J(t)
+ J T (t)QJ(t) − J T (t − τ )QJ(t − τ )
− γ(L ⊗ In )ṽ(t − τ ) .
(19)
Equation (18) can be written in matrix form as
= J T (t)[P M + M T P + Q]J(t)
˙ x̃
0
IN ⊗ In
x̃
+ 2J T (t)P N J(t − τ ) − J T (t − τ )QJ(t − τ ) .
˙ṽ = −βIN ⊗ In −βγIN ⊗ In
ṽ
From Lemma 2, we have
0
0
x̃(t − τ )
+
,
(20)
2J T (t)P N J(t − τ ) ≤ J T (t − τ )QJ(t − τ )
ṽ(t − τ )
−L ⊗ In −γL ⊗ In
+ J T (t)P N Q−1 N T P J(t) ,
and equation (18) can be written as the form
˙ = M J(t) + N J(t − τ ) ,
J(t)
(21) and then we can get
where
V̇ ≤ J T (t)[P M + M T P + Q + P N Q−1 N T P ]J(t) .
T
T T
J(t) = [x̃ , ṽ ] ,
From Lemma 1, the LMI (22) is equivalent to
0
IN ⊗ In
M=
,
P M + M T P + Q + P N Q−1 N T P < 0 .
−βIN ⊗ In −βγIN ⊗ In
This completes the proof of Theorem 3.
0
0
N=
.
The delay-independent is very conservative if the delay
−L ⊗ In −γL ⊗ In
is already known and small. In the following, we will proClearly, if eqaution (21) is asymptotically stable about its
zero solution, then the consensus state for protocol (5) is vide a delay-dependent condition for protocol (5), which
is less conservative than the delay-independent one.
asymptotically stable.
Based on the Lyapunov–Krasovskii functional method, Theorem 4 Suppose that the time-invariant delay τ ∈
convergence conditions for delay-independent asymptoti- [0, h] for some h < ∞, if there exist some symmetric macal stability and delay-dependent asymptotical stability in trices P > 0, Q > 0, such that
"
#
P (M + N ) + (M + N )T P + 2h(M T QM + N T QN ) P N
< 0,
(24)
1
N TP
− Q
h
No. 1
Second-Order Consensus of Multiple Agents with Coupling Delay
then the consensus states of consensus protocol (5) are
asymptotically stable for any time-delay τ ∈ [0, h].
Proof Choose a Lyapunov–Krasovskii functional
V = V1 + V2 + V3 ,
(25)
where
V1 = J T (t)P J(t) ,
Z τ
Z t
˙
V2 =
dβ
J˙T (α)QJ(α)dα
,
0
V3 = 2
t−β
Z
t
τ J T (α)N T QN J(α)dα .
t−τ
The following proof is similar to that of Theorem 2, and
is omitted here.
Remark 2 From Eqs. (22) and (23), the convergence conditions for second-order consensus with information feedback of delayed dynamical networks are determined by L,
β, and γ. Hence, the stabilization of protocol (5) is determined by the topological structure, coupling weight and
scaling factor.
4 Numerical Examples
The simulations are performed with four agents in the
plane. Initial values of xi and vi denote the position and
velocity vectors of the agents, which are randomly chosen
from box [0, 50]×[0, 50] and [0, 4]×[0, 4], respectively. Assume that the information graph is a cyclic graph, and the
adjacency matrix A of the information exchange topology
is


0 0 0 1
1 0 0 0


A=
.
0 1 0 0
0 0 1 0
105
4.1 Fundamental Second-Order Consensus of
Delayed Dynamical Networks
Based on the fundamental second-order consensus protocol with a coupling delay, when the weight strength
wij = 2, the scaling factor γ = 1, using Theorem 2, we can
get a bound for the time delay as τ = 0.096, guaranteeing
the asymptotical stability of the consensus states of the
protocol (4).
For example, if τ = 0.093 (< 0.096), by using the
MATLAB LMI Toolbox, we find that there exist two
positive-definite matrices,


101.04 40.69 33.82 20.93 −1.59 1.55
 40.69 98.52 29.6 13.79 23.27 −1.27


 33.82
29.6
90.8 12.11 22.7 25.49 


P =
⊗ I ,
 20.92 13.79 12.11 35.03 19.74 15.77  2


 −1.59 23.27 22.7 19.74 47.33 21.48 
1.55
−1.27
25.49 15.77
21.48
41.12

5.47
10.48 

17.16 

 ⊗ I2 ,
9.92 

11.04 
28.83 10.72 12.8 17.76 10.89
 10.72 29.52 10.51 10.76 22.83

 12.79 10.51 28.91 5.54 10.76

Q=
 17.76 10.76 5.54 22.42 11.15

 10.89 22.82 10.76 11.15 23.79
5.47 10.48 17.16 9.91 11.04 22.24
such that condition (15) holds. Figure 1 shows fundamental second-order consensus with a coupling delay on the
cyclic graph for the case τ = 0.093. Figures 1(a) and 1(b)
plot the curves of the values of xi , which are convergent
for x-axis and y-axis, respectively. Figures 1(c) and 1(d)
depict the curves of the values of vi , which are convergent
for x-axis and y-axis, respectively.

Fig. 1 Fundamental second-order consensus with a coupling delay on the cyclic graph (τ = 0.093).
106
SU Hou-Sheng and ZHANG Wei
Vol. 51
Figure 2 shows fundamental second-order consensus with a coupling delay on the cyclic graph for the case τ =
0.18 (> 0.096). Figures 2(a) and 2(b) plot the curves of the values of xi , which are divergent for x-axis and y-axis,
respectively. Figures 2(c) and 2(d) depict the curves of the values of vi , which are divergent for x-axis and y-axis,
respectively.
Fig. 2 Fundamental second-order consensus with a coupling delay on the cyclic graph (τ = 0.18).
4.2 Second-Order Consensus with Information Feedback of Delayed Dynamical Networks
In the following simulations, initial values of reference xd and vd are selected as (1,1) and (3,3) , respectively, and
v̇d = (0.1, 0.1). Based on the second-order consensus protocol (5), when the weight strength wij = 2, the scaling factor
γ = 1 and β = 5, by using the MATLAB LMI Toolbox, we find that there exist two positive-definite matrices for
Theorem 3,


1.31
0.51
0.05
0.51
0.18
0.03 −0.02 0.03
 0.51
1.31
0.51
0.05
0.03
0.18
0.03 −0.02 


 0.05
0.51
1.31
0.51 −0.02 0.03
0.18
0.03 




 0.51

0.05
0.51
1.31
0.03
−0.02
0.03
0.18

 ⊗ I2 ,
P =
0.03 −0.02 0.03
0.24
0.03 −0.03 0.03 
 0.18



 0.03
0.18
0.03 −0.02 0.03
0.24
0.03 −0.03 


 −0.02 0.03
0.18
0.03 −0.03 0.03
0.24
0.28 
0.03 −0.02 0.03
0.18
0.03 −0.03 0.03
0.24


0.98
0.11 −0.11 0.11
0.41 −0.13 −0.15 −0.13
 0.11
0.98
0.11 −0.11 −0.13 0.41 −0.13 −0.15 


 −0.11 0.11
0.98
0.11 −0.15 −0.13 0.41 −0.13 





 0.11 −0.11 0.11
0.98
−0.13
−0.15
−0.13
0.41
 ⊗ I2 ,

Q=
0.12 −0.11 0.12 

 0.41 −0.13 −0.15 −0.13 0.95


 −0.13 0.41 −0.13 −0.15 0.12
0.95
0.12 −0.11 


 −0.15 −0.13 0.41 −0.13 −0.11 0.12
0.95
0.12 
−0.13 −0.15 −0.13 0.41
0.12 −0.11 0.12
0.95
such that condition (22) holds. Figure 3 shows second-order consensus with information feedback and a coupling delay
on the cyclic graph for the case that β = 5 and τ = 1. Figures 3(a) and 3(b) plot the curves of the values of xi , which
No. 1
Second-Order Consensus of Multiple Agents with Coupling Delay
107
are convergent for x-axis and y-axis, respectively. Figures 3(c) and 3(d) depict the curves of the values of vi , which
are convergent for x-axis and y-axis, respectively.
Fig. 3 Second-order consensus with information feedback and a coupling delay on the cyclic graph (τ = 1, β = 5).
Figure 4 shows second-order consensus with information feedback and a coupling delay on the cyclic graph for the
case that β = 2 and τ = 1. Figures 4(a) and 4(b) plot the curves of the values of xi , which are divergent for x-axis
and y-axis, respectively. Figures 4(c) and 4(d) depict the curves of the values of vi , which are divergent for x-axis and
y-axis, respectively.
Fig. 4 Second-order consensus with information feedback and a coupling delay on the cyclic graph (τ = 1, β = 2).
108
SU Hou-Sheng and ZHANG Wei
Vol. 51
For the case that β = 2, Theorem 3 fails to verify that the consensus states of protocol (5) are asymptotically
stable. However, using Theorem 4, we can get a bound for the time delay as τ = 0.225, guaranteeing the asymptotical
stability of the consensus states of the protocol (5). For example, for τ = 0.2, by using the MATLAB LMI Toolbox,
we get the following positive-definite matrices for Theorem 4,


25.2 8.99 4.28 8.99 6.72 1.44 0.79 2.48
 8.99 25.2 8.99 4.28 2.48 6.72 1.44 0.79 


 4.28 8.99 25.2 8.99 0.79 2.48 6.72 1.44 




 8.99 4.28 8.99 25.2 1.44 0.79 2.48 6.72 

P =
 6.72 2.48 0.79 1.44 9.44 3.32 1.78 3.32  ⊗ I2 ,




 1.44 6.72 2.48 0.79 3.32 9.44 3.32 1.78 


 0.79 1.44 6.72 2.48 1.78 3.32 9.44 3.32 
2.48 0.79 1.44 6.72 3.32
1.78 3.32 9.44

8.28 −0.27 −0.8 −0.27
2.9
−0.9 −1.26 −0.74
 −0.27 8.28 −0.27 −0.8 −0.74
2.9
−0.9 −1.26 


 −0.8 −0.27 8.28 −0.27 −1.26 −0.74
2.9
−0.9 




 −0.27 −0.8 −0.27 8.28
−0.9 −1.26 −0.74
2.9 
 ⊗ I2 ,
Q=
 2.9
−0.74 −1.26 −0.9
4.94
0.98
0.04
0.98 




 −0.9
2.9
−0.74 −1.26 0.98
4.94
0.98
0.04 


 −1.26 −0.9
2.9
−0.74 0.04
0.98
4.94
0.98 
−0.74 −1.26 −0.9
2.9
0.98
0.04
0.98
4.94
such that condition (24) holds. Figure 5 shows second-order consensus with information feedback and a coupling delay
on the cyclic graph for the case that β = 2 and τ = 0.2. Figures 5(a) and 5(b) plot the curves of the values of xi ,
which are convergent for x-axis and y-axis, respectively. Figures 5(c) and 5(d) depict the curves of the values of vi ,
which are convergent for x-axis and y-axis, respectively.

Fig. 5 Second-order consensus with information feedback and a coupling delay on the cyclic graph (τ = 0.2, β = 2).
5 Conclusions
In this paper, we investigate a set of second-order consensus algorithms with coupling delay under general directed
information exchange topology. Based on the Lyapunov–Krasovskii functional method, a consensus condition for delaydependent asymptotical stability in terms of LMIs is derived for fundamental second-order consensus algorithm, and
No. 1
Second-Order Consensus of Multiple Agents with Coupling Delay
109
consensus conditions for both delay-independent and delay-dependent asymptotical stabilities in terms of LMIs are
derived for the second-order consensus algorithm with information feedback.
References
[1] E. Shaw, Natural History 84 (1975) 40; A. Okubo, Adv.
Biophys. 22 (1986) 1; T. Vicsek, A. Czirook, E. BenJacob, O. Cohen, and I. Shochet, Phys. Rev. Lett. 75
(1995) 1226; H. Levine and W.J. Rappel, Phys. Rev. E 63
(2001) 208; N. Shimoyama, K. Sugawara, T. Mizuguchi,
Y. Hayakawa, and M. Sano, Phys. Rev. Lett. 76 (1996)
3870; A. Mogilner and L. Edelstein-Keshet, Phys. D
89 (1996) 346; A. Mogilner and L. Edelstein-Keshet, J.
Math. Biology 38 (1999) 534; J. Toner and Y. Tu, Phys.
Rev. E 58 (1998) 4828; J. Toner and Y. Tu, Annals of
Phys. 318 (2005) 170.
[2] W. Ren, R.W. Beard, and E. Atlkins, IEEE Control
Systems Magazine 27 (2007) 71; A. Jadbabaie, J. Lin,
and A.S. Morse, IEEE Trans. Automat. Contr. 48 (2003)
988; R. Olfati-Saber, and R.M. Murray, IEEE Trans. Automat. Contr. 49 (2004) 1520; W. Ren, and R.W. Beard,
IEEE Trans. Automat. Contr. 50 (2005) 655.
[3] W. Ren and E. Atkins, Int. J. Robust Nonlinear Contr.
17 (2007) 1002; W. Ren, IET Contr. Theory and Applications 1 (2007) 505.
[4] D. Lee and M.W. Spong, IEEE Trans. Automat. Contr.
52 (2007) 1469.
[5] J. Hu and Y. Hong, Phys. A 374 (2007) 853.
[6] P. Lin and Y. Jia, Phys. A 378 (2008) 303.
[7] L. Moreau, IEEE Trans. Automat. Contr. 50 (2005) 169.
[8] F. Xiao and L. Wang, Int. J. Contr. 52 (2007) 1469.
[9] V.B. Kolmanovskii and J.P. Richard, IEEE Trans. Automat. Contr. 44 (1999) 984.
[10] C. Li and G. Chen, Phys. A 343 (2004) 263; Z.X. Liu,
Z.Q. Chen, and Z.Z. Yuan, Phys. A 375 (2006) 345.
[11] Y.M. Moon, P. Park, W.H. Kwon, and Y.S. Lee, Int. J.
Contr. 74 (2001) 1447.