Intelligent Control and Automation, 2011, 2, 1-7
doi:10.4236/ica.2011.21001 Published Online February 2011 (http://www.SciRP.org/journal/ica)
Distributed H Consensus of High-Order Multi-Agents
with Nonlinear Dynamics*
Jianzhen Li
School of Automation, Nanjing University of S cience and Technology, Nanjing, China
Received January 20, 2011; revised February 12, 2011; accept ed Fe bruary 13, 2011
Abstract
This paper deals with the distributed consensus problem of high-order multi-agent systems with nonlinear
dynamics subject to external disturbances. The network topology is assumed to be a fixed undirected graph.
Some sufficient conditions are derived, under which the consensus can be achieved with a prescribed
H

norm bound. It is shown that the parameter matrix in the consensus algorithm can be designed by solving
two linear matrix inequalities (LMIs). In particular, if the nonzero eigenvalues of the laplacian matrix ac-
cording to the network topology are identical, the parameter matrix in the consensus algorithm can be de-
signed by solving one LMI. A numerical example is given to illustrate the proposed results.
Keywords: Consensus, Multi-Agent Systems, Nonlinear Dynamics, External Disturbances
1. Introduction
The consensus problem of multi-agent systems has been
researched extensively in recent years. This is because of
its widely application in much areas such as flocking
[1-2], synchronization of coupled oscillators [3], forma-
tion control of mobile robots [4-5], distributed computa-
tion [6] and information fusion in wireless sensor net-
works [7]. The object of consensus control is to design
consensus protocol such that the group of agents can
asymptotically agree upon certain quantities of interest
based on information received from their neighbors.
Most of the work on the consensus problem focuses on
the multi-agent systems with first-order dynamics. In
particular, [8] deals with the first-order multi-agent sys-
tems with switching topolog ies and time delays in a con-
tinuous setting. The fist-order multi-agent systems with
switching topologies is investigated in [9] in a dis-
crete-time setting. The consensus problem has also been
investigated from many other aspects such as reference
signals [10], asynchronous sampling time [11], and so on.
Recently, the consensus problem of second-order multi-
agent systems has been investigated extensively [12-14].
In particular, the consensus problem of second-order
multi-agent systems with nonlinear dynamics was inves-
tigated in [15]. The nonlinear dynamics can be taken as
the potential functions or the desired final dynamics of
the agents. There is also some work on the consensus
problems of high-order multi- agent systems [16-17].
Generally speaking, the consensus cannot be achieved
accurately if there are external disturbances. To deal with
this problem, the
H

consensus problem is considered
[18-21]. It is shown that for undirected network topolo-
gies, the desired parameter matrix in the con sensus algo-
rithm can be designed by solving two LMIs, which relate
to the system matrix of the ag ents and the eigenvalue s of
the laplacian matrix corresponding to the network topol-
ogy. The 2
H
consensus problem was investigated in
[22].
In the aforementioned work on the
H
 or 2
H
con-
sensus problem, the nonlinear dynamics was not consid-
ered. As is mentioned in [14-15] much multi-agent sys-
tems have nonlinear dynamics. Motivated by this, this
paper considers the
H

consensus problem of high-
order multi-agent systems with nonlinear dynamics. To
the best of the author's knowledge, this problem has not
been considered in the literature. Some sufficient condi-
tions will be derived, under which the consensus can be
achieved with a prescribed
H
 norm bound. It will be
shown that the parameter matrix in the consensus algo-
rithm can be designed by solving two LMIs, which relate
to the system matrix of the agents and the smallest and
*This work was supported by the National Natural Scie nce Fou nda tio n
of P. R. China under Grant 60904022.

J. Z. LI
2
the biggest nonzero eigenvalue of the laplacian matrix
corresponding to the network topology. In particular, if
the nonzero eigenvalues of the laplacian matrix accord-
ing to the network topology are identical, the parameter
matrix in the consensus algorithm can be designed by
solving one LMI.
2. Preliminary Notations and Problem
Formulation
Let

,, be a weighted undirected graph of
order N, where

1,, N is the node set.
 is a set of unordered pairs of nodes, and 
is the adjacency matrix. An undirected path is a se-
quence of edges in a undirected graph of the form

12
v,v ,
ii

23
v,v ,
ii, where 12
,
ii
vv  An undi-
rected graph is called connected if for any two nodes of
the graph, there ex ists a p ath that fo llows the edge s of the
graph. The adjacency matrix is a nonnegative matrix
N
N
dij
a


 satisfying a0
ii  for any i

,
aa0
ij ji
, if

,ji , and a0
ij  if agents j
and i are not adjacent. The Laplacian matrix of the
graph is defined as
N
N
ij
l


  with ii ij
ji
la



and la, .
ijij ij We can see that  satisfies
01 and 0
T1 where

.1,,1T 1 For matri-
ces M and N,
M
N denotes their Kronecker product.
It is well known that if the undirected network topology
 is connected, the lapalacian matrix corresponding to
 has 1N positive eigenvalues and a simple zero
eigenvalue.
Consider a group of N agents with the following dy-
namics:

12, iii ii
x
AxBuBB fx  
 (1)
where

,
m
i
xt

p
i
ut are, respectively, the state,
the control input of agent i,

m
it
  is the external
disturbance which belongs to


20, , and


m
i
fx
is a nonlinear function.
Assumption 1: There exists a positive scalar

such
that





,
,.
T
ij ij
T
ij ij
m
ij
f
xfx fxfx
xx xx
xx




 

Remark 1: Assumption 1 is similar to the Assumption
1 in [14]. It is a Lipschitz-type condition satisfied by
many systems.
Definition 1: We say algorithm ui solves the con-
sensus problem if
10, , .
Nj
ij
x
xti
N



Definition 2: We say algorithm ui solves the con-
sensus problem with
H

norm bound

if the fol-
lowing two conditions are satisfied:
1) Algorithm ui solves the consensus problem if
0;


2) If 00,z

the following inequality is satisfied:
22
2
00
,zdt dt 



where
1
1
1
,
,
,
Nj
ii
j
T
TT
N
T
TT
N
x
zx N
zz z
 












and 0 z is the initial value of z.
The object of the
H

consensus control is to design
consensus algorithms such that the consensus problem is
solved for a presc ribed
H

norm bound.
Lemma 1: (Schur complement [23]) Let S be a sym-
metric matrix of partitioned form ij
SS

 with
11 ,
rr
S  ()
12 rnr
S


 and ()()
22 nr nr
S  Then,
0S

if and only if 11 0,S 1
2221 11120,SSSS

 or
equivalently 1
221112 2221
0, 0.


SSSSS
Lemma 2: For matrices A, B, C, D with appropriate
dimensi ons, o n e has
 
,
TTT
A
BABCDACBD
and


A
BCACBC


3. Results
In this section, the
H

consensus problem of multi-
agent systems with nonlinear dynamics will be investi-
gated. Considers the following state feedback consensus
algorithm:
 
1
N
iijij
j
utKa xx


 (2)
With (2), system (1) becomes


12
1,
N
iiijijii
j
x
AxBKaxxBBfx 



 (3)
which can be written in a compact form as

 

1
2
,
NN
N
xI ALBKxIB
IBf

 

 (4)
where 1 ,
N
TT
T
xx x





 and 1 () (). T
TN T
ffx fx

 
By the definition of z we have the consensus is
achieved if and only if 0z as .t It is easy to
see that

J. Z. LI
3

,
m
zHIx (5)
where
N
N
H

 with
1,
1,.
ij
Nij
N
H
ij
N








It can be seen that ,
T
N
H
IN11 2,
H
H
,
TT
N
N
H10
N
N
H10 and .HH
Lemma 3: There exists an orthogonal matrix
N
N
U  with last column NN1H such that
11 1
,.
*0 *0
Nn N TT
I
UHU ULU
 

 

 
 
00
From (4) and (5) we have





1
2
1
2
.
m
zHIx
HALBKxHB
HBf
HALBKzHB
HBf



 

 

 
(6)
Define 1N UU N



1
, from Lemma 3 we have
111
TN
UHU I  and 11.
T
ULU Define

Tm
UIz  
1,
T
TT
N



 we have

 




12
11 1
11
12
11
*0 *0
.
Tmm
TT
mm
Nn N
TT
TT
NN
UIHALBKUIz
UI HBUI HBf
IABK
UH UH
BBf




 

 
  








 


 

 

00
00

(7)
It can be seen that 0
N

. So 0

 if and only if
0
i

, 1, ,1.iNDefine 11 .
T
TT
N
 




From (7) we have


1
11 12
.
N
TT
IA BK
UHBUHB f






(8)
Note that the eigenvalues of  are 2,,.
N

 
There exists an orthogonal matrix

11NN
F
 
 so
that

2,, .
TN
FF diag   
Define

11
T
TTT
mN
FI 






 we have




12
11 12
,,
.
NN
TT TT
IAdiag BK
F
UHBFUHB f

 




 (9)
Noting that ,
TT TT
zz

   we conclude
that algorithm (2) solves the consensus problem with
H

norm bound

if and only if system (9) is asymp-
totically stable with ,T


 where T


denotes
the
H

norm of the transfer function matrix from

to

.
Theorem 1: Suppose the undirected graph  is
connected and the nonzero eigenvalues of  are
2,,
N

 . Using algorithm (2), the consensus is
achieved with
H

norm bound

if there exists a
symmetric positive definite matrix
X
and a matrix W
such that the LMIs
12
2
1
2
000,
00
1
00 1
T
Tm
m
m
BXBX
BI
BI
XI






 





 

(10)
2, ,iN

hold, where

.
TTT
i
A
XXA BWWB   
In this case, the parameter matrix in (2) can be chosen as
1.
K
WX


Proof: Assume that the undirected graph  is con-
nected, we have 20

. Suppose there exists a symmet-
ric positive definite matrix X and a matrix W such that
(10) hold. Define 1,WPX   1.
K
WX   Pre- and
post-multiply both sides of (10) by
000
000
00 0
000
m
m
m
P
I
I
I






one can get
12
2
1
2
000,
00
1
00 1
Tm
Tm
m
mm
PB BI
BP I
BI
II






 





 

(11)
2, , iN

hold, where

TTT
i
P
AAP PBKKBP   .
For ,
i

3, ,1,iN

 there exists 01
i

 such
that


21. iN

  It is easy to see that (11) also
holds for 3, ,1iN

. By Lemma 1 we know that (11 )
holds if and only if


22 1
2
1
10
Tm
Tm
BBI PB
BP I







(12)
holds. Define

J. Z. LI
4



11
2
11 1
,
N
T
NNm
IPB
IPB I 



 



where



 
2
1
122 1
,,
.
TT
N
Nm
TT
NN m
I
diagPBKKB P
IBBIPAAPI




 
 

From (12) we know that 0.
Next prove that the consensus is achieved if 0.


If 0

, (9) becomes




12
12
,,
.
NN
TT
IA diagBK
FUHB f

 



 (13)
Consider the Lyapunov function
 
1.
TN
VIP




Because P symmetric positive definite, we have


V

is symmetric positive definite with respect to .

Taking
derivative of

V

along (13), we have


  





 

11
12
112
1
2
12
2
2,,
2
,,
2.
TNN
TNN
TTT
N
TT
N
TT
N
TTT
VIPIA
IPdiagBK
IPFUHBf
IPAAP
diagPBKKB P
FUHB f
 




 






 
 






(14)
Because U is an orthogonal matrix, one has
11
T
TNN
N
IUUU U
NN




11
It follows that
111
.
TN
UU I  Then we have





12
11 22
2
122
2
.
TTT
TTT T
Tm
TTT
Nm
FUHB f
FUUF BB
fH If
I
BBf H I f












 

(15)
Notice that






1
1
1
1
Tm
NT
ij ij
iji
NT
ij ij
iji
fHIf
f
xfx fxfx
N
xx xx
N





 






.
Tm
TT
x
HIx
zz





 (16)
From (14)-(16) we have


 

 

 

 
1
2
122
122
2
,,
,,.
TT
N
TT
N
TT T
N
TTT
Nm
TT
N
V
IPAAP
diagPBKKB P
IBB
IPAAPBBI
diagPBKKB P





















(17)
It follows from (12) that

22 0,1
Tm
BB I  


which, together with (17) implies th at

V

 is negative
definite with respect to

. It then follows that 0


asymptotically. From the analysis above we know the
consensus can be achieved.
Assume that 0.


 Taking derivative of


V

along (9), we have





  
 







11
211
12
122
2
11
2
,,
,,
2.
TNN
TT
N
TT
TTT
Nm
TT
N
TTT
VIPIA
diagBKF UHB
FUHB f
IPAAPBB I
diagPBKKB P
FUH PB


 

 












(18)
Assume that 00z

, which implies that 00


,
where 0

is the initial state of

. It follows that





 



22
2
00
22
20
0
2
0
122
2
11
0
,,
2
,
TT
TTT
Nm
TT
N
TTT
T
zdt dt
VdtV V
IPAAPBBI
diagPBKKB P
FUH PBdt
dt

 
 













 












(19)
where ,
T
TT
 




 00V is the initial value of


V

, and



11
2
11
.
TT
T
Nm
FUH PB
HU FPBI 













J. Z. LI
5
By Lemma 1 we know that 0

 if and only if




 

2
1111
2
2
111
2
,,
1()
,,
1
0.
TT
N
T
TT
TT
N
T
N
diagPBKKB P
FUHPBHUF PB
diagPBKKB P
IPBBP





 



 



 

 


 (20)
Also by Lemma 1 we know that (20) is equivalent to
0 , which has been proved in the above analysis. So
we have that  is symmetric negative definite. It fol-
lows from (19) that
22
2
00
0.zdt dt 



Therefore, the consensus is achieved with
H

norm
bound

. The proof is completed.
Sometimes, the laplacian matrix has 1N identical
nonzero eigenvalues, i. e. 2 0
N



. Take the
complete graph for example. Consider the complete
graph wit h N nodes. The laplacian matrix is chosen as
111
11 1
.
11 1
N
NN N
N
NN N
N
NN N





















By some calculations we have the eigenvalues of 
are 11 0,, ,
11NN
. In this case, we have the fol
lowing corollary.
Corollary 1: Suppose the undirected graph  is
connected and the nonzero eigenvalues of  satisfy
2.
N

  Using algorithm (2), the consensus is
achieved with
H
 norm bound

if there exists a
symmetric positive definite matrix X and a matrix W
such that the LMI
12
2
1
1
2
000,
00
1
00 1
T
Tm
m
m
BXB X
BI
BI
XI






 








(21)
holds, where 1.
TTT
A
XXA BMMB  In this
case, the parameter matrix in (2) can be chosen as
1.
K
WX  
Proof: Assume that there exists a symmetric positive
definite matrix X and a matrix W such that (21) holds.
Define
2
1.X
M
W

 It follows that
12
2
1
2
000
00
1
00 1
T
Tm
m
m
BXBX
BI
BI
XI






 








Holds for 2.i

From Theorem 1 we have the con-
sensus is achieved with
H
 norm bound

, and the
parameter matrix can be chosen as 1.
K
WX  
Remark 2: From Corollary 1 one has that if the non-
zero eigenvalues of the laplacian matrix are identical, the
H

performance is determined by 2

and the system
matrices of the agents. It has no relationship with the
number of the agents.
4. A Numerical Example
Consider a multi-agent systems consisted of N nodes
with the following second-order

, sin0.5,
iiiiii i
x
vvu x    
where 1
sin 1
it





is the external disturbance. This
multi-agent system can be written in the form of (1) with
12
01 01000
, , , .
00 10101
ABB B

 
 

 

 
The communication topology is given in Figure 1.
The laplacian matrix is chosen as
1100
12 10
.
0121
001 1



















The eigenvalues of  are 0, 0.5858, 2 and 3.4142.
Solving the LMIs in (10) with 20.5858

, 4


3.4142 , 0.5


and 1.29


, we can get

0.04550.6688 , 0.12605.7.
0.6688 23.2996
XW






From Theorem 1 we know that K can be chosen as


12844.2193.5 .KWX 

Figure 2 shows the trajectory of the external distur-
bance. Figures 3 and 4 show, respectively, the position
and velocity responses of nodes 14.
Figure 1. The communication topology of nodes 1-4.

J. Z. LI
6
0.9
0 5 10 15
t(s)
Extemal dis t ur bance
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 2. Positions of nodes 1-4.
–
2
0 5 10 15
t
(
s
)
Positions (m)
–
3
–
4
–
5
–
6
–
7
–
8
–
9
N
ode 1
N
ode 2
N
ode 3
N
ode 4
Figure 3. Positions of nodes 1-4.
10
0 5 10 15
t(s)
Velocities (m/s)
N
ode 1
N
ode 2
N
ode 3
N
ode 4
–
15
–
10
–
5
5
0
Figure 4. Velocities of nodes 1-4.
5. Conclusions
The
H
 consensus problem has been investigated in
this paper, for the high-order multi-agent systems with
nonlinear dynamics. Sufficient conditions have been
given in the forms of LMIs, under which the
H

con-
sensus problem can be solved. The parameter matrix in
the consensus algorithm can be designed by solving two
LMIs. If the nonzero eigenvalues of the laplacian matrix
according to the network topology are identical, the pa-
rameter matrix in the consensus algorithm can be de-
signed by solving one LMI. The numerical simulation
confirmed the propose d res ults.
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