Consensus of Nonlinear Multi-Agent Systems with Exogenous
Disturbances
Xiaozhi Yu
1
, Zhen He
1
and Feng Yu
2
1
College of Automation Engineering, Nanjing University of Aeronautics and Astronautic, Yudao Street 29, Nanjing, China
2
College of Astronautics, Nanjing University of Aeronautics and Astronautic, Yudao Street 29, Nanjing, China
Keywords: Consensus, Disturbance Observer, Dynamic Gain Technique, Multi-Agent Systems.
Abstract: Most existing research concerning the consensus problem of multi-agent systems has been focused on linear
first-order or two-order systems without disturbances. However, in practice, most multi-agent systems are
complicated nonlinear system subjected to disturbances. In this paper, the coordinated tracking problem for
nonlinear undirected multi-agent systems with exogenous disturbances is studied in the framework of
consensus theory. The exogenous disturbances generated by both linear exosystems and nonlinear
exosystems are considered. Disturbance observers are developed to estimate the disturbances generated by
the linear exogenous systems. The Lyapunov stability theorem is used to prove the asymptotical consensus
of the systems. The dynamic gain technique is used to construct the disturbance observer for the disturbance
generated by a nonlinear exosystem. Based on the adaptive disturbance observer, a consensus protocol is
proposed for the nonlinear multi-agent system. Finally, the proposed design approaches are verified though
simulation examples.
1 INTRODUCTION
Recently the consensus problem of multi-agent
systems has attracted considerable research attention
due to the broad applications of consensus
algorithms in cooperative control of mobile vehicles
flocking (Tanner et al., 2007; Liu Y. et al., 2003),
deployment (Corts et al., 2005), formation control
(Hu et al., 2008; Konduri et al., 2013), and so on.
Multi-agent system is composed of multiple agents.
Each agent perceives the surrounding environment,
and communicates with other agents.
First-order consensus problems (Bliman, P. A.
and Ferrari, T. G., 2008), and second-order
consensus problems (Ren, W. and Atkins, E., 2007;
Xie, G. M. and Wang, L., 2007) have been studied
intensively. However, in reality, mobile agents may
be governed by more complicated intrinsic dynamics
subjected to disturbances. Examples include tethered
satellites, close formation flying and morphing
structures with distributed actuators. All these are
nonlinear multi-agent systems which are subjected
to exogenous disturbances. A few attentions have
been paid to the problems of the consensus of
nonlinear multi-agent systems (Wu, 2005; Yu et al.,
2010). So far, the research of the consensus of
nonlinear multi-agent system has been carried out
for several specific types of dynamics, such as
Euler-Lagrance model (Mei et al., 2011), unicycle
model (Sepulchre et al., 2008), rigid body posture
model (Ren, W., 2007) and some general nonlinear
model satisfying some certain conditions. For
general nonlinear systems, it is difficult to handle the
consensus control problem under a unified
framework.
Few works have considered the consensus
problem of nonlinear multi-agent systems with
exogenous disturbances. In (MA, G. F. and Mei, J.,
2011), the authors have studied the consensus
problem of nonlinear multi-agent systems, but they
have not considered disturbances. In (Yang et al.,
2011), the authors have considered linear multi-
agent systems with disturbances generated by linear
exogenous systems, and a disturbance observer
based protocol has been designed is using linear
matrix inequality method. In (Zhang, X. X. and Liu,
X. P., 2013), the authors have investigated the
consensus problem of linear systems with
disturbances generated by nonlinear exosystem by
utilizing the dynamic gain technique. In (Das, A. and
Lewis, F. L., 2011), Das and Lewis have studied the
synchronization of nonlinear systems with
disturbances by using a neural network based
281
Yu X., He Z. and Yu F..
Consensus of Nonlinear Multi-Agent Systems with Exogenous Disturbances.
DOI: 10.5220/0005566702810286
In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 281-286
ISBN: 978-989-758-122-9
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

method. However, they have not considered external
disturbances generated from nonlinear exogenous
systems. In this paper, the consensus problem of
nonlinear undirected multi-agent systems with
exogenous disturbances is investigated. New
disturbance observers for multi-agent systems are
derived based on pinning control to estimate external
disturbances. The influence of the disturbances is
compensated by using proper feedback. Based on
the disturbance observers, a consensus protocol is
proposed.
This paper is organized as follows. In Section 2,
some concepts and useful lemmas are briefly
outlined. The consensus of nonlinear multi-agent
systems with exogenous disturbances is studied in
Section 3. The utility of the new theoretical findings
is illustrated by simulation results in Section 4.
Finally, some conclusions are drawn in Section 5.
2 PRELIMINARIES
A weighted undirected connected graph
G
is
defined as a triple
()
,,GVB ε =
, where
V
denotes
the set of nodes,
()
{
}
,:,ij ij V ε ⊆∈
denotes the set
of edges. Denote
()
,ij ε ∈
if
j
is a neighbor of
i
.
The set of neighbors of
iV∈
is defined as
()
{
}
:,
i
NjVij ε =∈ ∈
.The adjacency matrix is
nn
ij
Bb R
×

=∈

with weighted adjacency elements
0
ij
b ≥
. It is clear that if
()
,ij ε ∈
, we have
0
ij ji
bb=>
, and if
ji≠
,
{
}
,ij ε ∉
we have
=0
ij ji
bb=
. Assume that there are no self-loops,
that is, for any
iV∈
,
()
,ii ε ∉
. A path between two
nodes is a sequence of edges by which it is possible
to move along the sequence of the edges from one of
the node to the other node. If there exists at least one
path from any node to any other node in
()
,,GVB ε =
, the graph is said to be connected. The
matrix
D
is a diagonal matrix with the elements
i
d

along the diagonal, where
,.
i
iij
jN
dbiV
∈
=∈


The
Laplacian matrix of the weighted graph is defined as
LDB=−
. In an undirected connected graph, the
Laplacian matrix is a symmetric matrix.
Consider a nonlinear multi-agent system over
an undirected graph
()
,,GVB ε =
with
n
nodes,
and the dynamics of agent
i
is written as:
(, ) (t,x)
(t, )
iiiii
ii
x
ftx u g d
yhx
=++
=

(1)
where
,
mm
ii
x
Ru R∈∈, and
m
i
dR∈ denote the
state, the control input, and exogenous disturbance,
respectively,
(, )
i
f
tx ,
()
,
i
g
tx
,and (, )
i
ht x are
smooth function in terms of
i
x
.
Remark 1: Assume the nonlinear dynamic
equation
()
,
f
tx
satisfies the Lipschitz condition,
that is
() ()
,,
f
tx f ty lx y−≤−
, where
0l >
(2)
We say the system described by Eq. (1)
asymptotically reaches consensus if
0
ij
xx−→
as
t →∞for all
,ij V∈
.
3 THE CONSENSUS OF
NONLINEAR MULTI-AGENT
SYSTEMS WITH EXOGENOUS
DISTURBANCES
Assume that the disturbance
i
d ,
iV∈
is generated
by the following linear exogenous system (Zhang,
X. X. and Liu, X. P., 2013)
() ()
() ()
ii
ii
tAt
dt C t
ξξ
ξ
=
=

(3)
where
i
m
i
R ξ ∈ is the internal state of the exogenous
system, and
ii
mm
A
R
×
∈ , and
i
mm
CR
×
∈
are the
coefficient matrices of the disturbance system.
Before proceeding further, we make the
following standard assumptions.
Assumption 1: The graph
()
,,GVB ε =
describing the interaction topology is connected.
Assumption 2: The matrix pair
()
,
A
C
is
observable.
The following disturbance observer (Chen, W.
H., 2004) is proposed to estimate the unknown
disturbance
i
d in system (1):
ˆ
ˆ
ˆ
(x )[x ( , ) ( , )d]
ˆ
ˆ
(x )
q(x )
ii ii ii i
ii
i
i
i
Aq ftxugtx
dC
p
x
ξξ
ξ
=+ − −−
=
∂
=
∂


(4)
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282

where
i
n
i
zR∈ is the internal state variable of the
observer,
ˆ
i
ξ
and
ˆ
i
d
are the estimation of
i
ξ
and
i
d .
(x )
i
p is a function, and the matrix
()
i
nn
i
qx R
×
∈
is
the observer gain to be designed.
Define the estimate error as
ˆ
iii
e
ξξ
=−
,
iV∈
(5)
Based on (1), (3) and (4), it is known that
()()
[gC]
iiii
eAqx xe=−

(6)
Then, the following proposition ensures that the
disturbance observer (4) can exponentially track the
disturbance.
Proposition 1
(Chen, W. H., 2004)
:
The
estimation error system (6) is globally and
exponentially stable if there exists a gain
K
such
that the following transfer function is asymptotically
stable and strictly positive real:
()
()
1
H
sCsIAK
−
=−
(7)
where
A
is as follows
()
0
A
AK C α =−
(8)
The Proof is similar to (Chen, W. H., 2004).
Theorem 1: Under the assumptions 1 and 2, with
the disturbance observer (4), the pinning control
protocol given by
00
[()()]
ˆ
(t, x )
i
iijijii
jN
ii
uaxxaxx
gd
α
∈
=− − + −
−

(9)
ensures the asymptotical consensus of the system
(1).
Proof. Let
0
1
1
() ()
() [ (), , ()]
( ) [ ( ), , ( )]
ii
TTT
n
TTT
n
xt xt x
xt x t x t
et e t e t
=−
=
=

 


where
0
x
is the constant of consistency. One can
obtain the following system
()
()
(, )
n
x
tHIxFtxe α =− ⊗ + +Ψ


(10)
()
(, )
x
tLxFtxe α =+ +Ψ


(11)
where
()
=- ,
p
LHI⊗
() ( )
()
()
()()
()
10
0
,[, , ,
,, , ]
T
T
T
n
Ftx f tx ftx
ftx ftx
=−
−


()
,,GtxCΨ=
()
12
, [g(t,x ) ,g(t,x ) , ,g(t,x ) ]
TT TT
n
Gtx = 
,
()
10 20 0
,,, ,
n
HLdiagaa a=+ 
L
is the Laplacian
matrix of the weighted graph,
⊗
is the Kronecker
product,
n
I
is a
n
-dimensional identity matrix,
L
is
Hurwitz.
According to the definition of Euclidean norm
and Lipschitz condition, we can know that
()
1
,[,,]
T
n
F
tx l x x lx≤=
 

(12)
Now, our aim is to show that
()
lim 0
x
xt
→∞
=

.
Define the Lyapunov function candidate as
following:
T
VxWx=

(13)
where the matrix
W
is a positive matrix satisfying
the following Lyapunov equation (Zhang, X. X. and
Liu, X. P., 2013)
T
WL L W I+=−
.
Taking the derivative of the Lyapunov function
yields
()()
()
()
()
()
2
2
max
22 2
max
2
max
=
22
22
2
2
TT
T
T
TT
VxWxxWx
Lx F e Wx x W Lx F e
xxWexWF
x
xW e W F x
xx Wlx
lWx
αα
α
αλ
αθ λ
αθ λ
=+
++Ψ + ++Ψ
=− + Ψ +
≤− + Ψ +
≤− + +
=− − −




 
 
 

(14)
for
()
()
2/xWe θ ∀≥ Ψ

,where
1
1
2
θ ≤<
.
When
()
max
2lW αθ λ >+
,the system (11) is
globally and exponentially stable.
Hence, by combining (6) and (11), one can
obtain the following system
()
()()
(, )
(gC)
iiii
x
tLxFtxe
eAqx xe
α

=+ +Ψ


=−





is globally and exponentially stable.
Finally, one obtains
()
lim 0
t
xt
→∞
=

, which
completes this proof.
Assume that the disturbance
i
d ,
iV∈
is
generated from following nonlinear exogenous
system (Zhang, X. X. and Liu, X. P., 2013)
()
() ()+
() ()
iii
ii
tAt
dt C t
ξξφξ
ξ
=
=

(15)
where
i
m
i
R ξ ∈ is the internal state of the exogenous
system,
ii
mm
A
R
×
∈ , and
i
mm
CR
×
∈
are the
ConsensusofNonlinearMulti-AgentSystemswithExogenousDisturbances
283

coefficient matrices of the disturbance system.
Remark 2:
Assume the nonlinear disturbance
()
i
φξ
satisfies the Lipschitz condition, that is
() ( )
12 12ii ii
c
φξ φξ ξ ξ
−≤−
, where
0c >
(16)
A disturbance observer
(
Chen, W. H., 2004
)
is
proposed to estimate the unknown disturbance
i
d
in
system (1)
()
()
1
ˆ
ˆ
ˆ
(, ) (, )
ˆ
+
ˆ
ˆ
ii ii ii ii
iii
ii
T
ii
A
qx x ftx u gtx d
Qe
dC
ee
ξξ
φξ ζ
ξ
ζ
−

=+ − −−

+
=
=



(17)
where
ˆ
iii
e
ξξ
=−
(18)
denotes the estimation error between
i
ξ
and
ˆ
i
ξ
.
()( )
()
()
()
1
,
ˆ
iiiiii
ii
eAqxgtxCeQe
ζ
φξ φξ
−
=− −
+−

(19)
Proposition 2
(Chen, W. H., 2004)
:
The estimation
error system (30) is globally and exponentially
stable if there exists a gain
K
such that the
following transfer function is asymptotically stable
and strictly positive real:
()
()
1
H
sCsIAK
−
=−
where
A
is as follows
()
0
A
AK C α =−
The Proof is similar to (Zhang, X. X. and Liu, X.
P., 2013).
Theorem 2: Under the assumptions 1 and 2, with
the disturbance observer (17), the pinning control
protocol given by
()
00
[()( )]
ˆ
,
i
iijijii
jN
ii
uaxxaxx
gtx d
α
∈
=− − + −
−

(20)
ensures the asymptotical consensus of the system
(1).
Proof. Now, it is only need to show that the
following system
()
()()
()
()
()
1
(, )
gC
ˆ
iiiiii
ii
xt Lx Ftx e
eAqxx eQe
α
ζ
φξ φξ
−

=+ +Ψ


=− −


+−





(21)
is globally and asymptotically stable.
It can be known that the matrix
K
satisfying
Proposition 2 can make the matrix
()()
gC
ii
Aqx x−
be Hurwitz, and
L
is also
Hurwitz.
So, we define a Lyapunov function as follows:
1
=+
n
i
i
VV V
=

Taking the derivative of the Lyapunov function
yields
()
(
)
()
()
1
22
max
2
2
1
222
max
22 2
2222
max
22
1
2
2
1
2
22
2
n
i
i
T
n
VV V
xl WxxW
ee
xl Wx x
We e
lW xWe e
αλ
λ
αλ
λ
αλ λ
=
=+
≤− + + Ψ
−++
≤− + +
+Ψ −

=− − − + Ψ −









Let
2
21W λ ≥Ψ+
, which leads to
()
22
max
1
2
2
VlWxe
αλ

≤− − − −




because of
()
max
2lW αθ λ >+
and
1
1
2
θ ≤≤
.
Finally, one obtains
()
lim 0
t
xt
→∞
=

, which
completes this proof.
4 SIMULATIONS
An undirected connected network with 6 agents is
shown in figure 1.The weighted values of the graph
are given randomly in [0 1]. The dynamic of the
multi-agents system is specified as follows:
sin( ) (x )
iiiii
x
xugd=++

Let the initial states of agents be [2 -2 6 4 -1 1],
and the expected consensus state be
0
3x =
, and the
controller coefficient is
=5 α
.
Figure 1: Switching graph of multi-agent systems.
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284

The linear exogenous system is specified as
follows:
() 0.2 ()
() 0.1 ()
ii
ii
tt
dt t
ξξ
ξ
=
=

The nonlinear exogenous system is
( ) 0.2 ( )+sin( )
() 0.1 ()
iii
ii
tt
dt t
ξξξ
ξ
=
=

Choose the coefficient matrix of system (1) as
()
1
i
gx =
, and the observer gain
(x ) 5
ii
px=
. It is
easy to verify that the gain matrix of disturbance
observer can be chosen as
()
5.
i
qx =
Then the same
Hurwitz matrix
()()
gC0.3
ii
Aqx x−=−
.
Simulation results are presented as follows:
Figure 2: Errors of the disturbances for linear systems.
Figure 3: States of multi-agent systems with disturbance
observer.
Figure 4: Errors of the disturbances for nonlinear systems.
Figure 5: States of multi-agent systems with disturbance
observer.
Figure 2 is the error between the disturbances of
exogenous linear system and the disturbances
estimated by disturbance observer. Figure 3 depicts
the final states of the multi-agent systems suffering
exogenous disturbances generated by linear
exosystems. Figure 4 shows the error between the
disturbances of exogenous nonlinear system and the
disturbances estimated by disturbance observer.
Figure 5 depicts the final states of the multi-agent
systems, sustaining exogenous disturbances genera-
ted by nonlinear exosystems. From Figures 2-3 and
figures 4-5, it can be seen that the consensus
algorithm proposed in the paper allows the agents to
reach consensus, in the case of exogenous
disturbances generated by linear or nonlinear
exosystems.
5 CONCLUSIONS
In this paper we have studied the consensus problem
of nonlinear multi-agent systems with exogenous
disturbances under an undirected topology. For the
case when the exogenous disturbance is generated
by linear exogenous system, we have shown that, the
global exponential convergence of the proposed
observer ensures the asymptotical consensus of the
nonlinear multi-agent systems with exogenous
disturbances. For the case when the exogenous
disturbance is generated by nonlinear exogenous
system, a disturbance observer with dynamic gain
has been designed. The disturbance observers are
integrated with the controller by replacing the
disturbance in the control law by its estimated value.
The numerical simulations on the asymptotical
consensus of nonlinear multi-agent systems with
disturbances demonstrate the effectiveness of the
proposed method.
0 5 10 15 20
-1
0
1
2
3
t/s
ei/mm
0 2 4 6 8
-2
0
2
4
6
t/s
xi/mm
0 1 2 3 4 5
-2
-1
0
1
2
3
t/s
ei/mm
0 1 2 3 4 5
-2
0
2
4
6
t/s
xi/mm
ConsensusofNonlinearMulti-AgentSystemswithExogenousDisturbances
285

ACKNOWLEDGEMENTS
This work was supported by National Natural
Science Foundation of China under Grant No.
61304139 and Grant No. 61203197.
REFERENCES
Bliman, P. A. and Ferrari, T. G., 2008. Average consensus
problems in networks of agents with delayed
communications. Automatic, vol. 44, pp.1985-1995.
Corts, J., Martinez, S. and Bullo, F., 2005. Spatially-
distributed coverage optimization and control with
limited-range interactions. ESAIM. Control,
Optimization and Calculus of Variations, vol. 11, pp.
691-719.
Chen, W. H., 2004. Disturbance observer based control for
nonlinear systems. Mechatronics, IEEE/ASME
Transaction on vol. 9, no.4, pp.706-710.
Das, A. and Lewis, F. L., 2011. Cooperative adaptive
control for synchronization of second-order systems
with unknown nonlinearities. International Journal of
Robust and Nonlinear Control, vol. 21, pp. 1509–1524.
Hu, J. P., Hong, Y. G. and Feng, G., 2008. Distributed
dynamic control for leaderless multi-agent consensus
with star-like topology. Asian Journal of Control, vol.
10, no. 2, pp.233-237.
Konduri, S., Pagilla, P. R. and Darbha, S., 2013. Vehicle
formations using directed information flow graphs.
American Control Conference (ACC), pp.3045, 3050,
17-19.
Khalil, H., 2002. Nonlinear Systems, Upper Saddle River,
USA: Prentice-Hall, 3
nd
edition.
Liu Y., Passino, K. M. and Polycarpou, M. M., 2003.
Stability analysis of m-dimensional asynchronous
swarms with a fixed communication topology. IEEE
Transactions on Automatic Control, vol. 48, no. 1,
pp.76-95.
Mei, J., Ren, W. and Ma, G. F., 2011. Distributed
coordinated tracking with a dynamic leader for
multiple Euler-Lagrange systems [J]. IEEE
Transactions on Automatic Control, pp. 1415-1421.
MA, G. F. and Mei, J., 2011. Coordinated tracking for
nonlinear multi-agent systems under directed network.
Control and Decision, vol. 26, no. 12, pp. 1861-1865.
Ren, W. and Atkins, E., 2007. Distributed multi-vehicle
coordinated control via locial information exchange.
International Journal of Robust and Nonlinear
Control, vol. 17, pp. 1002-1033.
Ren, W., 2007. Distributed attitude alignment in
spacecraft formation flying [J]. International Journal of
Adaptive Control and Signal Processing, pp. 95-113.
Sepulchre, R., Paley, D. A. and Leonard, N. E., 2008.
Stabilization of planar collective motion with limited
communication [J]. IEEE Transactions on Automatic
Control, pp. 706-719.
Tanner, H. G., Jadbabaie, A. and Pappas, G. J., 2007.
Flocking in fixed and switching networks. IEEE
Transactions on Automatic Control, vol. 52, no. 5, pp.
863-868.
Wu, C. W., 2005. Synchronization in networks of
nonlinear dynamical systems coupled via a directed
graph [J]. Nonlinearity, pp. 1057-1064.
Xie, G. M. and Wang, L., 2007. Consensus control for a
class of networks of dynamic agents. International
Journal of Robust and Nonlinear Control, vol. 17, pp.
941-956.
Yu, W. W., Chen, G. and Cao, M., 2010. Second-order
consensus for multi-agent systems with directed
topologies and nonlinear dynamics [J]. IEEE
Transactions on Systems Man and Cybernetics-Part,
pp. 881-891.
Yang, H., Zhang, Z. and Zhang, S., 2011. Consensus of
second-order multi-agent systems with exogenous
disturbances. International Journal of Robust and
Nonlinear Control, vol. 21, no. 9, pp. 945–956.
Zhang, X. X. and Liu, X. P., 2013. Further results on
consensus of second-order multi-agent systems with
exogenous disturbance. IEEE Transaction on Circuits
and Systems, vol. 60, no. 12, pp. 3215-3226.
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