Distributed Encirclement Control of Multi-agent Systems
Duan min
1
,Song yongduan
1
*
1. School of Automation, Chongqing University, Chonging 400044, P.R.China.
E-mail: duanmincqu@163.com
Abstract: In this paper, we investigate the distributed encirclement control problem of a group of multi-agent systems with
static and dynamic targets. A distributed encirclement control algorithm combined with a distributed estimator is proposed for
agents to encircle static or dynamic targets. It is shown that the distributed center of targets can be achieved in nite time and the
distributed encirclement problem can be solved with a bounded error. Finally, a numerical example is included to illustrate the
obtained theoretical results.
Key Words: Multi-agent system, Encirclement control, Distributed estimation
1 Introduction
In recent years, distributed coordination control problem
of multi-agent systems has drawn more and more attention
from various elds, such as physics, science and mathemat-
ics [1]-[14], and important practical applications including
unmanned air vehicles(UVA), satellite clusters and robotic
systems. One of the important research topics is distributed
encirclement problem. Distributed encirclement problem
means how to design a algorithm to make all agents encir-
cle the targets and track the targets . Most exisiting results
are concentrated on the case where all agents nally reach a
consensus with the leaders or converge to their convex hull
of the leaders. For example, in [10], Hong et al. investigated
the problem that tracking an active leader by a neighbor-
based local controller together with a neighbor-based state-
estimation rule. The work in [11] expanded the results in
[10] to the distributed observers design for a second-order
multi-agent system. [12] investigated the collective coordi-
nation of multi-agent systems guided by multiple leaders, the
agents will converge to the polytope formed by the multiple
leaders.
Currently, there have been rare works to solve the dis-
tributed encirclement control problem. In [13], a group
of UAVs encircling a target with a decentralized nonlinear
MPC was studied, but the model they considered has only
one target. In [14], two surrounding problems with station-
ary leaders were proposed , where a leader-follower frame-
work and with xed communication graph was used. In real-
ity, it is more meaningful and challenge to encircle multiple
second-order dynamic targets.
In this paper, we investigate the distributed encirclement
control problem of a group of multi-agent systems with mul-
tiple static and dynamic targets. Each agent can only com-
municate with the neighborhoods and get local measured in-
formation, to achieve the distributed center(the average po-
sitions of targets), we designed a dynamic averaging esti-
mator. To solve the distributed encirclement control prob-
lem, To solve the distributed encirclement control problem,
a nearest-neighbour rule is introduced based on the estimate
of distributed center. By graph theory and Lyapunov the-
ory, it is proved that the estimate of distributed center can
This work is supported by the Major State Basic Research Develop-
ment Program 973 (No. 2012CB215202) and the National Natural Science
Foundation of China (No. 61203080 and No.61134001)
be got and converge to the distributed center center of tar-
gets in nite time and distributed encirclement control can
be achieved with a bounded error.
2 Preliminaries
2.1 Graph Theory
Let G(V, ε,A) be an undirected graph of order n,where
V = {s
1
,s
2
, ···,s
n
} is the set of nodes, ε ∈ V ×V is the
set of edges, and A =[a
ij
] is a weighted adjacency matrix.
The node indexes belong to a nite set I = {1, 2, ···,n}.
An edge of G is denoted by e
ij
=(s
i
,s
j
). Since the graph
is undirected, for any (s
i
,s
j
) ∈ ε,then(s
j
,s
i
) ∈ ε.The
adjacency elements associated with the edges are positive,
i.e., the adjacency matrix is dened as a
ii
=0and a
ij
=
a
ji
≥ 0,wherea
ij
> 0 if and only if e
ij
∈ ε.Theset
of neighbors of node s
i
is denoted by N
i
= {s
j
∈ V :
(s
i
,s
j
) ∈ ε}. Correspondingly, the graph Laplacian with the
diagraph is dened as L =[l
ij
],wherel
ii
= −a
ij
,i = j.
If there is a path from every node to every other node, the
graph is said to be connected.
Lemma 1.
[15]
If the undirected graph G is connected, then
its Laplacian matrix L satises:
(1) Zero is one eigenvalue of L,and1
n
is the corresponding
eigenvector,i.e., L1
n
=0.
(2)The rest n − 1 eigenvalues of L all have positive real-
parts. And L is a symmetric matrix and the eigenvalues 0=
λ
min
= λ
1
≤ λ
2
···≤λ
n
= λ
max
.
Lemma 2.
[16]
Suppose there is a C
1
positive denite Lya-
punov function V (x, t) dened on U × R
+
where U ∈
U
0
is the neighborhood of the origin, and there are pos-
itive real constants c>0 and 0 <α<1, such that
˙
V ((x, t)+cV
α
(x, t) is negative semidenite on U.Then
V (x, t) is locally in nite-time convergent with a settling
time
T ≤
V
1−α
(x
0
(t))
c(1 −α)
2.2 Model and problem description
Suppose that the network system under consideration con-
sists of n agents. Each agent is regarded as a node in an
undirected graph G. Each edge (s
j
,s
i
) corresponds to an
available information link from the agent j to i. Moreover,
each agent updates its current state based upon the informa-
tion received from its neighbors.
Proceedings of the 33rd Chinese Control Conference
Jul
28-30, 2014, Nan
in
, China
1253