J Control Theory
Appl
2013
11
(4) 579-585
DOI
10.1007/s11768-013-2040
亿
Passivity-based decentralized stabilization of
multi-agent systems with uncertainty
and
mixed delays
Li
anzeng
MA
1
，针，
Xuebo
CHEN
2,
Huaguang
ZHANG
1
l.
School
of
lnformation
Science
and
Engine
巳
ring
，
Northeastern
University
,
Shenyang
Liaoning
110004
,
China;
2.School
of
Electronics
and
Information
Engin
巳
ering
，
University
of
Science
and
Technology
Li
aoning
,
Anshan
Liaoning
114051
,
China
Abstract:
In
this paper,
w
巳
investigate
a decentralized stabilization
probl
巳
m
ofunc
巳
rtain
multi-ag
巳
nt syst
巳
ms
with
rnixed delays in
c1
uding
discret
巳
and
distributed time-varying delays based
on
passivity stability.
W
巳
design
a decentralized
stat
巳
feedback
stabilization
sch
巳
me
such that the family of
c1
osed-loop feedback
subsyst
巳
ms
enjoys the delay-dependent
passivity stability for each subsystem. Then
,
by
巳
mploying
a
new
Lyapunov-Krasovskii function, a
Ii
near matrix inequality
(LMI) approach is
develop
巳
d
to establish
th
巳
delay-dependent
crit
巳
ria
for
th
巳
passivity
stability
of
multi-ag
巳
nt
systems. The
suffici
巳
nt
condition
is
given for
ch
巳
cking
th
巳
passivity
stability. The proposed LMI result is computationally
effici
巳 n
t.
An
巳
xample
is given to show the
effectiven
巳
ss
of
th
巳
method.
Keywords: Multi-agent systems; Passivity stability; Decentralized stabilization; Time-delay systems
1 Introduction
Multi-agent
systems
hav
巳
found
successful applications
in
many
areas
such
as
problem
solving,
multi-agent
simula-
tion
,
construction
of
synthetic worlds
and
collective robotics
etc.
[1]. Stability is a basic qualitative property
of
multi-
agent
systems
since
if
it is
not
present
,
it
may
be
impos-
sible for
the
multi-ag
巳
nt
to achieve
any
other
group
0
时巳
c
tive. Stability
analysis
of
multi-agent
systems
is still
an
open
problem
[2-5].
In
recent
y
巳
ars
，
it
has
b
巳
en
recognized
that
time
delay is
oft
巳
n
one
of
th
巳
main
sources
of
instability
and
poor
performance.
Sinc
巳
multi-agent
systems usually
have a spatial
nature
and
a variety
of
communication
chan-
nels,
it
is
desired
to
model
them
by
introducing
continu-
ously
distributed delays
over
a certainly duration
of
time,
such
that
distant
past
has
less
influenc
巳
compared
to the re-
cent
behavior
of
the
state
[6-9].
Therefor
巳，
when
we
model
multi-ag
巳
nt
systems
,
both
time-varying
discrete delay
and
distributed
delays
should
b
巳
taken
into
accoun
t.
In
practice
,
there
are
inevitably
some
uncertainties
due
to inaccuracy in
mathematical
model
and
paramet
巳
r
fluctuations.
Thus
,
it
is
of
great
importance
to investigate
the
stability
problem
for
unc
巳
rtain
multi-agent
syst
巳
ms
with
both
discrete
and
dis-
tributed
time-varying
delays.
It
is
known
that
th
巳
dynamics
of
multi-agent
systems are
very
complicated
,
and
it
is
hard
to
design
the
collective
behavior
by
studying
individual
agent
dynamics
and
vari-
able
intra-agent
couplings.
Dynamic
graphs
are
on
巳
of
the
巳旺
órts.
Referenc
巳
[10]
discussed
the
controllability
of
dy-
namic
graphs
of
multi-agent
networks
,
while
[11] consid-
ered
the
cas
巳
when
the
arcs
of
the
graphs
can
be
described
R
巳
ceived
3
March
2012;
revised
1
May
2012
by dynamics.
In
this
pap
巳
r
，
we
巳
xt
巳
nd
mathematical
model
in
[11]
and
analyze passivity stability properties
of
multi-
agent
systems.
The
passivity theory, intimately
related
to
circuit
analysis
method
,
has
received a
lot
of
attention
from
the
control
com-
munity
during
the
last
several
d
巳
cades
[1
2-13].
It
provides
a
nice
tool
for
analyzing
the
stability
of
systems.
In this
pap
町，
we
use
th
巳
passivity
theory to
analyze
th
巳
decentralized
stability
of
uncertain
multi-agent
systems
with
both
discret
巳
and
distributed
time-varying
delays. We
consider
an interconnection
of
ag
巳
nts
which
have local
con-
trol inputs
and
are nonlinearly
connected
to
each
other.
The
inputs are
used
to
stabiliz
巳
each
individual
agent
in
a
decen-
tralized manner.
Notations
Throughout
this
paper
,
]R
n
and
]R
nxm
de-
no
饨，
respectively, the
n-dim
巳
nsional
Eu
c1
idean
spac
巳
and
the
set
of
a11
n x m real matrices.
The
superscript
‘T'
d
巳
notes matrix transposition.
The
identity matrices
and
z巳
ro
matrices are
denoted
by
1
and
0,
r
巳
spectively.
The
notion
X
:):
Y
(resp
巳
ctively
，
(X
>
Y))
means
that
X
and
Y
ar
巳
symmetric
matrices,
and
that X - Y is positive
semidefinit
巳
(respectively, positive definite).
2 Problem formulation
Let
us
consider
a
multi-agent
system
which
is
composed
of
n
agent
(scalar
subsyst巳
ms)
，
and
each
subsystem
is
d
巳
scribed as follows:
Si
:白(肚
(αz
十
Aαz)zz(t)+;
二
(e;j
+
ße;j
)fj
(Xj
(t))
•
Corresponding
author.
E-mail:
m1zhxm@sina.com.
Te
l.:
+86-412-5929741;
fax:
+86-412-5929129.
This
work
was
suppo
口
ed
by
the
Nationa1
Natural
Sci
巳
nce
Foundation
of
China
(Nos.
60874017
,
50977008
,
60821063
,
61034005)
,
th
巳
National
High
T
，巳
chnology
Research
and
Deve10pment
Program
of
China
(No.
2009AA04Z127)
,
and
the
National
Basic
R
巳
search
Program
of
China
(No
2009CB32060
1).
@)
South
China
University
of
Techno1ogy
and
Academy
of
Mathematics
and
Systems
Science
,
CAS
and
Springer-Verlag
Berlin
Heide1berg
2013
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