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In this paper, we investigate a decentralized stabilization problem of uncertain multi-agent systems with mixed delays including discrete and distributed time-varying delays based on passivity stability. We design a decentralized state-feedback stabilizat
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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
580
L.
Ma
et
al.
/ J Control Theory Appl
2013
11
(4)
579-585
十三
(eL+AEi)93(Z30
一
T(t)))
十汇
(e:;
+ße:J
C
,.,
h
j
(Xj
(s))ds
十
Ui(
吗?
j=1'
'J
叮
Jt
一
σ
(t)
Yi(t)
=
且
(Xi(t))
,
i =
1
,
2
,...,风
j
=
1
,
2
,...,
η
,
+(E
3
+
ßE
刊二
(t)h(z(s))ds+
叫
U
以
(t
均)
=
f(
伊
x(t
例
t
功)川)
•
The following assumptions are made in this paper.
(4)
(1) A
l)
Tnhe
time-varying uncertainties
ßD
,
ßE
1
,
牛
E
2
and
ùEJ
are
of
出
e
form:
ßD
H
1
月
(t)
且
,
ßE
1
H
2
F
2
(t)E
2
,
ßE
2
= H3F3(t)E3,
ßE
3
H
4
F
4
(t)E4,
where
H1' H2'
H3
,
H4'
E1' E2'
E3
,
E4
are known constant
matrices that characterize
tl1
e structure
of
the uncertainties,
and
Fì
(t) , F
2
(t) ,
F
3
(t)
,
且
(t)
are unknown time-varying
matrices with Lebesgue measurable elements bounded by
Fl
(t
)Fi(t)
运
I
,
i
= 1,2,3,4.
orth
巳
whole
system is
S:
土
(t)
= (A +
ßA)x(t)
+
(E
1
+
ßE
1
)f(x(t))
+(E
2
+
ßE2)g(x(t
- T(t)))
+(E
3
+
ßE
3
)
r
t
,.,
h(x(s))ds + u(t),
Jt
σ
(t)
ν
(t)
=
f(x(t))
,
(2) A2) For
iε1
,
2
,
. . . ,n,
tl1
e subsystem nonlinear inter-
where
x(t) =
[X1(t)
X2(t)
...
xn(t)]T is
multi
吨
ent
state
vector
,
f(x(t))
=
[h(X1(t))
…
fn(xn(t))]T
εRn?
h(x(t)) = [h1(X1(t))
...
hn(xn(t))]T
εRn
,
g(x(t - T(t)))
= [g1(xdt - T(t)))
...
gn(xn(t -
T(t)))]T
ε
lR,
n
are the output nonlinear interconnected functions includ-
ing time-varying delay
, u(t) =
[U1(t)
...
un(t)
]T
ε
lR,
n
is multi-agent input vector,
y(t)
is
th
巳
output
vectoζ
A=
diag{α1
,
. . . , a
n
} is a diagonal matrix,
E
1
ε
;j
)nxn,
E
2
=
(ε;
,
)nxn and
E
3
ε
~
,
)nxn
are the interconnec-
lJ'
,
't
J
tion weight matrices representing the strength
of
coupling
between the individual subsystems.
ßA
,
ßE
1
,
ßE
2
and
ßE3
are time-varying parametric uncertainties, T(t) and
σ
(t)
denote the discrete time-varying delay and the dis-
tributed time-varying delay
, respectively, and are assumed
to satisfy 0
运
T(t)
~
T
m
, 0
运
σ
(t)
~σm
,
where T
m
and
σm
are constants.
The initial condition associated with system
(1) is
Xi(S)
=
队
(s)
,
where
if>
i(S) is bounded and continuously
differ
巳
ntialon
[一
ρ
,
0
],
ρ=
maX{T
m
,
σ
m}.
We
note that the subsystems are unstable and we use a
d
巳
cent
时
1Z
巳
d
state feedback
Ui
-(k
i
+ ßki)Xi + ri
to stabilize each subsystem independently.
ßk
i
is
tl1
e con-
troller gain perturbations.
We
choose the feedback gain
k
i
=
向
+
d
i
and
ßk
i
=
ß
向
+
ßd
i
,
leav
巳
the
reference
input
ri (bias)
fr.
巳
e
but constan
t.
Therefore, we obtain
th
巳
closed-Ioop
system
Si
:
土
i(t)
=
-(d
i
十
ßdi)Xi(t)
+ζ(EL+AeL)ι(
句
(t))
+
ζ
(吨巳吃
3
+ A
ε
气
ψ3υ
)g
幻蚓
3
川(伊
Z
句
j
(t
一
7
叫咐咐
(t
例
t
均)))
+主咿
A
叫
e
弓
ψ3
引
)ι
Lt_u(t
σ
叫州
(ο(t)
h
j
(
吟)
+r
η
i(t)
,
以
t)
= fi(Xi(t)) ,
connected functions in (1) are bounded and satisfy
5-;
~
fi(xd
- fi(X2)
也~
Jt\~~1
Jt\~
L.
1
~
5;
,
X1
-
X2
ν
g
'i(xd
-
gi(X2
)
~
1+
lrz
二《
If
,
X1
-
x2
m-;
~仇
(xd
- hi
(X2)
,、电
m
咽
X1
-
X2
Remark
1
Th
巳
constants
in A2) are allowed to be pos-
itive
, negative or zero. Lipschitz conditions are
just
出
e
spe-
cial cases
of
A2). Hence, the interconnected functions could
be non-monotonic and more general descriptions than
tl1
e
usual sigmoid functions. Note that assumption A2) is more
general than the sigmoid function which was used in
[1
1].
Wedenote
51
= diag{
51
5t
,
52
5
;t-
, .. . ,
5;
5
才},
r
占了十
5;
5:;
十占士
占一
+5i
、
52
=
diag{.:::..!.
一
-L
,
i
一~:l
,...,
""n
'"'n
1
2
h, m1 and h, m2 are similar
to
出
e
form
of
5
1
and
52
, re-
spectively.
This paper adopts the following widely accepted deÍÌni-
tion
of
passivity [12].
Definition 1 System
(4) is called passive
if
these exists
a scalar
γ~
0 such that
2fuT(t)T(t)dt
>
-47(t)r(t)dt
for
aH
t
p
注
o
and for all solution to (4) wi
tl1
Xo
=
O.
L{~mma
1
[8]
Let
α
,
b
E
三
lR,
n
,
and P be a positive deÍÌ-
nite matrix
, then
2α
Tb
~二
α
Tp-
1
α
+
bTpb.
Lt~mma
2 [8] For given matrices H , E and F with
FTF
运
1
and a scalar E > 0, the following holds:
HFE
+
(HFE)T
~ε
HH
T
+
ε
-1E
T
E.
Lt~mma
3 (Jensen inequality) For any constant matrix
W
E:
lR,
nxn,
W
T
二
W
> 0, scalar h > 0, vector func-
tionω
:
[O
,
h]
•
lR,
n such that
th
巳
integrations
concemed
f'
h …
f'
h
are well
deÍÌn
叫
then
(儿
ω(
圳
s)~W(
儿
ω
(s)ds)
~
f'
n …
h
儿
ω
~(s)W
ω
(s)ds
Lf~mma
4 (Schur complement) Given symmetric ma-
(3) trices
E1
,
E2
and matrix E
3
, where E
1
Et
and 0 <
or
S:
土
(t)
=
一
(D
+
ßD)x(t)
+
(E
1
+
ßE
1
)f(x(t))
+(E
2
+ ßE2)g(X(t - T(t)))
E
2
EJ
, then E
1
+
EJ
E
2
1
E3
< 0
if
and only
if
[叫
l[-M
< Oor I •
E3
-E
2
I I E
:f
E
1
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