TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 8, August 201
4, pp. 6144 ~ 6152
DOI: 10.115
9
1
/telkomni
ka.
v
12i8.476
4
6144
Re
cei
v
ed
De
cem
ber 1
4
, 2013; Re
vi
sed
March 28, 20
14; Accepted
April 15, 201
4
Observer-based State Feedback H-infinity Control for
Networked Control Systems
Li Yanhui
*
, Zhou Xiujie
Coll
eg
e of Elec
trical an
d Information En
gi
ne
erin
g, Northea
st Petroleum U
n
iversit
y
,
Daqi
ng, He
ilo
n
g
jia
ng Prov
inc
e
, 1633
18, P.R
.
Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: LY hui@
hot
mail.com
Abstra
ct
T
he
o
b
se
r
v
e
r-b
as
ed
H
∞
controll
er
design
pro
b
l
e
m is
con
s
id
er
ed
f
o
r the
net
wo
r
k
e
d
control
systems
w
i
th un
cer
t
ain
netw
o
r
k
-
i
ndu
ce
d de
la
ys
a
nd
pac
k
e
t
dr
opouts
.
By ta
king
consider
ation
of the
de
l
a
ys a
n
d
p
a
c
k
e
t
dr
opo
uts
e
x
isti
ng i
n
the
se
nsor-to-contr
o
l
l
e
r
a
nd
contr
o
ller
-
t
o
-
a
ct
uat
or
s
i
m
u
lt
ane
ous
l
y
an
d
in
troduci
n
g
an
ob
ser
v
er
with time-v
ar
ying
de
la
ys
,
a n
e
w
error
augmented
mo
de
l is
es
ta
b
lis
he
d,
w
h
ic
h can refl
ect
the
dela
y
s
an
d
pac
k
e
t dropouts
char
acter
i
st
ic
s
of the
actual
ph
ysica
l
systems
a
ppro
p
r
i
ately
.
B
a
s
ed
on th
e
dela
y-
d
e
pen
de
n
t
L
y
ap
uno
v
sta
b
ilit
y
theor
y
,
a
suf
f
icient con
d
iti
on is
pr
op
ose
d
to
gu
a
r
an
te
e
that t
he
c
l
o
s
ed
-loop
sy
ste
m
is
asymptotically
stab
l
e
an
d
h
a
s
H
∞
pe
rf
o
r
mance
le
v
e
l
γ
.
Since
the
ob
ta
ined
c
o
n
d
iti
on is
n
onli
n
ea
r
,
the si
ngu
lar
v
a
lu
e
d
e
c
o
m
p
ositi
on m
e
th
o
d
is ap
pli
ed
to
con
v
er
t
th
e
n
onli
n
e
a
r
i
n
e
q
u
a
lities
into LM
Is
.
T
he
dela
y-
de
pen
dent app
roach
sho
w
s
a less
c
onser
v
a
ti
v
e
resu
l
t
than
th
e
de
la
y
-
indepen
den
t
app
roac
h
.
A
nu
m
e
r
ic
al
e
x
am
ple
is gi
v
en
to
de
m
o
nst
r
a
t
e
the
h
i
gh
v
a
li
dity
a
nd
mer
i
t
of
the
p
r
opose
d
appro
a
ch
.
Ke
y
w
o
r
d
s
:
n
e
t
w
o
r
k
ed
control
syste
m
s
,
H-
infin
i
ty
control,
lin
ear
matr
ix
in
equ
al
i
t
y
,
s
i
ngul
ar v
a
l
ue
deco
m
position, d
ela
y-dependent
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
The
control
sys
tems
in
which
the
control
l
o
o
p
s
are
cl
osed
o
v
er
re
al-time
netw
o
r
k
are
called
the
netw
o
r
k
ed
control
sy
st
ems
(NC
S
s)
.
Recently
,
NCSs ha
v
e
b
e
en
widely ap
plied
in
au
tomotiv
e
,
aer
osp
a
ce
,
m
obile
sens
o
r
net
wor
k
s
and i
ndu
st
r
i
al
man
u
f
a
ctur
ing
fields
[1-4].
Ho
w
e
v
er
,
the
limitations
of
net
w
o
r
k
ban
dwi
d
th an
d the
d
a
ta
colli
sion
and
retr
ansmission
in the
proce
s
s
of
send
in
g
inf
o
r
m
ation
cause
the
e
x
i
s
te
nce
of inf
o
r
m
atio
n
tr
an
smi
ssion dela
ys
and
pac
k
e
t dropout
s.
The netw
o
r
k
t
i
me-dela
y
s
and
data
pa
c
k
e
t
dropou
ts
are
commo
nly e
x
isted and
u
n
a
v
oidab
le
in real-time
NCSs
, which
will
dec
rease
t
he
perf
o
r
m
ance
an
d e
v
en ma
k
e
systems uns
tab
l
e
.
He
nce
,
the control
system
s
wit
h
time-v
ar
y
i
n
g
del
a
ys
and
pa
c
k
et dropouts
a
r
e clo
s
er
to
the
actu
al
sy
ste
m
s
and the
research
on this kin
d
of
sys
tems
h
a
s
a
s
t
r
o
n
g
p
r
a
c
t
i
ca
l
b
a
ck
gr
oun
d
.
Network-ba
sed co
ntrol th
eorie
s an
d control meth
o
d
s a
r
e sp
rin
g
i
ng up in recent years,
su
ch a
s
syst
em modeli
ng,
stability anal
ysis [5] and
robu
st H-infini
ty control [6-8]. In fact, ti
me
delays
are
often encounted in pr
acti
cal
system
s, which may indu
ce instability. Literature [9]
prop
osed a
new d
e
layed
feedba
ck
co
ntrol de
sig
n
method fo
r u
n
ce
rtain
syst
ems
with time-
varying in
put
delay by int
r
odu
cin
g
so
me rel
a
xatio
n
matri
c
e
s
a
nd turning
p
a
ram
e
ters. Data
packet drop
outs al
so o
c
cur d
ue to node failu
re
s
or netwo
rk cong
estio
n
. Literature [10]
con
c
e
r
ne
d wi
th
the stabilit
y
and co
ntroll
er
d
e
si
gn
of
NCS
s
with p
a
cket d
r
op
out
s. However, t
he
probl
em
of st
ability or cont
rol l
a
w de
sig
n
for NC
S
s
h
a
s
not
been
f
u
lly investig
ated
so
far. It i
s
worth
notin
g
that not
all
of t
he state variable
s
ca
n
be
m
e
a
s
ure
d
for practi
cal
en
ginee
rin
g
system
s and
estimation is needed. Th
e introdu
ctio
n
of observe
r can avoid th
e seri
ou
s noi
se
disturban
ce
i
n
the
teal-tim
e me
asure
m
ent results,
b
u
t
it
al
so m
a
k
e
s t
h
e
sy
st
e
m
an
aly
s
i
s
m
o
r
e
compl
e
x and
even lead
s to
some no
nlin
ear p
r
obl
ems.
Many attentions have b
een
paid to the re
se
a
r
ch on the
stability anal
ysis an
d ob
se
rver-
based co
ntrol
l
er
d
e
si
gn of NCS
s with
d
e
l
ays
a
nd
dro
p
outs [1
1-14].
The
wo
rk [12]
discu
s
sed
the
observe
r-b
ased
H-infinity
co
ntrolle
r d
e
sig
n
p
r
obl
e
m
for
NCS
s with p
a
cket
dro
pout
s in
the
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Observe
r-b
ased State Fee
dba
ck
H-infini
ty Cont
ro
l for
Networked
Control System
s (Li Yan
hui)
6145
multiple
cha
n
nels case, an
d [13]
con
c
e
r
ned
with
th
e
contin
uou
s-ti
me n
e
two
r
ke
d control
syst
em
with ran
dom
sen
s
or-del
a
y
. Among th
ese works,
it should b
e
mentione
d that the Lyapunov
function
al ad
opted in [1
2] only ha
s pa
rtial inform
ation
of the clo
s
e
d
-
loop
state
s
. This
can
sim
p
lify
the analysi
s
and synth
e
si
s but also make the re
su
lt conservative for the ab
sen
c
e of del
ay
informatio
n in
the oth
e
r si
d
e
. No
wa
days, more
an
d
more
effort
s
are
directe
d
t
o
wa
rd
s
red
u
cing
the con
s
e
r
vatism, whi
c
h m
o
tivate this prese
n
t pape
r.
Most
of the
s
e p
r
eviou
s
works a
s
sume
d the
delay
s to b
e
con
s
t
ant an
d the
obtaine
d
results
we
re
delay-in
dep
e
ndent. In thi
s
pre
s
e
n
t pap
er, ou
r p
u
rp
o
s
e i
s
to d
e
si
gn an
ob
se
rver-
based control
l
er for a conti
nuou
s-tim
e
NCSs wi
th
con
s
ide
r
ation of
the netwo
rk-i
ndu
ced del
ays
and pa
cket drop
outs. It is sh
own tha
t
a new
aug
mented sy
stem model i
s
con
s
tru
c
ted
by
introdu
cin
g
a
n
ob
se
rver a
nd we ap
ply
an ap
pro
p
ri
ate delay
-dep
e
ndent Lya
pun
ov function
al
for
the sta
b
ility
analysi
s
. T
h
e imp
o
rtant
point i
s
t
hat
the
se
nsor-to-controller a
nd
cont
rolle
r-to
-
actuato
r
n
e
twork-ind
u
ced
delays an
d
pa
cket
dro
pouts are
consi
dered
a
nd tre
a
ted t
o
be
equivalent to
time-varyin
g
delay
s in th
is p
ape
r. Mo
reove
r
, the
singula
r
valu
e
de
comp
ositi
on
method is a
d
opted for the
obtaine
d nonl
inear
con
d
itio
n.
This will b
r
i
ng a less
con
s
ervative result.
The notatio
n
used
in this
pap
er
is
stand
ard. R
n
den
otes
the
n
-dimens
ional
real
Euclidean
space
,
R
n×
m
is the
set
of
n
×
m
real
matr
ices
.
N
denote
s
the natur
al
n
u
mbers
s
e
t. The
no
tation
X
T
and
X
−
1
d
eno
te
its
tr
anspos
e
and
in
v
e
rs
e
wh
en it
e
x
i
s
ts
,
respe
c
tiv
e
ly
.
Giv
en a
symmetr
ic
matr
ix
X
=
X
T
, the
notation
X
>
0
(
X
≥
0)
means
th
at
the matr
ix
X
is real
p
o
siti
v
e
de
finiten
e
ss
(semi-definiteness).
By
diag
we
de
not
e
a b
l
oc
k
diagonal
matr
ix with
its
i
nput
arg
u
m
e
nts
on
the
d
i
a
gona
l.
I
d
eno
tes
the
identity
matr
ix.
The symbol
∗
within a
matr
ix
represents
the
symmetr
ic
entr
i
es
.
∥
·
∥
stan
ds
fo
r
either the
Euclidean
v
e
ctor nor
m
o
r
its
indu
ced
matr
ix
2-n
o
r
m
.
2. Problem Statement
The net
work-i
ndu
ced d
e
lay
s
and
pa
cket drop
outs
are
the main feat
ure
s
of NCS. Firstly,
we will
do some
explanati
ons for these characteri
sti
cs
of the
sy
stem under
consid
eration in this
pape
r. The sensor i
s
time-driven an
d its si
gnal i
s
sam
p
led pe
riodi
cally (sam
pling
perio
d
h; h >
0)
at sam
p
ling i
n
stant
s. The
controlle
r an
d
actuato
r
a
r
e
event-d
riven.
That mea
n
s the sen
s
o
r
da
ta
arrive
s at th
e
cont
rolle
r, th
e co
ntrol
sig
n
a
l is
ca
l
c
ulate
d
. And the
ou
tput of the
co
ntrolle
r a
rrive
s
at the actuato
r
, the plant i
nputs are ch
an
ged imme
diat
ely.
In view of t
he n
e
two
r
k-i
ndu
ced
del
a
y
s
an
d pa
cket
dropo
uts, we use
th
e term
referring to t
he delay
s be
tween the
se
nso
r
and th
e
controller
and delay
s be
tween the
controlle
r an
d the a
c
tuat
or
, namely,
. Similarly, we u
s
e
d
referring to the con
s
e
c
uti
v
e packet dropout
s betwe
en the sen
s
o
r
and the co
ntrolle
r
and
drop
outs b
e
twee
n the co
n
t
roller a
nd the
actuato
r
na
mely,
.
Con
s
id
er a sy
stem de
scrib
ed as follo
ws:
(
1
)
Whe
r
e
x
(
t
)
∈
R
n
i
s
the
stat
e,
u
(
t
)
∈
R
m
is
the input,
y
(
t
)
∈
R
r
is the
measured
out
put of the
pla
n
t,
!
(
t
) and
z
(
t
) a
r
e the distu
r
ba
nce in
put and
controlled o
u
t
put, resp
ecti
vely. And
!
(
t
)
∈
L
2
[0
;
∞
). The
sy
st
em mat
r
i
c
e
s
A
,
B
,
B
1
,
C
1
,
C
2
,and
D
are con
s
tant matrices
with approp
riate di
mensi
o
n
s
.
This
pap
er views the
pa
cket d
r
op
outs as
a
delay
whi
c
h
grow beyon
d the
define
d
boun
ds. Then
th
e
total system d
e
lays den
oted
by
satisfy
, for any
, where
. The
n
the ori
g
inal
system
with d
e
lays
and
pa
cket d
r
opo
uts is
equival
ent
to a
syste
m
with time
-varying
delays.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 614
4 –
6152
6146
Rem
a
r
k
1.
By the defin
ition,
is a piecewi
s
e
co
ntinuou
s fun
c
tion, whi
c
h
chan
ge
s
whe
n
e
v
er th
ese
n
sor
sign
al rea
c
h
e
s th
e co
ntrolle
r.
The de
ri
vative of
is al
wa
ys eq
ual to
one, exce
pt at the transitio
n point.
Due
to that
not all
state
s
ca
n b
e
me
a
s
ur
ed, the
dy
namic ob
se
rv
er a
nd
co
ntroller
are
c
o
ns
tr
uc
te
d
as
fo
llo
ws
:
(
2
)
Whe
r
e
x
o
(
t
)
∈
R
n
is the
state
of the
ob
s
e
r
v
e
r
,
K
and
L
are
the
gains
of
the
controll
er
an
d
obs
er
v
er
, re
spectiv
ely
.
By introdu
cin
g
the
es
timation
erro
r
e
(
t
) =
x
(
t
)
−
x
o
(
t
),
w
e
get
the
following
augm
ented
system.
(
3
)
Whe
r
e,
(4)
This
pap
er
aims
to
de
s
i
g
n
an
obser
v
e
r-base
d
H
∞
control
la
w
such
that
the
closed-loop
system
(3)
sat
i
sf
ies
the f
o
llowin
g
prop
er
ties
s
i
m
u
lt
aneous
ly
:
a
)
T
h
e
c
l
os
e-
loop
s
y
s
t
e
m
(3
) i
s
asym
ptotica
lly stable;
b)
Subject to th
e ze
ro initial
con
d
ition
an
d
all non
zero
, the controlle
d output
sat
i
sf
ie
s .
Lemma 1
[1
5].
Fo
r
a
n
y
v
e
cto
r
s
a, b
a
nd
m
a
tr
ice
s
N,
X
, Y,
Z
with
approp
r
i
at
e
di
men
s
io
ns
,
where
X and
Z are sym
m
e
tric. If
then:
3. Result a
n
d Analy
s
is
Firstly, we
have the
H
∞
perform
ance analy
s
i
s
, a suffici
ent con
d
itio
n of the
asymptoticall
y
stability with
H
∞
perfo
rm
ance level
for the system i
n
(3) i
s
given
as follo
ws:
Theorem 1.
Con
s
id
er the
clo
s
ed
-loo
p system (3
) an
d for the give
n po
sitive co
nstant
s
γ
and
, K and L
are th
e gai
n
s
of the
co
ntroller a
nd
ob
server,
re
spe
c
tively. if there exist matri
c
e
s
with app
rop
r
i
a
te dimen
s
io
ns P > 0, Q > 0, R > 0 and
X
1
> 0,
X
2
, X
3
> 0,
Y
1
, Y
2
sa
t
i
sf
y
i
ng:
(
5
)
(
6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
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ISSN:
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Observe
r-b
ased State Fee
dba
ck
H-infini
ty Cont
ro
l for
Networked
Control System
s (Li Yan
hui)
6147
Whe
r
e,
Then the
sys
tem
(3)
is
asymptotic
a
l
ly
s
t
ab
le
with
an
H
∞
perf
o
r
m
an
c
e
le
v
e
l
γ
.
Pr
oof:
Constr
uct
a
L
y
apu
no
v-
Kr
as
o
v
s
k
i
i
functiona
l
as:
Whe
r
e
P >
0,
Q >
0,
R >
0
T
a
kin
g
the
time
de
r
i
v
a
tiv
e
of
V
to
obtain:
Define ,
combining
Le
ib
n
i
z-
Ne
w
t
on
f
or
m
ula
and
th
e
Lemma
1,
w
e
ha
v
e:
Combi
n
ing al
l
of
the
abo
v
e
,
we
obtain
:
Usi
ng the
Schur
complement
[16], th
e
inequality
(5)
implies
. Thus,
the closed l
oop
system
(3) i
s
asymptoti
c
a
lly stable.
And
then
we will di
scuss the
H
∞
per
for
m
an
ce
of
the
syste
m
.
Letting
:
(
7
)
Und
e
r z
e
ro initial
conditio
n
,
V
(0)
=
0 and
V
(
∞
)
≥
0.
Thus:
Fo
r
a
n
y
non
z
e
ro
ω
(
t
)
∈
L
2
[0
,
∞
), w
e
h
a
v
e
:
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Vol. 12, No. 8, August 2014: 614
4 –
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6148
Where
,
Applying the
Schu
r
complement
to (5) y
i
elds
Ξ
<
0.
Using
the
z
e
ro
initial
co
nditio
n
.
We ha
ve
namely,
the
system h
a
s
a pr
escribed
H
∞
performance
le
v
e
l
γ
The pro
o
f
is
completed.
Rem
a
r
k
2.
T
he g
a
ins
K
and L
are
gi
v
en
in
the
stability
analysis
,
condition
(5)
is
LMI which
can
be
solv
ed
easily
.
Ho
w
e
v
e
r
, when
con
s
ider
ing
the controll
er
desi
g
n,
the
pa
r
a
me
ters K
and L
beco
m
e
unkn
o
wn
va
r
i
a
b
l
e
s
,
namely
,
the m
a
tr
ix
in
equality
in
(5
)
is
nonlin
ea
r
.
Theref
o
r
e
,
w
e
cann
ot
solv
e
it
directly
.
This
pap
er
ex
t
e
n
d
s
the s
i
ng
u
l
ar
v
a
lu
e
deco
m
po
siti
on me
thod
propo
sed
in [17]
to
deal
with this
prob
lem.
Theorem
2.
F
o
r th
e
giv
e
n
positiv
e
con
s
ta
nt
s
γ
a
nd
η
¯,
K
and
L
are
th
e
g
a
ins
of
the co
ntrolle
r
and obser
v
e
r
,
respect
i
v
e
ly
.
T
he
closed-loop system
(3
) i
s
as
y
m
pt
o
t
ic
al
ly
stab
l
e
with
an
H
∞
perf
or
manc
e
le
v
e
l
γ
,
if
there
e
x
ist
matr
i
c
es
with
a
pprop
r
i
ate
d
i
m
e
ns
i
o
ns
W
>
0,
W
21
>
0,
W
22
>
0
,
Q
i
>
0,
S
i
>
0
,
R
i
>
0
a
nd
X
1
i
,
X
2
i
,
X
3
i
,
Y
1
i
,
Y
2
i
, M
,
N
(i
=
1, 2, 3
)
sa
ti
sfyi
n
g
the
matr
ix
i
nequ
alities
(8),
(9)
and
th
e
matr
ix
Equation
(10),
(11).
(8)
(
9
)
(
1
0
)
(
1
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
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ISSN:
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046
Observe
r-b
ased State Fee
dba
ck
H-infini
ty Cont
ro
l for
Networked
Control System
s (Li Yan
hui)
6149
Where
,
Furth
e
r
m
or
e, th
e
co
ntr
o
lle
r p
a
ra
me
ter
s
ar
e gi
ven
a
s
Pr
oof:
By using
the
Schu
r
compleme
nt,
(5) can
b
e
re
w
r
itten
as
follows
:
(
1
2
)
Where
,
Defi
ning
,
.
P
e
rf
o
r
ming
cong
r
uence
tr
ansf
o
r
m
ations
to (12)
b
y
∆
1
and
substi
t
u
ti
ng
(4)
int
o
(12
)
,
since
the mat
r
ix
is of
full
colum
n
r
ank,
w
e
d
enot
e:
Then w
e
can
obtain
(8).
Similar
l
y
,
(9) is giv
e
n
through
perf
o
r
m
ing
cong
r
u
e
n
ce tr
ansf
o
r
m
ations
to (6) b
y
∆
2
.
The
proof is
comple
ted.
Rem
a
r
k
3.
F
o
r solving
the
equation
co
nstr
ai
nt (10),
liter
ature
[17]
prop
osed
a
sing
ular v
a
lu
e
de
-
composition meth
od
(SVD). F
o
r
the
m
a
tr
ix
,
there
alw
a
ys
ha
v
e:
Where
and V are two orthogonal matrices and
Σ
is a
diag
on
al
matrix
with po
sitive diago
nal ele
m
ents.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
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046
TELKOM
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Vol. 12, No. 8, August 2014: 614
4 –
6152
6150
If matr
ix
W
s
a
tisf
yin
g
,
Whe
r
e W
21
> 0,
W
22
> 0
and
U
1
, U
2
are
de
fine
d abo
v
e
,
then there
ex
i
s
t
s
W
˜
2
sa
ti
sf
y
i
n
g
.
It is
e
quiv
a
len
t
to
:
Thus,
W
e
ha
v
e
the
gains
of
the
controller
an
d
ob
se
r
v
er.
Rem
a
r
k
4.
By
using
th
e
sing
ular
v
a
lu
e
d
e
c
om
pos
i
t
i
o
n m
e
t
hod
,
w
e
h
a
v
e
a
good
s
o
lution to
d
eal
with the
nonlin
ear
te
r
m
s
B
K
W
a
nd
L
C
1
W. This
implies
t
hat
the
f
easib
le
va
l
u
e
s
of
K
and
L
can
be
obtained
easily and
the
ob
se
r
v
e
r
-b
ased
H
∞
controll
er is
design
ed.
4. A
Numeri
cal Exampl
e
In
this par
t,
w
e
will use
a
n
u
mer
i
cal
e
x
a
m
ple
to
de
mo
nstr
a
t
e
the v
a
lidity
of
the
prop
osed
ap
pro
a
c
h
. C
o
nsi
d
er
s
yst
em
(1) with:
A
cco
rdi
ng t
o
Remar
k
3,
w
e
u
s
e the
matr
ix
sing
ular v
a
lu
e
decompositio
n
and
obtain:
Su
pp
ose
that the
sam
p
ling
inte
r
v
al
i
s
h
=
0
.
1 a
nd
γ
=
0
.
62
5. Wh
en
th
e
time
dela
y
v
a
r
i
a
b
l
e
initial condit
i
on
are
giv
e
n
with
η
¯
=
0
.
9 and
x
0 =
[0
.
5 1]
T
, the
control
g
a
in
K
and
obs
er
v
e
r
gain
L
ca
n
be
obtained
b
y
ap
plying
Theorem
2
as:
(13)
The
distu
r
b
ance
i
nput
pres
en
ted
in
th
i
s
e
x
am
p
l
e
is
ω
(
t
)
=
e
−
t
. Figure
1
sho
w
s
the
ma
xi
m
um
singula
r
v
a
lue
plot
of
the
closed
-loo
p syste
m
b
y
the
obtained
controlle
r an
d
ob
s
e
r
v
e
r
(13)
an
d
Fi
gure 2
sho
w
s
the
state re
spo
n
se
of the
clo
s
ed
-lo
op
sy
ste
m
.
It is
c
l
ear that
the
states be
co
me
con
v
ergent
to z
e
ro
and
the
sy
stem
ca
n w
o
r
k
w
e
ll with
the
prop
osed
m
e
thod in
thi
s
pape
r
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
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046
Observe
r-b
ased State Fee
dba
ck
H-infini
ty Cont
ro
l for
Networked
Control System
s (Li Yan
hui)
6151
Figure 1.
Maxim
u
m
Singu
lar
V
a
lue of
the
System
with
Obser
v
er-ba
s
ed Contro
l
Figure 2.
The
State
Respon
se
of the
System
with O
bser
v
e
r-ba
s
ed Contro
l
5.
Conc
lusion
The
desi
g
n
pro
b
lem of
a
n
obs
er
v
e
r-
base
d
H
∞
controlle
r for
a linea
r
contin
uous-
time
NCSs with
time
de
l
a
ys
and
pac
k
e
t d
r
opou
ts
ha
s
b
een
in
v
e
stigate
d
in this
p
ape
r
.
It is
sho
wn
that a
ne
w
augmented system
model is
co
nstr
u
c
ted
b
y
introdu
cing
an
obser
v
e
r
.
Based
on
the
d
e
la
y-d
epe
nden
t
L
y
apun
o
v
-K
r
a
so
vskii
functional, a sufficient
con
d
ition i
s
d
e
r
i
v
e
d to
gu
ar
ante
e
the
c
l
o
s
ed-lo
op
sys
te
m
a
symptotic
ally
stab
le
with
H
∞
perf
o
r
m
ance
le
v
e
l
γ
.
Th
e SVD
m
e
th
od
is
used
to deal
with
t
he
nonlinear
prob
lem e
x
isted
in the
obt
ained
conditio
n
.
Co
mpare
d
with
the
e
x
isting
re
sults
,
w
e
e
m
plo
y
tighter
bounding
of the
cross
ter
m
s in
der
iving
stability
condition
an
d
obtain a
d
e
la
y-dep
enden
t
result. The
m
e
thod
pr
o
pose
d
in thi
s
p
aper
is
l
e
ss
conser
v
a
tiv
e
and
a
nu
me
r
i
c
a
l
e
x
am
pl
e
has sho
w
n
its
simpli
city
and
eff
ectiv
eness
.
Ref
ere
n
ce
s
[1] MB
G
C
l
oo
ster
man,
N v
a
n
de
W
o
u
w
,
W
P
MH
Heeme
l
s
,
H
N
ijme
ijer
.
S
t
abilit
y
of
n
e
t
w
or
k
e
d
control
sy
st
em
wi
t
h
unce
r
ta
in
tim
e
-
v
ar
ying
dela
ys
.
IEEE
T
r
a
n
s
a
cti
ons
on
A
utomatic
C
ontr
o
l
.
2
009;
54
;
1
575-
158
0.
[2] Y
Tipsuw
an,
MY
Cho
w
.
Contro
l
m
e
tho
dol
ogi
es
in
n
e
t
w
or
k
e
d
control
systems
.
Co
ntrol
En
gin
eer
in
g p
r
actice
.
2
003
;
11:
10
99-1
111.
[3] M
T
r
iv
ellato
,
N
Ben
v
e
n
ut
o
.
S
t
ate control
i
n
n
e
t
w
or
k
e
d
control
systems
un
der
pac
k
e
t drops
and
limited
tr
ans
mission.
IEEE
T
r
ans
acti
on
s
on
Comm
un
ica
t
ions
.
201
0;
58:
6
11-6
22.
[4] T
C
Y
ang.
Net
w
o
r
k
ed
c
ontro
l
sys
tem:
A b
r
i
e
f
su
r
v
e
y
.
IEE
Proc
eed
ing
s
of Co
ntrol
Theor
y
Appl
icatio
n
.
20
06
;
153:
403
–4
12.
[5] H
J
Ga
o
,
XY
Meng,
T
C
hen. Stabiliz
ation
of
netw
o
r
k
ed
control
sys
t
em
s
wi
t
h
a
n
e
w
dela
y
char
acte
r
izati
on.
I
EEE
T
r
an
s
a
cti
ons
on
A
u
t
o
matic
control
.
2
0
08;
53
:
21
42
-2
14
8
.
[6]
J
W
u
, T
Chen
,
L
W
ang.
De
la
y
-
dependent
rob
u
st stab
ilit
y
and
h-c
ontro
l
fo
r
jump lin
e
a
r
sy
s
te
m
s
wi
t
h
dela
ys
.
S
y
ste
m
s
an
d Contro
l
Letters
.
2
0
06;
51:
511-
518.
[7] W
Z
hang,
MS
Br
anic
k
y
,
S
M
Phil
lip
s
.
On i
m
pr
o
v
e
d
rob
u
s
t
stabili
z
a
ti
on
fo
r
unce
r
ta
in sys
tem
s
with
u
n
k
no
wn
i
n
pu
t
dela
ys
.
Pr
oce
edi
ngs
of Amer
ican
Co
ntrol
Co
nf
er
en
c
e
.
2
011;
16
56
-
16
61
.
[8] D
Y
u
e
,
QL
Han,
J
Lam.
Netw
or
k-ba
sed
ro
b
u
st h-
control
of
systems
wi
t
h
uncer
ta
in
ty
.
A
u
tomatica
.
20
05;
41
:
9
99-1
007.
[9]
D
Y
u
e
,
QL
Han. Dela
y
e
d
f
ee
db
ac
k
co
ntrol
of
uncer
tain
syste
ms
w
i
t
h
time-v
ar
y
i
ng
in
put
dela
y
.
A
u
tomatica
.
20
05;
41
:
2
33-2
40.
[10] W
T
Chen.
Des
i
gn
of
n
e
t
w
or
k
e
d
control
systems
wi
t
h
p
a
cke
t
d
r
opouts
.
IEEE
T
r
ansactions
on
A
u
tom
a
tic cont
rol
.
20
07;
52:
1
314-
131
9.
[11]
D
Y
u
e
,
QL
H
a
n, C
P
eng
.
State
f
eed
ba
c
k
controler
design
of N
e
t
w
or
k
ed
co
ntrol
systems
.
IE
EE
T
r
ansactions
on
Circu
its
Sy
ste
m
s:
II
.
2004;
51:
640-
644.
[12] W
W
Che
,
J
L W
ang,
GH
Y
a
ng.
Obser
v
er-based
h-i
n
fin
i
ty c
ontrol
in
multipl
e
ch
a
nne
l
ne
tw
o
r
k
ed
control
systems
w
i
th
r
a
ndo
m
p
a
cke
t
dro
pouts
.
Jo
ur
n
a
l of
Co
ntrol
T
heor
y
a
nd
Ap
plic
atio
ns
.
2
010;
8:
35
9-3
67.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 614
4 –
6152
6152
[13] C
Lin,
Z
W
a
n
g
,
F
Y
ang.
Ob
s
e
r
v
er
-
b
a
s
ed
net
wor
k
e
d
c
ontrol
fo
r
cont
in
uous-time
systems
wi
t
h
r
a
n
dom
se
nsor
dela
ys
.
A
u
tomatica
.
200
9;
45:
57
8-58
4.
[14]
Y
L
W
a
n
g
,
QL Han.
Obse
r
v
e
r
-
based
con
t
in
uous-time
ne
tw
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