TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 11, Novembe
r
2014, pp. 77
0
5
~ 771
5
DOI: 10.115
9
1
/telkomni
ka.
v
12i11.60
06
7705
Re
cei
v
ed Ma
rch 2
2
, 2014;
Re
vised June
12, 2014; Accepte
d
Jul
y
6
,
2014
Robust Controller Design for Networked Control
Systems Based
on State Estimation
Fengxia Xu
1
, Qunhong Ti
an*
2
, Junke
Wang
3
, Men
g
kun Li
4
1-
4
Colle
ge of C
o
mputer a
nd C
ontrol En
gi
neer
ing, Qiqi
har
Un
iversit
y
, Qiq
i
h
a
r
, Heilo
ngj
ian
g
161
00
6, Chin
a
2
Colle
ge of Ch
emical E
ngi
ne
erin
g, Chin
a U
n
iversit
y
Of Petrole
u
m, Qingd
ao, Shan
do
ng,
266
58
0, Chin
a
*Corres
p
o
ndi
n
g
suthor, e-mai
l
: xufe
ng
xi
a_
hit
@
16
3.com
1
, tianq
un
hon
g@1
63.com
2
A
b
st
r
a
ct
For the networked control system
s wi
th uncertain time delay in
both sensor-to-c
ontr
o
ller
and
control
l
er-to-ac
tuator cha
n
n
e
l
s
,
and d
a
ta
drop
outs in
s
ensor-to-c
ontro
ller ch
an
nel,
a new
mod
e
l
of
networked c
o
ntrol system
s based on state estim
a
tion
is
proposed using sing
le-exponential s
m
oothing
meth
od
to
pre
d
ict the
state v
a
ria
b
les. After
ado
ptin
g Ly
ap
unov
stab
ility t
heory, th
e
asy
m
ptotic
stab
ilit
y for
the syste
m
ha
s be
en
prov
e
d
, an
d th
e
R
obust
Co
ntro
ll
er
h
a
s bee
n desi
gne
d by u
s
ing
li
ne
ar ma
trix
ine
qua
lity. In the si
mulati
on
exper
iment, c
o
mpari
ng th
e
state respo
n
s
e
s un
der
diffe
rent situati
ons
of
divers
e time de
lay an
d data dr
opo
uts, the resu
lts show
that the metho
d
is e
ffective.
Ke
y
w
ords
:
singl
e-ex
po
nen
tial s
m
o
o
thin
g
meth
od, stat
e es
ti
mati
on,
Lyap
un
ov fun
c
tion, li
near
matri
x
ine
qua
lity, robu
st controller
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
Networked
co
ntrol syste
m
s (NCS
s) a
r
e
distrib
u
ted sy
stem
s in whi
c
h the comm
u
n
icatio
n
betwe
en se
n
s
ors, actu
ato
r
s, and
controllers is s
upp
orted by a share
d
real
-ti
m
e netwo
rk [1-3].
Comp
ared wi
th the traditional point-to
-
p
o
int wi
rin
g
, the use of the comm
uni
cati
on ch
annel
s
can
redu
ce the costs of ca
ble
s
and po
we
r, simplify
the
installatio
n
and mainten
a
n
ce of the whole
system, and i
m
prove the reliability [4-6]
.
But at
the same time, tim
e
delay
s and
packet dropouts
will inevitably degra
de the
control p
e
rf
orma
nce
of NCS
s, or ev
en ca
use the system to be
unsta
ble [7, 8], which com
p
licate the an
alysis
an
d co
mpre
hen
sive
ness of the networke
d
co
n
t
rol
system
s. Th
e
r
efore, time
d
e
lay an
d d
a
ta
dro
pout
s a
s
the main
p
r
o
b
lems of n
e
t
w
orke
d
syste
m
s
have re
ceive
d
wide attenti
on from many
sch
olars [9].
Aiming at the netwo
rked
contro
l syste
m
s with time
delay
and d
a
ta drop
outs,
in [10],
whe
n
the se
n
s
or,
th
e controller and
th
e actuato
r
a
r
e
all clo
c
k d
r
iven, multi-rate
NCS with
bo
th
sho
r
t time del
ay and pa
cke
t
dropout
s is
con
s
tru
c
ted a
s
a switch
ed
system mo
de
l. Based on t
h
e
approa
che
s
of swit
che
d
system
a
n
d
Lyapun
ov function
s, the ne
ce
ssary condition
s for
asymptotical stability
for multi-rate NCS
with
both
sho
r
t time d
e
lay and p
a
cket dropo
uts are
given. In [11], a novel
control law m
odel i
s
pr
opo
se
d to
take the
net
work-ind
uced
delay, ran
d
o
m
packet d
r
op
o
u
ts and
packet-d
r
o
pout
s comp
en
sati
on into co
nsideratio
n sim
u
ltaneo
usly. By
con
s
tru
c
ting
a network-st
a
tus-dep
end
e
n
t Lyapunov
function, a
sufficie
n
t condition fo
r
the
existen
c
e of
the
H
output fe
edba
ck
controller i
s
fo
rmu
l
ated in
the f
o
rm
of no
nco
n
vex matrix
inequ
ality, an
d the
co
ne
co
mpleme
ntarit
y lineari
z
at
io
n (CCL) p
r
o
c
edure i
s
exploited to
solve
the
non
convex fe
asibl
e
p
r
obl
e
m
. But, the n
e
twork statu
s
is
assu
med t
o
vary in
a
M
a
rkovian fa
sh
ion
satisfying
a certain tran
sition p
r
oba
bility matrix. In [12], the pape
r is con
c
e
r
ne
d with the
H
control issue
for a class of networke
d
control
syst
ems with pa
cket dro
pout
s and time-va
r
ying
delays. T
he
a
ddre
s
sed
NCS is m
odel
ed
as
a Ma
rkov
i
an di
screte
-ti
m
e switched
system
with t
w
o
sub
s
ystem
s
;
by usin
g the
averag
e d
w
el
l time metho
d
, a sufficient
con
d
ition i
s
obtaine
d for t
he
mean
sq
ua
re
expon
ential
stability of t
he clo
s
ed
-lo
op NCS with
a
d
e
si
red
H
disturban
ce
attenuation l
e
vel. The de
sire
d
H
contro
ller is o
b
tain
ed by solvin
g a set of li
near
matrix
inequ
alities. In [13], the paper
inve
stigat
es the ob
se
rver-b
ased
H
fu
zzy control problem for
a
cla
ss
of discrete-time fu
zzy mixed dela
y
system
s
wi
th rand
om communi
catio
n
pa
cket losse
s
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 77
05 – 771
5
7706
and multipli
cative noises,
whe
r
e the
mixed delay
s com
p
ri
se b
o
th discrete
time-varying
and
distrib
u
ted d
e
lays. The
random
pa
cket losse
s
are
descri
bed
b
y
a Berno
u
lli
distrib
u
ted
white
seq
uen
ce th
at obeys a
condition
al probability dist
ribution, and t
he multipli
cat
i
ve disturban
ce
s
are i
n
the fo
rm of a
scalar Gau
ssi
an
white noi
se
with unit vari
an
ce. In the
prese
n
ce of mi
xed
delays, ra
ndo
m packet losse
s
and multi
p
licative noi
se
s, suffici
ent con
d
ition
s
for the existence of
an ob
serve
r
-based fuzzy feedba
ck co
n
t
roller a
r
e de
rived, after that, a linear
matrix inequ
ality
approa
ch fo
r
desi
gning
su
ch a
n
ob
se
rv
er-ba
s
ed
H
fuzz
y c
o
ntroller is
pres
ented. In [14], the
study is concerne
d
with th
e optimal li
ne
ar e
s
ti
mation
probl
em fo
r li
near di
screte
-time sto
c
h
a
stic
system
s with
possibl
e multiple ran
dom
measur
ement
delays an
d p
a
cket dropout
s. The mo
del
is
con
s
tru
c
ted
t
o
de
scrib
e
t
he p
hen
ome
na of
multipl
e
rando
m d
e
l
ays a
nd
pa
cket d
r
o
pout
s b
y
employing some random v
a
riabl
es of B
e
rnoulli di
stri
bution. By
state augmentation, the sy
stem
with ran
dom
delays an
d p
a
cket dro
pou
ts is tran
sferred to a syste
m
with rand
o
m
param
eters
.
The e
s
timato
rs a
r
e
re
cursively com
p
u
t
ed in term
s of the solut
i
ons of a
Ri
ccati differen
c
e
equatio
n a
n
d
a
Lyapu
nov differe
nce e
quation. In
[1
5],
Aiming at
the networked system
s with
sen
s
o
r
delay
s, missi
ng m
easure
m
ent
s and pa
cke
t
drop
outs, a d
e
sig
n
method
about adapti
v
e
Kalman filter is
pro
p
o
s
ed.
Two
differen
t
adaptive
filt
ers a
r
e
co
nsi
dere
d
to
esti
mate un
kn
own
para
m
eter v
e
ctor a
s
so
ci
ated wi
th the system m
a
trice
s
, then,
the estimation of state and
para
m
eters o
f
the system
based on
th
e
minimizatio
n
of squa
re
of
the output p
r
edictio
n erro
r is
adopte
d
in bootstra
p man
ner. An estim
a
tor-ba
sed ro
bust controll
e
r
desi
gn ha
s been p
r
op
osed
for asymptoti
c
stability of the syste
m
wh
ose
p
a
ra
mete
rs
can vary wi
thin a kno
w
n
boun
d.
It needs to po
int out, in the
above refe
re
nce
s
, the sig
n
ificant limitations have b
e
en found
in the m
e
tho
d
s
of co
ntroll
er d
e
si
gn fo
r the net
wo
rked
control
system
s with
time del
ay wit
h
rest
raint th
at l
e
ss tha
n
a
sa
mpling
pe
riod
. Mean
while
,
it is
difficult to
achieve
in
practical p
r
oje
c
t
to
create
the con
d
ition
s
with
time delay or data
dro
p
o
u
ts that
re
aches ce
rtain
known stocha
stic
distrib
u
tion. Ho
wever, the
system de
si
gn with
time delay and pa
cket drop
outs only under the
situation of sensor p
r
e
s
en
ce is la
ck of
comp
reh
e
n
s
ive factors. In this pape
r,
base
d
on the
method
of sta
t
e estimatio
n
, a robu
st con
t
rolle
r
i
s
de
si
gned
fo
r networked co
ntrol system
s whi
c
h
have un
ce
rtai
n time delay
in both
sen
s
or-to
-
cont
roll
er an
d contro
ller-to
-
a
c
tuato
r
ch
ann
els, a
n
d
data dropout
s in se
nsor-to-c
ontroller
chann
el. In this de
sign, th
ere a
r
e no
re
stri
ctions
with
the
length
or
dist
ribution
of ti
me del
ay. Co
nsid
erin
g the
co
mplex
situ
ation of
mod
e
ling
un
certai
nty
and extern
al disturban
ce
s,
the dr
iving m
e
thod of co
ntrolle
r whi
c
h i
s
combin
ed wi
th event-drive
n
and time-driv
en is ad
opted
to design the
robu
st cont
ro
ller ba
sed o
n
state estim
a
tion.
2. Model of Con
t
rol Sy
st
em Base
d on State Es
timation
The m
odel
of
networked
control
syste
m
s b
a
s
ed
on
state estim
a
tion is sho
w
n i
n
Figu
re
1. Senso
r
an
d controller a
r
e co
nne
cted
through
com
m
unication n
e
twork, so i
s
the conne
cti
on
betwe
en con
t
roller and a
c
tuator. He
re
,
s
c
is the n
e
tworke
d un
ce
rt
ain d
e
lay i
n
se
nsor-to-
controlle
r ch
annel, and
ca
is the
net
wo
rked
un
ce
rtai
n del
ay
in co
ntrolle
r-to
-
a
c
tuator
chann
e
l
.
Comp
ared wi
th
s
c
or
ca
, the networke
d
del
ay in plant-to
-
sens
or o
r
a
c
tu
ator-to
-
pla
n
t cha
nnel
i
s
so short that
it can be ignore
d
. There
f
ore,
the networke
d
un
ce
rtain
delay in
the networked
cont
r
o
l sy
st
e
m
s is
ks
c
c
a
. For c
o
nvenienc
e
, the ass
u
mptions
are as
follow:
(1) Sen
s
o
r
is time-driven,
and the samp
ling time is T.
(2)
Controlle
r ad
opts the
comp
oun
d d
r
i
v
ing mod
e
. T
he
controlle
r i
s
time
-drive
n
whe
n
it
gene
rate
s th
e data
ba
sed
on
state e
s
ti
mation at
eve
r
y sa
mpling
p
e
riod; th
e
co
ntrolle
r i
s
eve
n
t-
driven when t
he data tran
smit from sen
s
or
to co
ntrolle
r after the time delay of
s
c
.
(3) T
he sequ
ence disord
er doesn’
t exist in the pro
c
e
s
s of data tran
smissio
n
fro
m
sen
s
o
r
or co
ntroll
er.
s
c
ca
Figure 1. The
Model of Net
w
orke
d Co
nt
rol Systems B
a
se
d on State Estimation
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rob
u
st Controller Desi
gn for Net
w
o
r
ked
Control S
yst
em
s Based o
n
State… (Fe
ngxi
a
Xu)
7707
The sequ
en
ce diag
ram
s
o
f
state estim
a
tion
co
ntrol
system
whi
c
h exist time delay is
sho
w
n
in Fi
g
u
re
2. When
the time d
e
la
y
k
is smalle
r t
han
a con
s
ta
nt of a p
e
ri
o
d
of
sampling
time, se
nsor
sampl
e
s data
at
kT
time. Me
anwhile, the
output of
co
n
t
roller i
s
[(
)
]
c
ux
k
, bas
e
d
on estim
a
te state variable
s
. The output
of controller i
s
[(
)
]
ux
k
based o
n
sampli
ng val
ue
()
x
k
whi
c
h is t
r
an
sferre
d to the
controlle
r after time delay
s
c
.
[(
)
]
c
ux
k
, the output of
controller
at
kT
time is tran
sf
erred to actu
ator after tim
e
delay
ca
. The
output
[(
)
]
ux
k
of co
ntrolle
r at
s
c
kT
time is tran
sferre
d to act
uator after ti
me delay
ca
.
Therefore, During
a sam
p
ling
pe
riod
[,
(
1
)
)
kT
k
T
, the control
inputs of t
he control
system ca
n
be divided i
n
to three
p
a
rts
respe
c
tively as follo
wing:
(1) th
ere i
s
a netwo
rke
d
time delay
ca
in cont
roll
er-to
-
a
c
tuato
r
cha
nnel, so
durin
g
[,
)
ca
kT
k
T
, the c
ontrol inp
u
t of the system is
[(
1
)
]
ux
k
. (2) D
u
ring
[,
)
ca
k
kT
k
T
, the control input of the system is
[(
)
]
c
ux
k
. (3)
Duri
ng
[,
(
1
)
)
k
kT
k
T
, the
control input
of the system
is
[(
)
]
ux
k
.
s
c
(1
)
xk
ca
kT
(1
)
kT
(2
)
kT
s
c
ca
s
c
ca
(2
)
xk
()
x
k
()
c
x
k
()
x
k
s
c
kT
s
c
s
c
(1
)
c
xk
(1
)
x
k
(2
)
c
x
k
(2
)
xk
(1
)
kT
(2
)
kT
ca
s
c
kT
(1
)
kT
(2
)
kT
[(
)
]
c
ux
k
[(
)
]
uxk
[(
1
)
]
c
ux
k
ca
s
c
[(
1
)
]
uxk
[(
2
)
]
c
ux
k
ca
kT
(1
)
kT
(2
)
kT
[(
1
)
]
uxk
ca
[(
)
]
c
ux
k
ca
[(
)
]
uxk
[(
1
)
]
c
ux
k
s
c
ca
[(
1
)
]
uxk
s
c
s
c
[(
2
)
]
c
ux
k
kT
(1
)
kT
(2
)
kT
(2
)
k
TT
ca
[(
1
)
]
uxk
ca
[(
)
]
c
ux
k
s
c
[(
1
)
]
c
ux
k
ca
[(
1
)
]
uxk
s
c
[(
2
)
]
c
ux
k
(0
)
k
T
Figure 2. The
Sequen
ce Di
agra
m
s of State
Es
timation Control System which Exis
t Time Delay
Duri
ng th
e
sampling
time
of
[,
(
1
)
]
kT
k
T
, if the time del
ay
k
is long
er th
an
one
sampli
ng
peri
od b
u
t short
e
r tha
n
o
r
e
qual to
two
sampli
ng
peri
ods,
namely
2
k
TT
,T
he
situation i
s
si
milar to the
a
nalysi
s
of the
time delay th
at
k
is
smalle
r than on
e sa
mpling p
e
ri
od.
In
this ca
se,
t
he control
in
puts of
the control syst
em c
a
n
be
d
i
vide
d
in
to tw
o pa
r
t
s r
e
s
p
e
c
tive
ly
as
following:
(1) During
[,
)
ca
kT
k
T
, the control in
put of the
sy
stem i
s
[(
1
)
]
ux
k
. (2)
Duri
ng
[,
(
1
)
)
ca
kT
k
T
, the control i
nput of the system is
[(
)
]
c
ux
k
.
Whe
n
the time delay meet
s
k
nT
(n is a
n
integer la
rg
er th
an 1), by that analogy, the
control in
puts of the net
wo
rke
d
control
system
s can be
obtai
ned. Whe
n
the
r
e are data
p
a
cket
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 77
05 – 771
5
7708
drop
outs in
sensor-to-cont
rolle
r c
han
nel
, the sa
me m
e
thod i
s
u
s
e
d
to estim
a
te t
he
state vari
a
b
le
whi
c
h isn’t transferre
d
to the controlle
r.
The predi
ctio
n model ba
se
d on state e
s
timation is a
s
follow:
(1
)
(
)
(
1
)
(
)
cc
kk
x
k
xx
(
1
)
Whe
r
e
(1
)
c
k
x
is the predi
cted
value of state va
riable
of control sy
stem at time
(1
)
kT
,
()
c
k
x
is the
pred
icted valu
e of
state va
riabl
e of control
system at tim
e
kT
,
()
x
k
is
the actual i
n
put value of controlle
r at time
0
[(
1
)
]
kT
(where,
0
0
). That is, wh
en time
delay is sm
al
ler than a sa
mpling time,
()
x
k
is
c
o
ns
id
er
ed
a
s
th
e
co
ntr
o
lle
r
inp
u
t; w
h
en
time
delay is la
rg
er than
a sa
mpling time
or data
drop
outs,
()
c
x
k
is co
n
s
ide
r
ed
as th
e co
ntrolle
r
input.
Firstly, con
s
id
ering a
bout th
e model of ge
neral
contin
u
ous
system:
()
()
()
()
(
)
tt
t
tx
t
xA
x
B
u
zC
(2)
Her
e
()
p
t
xR
is the state,
()
q
t
uR
is the
control inpu
t,
()
m
t
z
R
is the con
t
rolled
sign
al o
u
tput
.
,,
AB
C
are
matrixe
s
whi
c
h
hav
e ap
pro
p
ri
ate dime
nsi
on,
,,
pq
m
a
r
e
po
sitive
integers.
Con
s
id
erin
g about the time delay and d
a
ta dr
op
outs i
n
the networked cont
rol sy
stem
s, a
contin
uou
s ti
me state fe
ed
back la
w
()
()
tx
t
uK
is propo
sed. Acco
rding to S
equ
ence dia
g
ra
m
2,
the Equation
(2) i
s
discretized into Equ
a
t
ion (3).
12
0
ˆ
ˆˆ
(1
)
(
)
(
1
)
(
)
(
)
()
()
c
kk
k
k
k
kx
k
xA
x
B
K
x
B
K
x
B
K
x
zC
(3)
Whe
r
e
AT
A
=e
，
1
0
ˆ
ca
As
ed
s
B=
B
，
2
ˆ
k
ca
T
As
ed
s
B=
B
，
0
ˆ
k
T
As
ed
s
B=
B
. Ac
c
o
rding to
matrix theory,
11
1
ˆ
()
k
B=
B
E
F
H
,
22
2
ˆ
()
k
B=
B
E
F
H
，
00
0
ˆ
()
k
B=
B
E
F
H
,
()
k
F
is a
time-de
pen
de
nt quantity base
d
on un
certain time d
e
lay
k
, and it meets
T
()
()
kk
FF
I
,
12
0
()
()
()
0
kk
k
EF
H
E
F
H
EF
H
.
Con
s
id
er mo
deling u
n
cert
ainty and external
di
sturban
ce in th
e netwo
rked
control
sy
st
em
s,
t
h
e
n
,
12
1
2
(
1
)
(
)
(
1
)
()
()
()
()
(
)
c
kk
x
k
x
k
k
kx
k
k
xA
x
B
K
B
K
F
zC
F
(4)
Whe
r
e
00
+
A
AB
K
B
，
11
1
BB
B
，
22
2
BB
B
.
00
0
()
k
BE
F
H
K
B
，
11
1
()
k
BE
F
H
B
，
22
2
()
k
BE
F
H
B
are
the tot
a
l mod
e
ling
uncertainty a
nd me
ets
00
BG
F
E
，
11
BG
F
E
，
22
BG
F
E
.
01
2
,,
p
ll
p
GR
,
E
E
E
R
are
kn
own
matrixes,
ll
FR
is unkno
wn
matrix and m
eets
T
F
FI
.
0
B
，
1
B
，
2
B
are unkno
wn m
odel erro
rs,
()
r
kR
are
the di
st
urba
nce in
pu
ts, whi
c
h
bel
ong to
2
()
[
0
,
)
kL
, w
her
e
r
is a
po
sit
i
ve
integer.
Definition: Gi
ven
as a positive consta
nt, if
the cont
rol
system (4
) with the unce
r
tainty
whi
c
h me
ets
T
FF
I
has
cha
r
a
c
ters
as follo
w:
1) System
is asymptotical
ly stable; 2)
Und
e
r
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rob
u
st Controller Desi
gn for Net
w
o
r
ked
Control S
yst
em
s Based o
n
State… (Fe
ngxi
a
Xu)
7709
zero initial condition,
22
()
()
kk
zw
for any
2
()
[
0
,
)
kL
w
. Then we call the syste
m
(4) ha
s
the performa
n
ce
of
H
.Where
22
()
()
kk
zw
resp
ond
s the interference sup
p
re
ssi
on
capabilities for cont
rol sy
stem, so, the smaller of
, the
better of the system's p
e
rfo
r
man
c
e.
3. Design of
Robu
st Controller Base
d on State Esti
mation
Lemma
[16]: Given matri
c
es
Y
,
H
,
E
and
R
of app
rop
r
iate dime
nsi
ons a
nd with
Y and R
symmetri
c
al and
0
R
, then:
TT
T
0
YH
F
E
E
F
H
For all F that satisfying
T
FF
R
, if a
nd only if there exists some
0
s
u
ch that:
2T
2
T
0
YH
H
E
R
E
Theorem
:
Fo
r a given
0
, the control syst
em (4) ha
s the perfo
rma
n
c
e
of
H
if
there a
r
e po
si
tive-definite matrices
12
Q,
S
,
S
,
M
and con
s
tant
0
s
u
ch that.
TT
11
1
0
2
1
TT
T
T
T
T
0
0
T
T
22
2
2
T2
T
1
2
2
1
T
02
1
1
1
00
0
0
00
0
00
0
0
0
00
0
0
0
0
0
0
00
0
0
0
00
0
0
0
00
0
0
0
0
00
0
0
0
QG
G
B
S
B
A
Q
B
M
B
F
QA
M
B
Q
C
M
Q
E
BS
E
S
FI
F
CQ
F
I
KQ
S
EQ
E
I
E
S
E
Proof:
Fi
rstly
,
a Lyapunov function i
s
de
fined as:
TT
T
T
12
(
)
()
()
(
1
)
(
1
)
+
(
1
)
(
1
)
T
cc
kk
k
k
k
k
k
Vx
P
x
x
K
R
K
x
x
K
R
K
x
Whe
r
e
12
,
PR
,
R
are real symmet
r
y matrixes. Taki
n
g
the d
i
fference of the Lyapu
nov
function,
TT
T
T
T
11
TT
T
T
22
()
(
1
)
(
)
(
1
)
(
1
)
()
(
)
()
(
1
)
(
1
)
()
(
)
(
1
)
(
1
)
cc
c
c
kk
k
kk
k
k
k
k
kk
k
k
VV
x
V
x
xP
x
x
P
K
R
K
x
x
K
R
K
x
xK
R
K
x
x
K
R
K
x
(6)
Whe
n
()
0
k
w
,
T
1
()
(
)
(1
)
(
1
)
()
0
()
(
)
(1
)
(
1
)
cc
cc
kk
kk
k
xk
x
k
xk
x
k
xx
Kx
K
x
VU
KK
KK
(7)
Whe
r
e
1
U
is defined:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 77
05 – 771
5
7710
TT
11
2
TT
T
11
1
1
1
2
1
TT
T
22
1
2
2
2
2
0
0
0
00
0
AP
A
P
K
R
K
A
P
B
A
P
B
B
P
A
B
PB
R
B
PB
U
BP
A
B
P
B
BP
B
R
R
Whe
n
()
0
k
w
,
T
2
()
(
)
(1
)
(
1
)
()
()
(
)
0
(1
)
(
1
)
()
(
)
cc
cc
kk
kk
kx
k
x
k
xk
xk
kk
xx
Kx
K
x
VK
U
K
KK
(8)
Whe
r
e
2
U
is defined:
TT
T
T
T
11
2
1
TT
T
T
11
1
1
1
2
1
1
TT
T
T
2
22
1
2
2
2
2
1
2
TT
T
T
11
1
1
2
1
1
0
0
0
00
0
0
0
A
P
A
P
K
R
K
A
PB
A
P
B
A
PF
BP
A
B
P
B
R
B
P
B
BP
F
U
BP
A
B
P
B
BP
B
R
BP
F
R
F
PA
F
P
B
F
PB
F
P
F
Whe
n
mat
r
ix inequ
ality (8)
is
satisfied,
cl
os
e
d
-cont
rol system (4
)
i
s
asymptotic
st
ability.
It's clea
r that if inequality is satisfie
d, the inequ
ality (7) i
s
ce
rtain tru
e
.
In the next place, a
s
sume
the zero in
itia
l conditio
n
an
d let us introd
uce:
T2
T
0
()
(
)
()
()
k
kk
k
k
Jz
z
(9)
Whe
r
e
0
is con
s
tant, then:
T2
T
0
()
()
()
(
)
()
k
kk
k
k
k
Jz
z
V
(10)
Pluging ine
q
u
a
lity(8) an
d e
quality(9
)
into
the inequalit
y (10).
T
3
()
()
(1
)
(
1
)
()
()
(1
)
(
1
)
()
(
)
cc
cc
kk
kk
kk
kk
kk
xx
Kx
Kx
JK
x
U
K
x
Kx
Kx
(11)
Whe
r
e
3
U
is defined:
TT
T
T
T
T
T
11
2
1
2
TT
T
T
11
1
1
1
2
1
1
TT
T
T
3
22
1
2
2
2
2
1
2
TT
T
T
T
2
12
1
1
1
2
1
1
2
2
0
0
0
0
00
0
0
0
T
AP
A
P
K
R
K
+
C
C
AP
B
A
P
B
AP
F
C
F
B
P
A
B
PB
R
B
PB
B
P
F
U
BP
A
B
P
B
BP
B
R
BP
F
R
FP
A
F
C
F
P
B
FP
B
F
P
F
I
F
F
(12)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rob
u
st Controller Desi
gn for Net
w
o
r
ked
Control S
yst
em
s Based o
n
State… (Fe
ngxi
a
Xu)
7711
If
3
0
U<
is sati
sfie
d, unde
r ze
ro initial con
d
i
t
ion, there is
22
()
()
kk
zw
sat
i
sf
ied.
So, when i
n
e
quality(12
) i
s
satisfie
d, the de
sign
ed system co
ntrol
l
er ma
ke
s
co
ntrol sy
stem
(4)
has the p
e
rfo
r
man
c
e
of
H
.
Applying Sch
u
r co
mplem
e
nt, inequality (12
)
is tran
sfo
r
med into:
1
12
1
TT
T
T
11
T
22
2
T2
T
12
2
1
1
00
0
00
0
0
00
0
0
0
0
00
0
0
0
0
0
00
0
0
0
0
0
00
0
0
0
00
0
0
0
00
0
0
0
0
PA
B
B
F
AP
C
K
BR
BR
R
FI
F
CF
I
KR
(13)
Plug
00
A
AB
K
+
B
，
11
1
BB
B
，
22
2
BB
B
into inequ
ality(13) , Adopt
Lemma
,
inequ
ality (13
)
is tran
sfo
r
m
ed into
1T
01
2
1
TT
T
TT
T
0
0
T
T
11
1
T
T
22
2
2
T2
T
1
2
2
1
1
01
2
00
0
0
00
0
0
00
0
0
0
0
00
0
0
0
0
0
00
0
0
0
0
0
0
00
0
0
0
0
00
0
0
0
0
00
0
0
0
0
0
00
0
0
0
PG
G
A
B
K
B
B
F
AK
B
P
C
K
E
BR
E
BR
E
R
FI
F
CF
I
KR
EE
E
I
(14)
By using th
e elem
entary
matrix tra
n
s
form
ati
on a
nd Schur co
mpleme
nt, inequality (14) is
equivalent to:
1T
1
T
11
1
0
2
1
TT
T
T
T
T
0
0
T
T
22
2
2
T2
T
1
2
2
1
1
1T
02
1
1
1
00
0
0
00
0
00
0
0
0
00
0
0
0
0
0
0
00
0
0
0
00
0
0
0
00
0
0
0
0
00
0
0
0
PG
G
B
R
B
A
B
K
B
F
AK
B
P
C
K
E
BR
E
R
FI
F
CF
I
KR
EE
I
E
R
E
(15
)
Two
side
s of Equation (15) are mult
iplie
d prop
er mat
r
ix as follow:
1
0
000
000
0
000
000
0
0
00
000
000
0
0
0
0
000
0
0
0
0
0
0
000
00
0
0
000
0
0
0
0
000
00
I
P
I
I
I
I
I
I
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 77
05 – 771
5
7712
The re
sult ca
n be written:
1T
1
T
1
1
11
1
0
2
1
1T
1
T
T
1
T
1
T
1
T
0
0
T
T
22
2
2
T2
T
1
2
1
2
11
1
1
1T
02
1
1
1
00
0
0
00
0
00
0
0
0
00
0
0
0
0
0
0
00
0
0
0
00
0
0
0
00
0
0
0
0
00
0
0
0
PG
G
B
R
B
A
P
B
K
P
B
F
PA
P
K
B
P
C
P
K
P
E
BR
E
R
FI
F
CP
F
I
KP
R
EP
E
I
E
R
E
(16
)
Usi
ng some
cha
nge
s of variabl
es,
11
1
11
2
2
,,
PQ
,
R
S
R
S
K
P
M
，
inequality(1
6
)
is ch
ang
ed to
inequality(5
)
.
4. Numerical
Example
Con
s
id
er a sy
stem which m
eets the sy
stem (2):
0
.
69
314
7
0
.3
794
685
0
0
.4
054
65
1
A
，
10
01
B
，
The sa
mpling time
is set as
T=1
s
, the
co
ntrol pl
ant i
s
discreti
zed, t
he p
a
ra
meters of th
e di
screte sy
stem(4
) with
un
ce
rt
ain
time delay, data drop
outs,
modelin
g un
certainty and e
x
ternal distu
r
ban
ce a
s
follow:
2.
00
0
0
0.
65
95
0
1
.5
00
0
A
，
0.
1
1
0
0.1
03
5
0
.0
02
0
0
0
.10
21
As
ed
s
B=
B
，
0.
4
2
0.
1
0.
35
74
0.
03
32
0
0
.3
32
2
As
ed
s
B=
B
，
1
0
0.4
0.
981
7
0.24
12
0
0
.798
9
As
ed
s
B=
B
，
Suppo
se the
other pa
ram
e
ters of the sy
stem (4
) a
s
follow:
0
0.08
0
0
0
.18
B=
，
1
0
B
，
2
0
B
，
1
0.
1
0
.
2
0.
2
0.1
F
，
2
0.
1
0
.
1
0.
1
0
.
1
F
，
10
01
C
。
A
cco
rdi
ngly
,
10
01
G
，
0
0.
1
0
00
.
2
E=
，
0.
8
0
00
.
9
F
，
He
re,
tak
i
ng
1
Ts
,
0.8
，
0.
3
，
using
softwa
r
e
MATLAB, the
discrete time
state fe
edb
a
c
k gai
n
is
0.
94
89
0
.23
90
0.
23
80
0
.
7
5
3
6
K
。
For th
e
ca
se
that the
r
e
e
x
ist data
dro
pout
s an
d ti
me del
ay wh
ich i
s
l
a
rg
er
than a
sampli
ng tim
e
in the networked sy
stem
s, the conve
n
t
ional method
of controlle
r
desi
gn is that
th
e
actuato
r
ta
ke
the value
of l
a
st
cycle
a
s
t
he
cu
rre
nt value. Usin
g the
same value
of K,
this pap
er
made th
ree
different an
al
ysis a
nd the
simulatio
n
s
to the metho
d
s of
conven
tional controll
er
desi
gn an
d controlle
r de
si
gn ba
sed o
n
state estim
a
tion.
In the firs
t
c
a
s
e
, the initial
s
t
ate is
0
1
1
x
, the time del
ay is
smalle
r tha
n
a sa
mplin
g
time (
k
T
), there is impul
se
interfere
n
ce
signal such
as
1
()
1
k
at the fifth and sixth
sampli
ng tim
e
, the data d
r
opo
uts in
se
nso
r-to
-
c
ontroller
cha
nnel
at the tenth
sampli
ng ti
me.
Usi
ng the con
v
entional met
hod of co
ntrol
l
er de
sign
of the co
ntrol sy
stem,
the re
spon
se
s of sta
t
e
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Rob
u
st Controller Desi
gn for Net
w
o
r
ked
Control S
yst
em
s Based o
n
State… (Fe
ngxi
a
Xu)
7713
variable
s
of
1
x
and
2
x
a
r
e
sh
o
w
n i
n
Fi
gure
3. Applying t
he met
hod
of
co
ntrolle
r
de
sign
ba
se
d
on state e
s
timation, the re
spo
n
ses of st
ate variable o
f
1
x
and
2
x
sho
w
n i
n
Figure 4.
Figure 3. The
Respon
se
Curve of State
Variabl
e
1
x
whe
n
the Time Delay Meets
0
k
T
Figure 4. The
Respon
se
Curve of State
Variabl
e
2
x
whe
n
the Time Delay Meets
0
k
T
In the
se
con
d
case, the i
n
itial state i
s
0
1
1
x
, and
the i
m
pulse inte
rference
signal
su
ch as
1
()
1
k
has
been
ap
plied
at the fifth a
n
d
sixth
sampli
ng time.
The
time d
e
lay i
s
l
a
rge
r
than a sa
mpli
ng time (
2
k
TT
) at the fifth and si
xth sampling
time, while it is sm
aller tha
n
a
sampli
ng tim
e
at othe
rs sa
mpling time
s.
The
dat
a dropout
s
in se
n
s
or-to-co
ntroll
er cha
nnel ha
ve
been fou
nd a
t
the tenth sa
mpling time.
Usi
ng t
he
co
nventional m
e
thod of cont
rolle
r de
sign
of
the co
ntrol
sy
stem, the respon
se
s of sta
t
e variable
s
o
f
1
x
and
2
x
are sho
w
n
in Figu
re 5.
Applyin
g
the metho
d
o
f
controlle
r d
e
sig
n
ba
se
d
on
state e
s
ti
mation, the
resp
on
se
of st
ate varia
b
le
o
f
1
x
and
2
x
is sho
w
n
in Figure 6.
Figure 5. The
Respon
se
Curve of State
Variabl
e
1
x
whe
n
the Time Delay Meets
2
k
TT
Figure 6. The
Respon
se
Curve of State
Variabl
e
2
x
whe
n
the Time Delay Meets
2
k
TT
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 11, Novem
ber 20
14: 77
05 – 771
5
7714
In the third
case, the initial state i
s
0
1
1
x
, the impul
se inte
rfere
n
ce si
gn
al su
ch
as
1
()
1
k
wa
s a
pplie
d
at the fifth a
nd
sixth
sam
p
ling tim
e
, th
e time
delay
is la
rg
er than
two
s
a
mpling times
but s
m
aller than three s
a
mpling times
(
23
k
TT
) at th
e fifth, sixth and
seventh
sam
p
ling time
s, the time dela
y
is sm
alle
r
than a sampl
i
ng time at o
t
hers
sa
mpli
ng
times, the da
ta dropo
uts i
n
sen
s
o
r-to
-
contro
lle
r ch
a
nnel ha
s bee
n found at the tenth samp
ling
time. Using th
e conve
n
tion
al method of controlle
r
de
sign of the con
t
rol system, the re
spo
n
ses of
state vari
abl
e of
1
x
and
2
x
are
sh
own in
Fi
gure
7. Appl
yi
ng the m
e
thod
of controller
de
sig
n
based on
state estimation,
the re
spon
se
s of state vari
able of
1
x
and
2
x
is shown in Figure 8.
Figure 7. The
Respon
se
Curve of State
Variabl
e
1
x
whe
n
the Time Delay Meets
23
k
TT
Figure 8. The
Respon
se
Curve of State
Variabl
e
2
x
whe
n
the Time Delay Meets
23
k
TT
Comp
ari
ng the thre
e cases, for the
control
sy
stem
with extern
al distu
r
ba
nce or tim
e
delay
whi
c
h
is la
rge
r
th
a
n
a
sam
p
ling
time fro
m
t
he fifth sam
p
ling time,
after ap
plying
the
conve
n
tional
method, th
e resp
on
se
s of t
he
state va
ria
b
le of
1
x
and
2
x
ha
ve fiercely ju
mp, but th
e
respon
se
s of
the state va
riable of the
system
ba
se
d on
state esti
mation
a
r
e smooth.
Comp
are
to the
ca
se
which
ha
s th
e
data d
r
op
out
s at th
e
tenth
sa
mpling
tim
e
, the
re
spon
se
s of th
e
sta
t
e
variable
al
so
have th
e
si
gnifica
nt imp
r
ovemen
t. O
b
viously, ap
pl
ying the m
e
thod
of controller
desi
gn state
estimation,
the reg
u
latio
n
time
and
overshoot a
r
e obviou
s
ly sho
r
ter. Cont
rol
pre
c
isi
on of the networked
system
s is e
nhan
ce
d;
the influen
ce
s of the c
ontrol sy
stem caused
by
modelin
g un
certainty and e
x
ternal distu
r
ban
ce are validly restraine
d
.
5. Conclusio
n
Aiming at the
netwo
rked
control
system
s with
un
ce
rtain time del
a
y
in both sen
s
or-to-
controlle
r a
n
d
co
ntroll
er-to-actuat
o
r
ch
an
nels, and
d
a
ta
d
r
op
outs
in
sensor-to-co
ntrolle
r cha
n
n
e
l
in this
pap
er,
a ne
w meth
o
d
of ro
bu
st control
de
sign
based o
n
state estim
a
tion i
s
p
r
op
osed.
To
get over the effectivene
ss
of data drop
o
u
ts and t
he d
e
lay of data transfe
r ca
use
d
by time delay,
A state predi
ction arithmeti
c
is ad
ded in
the end of
co
ntrolle
r input. The arithm
etic refe
rs to ap
ply
singl
e-exp
o
n
ential sm
ooth
i
ng metho
d
a
s
the p
r
edi
cti
on mod
e
l of
control sy
ste
m
, and p
r
o
s
p
e
ct
the state at every sam
p
li
ng time. To increa
se the
rapidity of the drive of
cont
rolle
r, cont
roll
er
applie
s the
way of
com
posite
-
d
r
iven
of time
-d
riv
en an
d eve
n
t-drive
n
. Ad
opting Lya
p
u
nov
function
an
d l
i
near matrix
inequ
ality, the de
sign
of ro
bust co
ntrolle
r
b
a
se
d on st
ate
e
s
timatio
n
is
compl
e
ted. T
he m
e
thod
o
f
rob
u
st
co
ntrolle
r d
e
si
gn
ba
sed
on
st
ate e
s
timatio
n
do
esn’t ha
ve
rest
rictio
ns o
n
distri
bution
or length
of time del
ay for t
he syste
m
. Simulation resu
lts demo
n
stra
te
that the
desi
g
ned
rob
u
st
controlle
r
ba
se
d on
stat
e
e
s
timation n
o
t o
n
ly re
solve
s
t
he p
r
obl
em
s
of
Evaluation Warning : The document was created with Spire.PDF for Python.