TELKOM
NIKA
, Vol.11, No
.11, Novemb
er 201
3, pp. 6737
~6
745
e-ISSN: 2087
-278X
6737
Re
cei
v
ed Ap
ril 18, 2013; Revi
sed
Jul
y
6, 2013; Accept
ed Jul
y
27, 2
013
Linear Unbiased Optimal Filter for Discrete-
Time
Systems with One-Step Random Delays and
Inconsecutive Packet D
r
opouts
Jian Ding,
Shuli Sun*
Schoo
l of Elect
r
ical En
gin
eeri
ng, Hei
l
on
gji
a
n
g
Univ
ersit
y
, H
a
rbi
n
150
08
0, Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: sunsl@
hlj
u
.e
du.cn
A
b
st
r
a
ct
T
h
is pa
per is
concer
ned
w
i
th the li
ne
ar
unb
iase
d
mi
ni
mu
m v
a
ria
n
ce
estimatio
n
pr
obl
e
m
for
discrete-ti
me stochastic li
ne
a
r
control syste
m
s w
i
th one-s
t
ep rand
o
m
d
e
lay a
nd i
n
co
nsecutiv
e pac
ket
drop
out. A ne
w
mode
l is de
velo
ped to d
e
s
c
ribe the
phe
n
o
men
a
of the one-ste
p
de
lay
and i
n
cons
ec
utive
packet dr
op
out
by e
m
p
l
oyi
ng
a Bern
oul
li
dist
ribute
d
stoch
a
s
tic varia
b
le. B
a
sed
on th
e
mode
l, a rec
u
rsi
v
e
line
a
r u
nbi
ase
d
opti
m
al filte
r
in the l
i
n
ear
mi
ni
mu
m v
a
r
i
anc
e sens
e i
s
desi
gne
d b
y
the metho
d
of
compl
e
ting th
e
square. T
he
soluti
on to the
line
a
r filt
er is given by thre
e equ
atio
ns in
cludi
ng a R
i
cc
ati
equ
atio
n, a
Ly
apu
nov
eq
uati
on
and
a s
i
mpl
e
differ
ence
e
q
uatio
n. A suffic
i
ent c
ond
it
io
n f
o
r the
existe
nc
e of
the steady-stat
e
filter is give
n. A simul
a
tio
n
s
how
s the effectiven
ess of the prop
osed
alg
o
r
i
thm.
Ke
y
w
ords
:
li
n
ear un
bias
ed fil
t
er, rando
m de
l
a
y, incons
ecuti
v
e packet dro
p
outs, steady-state filter
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
In recent yea
r
s, the re
se
arch on n
e
two
r
ked system
s and
sen
s
or n
e
tworks
ha
s gaine
d
lots of interests due to wid
e
application
s
in co
mm
uni
cation, co
ntro
l and sign
al processin
g
[1-3].
In networke
d
system
s, the time delays
and pa
cket drop
out
s are un
avoidabl
e in data
transmissio
n throug
h un
rel
i
able commu
nicatio
n
s fro
m
sen
s
o
r
s to
a pro
c
e
ssi
ng
cente
r
. The
data
available in th
e pro
c
e
s
sing
cente
r
may n
o
t be re
al
time due to the
delays o
r
p
a
cket dropo
uts.
So
estimation a
n
d
control in the netwo
rk
ed
system
s are very challe
ngi
ng [4].
In wireless
networks
, the s
y
s
t
ems
with s
t
och
a
sti
c
d
e
lays, pa
cket dropo
uts an
d missi
ng
measurement
s can b
e
de
scrib
ed by
a
st
och
a
sti
c
pa
ra
meter
system
[5-8]. Yaz et
al. [5] desig
n
s
the filtering problem in the least mea
n
sq
uare
sen
s
e. Ho
wev
e
r, the
filters
are not
optimal since
a
colo
red n
o
ise
induced by a
ugmentatio
n is treate
d
as
a white noi
se
. The estimati
on problem f
o
r
system
s
with
missin
g m
e
asu
r
em
ents i
s
studie
d
in [6
], w
h
e
r
e
s
e
ns
or
d
a
t
a a
r
e
on
ly th
e
measurement
noise
s at some sa
mple
s. Ray et
al. [7] present
s a linear un
biased minim
u
m
varian
ce
state estimato
r to accomm
od
ate the e
ffect
s of ra
ndom
delays in
dat
a arrival
at the
controlle
r. In [8], the state
e
s
timation fo
r
discre
te
-time linear
sy
stem
s wi
th sto
c
h
a
s
tic
paramete
r
s
is tre
a
ted. A
re
cu
rsive
le
ast-squ
a
res l
i
near
e
s
timat
o
r i
s
d
e
si
gne
d for
ra
ndom
delay
s by t
h
e
covari
an
ce in
formation
ap
proa
ch
in [9]
.
Studying
the ro
bu
st H-i
n
finite filter for
system
s
with
rand
om d
e
la
ys and
missi
ng me
asure
m
ents [1
0]. The o
p
timal
H
2
filtering f
o
r
system
s
with
rand
om d
e
la
ys, pa
cket dropout
s an
d u
n
ce
rtain
ob
servation
s
i
s
pre
s
ente
d
b
a
s
ed
on
a u
n
i
f
ied
stocha
stic pa
ramete
rized model
i
n
[11].
For
syste
m
s
with infinite
a
nd finite p
a
cket dro
pout
s, the
optimal lin
ea
r estim
a
tors a
r
e
develop
ed
in the
line
a
r
minimum va
ri
ance
sen
s
e
b
y
an in
novati
o
n
analysi
s
app
roach in
[12] a
nd [13],
re
sp
ectively.
Ho
wever, the
ra
n
dom
delays a
r
e
not ta
ken
i
n
to
con
s
id
eratio
n
in [12, 13].
Investigate
s
the opt
imal li
near e
s
timati
on pro
b
lem for syste
m
s
with
rand
om dela
y
s and pa
cket dropo
ut
s, however, whi
c
h may brin
g
network con
gestio
n
sin
c
e
a
sen
s
o
r
pa
cket is sent se
veral times t
o
avoid
loss [14]. Studie
s
the optima
l
linear filter for
system
s with
one-step
ran
dom delay an
d
comp
en
sati
on of packet
drop
outs [15].
In this pape
r,
we co
nsi
d
e
r
the linear
unb
iase
d optimal
filtering pro
b
l
e
m for sy
ste
m
s with
the po
ssi
ble
one-step
ra
n
dom d
e
lay an
d incon
s
e
c
uti
v
e packet
dro
pout. A mod
e
l is d
e
velope
d to
descri
be the
phen
omen
a by a Bernoul
li rando
m variable with a
kno
w
n p
r
ob
a
b
ility. A sensor
packet is o
n
ly sent on
ce to avoid
the net
work co
nge
stion
,
and the packet dro
p
o
u
t is
inco
nsecutive. A recursive l
i
near
unbia
s
e
d
optimal filter is obtai
ned
by the metho
d
of compl
e
ting
the squ
a
re.
The sol
u
tion
is given in term
s of on
e
Riccati, one
Lyapunov a
nd one si
mp
le
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No
. 11, Novemb
er 201
3: 673
7 – 6745
6738
differen
c
e e
q
uation. The
steady-state
prop
erty
is
analyzed. A sufficie
n
t condition fo
r the
existen
c
e of the stea
dy-sta
te filter is given.
2. Problem
Formulation
Con
s
id
er the
discrete time
-invar
iant
line
a
r st
o
c
ha
st
ic
sy
st
em:
(1
)
(
)
(
)
(
)
x
tx
t
B
u
t
w
t
(1)
(
)
()
()
zt
H
x
t
v
t
(2)
Whe
r
e
n
R
t
x
)
(
is the state,
()
h
ut
R
is the input,
m
R
t
z
)
(
is the measure
d
o
u
tput,
r
R
t
w
)
(
and
m
R
t
v
)
(
are th
e
pro
c
e
s
s and
measurement
noises,
re
sp
ectively, and
,
B
,
an
d
H
are con
s
tant matrices
with suitabl
e dime
nsio
ns.
In netwo
rked
system
s,
the
measurement
()
zt
of a
sen
s
o
r
i
s
sent to a
proce
s
sing
ce
nter
throug
h the
unreli
able
co
mmuni
cation
s with r
and
o
m
delays a
n
d losse
s
. To
avoid network
con
g
e
s
tion,
we
assum
e
t
hat a
pa
cket i
s
o
n
ly
sent o
n
ce.
He
re,
we only
deal
with one
-ste
p d
e
lay
and i
n
co
nsecutive pa
cket drop
out. We
adopt th
e
foll
owin
g
mo
del for
the mea
s
urem
ent
rece
ived
by the processing
cente
r
.
()
()
()
(
1
()
)
(
1
(
1
)
)
(
1
)
yt
t
z
t
t
t
z
t
(
3
)
Whe
r
e
)
(
t
is a
Bernoulli
random va
ria
b
le with th
e pro
babiliti
e
s
Pr
o
b
{
(
)
1
}
t
and
Pr
ob
{
(
)
0
}
t
1
with
01
, and i
s
un
co
rrelate
d
with
ot
her random
varia
b
les. T
able
1
sho
w
s t
he ca
se of
dat
a t
r
a
n
smi
ssi
on:
Table 1. Data
Tran
smi
ssio
n
in Network
t
1 2
3
4
5
6 7
8
9
10
ξ
(
t
)
1 0
1
0
0
1 0
0
0
1
y
(
t
)
z(
1
)
0
z(
3
)
0
z(
4
)
z(
6
)
0
z(
7
)
z(
8
)
z(
1
0)
From T
able
1
,
we
can
se
e
that z(1),
z(3
)
, z(6
)
a
nd
z(1
0
) a
r
e
re
ceiv
ed on tim
e
, z(2), z(5
)
and
z(9
)
a
r
e
lost, z(4),
z(7
)
and
z(8
)
are delaye
d
. It is kno
w
n that
the on
-time
arrivin
g
rate i
s
Prob
{
(
)
1
}
t
=
, one-step
delay rate i
s
Prob
{
(
)
0
,
t
(1
)
0
}
t
2
(1
)
and p
a
cket
drop
out rate
is
Prob
{
(
)
0
,
t
(1
)
1
}
t
(1
)
for the
data at
t
instant. So, model (3)
descri
b
e
s
po
ssi
ble on
e-step tran
smi
ssi
on del
ay an
d inco
nsecutive packet dro
p
o
u
ts.
In this pape
r,
the expectati
on E operate
s
on
)
(
t
and/or
)
(
t
w
and
)
(
t
v
.
I
and 0 are an
identity matri
x
and a
zero matrix with
suitabl
e dim
ensi
o
n
s
, re
spectively. Also, the followi
ng
assumptio
n
s are
u
s
ed.
Assump
tion
1.
)
(
t
w
and
)
(
t
v
a
r
e un
co
rrelate
d
white n
o
ises
with
ze
ro
mean
s and
v
a
rian
ce
s
0
w
Q
an
d
0
v
Q
.
Assump
tion 2.
The initial state
)
0
(
x
is un
co
rrel
a
ted with
)
(
t
w
and
)
(
t
v
, and
0
)]
0
(
Ε
[
x
,
0
T
0
0
]
)
)
0
(
)(
)
0
(
[(
Ε
P
x
x
(
4
)
Our aim
is to
find the
re
cu
rsive lin
ea
r u
n
b
iased
optim
al filter
of the
followin
g
Kal
m
an-li
ke
form:
ˆˆ
(1
)
(
)
(
)
(
)
(
)
(
)
(
1
)
xt
F
t
x
t
G
t
u
t
K
t
y
t
(
5
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
e-ISSN:
2087
-278X
Linea
r Un
bia
s
ed O
p
tim
a
l
Filter for Di
screte-Tim
e
System
s with O
ne-Step
Ran
dom
… (Jia
n Ding
)
6739
With the initial value is
ˆ
(0
)
x
0
. We will
solve the gain
matri
c
es
()
F
t
,
()
Gt
and
()
Kt
su
ch that th
e linear filter (5)
satisfie
s unbia
s
ed
ne
ss a
nd lea
s
t
mean squa
re criteri
on, i.e.
ˆ
E[
(
)
]
E
[
(
)
]
x
tx
t
and min
T
ˆˆ
E
[
((
)
(
)
)
((
)
(
)
)
]
x
tx
t
x
t
x
t
. No
te that we wil
l
desig
n the l
i
near filte
r
(5
)
depe
ndent o
n
the proba
bility
, which can
be com
puted
offline.
Rem
a
r
k
1.
From the
di
stributio
n of
)
(
t
, we have
E[
(
)
]
t
,
Cov
[
(
)
]
(
1
)
t
,
2
E[
(
)
]
,
t
2
E[(1-
(
)
)
]
1
,
t
0
))]
(
-
)(1
(
E[
t
t
,
E[
(
)
(1
-
(
))
]
(
1
)
,
kt
t
k
.
3.
Linear Un
bia
sed Op
timal Filter
In this se
ctio
n, a linea
r un
biased optim
al filt
er as
(5
) will be d
e
si
g
ned for
syste
m
(1)-(3).
Theo
rem 1 a
nd Theo
rem
2 give the results.
Theorem 1.
For
system (1)-(3
) with A
s
sumpti
on
s
1 and 2, the
state se
co
n
d
-o
rde
r
moment matri
x
T
()
E
[
()
()
]
qt
x
t
x
t
is co
mpute
d
by the following Lya
pun
ov equation:
TT
T
T
(1
)
(
)
(
)
(
)
w
qt
qt
B
u
t
u
t
B
Q
(
6
)
With the initial value
T
00
0
(0
)
qP
.
The mea
n
()
E
[
()
]
x
tx
t
of the state
()
x
t
satisfies the follo
wing differen
c
e eq
uation:
(1
)
(
)
(
)
x
tx
t
B
u
t
(7)
With the initial value
0
(0
)
x
.
Proof
.
Subst
i
tute (1) into
the definition
(1
)
qt
T
E[
(
1
)
(
1
)
]
xt
x
t
, and it
yields (6
).
Equation (7)
dire
ctly follows from taki
ng
expectatio
n
o
n
(1).
Theorem 2.
For sy
stem (1)-(3
) with Assumption
s 1
and 2, the gain matri
c
e
s
of the
linear u
nbia
s
ed optimal filter (5
) are co
mputed by:
()
()
F
tK
t
M
(8)
2
(1
)
M
HH
(9)
()
()
Gt
B
K
t
H
B
(10)
1
()
()
()
K
tt
t
(11)
Whe
r
e:
T
()
=
(
)
tP
t
M
2
(1
)
(
1)
[
(
1
)
]
K
tH
I
TT
(1
)
+
qt
H
2T
T
[(
1
)
(
1
)
]
w
K
tH
I
Q
H
(12)
T
()
(
1
)
{
()
tH
q
t
2
(1
)
[
1
(
1
)
]
(
)
qt
TT
(1
)
(
)
(
1
)
(
)
}
qt
q
t
H
T
()
MP
t
M
TT
T
(1
)
(
)
(
)
HBu
t
u
t
B
H
TT
T
T
(
1
)
{
()
()
()
()
H
x
tu
t
B
B
u
tx
t
TT
(1
)
(
)
(
)
xt
u
t
B
TT
(1
)
(
)
(
)
}
Bu
t
x
t
H
TT
2
(1
)
wv
v
HQ
H
Q
Q
2T
T
T
T
(1
)
(
1)
(
1
)
Hq
t
H
K
t
M
2T
T
(1
)
(
1)
(
1
)
MK
t
H
q
t
H
3T
T
T
(1
)
(
1
)
(
1
)
Hq
t
H
K
t
M
3T
T
(1
)
(
1
)
(
1
)
MK
t
H
q
t
H
2T
T
T
T
(1
)
(
1
)
w
HQ
H
K
t
M
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No
. 11, Novemb
er 201
3: 673
7 – 6745
6740
2T
T
(1
)
(
1
)
w
M
Kt
H
Q
H
(13)
The filtering e
rro
r varia
n
ce matrix is give
n by:
TT
T
(1
)
(
)
(
)
(
)
(
)
w
Pt
P
t
Q
K
t
t
K
t
(
1
4
)
Whe
r
e
()
Pt
is the filtering error covaria
n
ce
matrix with the initial value
0
(0
)
PP
.
Proof
.
Subst
i
tuting (2) int
o
(3) an
d using (1
) a
nd (5), we have
the filtering error
equatio
n.
(1
)
[
(
)
(1
)
(
)
xt
F
t
t
K
t
H
(1
(
1
)
)
(1
(
)
)
(
)
]
(
)
(
)
(
)
tt
K
t
H
x
t
F
t
x
t
[(
)
(
1
)
(
)
]
(
)
BG
t
t
K
t
H
B
u
t
(
)
(
1
)(
)
(
)
(
+
1
)(
)
(
1
)
wt
t
K
t
H
wt
t
K
t
v
t
(1
(
1
)
)
(
1
(
)
)
(
)
(
)
tt
K
t
v
t
(15)
Whe
r
e the filtering e
r
ror
ˆ
(
)
()
()
x
tx
t
x
t
. From the un
bia
s
ed
ne
ss, it re
quire
s that
(0
)
0
x
and
E[
(
)
(
1
)
(
)
Ft
t
K
t
H
(1
(
1
)
)
(1
(
)
)
(
)
]
0
tt
K
t
H
(16)
And,
E[
(
)
(
1
)
(
)
]
0
BG
t
t
K
t
H
B
(17)
Then it follows from (16) a
nd (17
)
that:
()
()
[
Ft
Kt
H
2
(1
)
]
H
(18)
()
()
Gt
B
K
t
H
B
(19)
Whi
c
h giv
e
(8
)-
(10
)
.
Substituting (18) an
d (1
9) i
n
to (15
)
yield
s
:
(1
)
xt
()
{
(
(
1
)
)
Kt
H
t
2
[
(
1
)
(
1
(
1
))(
1
(
))
]
}
(
)
tt
I
x
t
()
()
(
(
1
)
)
(
)
(
)
F
tx
t
t
K
t
H
B
u
t
()
(
1
)
(
)
(
)
wt
t
K
t
H
wt
(1
)
(
)
(
1
)
tK
t
v
t
(1
(
1
)
)
(1
(
)
)
(
)
(
)
tt
K
t
v
t
(20)
From (20
)
, we have the filtering e
r
ror va
rian
ce a
s
:
T
(1
)
(
1
)
(
)
{
(
)
Pt
K
t
H
q
t
2
(1
)
[
1
(
1
)
]
(
)
qt
(1
)
(
)
qt
TT
T
(1
)
(
)
}
(
)
qt
H
K
t
T
()
()
()
F
tP
t
F
t
TT
T
T
(1
)
(
)
(
)
(
)
(
)
K
t
H
B
u
tu
tB
H
K
t
TT
T
T
(
1
)(
){
(
)
(
)
(
)
(
)
K
t
H
x
tu
tB
B
u
tx
t
TT
(1
)
(
)
(
)
xt
u
t
B
TT
T
(1
)
(
)
(
)
}
(
)
Bu
t
x
t
H
K
t
TT
T
T
()
(
)
ww
QK
t
H
Q
H
K
t
TT
T
T
()
()
ww
K
tH
Q
Q
H
K
t
T2
T
()
()
(
1
)
(
)
(
)
vv
K
tQ
K
t
K
t
Q
K
t
2T
T
T
T
(1
)
(
)
(
1
)
(
1
)
(
)
K
tH
q
t
H
K
t
F
t
3T
T
T
(1
)
(
)
(
1)
(
1
)
(
)
K
tH
q
t
H
K
t
F
t
2T
T
T
T
(1
)
(
)
(
1
)
(
)
w
K
tH
Q
H
K
t
F
t
2T
T
T
(1
)
(
)
(
1)
(
1
)
(
)
F
tK
t
H
q
t
H
K
t
3T
T
T
(1
)
(
)
(
1
)
(
1
)
(
)
F
tK
t
H
q
t
H
K
t
2T
T
T
(1
)
(
)
(
1)
(
)
w
F
tK
t
H
Q
H
K
t
(
2
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
e-ISSN:
2087
-278X
Linea
r Un
bia
s
ed O
p
tim
a
l
Filter for Di
screte-Tim
e
System
s with O
ne-Step
Ran
dom
… (Jia
n Ding
)
6741
Whe
r
e the
un
correl
ation of
()
x
t
and
()
wt
,
(1
)
vt
,
()
vt
and the un
co
rrel
a
tion of
()
x
t
and
()
wt
,
(1
)
vt
are used. Su
bstituting (8
) i
n
to (21
)
and
comp
l
e
ting th
e squ
a
re, we can rewrite (2
1) as:
TT
(1
)
(
)
+
+
w
Pt
Pt
Q
1
[(
)
(
)
(
)
]
(
)
K
tt
t
t
1T
[(
)
(
)
(
)
]
K
tt
t
1T
()
()
()
tt
t
(22)
Whe
r
e
()
t
and
()
t
are defin
ed b
y
(12) an
d (1
3).
To minimize the right ha
nd
side of (22
)
, the filtering g
a
in
K
(
t
) only need
s to sati
sfy (11)
whi
c
h furthe
r
lead
s to (14
)
.
Rem
a
r
k
2.
It is wo
rthwhil
e
noting that
the gain
an
d varian
ce
matrices of t
he filter
desi
gne
d in
Theo
rem 2
a
r
e affe
cted b
y
the input
u
(
t
), whi
c
h i
s
different from
the stan
dard
Kalman filter [16]. The rea
s
on is that there ar
e rand
om
delay and pa
cket drop
out. So the steady
-
state filter do
es not exist g
ener
ally. In next section, we will study th
e steady-stat
e
prop
erty.
4. Stead
y
-
State
Propert
y
In the
se
ctio
n 3, the li
ne
ar u
nbia
s
e
d
optim
al filter in the finite
hori
z
o
n
h
a
s bee
n
desi
gned. In this
secti
on,
we will
study the ste
ady
-state
property in the infi
nite horizon
for
01
.
Theorem 3.
For sy
stem (
1
)-
(3
), if the matrix
is sta
b
le and the i
n
put
()
ut
is co
nsta
nt,
the sol
u
tion
s
()
qt
and
()
x
t
of equ
ations
(6
) an
d (7)
with any ini
t
ial con
d
ition
s
)
0
(
q
and
(0)
x
will
conve
r
ge to
the uniqu
e p
o
sitive se
mi-definite sol
u
tions
q
and
x
of the followi
ng alge
brai
c
Lyapun
ov eq
uation an
d differen
c
e eq
uat
ion:
TT
T
T
w
qq
B
u
u
B
Q
(23)
And,
x
xB
u
(
2
4
)
Proof.
Let
the matrix
A
whe
r
e
is the K
r
onecker product, from the
st
ability of
, it can be e
a
sily kno
w
n t
hat
()
1
A
, where
()
A
is the spe
c
trum radiu
s
of the matrix
A
.
Also the i
nput
()
ut
is con
s
tant, then
)
(
lim
t
q
q
t
sat
i
sf
ie
s (
23)
[
10]
.
From the
stabili
ty of
and
the con
s
tant i
nput
u
, then
li
m
(
)
t
x
xt
sat
i
sf
ies
(24
)
.
Theorem 4.
For sy
stem (
1
)-
(3
), if the matrix
is sta
b
le and the i
n
put
()
ut
is co
nsta
nt,
the solutio
n
()
Pt
of Equation (14)
with any initial con
d
itio
n
(0
)
0
P
will converges to the uni
que
positive semi-definite sol
u
tion
of the followin
g
algeb
raic Ri
ccati eq
uation:
T
(1
)
{
KH
q
2
(1
)
[
1
(
1
)
]
q
(1
)
q
TT
T
(1
)
}
qH
K
T
()
()
KM
KM
TT
T
T
(1
)
KH
Bu
u
B
H
K
TT
T
T
(1
)
{
KH
x
u
B
B
u
x
TT
(1
)
xu
B
TT
T
(1
)
}
Bux
H
K
TT
T
T
ww
QK
H
Q
H
K
TT
T
T
ww
KH
Q
Q
H
K
T2
T
(1
)
vv
KQ
K
K
Q
K
2T
T
T
T
(1
)
(
)
KH
q
H
K
K
M
3T
T
T
(1
)
(
)
KH
q
H
K
K
M
2T
T
T
T
(1
)
(
)
w
KH
Q
H
K
K
M
2T
T
T
(1
)
(
)
KM
KH
q
H
K
3T
T
T
(1
)
(
)
KM
KH
q
H
K
2T
T
T
(1
)
(
)
w
K
MK
H
Q
H
K
(
2
5
)
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No
. 11, Novemb
er 201
3: 673
7 – 6745
6742
Then we hav
e
li
m
(
)
t
K
Kt
and
li
m
(
)
t
PP
t
. More
over, the stea
dy-state filter:
ˆˆ
(1
)
(
)
(
)
x
tK
M
x
t
()
(
)
(
1
)
IK
H
B
u
t
K
y
t
(26)
Is asymptoti
c
ally stable.
Proof.
Fro
m
Theo
rem 3,
we have
)
(
lim
t
q
q
t
an
d
li
m
(
)
t
x
xt
. Moreover,
the stability
of
means th
at the system
is detecta
ble
and stab
ili
za
ble. Then, fro
m
Kalman filtering the
o
ry
[16], it is known that the
solution
()
Pt
of equation (1
4) with any i
n
itial con
d
ition
(0
)
0
P
conve
r
ge
s to
the unique
positive se
mi
-definite solu
tion
of (25), and
K
M
is a stable
matrix, which implies the
stability of
the steady-state fil
t
er (26
)
.
5. Simulation
Example
Con
s
id
er a time-inva
r
iant
example:
0.8
0
1
0
.6
(1
)
(
)
(
)
(
)
0.9
0
.2
1
0
.5
x
tx
t
u
t
w
t
(
2
7
)
()
[
1
1
]
()
()
zt
x
t
v
t
(28)
()
()
()
(
1
()
)
(
1
(
1
)
)
(
1
)
yt
t
z
t
t
t
z
t
(
2
9
)
In the s
i
mulation, we tak
e
()
s
i
n
(
4
/
)
ut
t
N
,
0.5
and the initi
a
l values
T
ˆ
(0
)
[
3
,
3
]
x
and
2
0
1
.
0
I
P
, where
2
I
is the identity matrix.
We take
N
=100 sa
mplin
g data. Applying
Theo
rem
s
1
and 2, we
ca
n obtain t
he l
i
near u
nbia
s
e
d
optimal filter
ˆ
()
x
t
. The filter is sh
own in
Figure 1. Figure 2 sho
w
s t
he filtering e
r
ror vari
an
ce. It can be se
e
n
that the steady state values
do n
o
t exist
sin
c
e th
e va
ri
ance i
s
affe
ct
ed by t
he
tim
e
-varyin
g
in
p
u
t. To ve
rify the
steady
-sta
te
prop
erty, we
set the inp
u
t
()
0
.
2
ut
. The filter is
sho
w
n in Fi
g
u
re 3. Th
e co
rre
sp
ondi
ng filtering
error va
rian
ce is
sho
w
n i
n
Figure 4. It can
be
see
n
that the stea
dy-state valu
es exi
s
t, whi
c
h is
con
s
i
s
tent to the theory an
alysis. The compa
r
ison
of the steady-st
a
te filter
ing erro
r varian
ce
s in
this pap
er, [6, 7] and [15] for
0.1
1
and
u
(
t
)=0
is sho
w
n in
Figure 5. Fro
m
Figure 5, we can
see th
at our f
ilter ha
s the
better a
c
cura
cy than
[6] si
nce o
u
r filter
has p
o
ssibl
e
one-step
del
ay
but [6] is o
n
ly noise when t
he p
r
e
s
ent p
a
cket is l
o
st.
While
our filter ha
s the l
o
wer a
c
curacy t
han
that in [7] since [7] only ha
s ra
ndom d
e
l
a
y. Compa
r
e
d
with [15] where the
r
e i
s
comp
en
satio
n
of
packet d
r
o
p
o
u
t, our filter h
a
s b
e
tter
accura
cy at
the l
o
we
r a
rrival
rate whil
e
worse a
c
cu
ra
cy at
the highe
r arrival rate than [15].
0
50
100
-10
-5
0
5
10
0
50
100
-10
-5
0
5
10
T
r
ue value and f
ilter
T
r
ue value and f
ilter
T
r
ue value
Filter
T
r
ue value
Filter
Figure 1. Line
ar Un
bia
s
ed
Optimal Filter with
0.
5
and
(
)
sin(
4
/
)
ut
t
N
(a) F
iltering for
the first state compo
nent
(b) Filtering
for the second st
ate
t/step
t/step
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
e-ISSN:
2087
-278X
Linea
r Un
bia
s
ed O
p
tim
a
l
Filter for Di
screte-Tim
e
System
s with O
ne-Step
Ran
dom
… (Jia
n Ding
)
6743
Figure 3. Line
ar Un
bia
s
ed
Optimal Filter with
0.
5
and
u(t)
=
0.2
(a) F
iltering for
the first state compo
nent
(b) F
iltering for
the secon
d
state compo
n
e
n
t
t/step
t/step
0
50
100
-1
0
1
2
3
0
50
100
-1
0
1
2
3
4
T
r
ue value and f
ilter
T
r
ue value and f
ilter
T
r
ue value
Filter
T
r
ue value
Filter
0
50
100
0
0.1
0.2
0.3
0.4
0.5
0
50
100
0
0.1
0.2
0.3
0.4
0.5
Variances
Variances
Figure 4. Filtering Error Va
riances
with
0.
5
and
()
0
.
2
ut
(a) Filtering
error variances
for the first stat
e component
(b) Filtering
error variances
for the secon
d
state compon
e
n
t
t/step
t/step
Figure 2. Filtering Error Va
riances
with
0.
5
and
(
)
sin(
4
/
)
ut
t
N
(a) Filtering
error variances
for the first stat
e component
(b) Filtering
error variances
for the secon
d
state compon
e
n
t
t/step
t/step
0
50
100
0
0.2
0.4
0.6
0.8
0
50
100
0
0.2
0.4
0.6
0.8
1
Variances
Variances
Evaluation Warning : The document was created with Spire.PDF for Python.
e-ISSN: 2
087-278X
TELKOM
NIKA
Vol. 11, No
. 11, Novemb
er 201
3: 673
7 – 6745
6744
6. Conclusio
n
For the discrete
-time li
near
stocha
stic
control
systems
with one-step
rando
m
transmissio
n delay and in
con
s
e
c
utive
packet d
r
opo
ut, we have
derived the
recu
rsive lin
e
a
r
unbia
s
e
d
opti
m
al filter in the linear mi
ni
mum va
rian
ce sen
s
e, which depe
nd
s on
the data arri
val
rate, state
m
ean, stat
e se
con
d
-o
rd
er m
o
ment a
nd
co
ntrol in
put. T
he solution
is given in te
rm
s of
three equ
atio
ns
i
n
cl
uding
one Ri
ccati,
one Ly
apun
o
v
and
a
sim
p
le differen
c
e eq
uation.
The
asymptotic
stability of the
pro
p
o
s
ed
filter
ha
s
bee
n
analy
z
ed. A
suffici
ent
co
ndition fo
r th
e
existen
c
e of the stea
dy-sta
te
filtering ha
s bee
n given.
Ackn
o
w
l
e
dg
ements
This wo
rk wa
s sup
porte
d by
Natu
ral
S
c
ien
c
e
Fou
n
d
a
tion
of Chi
n
a
un
der
G
r
an
t
NSFC-
6117
4139, P
r
og
ram for
New Century
Excellent Ta
l
ents in University un
der
Grant
NCET
-
10-
0133, 11
54
-NCET
-
01, Progra
m
for
High-q
ualif
ied
Talents
und
er G
r
ant Hd
td2010
-03, a
nd
Province Key Laboratory.
Referen
ces
[1]
H Gao,
T
Chen.
H
estimatio
n
for uncerta
in
s
y
stems
w
i
th
limited c
o
mmu
nicati
on ca
paci
t
y
.
IEEE
T
r
ansactio
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