TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol. 12, No. 8, August 201
4, pp. 6164 ~ 6172
DOI: 10.115
9
1
/telkomni
ka.
v
12i8.568
7
6164
Re
cei
v
ed
Jan
uary 27, 201
4
;
Revi
sed Ma
rch 3
1
, 2014;
Acce
pted April 14, 2014
A New Particle Filter Algorithm with Correlative Noises
Qin Lu-fang
1
, Li Wei*
1,2
,
Sun Tao
1,3
, Li
Jun
1,2
, Cao Jie
2
1
Jiangsu Ke
y
Lab
orator
y of L
a
rge En
gi
neer
i
ng Equ
i
pm
ent Detectio
n an
d Contro
l,
Xuz
h
o
u
Institute of
T
e
chnol
og
y,
Xuzh
ou, 22
1
000, Jia
ngs
u, Chin
a
2
Colle
ge of El
ectrical a
nd Informatio
n
Engi
n
eer
i
ng, La
nzho
u Univ
ersit
y
of T
e
chnolog
y,
Lanz
ho
u, 730
0
50, Chi
n
a
3
Colle
ge of Me
chan
ical a
nd e
l
ectrical e
ngi
ne
erin
g
Nan
jin
g Univers
i
t
y
of
aer
on
autics a
n
d
astronautics,
Nanj
in
g, 210
00
0, Jiangs
u, Chi
n
a
A
b
st
r
a
ct
The stand
ard
particl
e filter (
SPF) require
ments system n
o
ise a
nd
me
a
s
ure
m
e
n
t nois
e
must b
e
ind
epe
nd
ent. In ord
e
r to
ov
erco
me th
is l
i
m
it, a
new
ki
nd of c
o
rrel
a
ti
ve no
ise
parti
cle filter (
CN-
PF
)
alg
o
rith
m
is pr
opos
ed. In
this
new
a
l
gor
ith
m
, system stat
e
mo
de
l w
i
th cor
r
elativ
e n
o
ise
i
s
estab
lish
ed,
an
d
the nois
e
rel
a
ted pro
pos
al dis
t
ributio
n
functio
n
character
i
stic
s w
e
re analy
z
e
d
in det
ail. At last, the concre
te
form of
the bes
t
prop
osal distri
butio
n functi
on
is der
ived
b
a
se
d on
the c
o
n
d
iti
on of th
e
mi
ni
mu
m vari
ance
of
importa
nce w
e
ight w
i
th t
he
a
ssumptio
n
of
g
aussi
an
no
ise.
T
heor
etical
a
nalysis
a
n
d
ex
peri
m
e
n
tal
res
u
lts
show
the effectiven
ess of the prop
osed
new
alg
o
rith
m.
Ke
y
w
ords
:
no
nlin
ear syste
m
,
correlativ
e
noi
ses, parti
cle filt
er, propos
al di
stributio
n functi
on
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
Particle filter
(PF) i
s
a n
e
w
ki
nd of no
nlinea
r filterin
g method. T
h
e co
re id
ea o
f
it is to
use
the
weig
ht whi
c
h
give
n a
se
rie
s
of
corre
s
p
ondin
g
information
of ra
ndom
sampling
pa
rticl
e
s
to approxim
a
t
e the syste
m
state of a poste
rio
r
i
pro
bability den
sity function wi
th weighte
d
sum
method
[1-2].
The
sy
stem
state
estim
a
tion is
realized with
the
minimum me
an squ
a
re
e
r
ror
crite
r
ion
impl
ementation.
PF method
n
o
ne
ed th
e
assumptio
n
t
hat the
ch
ar
act
e
ri
st
ic
s of
t
h
e
system
are
linear and
g
auss di
stri
bu
tion co
mpa
r
e
with the
cu
rre
nt wid
e
ly use
d
Extend
ed
Kalman
Filter (EKF) an
d
Un
scented
K
a
lman
Filt
er
(UKF
)
whi
c
h
use lin
ea
r a
pproxim
ation
of
nonlin
ear filtering m
e
thod
s. So, the PF can
ad
apt
to any non-linear n
on-ga
uss system
s in
theory. In re
cent years, wit
h
increa
sin
g
ability
of the comp
uter p
r
o
c
e
ssi
ng, the
PF algorithm
has
been
widely u
s
ed in the fiel
d of target tra
cki
ng [3-8].
Traditional P
F
algorithm
usually
chooses o
ne step sy
stem state
tr
ansition probability
as
the prop
osal distrib
u
tion fu
nction to sa
m
p
le. The
mai
n
goal of the method is for convenie
n
ce
o
f
sampli
ng an
d
calcul
ating. Although this
method is ea
sy to implement; but its filtering p
r
e
c
isi
o
n is
heavily dep
e
ndent
on the
system
mo
d
e
l. Espe
cially
wh
en the
m
odel e
r
ror is large,
due
to
the
lack of th
e la
test ob
se
rvati
on info
rmatio
n corre
c
ti
on
prop
osal di
stribution fu
ncti
on, it is e
a
sy
to
cau
s
e
the sy
stem
m
odel
mismat
ch error
in
crea
se
s
after ma
ny iteration
s
. Eve
n
tually produ
ce
a
so-call
ed "pa
r
ticle
weig
ht degradatio
n probl
em
s" an
d the filter e
s
timation p
r
e
c
isi
on is
gre
a
tly
redu
ce
d, or e
v
en diverge
n
c
e. The
r
efore
,
how to
sele
ct good p
r
op
osal di
strib
u
tion functio
n
is a
core co
ntent in the re
sea
r
ch of the algori
t
hm.
In
r
e
ce
n
t
yea
r
s
,
liter
a
t
u
r
es
[2
-
4
] pr
es
en
t
a se
rie
s
o
f
improved
algorithm
s in o
r
de
r to
solve the p
r
oblem of the
prop
osal di
stribut
io
n fun
c
tion
sele
ctio
n. Although
these im
prov
ed
algorith
m
we
ak the de
gra
dation p
r
oble
m
s of sa
mpli
ng to a ce
rtain extent, and increa
se t
h
e
overall
accu
racy of th
e al
gorithm
in th
e con
c
rete
a
pplication; b
u
t
these
stu
d
ie
s o
n
ly are u
s
eful
unde
r the assumptio
n
of gau
ss
white
noise with
th
e system n
o
ise an
d mea
s
urem
ent noi
se is
indep
ende
nt
of each othe
r. In a real
en
vironm
e
n
t, the inde
pend
en
t of system a
nd mea
s
u
r
em
ent
noise for ea
ch othe
r is very difficult to sa
tisfy the conditio
n
s becau
se of
the discre
zation
pro
c
e
ssi
ng
of mea
s
u
r
ing
i
n
formatio
n [9
]. Therefo
r
e, i
t
is m
eanin
g
ful in th
eoretical an
d p
r
a
c
tical
in pra
c
tice to
develop the n
o
ise related
case
s PF algo
rithm.
Based
on thi
s
, this
pap
er pro
p
o
s
ed
a
new ki
nd of
Co
rrel
a
tive Noi
s
e
s
Parti
c
le Filter
(CNPF).
The
remai
nde
r of
this p
ape
r i
s
orga
nized
as
follows. Se
ction 2
give
s th
e ba
ckg
r
oun
d
of
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TELKOM
NIKA
ISSN:
2302-4
046
A New Parti
c
l
e
Filter Algori
t
hm
with Correlative
Noi
s
e
s
(QI
N
Lu-fa
n
g
)
6165
the pro
b
lem.
The p
r
op
ose
d
ne
w algo
rit
h
m with
co
rrelative noises is de
rived in
detail in sect
ion
3. Experimen
tal results a
nd analy
s
is
are repo
rted
in se
ction 4
.
We co
ncl
u
de this p
ape
r in
se
ction 5.
2. Backg
rou
nd of the Pr
oblem
2.1. The Sy
st
em Model
For filtering p
r
oble
m
of non
linear
system
s,
usu
a
lly ado
pt nonline
a
r d
i
screte
syste
m
s a
s
s
h
own in the
following [10]:
1
(,
)
()
kk
k
k
k
k
kk
k
k
x
fx
u
w
yh
x
v
(
1)
Whe
r
e,
k
x
and
k
y
denote the sy
stem state an
d measur
e va
lue at k. Dyna
mic functio
n
()
f
and
()
h
determin
e
the overall dyn
a
mic mo
del o
f
the system with the initial state
0
x
of the
system.
k
u
is the
co
ntrol
input ve
ctor
of the sy
stem
. Whe
r
e
k
w
and
k
v
denote
the
system p
r
o
c
e
s
s a
nd
measurement
noi
se
re
spe
c
tively.
k
denot
es th
e in
put
matrix of p
r
o
c
e
s
s noi
se.
This
arti
cle
mainly aime
d
at the
re
sea
r
ch
of correlati
v
e noi
se
filteri
ng meth
od,
so he
re fi
rst gi
ve the follo
wi
ng
two hypothe
ses.
Hypothe
si
s 1: Noise sati
sf
y the following
feature
s
:
()
,
(
,
)
()
,
(
,
)
(,)
T
kk
k
j
k
k
j
T
kk
k
j
k
k
j
T
kj
k
k
j
Ew
q
C
o
v
w
w
Q
Ev
r
C
o
v
v
v
R
Co
v
w
v
S
(
2)
Whe
r
e
k
Q
and
k
R
rep
r
e
s
ent t
he sy
stem
pro
c
e
s
s an
d
mea
s
ureme
n
t noise
co
varian
ce
r
e
spec
tively
，
and he
re
kj
meets the followi
ng value
:
1,
0,
ij
ij
ij
(
3)
Hypothe
si
s 2: The sy
stem
initial state
0
x
is unrelated
with
k
w
and
k
v
, an
d meets the
following features
.
00
0
0
00
00
ˆ
()
ˆˆ
(
)
|
[
()
()
]
|
T
xE
x
PC
o
v
x
E
x
x
x
x
(
4)
T
h
is
pa
p
e
r
de
sc
r
i
be
th
e
no
n
lin
ea
r
mode
l o
f
noise filtering p
r
obl
e
m
of related
ca
se
s in
the premi
s
e o
f
the above assumption
s o
f
Equation (1).
2.2. The Sta
ndard Par
t
icle Filter Algorithm
In view of th
e
system
state
equatio
n d
e
scri
b
ed
by type (1
), we
can
sum
m
ari
z
e t
he SPF
as "fore
c
a
s
t" and "upd
ate" two step
s [2].
0:
1
,
N
ii
kk
i
x
denote the
sampling p
a
rti
c
le
s coll
ectio
n
of
the system
posterior
probability densi
ty
0:
(|
)
kk
p
xY
, where,
0:
1
N
i
k
i
x
denote th
e
sampli
ng
particl
es colle
ction
whi
c
h
g
i
ven to the
correspon
di
ng
wei
ght info
rmation, an
d t
he
weig
hts
m
eet
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046
TELKOM
NI
KA
Vol. 12, No. 8, August 2014: 616
4 –
6172
6166
1
1
N
i
k
i
,
0:
0
1
{
,
,
...,
}
kk
x
xx
x
just
den
ote
the
colle
ctio
n of th
e
sy
stem
state at
k. Base
d o
n
the idea of SPF, when we get measure
m
ent informat
ion
12
{
,
,
...,
}
kk
Yy
y
y
, we have:
0:
0:
0:
1
(|
)
(
)
N
ii
kk
k
k
k
i
p
xY
x
x
(
5
)
Whe
r
e:
1
1
1
(|
)
(
|
)
(|
,
)
ii
i
ii
kk
kk
kk
ii
kk
k
p
yx
p
x
x
qx
x
z
(
6)
(|
)
k
qx
denote
s
the p
r
opo
sal di
stri
bution fun
c
tio
n
, and us
uall
y
, Equation (7) is ta
ke a
s
the prio
r
distrib
u
tion of
the propo
sal
distrib
u
tion fu
nction,
11
(|
,
)
(|
)
ii
ii
kk
k
k
k
qx
x
z
px
x
(
7)
Bring (7
) into
(6)
，
we have
：
1
(|
)
ii
i
kk
k
k
p
yx
(
8)
3. Particle Filter Algorith
m
w
i
th
Corre
lativ
e
Noises
3.1. Situatio
n Analy
s
is o
f
Corr
elativ
e
Noises
In the research of algorithm
, we usu
a
lly
consi
dered the
obse
r
vation
noise as a
ddi
tive
noise. For the
sake of simpl
i
city, here we
rewrite the dy
namic m
odel
of (1) a
s
:
1
(,
)
()
kk
k
ll
l
x
fx
v
yh
x
e
(
9
)
The ob
se
rved
quantity and pro
c
e
ss
con
d
i
tion betwe
en
the joint post
e
rio
r
pro
babili
ty
den
sity
(|
)
kk
p
XY
can be rep
r
e
s
e
n
ted as:
11
1
1
1
(|
)
(
|
,
)
(
|
,
)
(
|
)
kk
k
k
k
k
k
k
k
k
p
XY
p
y
X
Y
p
x
X
Y
p
X
Y
(
10
)
For th
e
stan
dard
of
Markov mod
e
ls, i
n
a
se
pa
rate
process noi
se and
me
asurem
ent
noise w
e
have:
(,
)
(
)
(
)
ij
i
j
p
v
e
pv
pe
(
11
)
11
1
(|
,
)
(|
)
kk
k
k
k
p
xX
Y
p
x
x
(
12
)
1
(|
,
)
(|
)
kk
k
k
k
p
yX
Y
p
y
x
(
13
)
Where, th
e dynami
c
system of (9) can
be
sh
o
w
n in
Figu
re
1 with th
e
evolution of
the
grap
hics.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A New Parti
c
l
e
Filter Algori
t
hm
with Correlative
Noi
s
e
s
(QI
N
Lu-fa
n
g
)
6167
Figure 1. State Space M
o
d
e
l
From Fi
gure
1, we
can
get the re
lationship be
tween th
e
pro
c
e
ss
noi
se and
measurement
noise which can be d
enot
ed by Figure
2 [11]. As we can se
e fro
m
Figure 2, the
correl
ation m
a
in p
e
rfo
r
ma
nce
on
the ti
me a
s
so
ciation. Th
e mai
n
pu
rpo
s
e
of
co
nsi
deri
ng
th
e
correl
ation of
1
k
v
and
1
k
e
is to find the noise ap
propri
a
te deco
m
po
sition form o
f
joint
probability density function
(,
)
ij
p
ve
.
Figure 2. Pro
c
e
ss a
nd Me
asu
r
em
ent Noise
Correl
ation Dia
g
ra
m
Assu
ming
tha
t
the noi
se
ve
ctor seque
nce
11
(,
)
T
kk
ve
is ind
epe
nd
ent, acco
rdin
g to the
relation
shi
p
o
f
the Figure 2
sho
w
s, the
r
e
are:
11
1
1
(|
,
)
(|
,
)
kk
k
k
k
k
p
xX
Y
p
x
x
y
(
14
)
1
(|
,
)
(|
)
kk
k
k
k
p
yX
Y
p
y
x
(
15
)
Then the
proce
s
s and
measurement
noise j
o
int
prob
ability d
ensity fun
c
tio
n
ca
n be
decompo
se
d as follo
ws:
11
1
1
1
(,
)
(
|
)
()
kk
k
k
k
p
ve
p
v
e
p
e
(
16
)
3.2. Deriv
e
o
f
Optimal Proposal Dis
t
ri
bution Fun
ction
w
h
e
n
No
ise Related
For the co
nvenien
ce of formula de
rivati
on,
here de
scrib
ed the sta
t
e spa
c
e mod
e
l of (9)
as (1
7).
1
()
()
kk
k
k
k
kk
k
k
x
fx
G
v
yh
x
e
(
17
)
Whe
r
e,
k
v
and
k
e
is correl
ate
d
. Acco
rdin
g
to Figure 2,
we ca
n re
prese
n
t the de
pend
en
ce
relation of noi
se
s as
1
(,
|
)
kk
k
p
yx
x
furthe
r, then:
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TELKOM
NI
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Vol. 12, No. 8, August 2014: 616
4 –
6172
6168
11
(
,
|)
(
|
)
(
|)
kk
k
k
k
k
k
p
yx
x
p
x
x
p
y
x
(
18
)
In the situatio
n of giving
k
x
,
k
y
and
1
k
x
are ind
epen
dent. In the SPF, the
form of a
prop
osal di
stribution fun
c
ti
on is
1
(|
,
)
kk
k
qx
X
Y
, and we ca
n get (19
)
acco
rdi
ng t
o
the de
pend
ent
variable
1
k
Y
and
k
x
[12].
11
(|
,
)
(|
,
)
kk
k
k
k
k
qx
X
Y
qx
x
y
(
19
)
Acco
rdi
ng the
interdep
end
e
n
ce of
1
k
y
and
k
x
,
we can get (2
0) as follo
w:
11
1
(|
,
)
(|
,
,
)
kk
k
k
k
k
k
qx
X
Y
qx
x
y
y
(
20
)
There are
：
Theorem 1
：
Whe
n
noi
se related PF the
optimal pro
p
o
sal di
strib
u
tion functio
n
is
：
11
11
11
(|
)
(
|
,
)
(|
,,
)
(|
,
)
kk
k
k
k
kk
k
k
kk
k
py
x
p
x
x
y
qx
x
y
y
py
y
x
(
21
)
Proof:
Acco
rding
to the
rule
of b
a
yesia
n
infe
ren
c
e, the
advi
c
e
of the p
o
steri
o
r
distrib
u
tion can be di
stribu
tion
function
can b
e
expre
s
sed a
s
:
11
1
1
(|
,
,
)
(
|
,
,
)
kk
k
k
kk
k
k
qx
x
y
y
p
x
x
y
y
=
11
11
(,
|
,
)
(|
,
)
kk
k
k
kk
k
p
xy
y
x
py
y
x
=
11
11
(|
)
(
|
,
)
(|
,
)
kk
kk
k
kk
k
py
x
p
x
x
y
py
y
x
(
22
)
Theo
rem 1 is
proven.
3.3. Optimal Proposal Dis
t
ribution Fu
nction of
Ga
ussian Nois
e
Usually, the system and m
easure
m
ent
noise
meet th
e assumptio
n
of gaussian,
and the
noise ada
pte
d
by SPF meets the follow
need
s ju
st as (23).
01
|
0
1
|
0
ˆ
~(
,
)
x
Nx
P
(
23
)
0
0,
k
kk
T
kk
k
v
QS
N
SR
e
(
24
)
And can al
so
be expre
s
sed
as
：
1
()
|,
()
T
kk
kk
k
k
k
k
TT
kk
kk
k
xf
x
GQ
G
G
S
px
N
yh
x
SG
R
(
25
)
Acco
rdi
ng to (25
)
, we can
give the rule
of judging the
noise
correl
a
t
ion as:
0,
0,
k
k
Sn
o
Sy
e
s
(26)
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6169
Theorem 2
：
The gau
ssian
model optim
al prop
osal di
stributio
n fun
c
tion of type (25) can
expres
s
as
follows
:
11
11
1
1
1
1
11
1
1
1
1
1
1
(
|
,
,
)
(
(
)
(
(
)),
(
)
)
(
(
)
,
)
TT
k
k
k
k
k
k
kk
k
k
k
k
kk
k
k
k
k
qx
x
y
y
N
f
x
G
S
R
y
h
x
G
Q
S
R
S
G
N
h
x
R
(
27
)
Proof:
A
c
cording to the
rule of (21),
we can de
comp
osed th
e optimal p
r
opo
sal
distrib
u
tion fu
nction
into two facto
r
s a
s
(27
)
, an
d
(|
)
kk
p
yx
ca
n be
got by t
he me
asure
m
ent
model of (9
), and
11
(|
,
)
kk
k
p
xx
y
can be g
o
t by the following lem
m
a
1.
Lemma 1
：
A
s
sume that th
e vector
X
and
Y
is joint gau
ssi
an distri
butio
n, and there
are
(28) as
follows
:
~,
,
xx
xx
xy
T
xy
yy
yy
uu
PP
X
NN
P
PP
uu
Y
(
28
)
Whe
n
get th
e mea
s
urem
ent
Yy
value, then the condit
i
onal di
strib
u
tion ca
n be
e
x
press a
s
the followin
g
form
s of gau
ssian di
strib
u
tion.
11
(|
)
~
(
(
)
,
)
x
xy
yy
y
x
x
x
y
y
y
y
x
X
Yy
N
u
P
P
y
u
P
P
P
P
(
29
)
Let
1
|
kk
Xx
x
，
11
|
kk
Yy
x
, and combine
the j
o
int dist
ributi
on of X a
nd
Y as
(28
)
, th
ere
are:
11
(|
,
)
kk
k
px
x
y
11
11
1
1
1
1
1
1
1
1
1
1
((
)
(
(
)
)
,
(
)
)
TT
k
k
kk
k
k
k
k
kk
k
k
Nf
x
G
S
R
y
h
x
G
Q
S
R
S
G
(
30
)
Then:
11
(|
,
)
kk
k
qx
x
y
11
1
1
11
1
1
1
1
11
1
1
((
)
(
(
)
)
,
(
)
)
TT
k
k
kk
k
k
k
k
kk
k
k
Nf
x
G
S
R
y
h
x
G
Q
S
R
S
G
(
31
)
4. Simulation Anal
y
s
is
In this pape
r, we adapt t
he followi
ng
model ju
st a
s
literatu
r
e [10] to simul
a
te the
perfo
rman
ce
of the new m
e
thod. The m
odel ju
st as follows:
1,
2,
3
,
1,
1
2
,
12
,
1
1
,
3
,
0.
5
3,
1
2,
3
,
3
1,
3s
i
n
(
)
1
1
1
0.
2
(
)
k
kk
kk
kk
k
k
k
x
k
kk
xx
kk
k
xx
xx
x
x
Ex
x
zx
E
v
x
(
32
)
We can
see
(32) h
a
s hi
gh
nonlin
ear. He
re
k
and
k
v
are
all
white gau
ssi
an noi
se, an
d
t
he st
at
ist
i
c
a
l cha
r
a
c
t
e
ri
st
ic
s
of them me
ets the followi
ng:
0.
2
,
0.
04
,
0
.
3
,
0
.
0
9
kk
k
k
qQ
r
R
(
33
)
The initial parameter va
lue
of state is se
t to:
0
[
0
.
7
,1
,1
]
T
x
(
34
)
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TELKOM
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Vol. 12, No. 8, August 2014: 616
4 –
6172
6170
00
ˆ
[
0
.
7
,1
,1
]
,
T
x
PI
(
35
)
4.1. Analy
s
is
of Independ
ent Noise
We
can
kno
w
that the un
correl
ated of
k
and
k
v
can de
notes a
s
0
k
S
. In
order to
comp
are the
perfo
rman
ce
of this
pap
er
new metho
d
,
traditional
PF
and
the
ne
w method
of
CN
-
PF was u
s
e
d
to estimate the state of
1
x
resp
ectively in
experime
n
ts.
The estimat
e
results ju
st
as Figu
re 3,
Figure 4 and
Figure 5.
The exp
e
rim
e
nt re
sults de
n
o
te that the
tradi
tional PF
and
CN - PF
can
tra
c
k the
system
state effe
ctively whe
n
the
noi
se i
s
in
d
epen
dently. Both metho
d
s
to
kee
p
th
e goo
d tracking
pre
c
isi
on an
d
less
error. T
he filtering p
r
eci
s
ion
of the
two method
s are almo
st the sa
me in the
ca
se of noi
se
are ind
epe
n
dent of ea
ch
other,
which can b
e
seen i
n
figure 5.
Ca
n be un
derst
ood
as in the case of noise in
d
epen
dent, this method
i
s
a
pproxim
ate to the traditiona
l PF algorith
m
.
Figure 3. State Estimation
Curve of
1
x
Figure 4. The
Trackin
g
Error Cu
rve of
1
x
Figure 5. Mean Squa
re Error
Curve of
1
x
4.2. Noise Related Situ
ation Simulation Analy
s
is
We can kn
o
w
that
the correlated of
k
and
k
v
can de
notes a
s
0
k
S
. In
order to
comp
are the
perfo
rman
ce
of this p
ape
r
new
metho
d
, the value i
s
0.
1
k
S
The es
timate
res
u
lt
s
just a
s
Figu
re 6, Fi
gure 7
and
Figu
re
8. We
can
see that th
e
system mo
del
erro
r in
crea
se
s
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TELKOM
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ISSN:
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046
A New Parti
c
l
e
Filter Algori
t
hm
with Correlative
Noi
s
e
s
(QI
N
Lu-fa
n
g
)
6171
grad
ually wh
en
the
noi
se
correl
ation which
can be
g
e
t from
Figu
re 6
and
Figu
re 7. T
he tracking
errors of
sta
t
e
3
x
in
cre
a
se
s g
r
ad
ually,
but this pa
p
e
r m
e
thod
can
kee
p
a
g
ood trackin
g
perfo
rman
ce.
The me
an
sq
uare
erro
r of the traditio
nal
PF al
go
rithm
with a
cum
u
lative as the i
n
crea
se
of time, but th
is pa
pe
r ha
s
maintaine
d
a
good t
r
a
c
k effect a
nd the
root mea
n
squ
a
re
error curv
e
grad
ually co
nverge to ze
ro, whi
c
h ca
n be se
en in
the Figure
8. The re
sult
s sh
ow that
the
prop
osed me
thod name
d
CN - PF ha
ve fast c
onv
erge
nce rate,
and high p
r
eci
s
ion, stron
g
stability, whe
n
the noi
se i
s
related,
and
fully
proves the fea
s
ibility and
effe
ctiveness of the
n
e
w
method.
Figure 6. State Estimation
Curve of
3
x
Figure 7. The
Trackin
g
Error Cu
rve of
3
x
Figure 8. Mean Squa
re Error
Curve of
3
x
5. Conclusio
n
Aimed at
the
limitation
of the tradition
al
PF
algo
rithm u
nde
r th
e conditio
n
of noi
se
related, thi
s
pape
r p
r
op
osed a
noi
se
related p
a
rti
c
l
e
filter alg
o
rit
h
m, mainly d
o
the follo
wi
ng
s
e
ve
ra
l as
pec
ts
: 1
)
Th
e system state model of
noi
se relate
d situation
i
s
est
ablished, and
the
nature
of th
e
pro
p
o
s
ed
di
stributio
n fun
c
tion
wh
en
th
e noi
se
corre
l
ation i
s
d
edu
ced
in
detail;
2
)
Gives th
e de
comp
ositio
n
expre
ssi
on
of joint p
r
ob
abil
i
ty density in t
he related
noi
se
ca
se; 3
)
Th
e
optimal prop
osal di
strib
u
tion functio
n
in t
he noise
related
ca
ses was d
e
d
u
ce
d und
er
the
con
d
ition
s
of importa
nce weig
ht minim
u
m va
rian
ce
signifi
can
c
e
based on th
e
gau
ssia
n
no
ise
backg
rou
nd,
and the
effe
ctivene
ss of
the ne
w al
go
rithm i
s
verifi
ed by
com
p
uter
simulati
on.
Becau
s
e
of t
he p
r
o
posed
method
is a
expan
sion
of
the
scope
of
the tradition
al PF
algo
rith
m,
therefo
r
e, it i
s
e
a
sy to
co
mbine th
e o
p
t
imizati
on
of the
curre
n
t ex
isting
appli
c
a
t
ion a
c
cura
cy
of
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Vol. 12, No. 8, August 2014: 616
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6172
6172
the algorithm
in different fields. In the next step of
study, the optimal filtering pro
b
lem in the case
of unkn
o
wn system noise statistical c
haracteri
stics will
be further
studied.
Ackn
o
w
l
e
dg
ements
This work was suppo
rte
d
by the key project
s
Ji
ang
su Key Labo
rato
ry of Large
Enginee
ring
Equipme
n
t Detection a
nd
Control
(
JSKLEDC201
202
)
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il Gordo
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