TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5305 ~ 53
1
5
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.529
0
5305
Re
cei
v
ed
De
cem
ber 5, 20
13; Re
vised Janua
ry 1
8
, 20
14; Accepted
February 12,
2014
Anti-interference Tracking Methods of Maneuvering
Target for Struc
t
ure Random Jump Systems
Jianfe
ng Wu
*, Shucai Hu
ang, Xiao
y
a
n Wu
, Yu Zhong, Hong
xia Kang, Che
ngjing Li
Air and Miss
ile
Defens
e Col
l
e
ge, Air F
o
rce Engi
neer
in
g Uni
v
ersit
y
1 # Cha
n
g
l
e E
a
st Road,
Xi’
a
n
,
71005
1, Chi
n
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
w
j
f1
33
1@1
6
3
.com
A
b
st
r
a
ct
In this article, a study of anti
-
interfere
n
ce tra
ckin
g
method
of mane
uver
i
ng target for n
onli
n
e
a
r
structure rand
o
m
ju
mp systems (SRJSs) w
i
th
rando
m in
terf
e
r
ence w
a
s inve
stigated. Invie
w
of the rando
m
interference
pr
oblem
of the tracki
ng system
, the nonlinear
Gaussian a
ppr
oxim
ation filter
ing (NGAF) was
app
lie
d to achi
eve anti-i
n
terfe
r
ence tr
acki
ng
of man
euv
erin
g target in obs
ervatio
n
nois
e
s
environ
ment
w
i
th
the pip i
n
terfer
ence si
gna
l. Inview
of
the defects of the NG
AF
algorith
m
, bootstrap f
ilter
i
ng (BSF
) algor
i
t
hm
of SRJSs w
a
s
app
li
ed to
av
oid th
e l
o
ss of
infor
m
ati
on c
ause
d
by
ne
gl
ecting t
he
hig
her-or
der ter
m
s. A
me
an
ingfu
l
exa
m
p
l
e of rad
a
r/IR dual-
m
od
e compo
und se
ek
er is presente
d
to illust
rate the effectiveness
of
the a
u
thors
’
me
thods, th
e p
e
rformanc
es
of
e
x
tende
d K
a
l
m
a
n
filteri
n
g
(E
K
F
), NGAF
and
BSF
in
ter
m
s
of
stability,
accur
a
cy a
nd c
o
mp
utation
a
l c
o
mp
lexity w
e
re
co
mp
are
d
. The
purp
o
se
of thi
s
pa
per w
a
s
t
o
de
mo
nstrate th
e effectiven
ess
of applyi
ng the
NGAF
and
BSF
on anti-interf
erenc
e target trackin
g
prob
le
ms
of SRJSs, w
h
ic
h in
the
past th
e factors of r
a
n
d
o
m
n
e
ss
a
nd s
t
ructure u
n
cert
ainti
e
s ch
aract
e
ristics h
ad
be
e
n
rarely c
onsi
der
ed for th
e studi
es of
man
euv
e
r
ing tar
get
trac
king,
mostly th
e stru
ctures a
n
d
the
para
m
ete
r
s
of the trackin
g
system w
e
re i
n
varia
n
t and c
e
r
t
ainty,
and
ha
d
typically
be
en
solve
d
by Ka
l
m
a
n
or
extend
ed
Kal
m
a
n
filters.
Ke
y
w
ords
:
structure rand
om j
u
mp sys
tems (SRJSs)
,
target tacking, anti-
i
n
terferenc
e, nonl
i
near
Gaussia
n
ap
pr
oxi
m
ate filter
in
g
(NGAF
)
, bootstrap filterin
g (BSF
)
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
Structu
r
e ra
n
dom jump
sy
stem
s (SRJS
s
) exte
n
s
ivel
y exist in prac
tice, whic
h involv
e
both ra
ndom
ness a
nd st
ructure un
ce
rt
ainties
cha
r
a
c
teri
stics [1]. SRJS
s hav
e been
used
to
model th
e
system
with v
a
riabl
e p
a
ra
meters a
nd
stru
ctures ca
use
d
by
su
d
den
enviro
n
m
en
t
cha
nge
s, wo
rking mo
de
s switch, interfe
r
ence ex
ists, faults o
c
curre
d
in comp
one
nts and sudd
en
target m
o
tori
ze
s et
c. in m
any field
s
su
ch
as targ
et trackin
g
[2, 3]
, pro
c
e
s
s mo
nitoring
an
d f
ault
detectio
n
[4]. However, in
the past, the
studie
s
of
m
aneuve
r
ing t
a
rget tra
c
kin
g
rarely co
nsider
factors above
mentions, m
o
stly t
he stru
cture
s
an
d the param
eter
s of the tracki
ng system a
r
e
invariant and
certai
nty.
Moreover,
info
rm
ation
p
r
o
c
e
s
sing p
r
obl
em i
n
inthe
r
feren
c
e
enviro
n
me
nt
also i
s
one of
the hot spots in the field of rando
mne
s
s system [5].
For th
e ma
n
euverin
g targ
et trackin
g
p
r
oblem, the
Kalman filter (KF) meth
od i
s
o
ne of
the most po
p
u
lar tool
s u
s
e
d
to estimate
states fro
m
system
s [6, 7]. It may be applied o
n
line
a
r
dynamic sy
stems i
n
the
p
r
esen
ce
of G
aussia
n
white noi
se, a
n
d
it provid
es
an el
egant
a
nd
statistically optimal solution by minimi
zing t
he m
e
a
n
-squa
re
d e
s
timation erro
r. Ho
weve
r,
in
pra
c
t
i
ce,
all
sy
st
em
s in n
a
t
u
re a
r
e in
f
a
ct
nonlin
ea
r, esp
e
cially
in mane
rving
target tracki
ng
system, su
ch
that linear estimation techniqu
es
may
not be used
to provide optimal solutio
n
s.
For thi
s
rea
s
on, subo
ptimal techniq
ues may be
ap
p
lied to
handl
e the
nonlin
e
a
rities, it
s m
a
in
idea i
s
linea
r
filtering for
n
online
a
r
syst
em. Such
te
chniqu
es in
clu
de the exten
ded Kalm
an f
ilter
(EKF), it is a
popul
ar
exten
s
ion
of the K
F
and
is com
m
only u
s
ed
i
n
targ
et tra
c
ki
ng [8, 9]. It u
s
e
s
partial de
rivatives of the nonlinea
rities in
the
state dynamic an
d me
asu
r
em
ent model
s, such that
lineari
z
e
d
ap
proximatio
ns
are obt
ained
and then u
s
e
d
in the estim
a
tion pro
c
e
ss [10]. But due to
the informatio
n loss in the l
i
neari
z
atio
n process, the perform
an
ce
s
of EKF algorithm are difficult
to satisfy the
actual
req
u
ire
m
ents in
som
e
appli
c
at
ion
s
, and m
o
re i
m
porta
nt is that it difficult to
c
o
pe
w
i
th
r
and
o
m
in
te
r
f
er
en
c
e
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5305 – 53
15
5306
In actual
pro
c
e
ss
of man
euverin
g targ
et tr
ackin
g
, the target di
scha
rge
s
th
e
rand
om
in
te
r
f
e
r
e
n
c
e
(
s
uc
h as
pho
to
e
l
ec
tr
ic inte
r
f
e
r
en
ce, a
c
tive or pa
ssive inte
rference et
c), t
h
e
informatio
n o
f
target tracki
ng brea
ks off
or re
su
m
e
s,
the detectin
g
sen
s
o
r
s switch ea
ch oth
e
r
and so on, all above men
t
ion situation
s
ca
n lead
to
sharp jump
of t
he syste
m
stru
cture
and
pare
m
eters at
ra
ndom
times, whi
c
h
make
s
great
ch
allen
ge i
n
solving th
e target tra
c
king
probl
em
s. In t
h
is
case, the
perfo
rmances of cl
assi
cal
Ka
lman filter
will declin
e
rapidly, and ev
en
has th
e ph
e
nomen
on of
diverge
n
ce. Ho
wever,
n
online
a
r G
a
u
ssi
an a
pproximation filteri
ng
(NGAF) al
gorithm of SR
JSs
has satisf
actory per
formances in accuracy
and
stability, and the
ca
culatio
n
is much le
ss t
han the
opti
m
al filter
ing
algorith
m
of
the discrete-t
i
me SRJSs [
11].
Therefore, NGAF algo
rithm is a kind al
gorithm
of mo
re suita
b
le for practi
cal en
g
i
neeri
ng.
Although th
e
rand
om inte
rf
eren
ce
probl
ems
are
solv
ed by the
ant
i-interfe
r
en
ce
trackin
g
algorith
m
ba
sed
on NGA
F
algo
rithm, but it has
th
e defe
c
ts in
solving the
system analy
s
is
probl
em
s [11, 12]. Since the unco
nditional posterior probability den
sity functions (PDFs) of the
system stat
e
s
are the
wei
ghted
s
u
m of
t
he st
ruct
u
r
e
jump v
e
ct
ors, so even th
e sub
s
ystem
is
linear
und
er each st
ru
cture
state,
b
u
t its initial
states
and t
he noi
se
distribution
s
a
r
e
the
Gau
ssi
an di
stribution, the
uncondition
al and
co
ndi
tional PDF
s
of the syste
m
state
s
are
no
longe
r the G
aussia
n
distri
bution.
If the Gaussia
n
di
stributio
n is
app
lie
d to ap
proximate, it will
inevitably lead to the decli
ne of the performan
ce
s of
the filter. Inview of
the de
fects existin
g
in
the NGAF
m
e
thod, Bootst
rap filt
eri
ng (BSF) is a
ne
w no
nlinea
r fi
ltering m
e
tho
d
ba
sed
on S
m
ith
sampli
ng the
o
rem, which i
s
not st
rict li
mited by
the system initial
state
and
noi
se di
strib
u
tio
n
for
S
R
JS
s.
Inspired by th
e above m
o
tivations, in thi
s
pa
per, thre
e filters (t
he
comm
only u
s
ed EKF
,
NGAF, a
n
d
the
relatively
new BSF) are ap
plied
to
deal
with
ant
i-interfe
r
en
ce
targ
et tra
c
ki
ng
probl
em of radar/IR
dual
-mode
com
p
o
und
see
k
e
r
[13], and the
perfo
rman
ces in te
rm
s
of
stability, accuracy and co
m
putation are compared.
The
org
ani
za
tion of thi
s
p
aper is a
s
foll
ows.
In Se
cti
on 2,
we
de
scrib
e
NGAF
algorith
m
to deal
with
random
inte
rferen
ce
p
r
obl
em. In Se
ctio
n 3,
we
prop
ose
BSF al
go
rithm to
deal
with
the loss of information
ca
use
d
by negl
ecting the hi
gher-o
rde
r
terms
in NGAF
algorithm. T
h
e
obje
c
tive of
Section
4 i
s
to demo
n
st
ra
te the
effe
ctiveness
of the metho
d
s with a si
mulati
on
example. Fin
a
lly, the concl
u
sio
n
s a
r
e d
r
awn in Se
ctio
n 5.
2. Nonlinear
Gaussi
an Ap
proximate Fi
ltering
2.1. Gaussia
n
Appro
x
imate Filtering
For the
ca
se of linear
system with
Gau
ssi
an wh
ite noise, th
e state equ
a
t
ion an
d
observation e
quation of the
target motion
may be described
sep
a
rately as follows:
(1
)
(
,
)
(
)
(
,
)
(1
,
)
(
1
)
(
,
1
)
(
1)
(
,
1)
ks
k
k
s
k
s
SM
ks
k
k
s
k
XF
X
w
ZH
X
v
(1)
Whe
r
e
()
k
X
is the
state vecto
r
and
()
k
Z
is the correpo
nding
observation v
e
ctor
at time
k
,
(,
)
sk
w
and
(,
)
sk
v
denote pro
c
e
ss n
o
ise and ob
se
rvation noi
se,
s
is the structu
r
e label of the
system and
d
e
scrib
ed by
condi
tion
al
Ma
rkov ch
ain wi
th
M
finite s
t
ates
,
(,
)
sk
F
and
(,
)
sk
H
are kno
w
n fu
nction mat
r
ix.
Assu
me that
all noi
ses
are
zero-m
ean
G
aussi
a
n
ra
nd
om se
que
nce
s
an
d ind
epe
ndent of
each othe
r. In additio
n
, th
e initial value
s
of the ta
rg
e
t
state al
so o
bey Gau
s
sia
n
dist
ributio
n, and
are ind
epe
nd
ent with all no
ise
s
, namely:
(
,
)
[
(
,
)
(
,)
,
(
,)
]
(
,
)
[
(
,
)
(
,)
,
(
,)
]
[,
]
s
ks
k
s
k
s
k
sk
s
k
sk
s
k
w
v
(s
)
(
s
)
(s
)
(
s
)
00
0
0
wN
w
vN
v
XN
X
m
(2)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Anti-interfe
re
nce T
r
a
cki
ng
Method
s of Maneu
ve
ring T
a
rget for Stru
cture
…
(Jianf
eng Wu)
5307
Whe
r
e
(s
)
0
X
is the target i
n
itial state
of
s
st
r
u
ct
ur
e,
()
(
)
tE
t
mX
,
()
()
(
)
()
(
)
T
tE
t
t
t
t
Xm
Xm
, the func
tion
,
N
α
denote
s
th
at the
rand
o
m
vectors obey
-mea
n Gau
ssian distri
butio
n with a varia
n
ce of
α
.
In order to
si
mplicity for writing, th
e ti
me
k
is
omitted, thus
assume that
(1
)
ss
k
,
()
rs
k
,
1
(1
)
k
k
zZ
,
[(
0
)
,
(
1
)
,
,
(
1
)
,
(
)
]
k
kk
ZZ
Z
Z
Z
.
Acco
rdi
ng
to Equation (1)~(2),
the
co
ndi
tional tran
siti
on PDF
[(
1
)
,
]
f
kr
XX
of the
target motion
state
(1
)
k
X
is defin
ed as follo
ws:
[(
1
)
,
]
[(
1
)
(
)
,
(
)
]
[(
1
)
(
,
)(
,
)
(
,
)
,
(
,
)
]
fk
r
f
k
k
s
k
k
r
k
k
rk
r
k
rk
w
XX
XX
NX
F
X
(3)
The
con
d
ition
a
l tran
sition P
D
F
1
[(
1
)
,
]
k
f
ks
zX
of the ob
servatio
n ve
ctor
(1
)
k
Z
is
defined a
s
fol
l
ows:
11
[
(
1
)
,
]
[
(
,1
)
(
1
,
)
(
,1
)
,
(
,
1
)
]
kk
f
k
s
s
k
k
sk
sk
sk
v
zX
N
z
H
m
(4)
Assu
me that
the proces
s
of stru
cture j
u
mp h
a
s
not
hing to
d
o
wi
th the sy
ste
m
state, it
only depe
nd
s on the forme
r
stru
ctu
r
e st
ate of t
he system, and th
us the tra
n
siti
on pro
bability
o
f
stru
cture stat
e has the foll
owin
g relatio
n
establi
s
h
e
d
.
()
[,
1
(
)
,
,
]
[,
1
,
]
(
1
,
)
sr
qs
k
k
r
k
qs
k
r
k
q
k
k
X
(5)
Since the
ra
ndom jum
p
a
nd switchi
ng
of the
syste
m
stru
cture, t
he distri
butio
n of the
target motion
state no long
er obei
es
Ga
ussian type, so the co
nditi
onal PDF
[(
)
,
]
k
fk
r
xZ
may
be obtain
ed b
y
Gaussian a
pproxim
ate method a
s
follo
ws:
[(
)
,
]
[
(
)
(
,
)
,
(
,
)
]
k
f
kr
k
k
r
k
k
r
k
XZ
N
X
m
(6)
Therefore, G
aussia
n
app
roximate filtering
equ
ation
s
can b
e
de
ri
ved by usin
g
optimal
filtering equ
ations of di
scre
te time SRJS
s as follo
ws [
14].
(a) the
state predi
ction:
(
1
,)
(
,
)
(
,)
(
,
)
k
r
k
r
k
k
rk
rk
mF
m
(7)
(b) the
covari
ance pre
d
icti
on:
(
1
,
)
(,
)
(
,
)
(,
)
(
,
)
T
k
r
k
r
k
k
rk
rk
rk
w
FF
(8)
(c) the mixed
st
ate predicti
on:
11
()
()
(
,
1,
)
(
1,
)
kk
s
rs
r
sk
r
k
k
r
k
mm
K
e
(9)
(d) the mixed
covari
an
ce p
r
edictio
n:
()
1
(,
1
,
)
[
(,
1
)
]
(
1
,
)
sr
k
sk
r
k
sk
k
r
k
IK
H
(10)
Evaluation Warning : The document was created with Spire.PDF for Python.
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02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5305 – 53
15
5308
(e) the
state estimation:
()
(
)
()
11
1
()
(
)
()
11
1
(1
,
)
[
0
,
]
(
)
(
,
1
,
)
(1
,
1
)
(1
,
)
[
0
,
]
(
)
M
sr
sr
sr
k
ke
k
r
M
sr
sr
sr
k
ke
k
r
qk
k
f
r
s
k
r
k
ks
k
qk
k
f
r
Ne
Z
m
m
Ne
Z
(11)
(f) the c
o
varianc
e
es
timation:
(
)
()
()
1
11
1
(
)
()
()
1
11
1
(1
,
)
[
0
,
]
(
)
(
1
,1
)
(
,1
,
)
(1
,
)
[
0
,
]
(
)
[(
,
1
,
)
(
1
,
)
]
[
(
,
1
,
)
(
1
,
)
]
M
sr
sr
sr
k
ke
k
r
M
sr
sr
sr
k
ke
k
r
T
qk
k
f
r
ks
k
s
k
r
k
qk
k
f
r
sk
r
k
k
s
k
s
k
r
k
k
sk
Ne
Z
Ne
Z
mm
mm
(12)
(g)
synthe
size the state estimation:
1
1
(1
)
(
)
(
1
,
1
)
M
k
s
kf
s
k
s
k
mZ
m
(13)
(h)
s
y
nthes
i
ze the c
o
varianc
e
es
timation:
1
1
(
1
)
(
)
(
1,
1
)
[
(
1,
1
)
(1
)
]
[
(1
,
1
)
(
1
)
]
M
k
s
T
kf
s
k
s
k
k
s
k
kk
s
k
k
Zm
mm
m
(14)
(i) the conditi
onal PDF of t
he syste
m
structure state:
()
()
(
)
11
1
1
()
()
(
)
11
11
(1
,
)
[
0
,
]
(
)
()
(1
,
)
[
0
,
]
(
)
M
sr
sr
sr
k
ke
k
k
r
MM
s
rs
r
s
r
k
ke
k
sr
qk
k
f
r
fs
qk
k
f
r
Ne
Z
Z
Ne
Z
(15)
Whe
r
e,
()
11
()
1
()
(
)
1
11
(,
1
)
(
1
,
)
(,
1
)
(,
1
)
(
1
,
)
(,
1
)
(
,
1
)
(1
,
)
(
,
1
)
[
]
sr
kk
sr
T
ek
sr
T
s
r
ke
k
sk
k
r
k
s
k
sk
k
r
k
s
k
s
k
kr
k
s
k
v
ez
H
m
HH
KH
(16)
The initial co
ndition
s are
calcul
ated a
s
follows:
(
)
()
()
00
0
()
(
)
00
(
)
()
()
00
0
()
()
00
()
()
()
1
00
0
()
()
0
0
()
()
0
0
(0
)
(
,
0
)
(
,
0)
(
,
0)
(
,
0)
(0
,
0
)
(0
,
0
)
[
(
,
0
)
]
(,
0
)
[
]
[(
0
)
]
[
0
,
]
[(
0
)
]
[(
0
)
]
[
0
,
ss
s
ss
T
e
ss
s
ss
ss
T
s
e
s
s
e
s
s
e
s
ss
s
s
ss
s
qs
i
fs
i
qs
j
v
eZ
H
m
HH
mm
K
e
IK
H
KH
Ne
Ne
1
(1
,
2
,
,
)
]
N
j
iM
(17)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
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ISSN:
2302-4
046
Anti-interfe
re
nce T
r
a
cki
ng
Method
s of Maneu
ve
ring T
a
rget for Stru
cture
…
(Jianf
eng Wu)
5309
2.2. Impro
v
e
d
Gaussia
n
Appro
x
imate
Filtering for
Nonlinear sy
stems
The ob
se
rvation equ
ation is linea
r abov
e disc
u
s
sed a
l
gorithm. However, the observation
equatio
ns of
rad
a
r/IR du
al-mo
de com
poun
d
see
k
e
r
are n
onlin
ear [1
3], here Taylor seri
es
expan
sion m
e
thod i
s
u
s
e
d
by ce
ntere
d
on
ˆ
(1
,
)
kr
k
X
, and the hig
her-o
rd
er term
s
(mo
r
e
than two-orde
r) a
r
e negl
ect
ed [15].
ˆ
(1
,
)
(1
)
(
,
(
1
)
)
(
,
1
)
ˆ
ˆ
(,
(
1
,
)
)
(
(
1
)
(
1
,
)
)
(
,
1
)
kr
k
ks
k
s
k
s
k
rk
k
k
rk
s
k
XX
ZH
X
v
H
HX
X
X
v
X
(18
)
So the difference betwee
n
the mea
s
u
r
eme
n
t value
and the p
r
e
d
iction valu
e
in polar
c
o
ordinate is
as
follows
:
ˆ
(1
,
)
ˆ
(1
)
(
1
)
(
1
)
ˆ
((
1
)
(
1
,
)
)
(
,
1
)
kr
k
kk
k
k
kk
r
k
s
k
XX
ZZ
Z
H
XX
v
X
(19
)
From a
bove
disscu
ssed,
we can se
e that
NGA
F
algorith
m
is a man
e
vui
ng targ
et
tracking
meth
od
b
a
sed on “soft swithing
”
b
e
twe
en
th
e different m
odel
s, it bel
o
ngs to n
onlin
ea
r
para
m
eters self-a
dptive
filteri
ng alg
o
ri
thm, the outputs of th
e
s
e model
s are synthetical
ly
caculated by
the pr
obability whight
s, thi
s
will improve the
estimation accuracy
and
convergence
of the filtering algorithm.
3. Bootstr
a
p Filtering
The mai
n
ide
a
of Bootstra
p filtering m
e
thod
is th
at PDFs
are
re
pre
s
ente
d
to
a set of
rand
om sam
p
les, the Bo
otstrap filter
use
s
ra
ndo
m
samplin
g to transmit an
d update the
s
e
sampl
e
s, an
d
to ensure that these sam
p
les
a
r
e con
c
entrated in
the high pro
bability densi
t
y
rang
e [11, 12
].
3.1. Smith Sampling The
orem
Smith sa
mpl
i
ng the
o
re
m
is de
scrib
e
d
a
s
follo
ws. Assume
the
ran
dom
sampl
e
s
{(
)
,
1
,
}
i
ki
N
X
can be obta
i
ned from th
e contino
u
s
PDFs
()
x
, and these sam
p
le
s are
prop
ortio
n
to
()
()
L
xx
in acco
rd
a
n
ce
with the
requi
rem
ent
s of PDF
s
,
whe
r
e
()
L
x
is a
kno
w
n fun
c
ti
on. A sampl
e
is obtai
ned
by the discrete distri
buti
on of
{(
)
,
1
,
}
i
ki
N
X
, the
corre
s
p
ondin
g
prob
ability factors of
()
i
k
X
may be caculate
d as follo
ws:
1
[(
)
]
()
[(
)
]
i
i
N
j
j
k
qk
k
LX
LX
(20
)
From
Smith
sampling
theo
rem, it can
be
se
en
that a
s
N
tend
s to
in
finity, the dist
ribution
tends to be n
eede
d pro
b
a
b
ility density. The ra
ndom
sampl
e
s
{(
)
,
1
,
}
i
ki
N
X
are
obtaine
d by
rand
om sam
p
ling PDF
s
[(
)
]
k
fk
XZ
,
and the forecasting
sampl
e
s a
r
e obtain
ed acco
rdin
g
to
kno
w
n
state
equation,
then acco
rding to
th
e app
roxim
a
tion dist
rib
u
tion of PDFs
1
[(
1
)
]
k
fk
XZ
, the random
sampl
e
s
{(
1
)
,1
,
}
i
ki
N
X
are
obtaine
d after weig
hted an
d
update.
The state e
q
u
a
tion and o
b
servation eq
ua
tion of SRJSs are se
pa
ratel
y
as follows:
(1
)
[
(
)
,
(
)
,
(
)
]
(
1
)
[
(1
)
,
(1
)
,
(
1
)
]
kk
s
k
k
kk
s
k
k
Xf
Xw
Zh
X
v
(21
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
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046
TELKOM
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KA
Vol. 12, No. 7, July 201
4: 5305 – 53
15
5310
Whe
r
e
()
k
X
denot
e the state v
e
ctor,
()
k
w
and
()
k
v
denote
syste
m
noise an
d
observatio
n
noise,
()
F
and
()
H
a
r
e give
n n
onli
near fun
c
tion,
()
1
,
sk
S
M
is Ma
rkov chain
with
M
finite states, its tran
sition p
r
obability is a
s
follows:
[,
1
(
)
,
,
]
(
,
1
,
)
sr
qs
k
k
r
k
q
k
k
XX
(22)
3.2. The step
s of Boo
t
s
t
r
a
p Filtering
Bootstra
p filtering of SRJSs mainly con
s
i
s
ts
of s
e
ven steps
as
follows
[11, 14].
Step 1: the
rand
om
sam
p
les
{(
0
)
,
1
,
}
i
iN
X
are o
b
tained
by random
samp
ling
according to known initial PDF
[(
0
)
]
q
X
of the system
sta
t
e vect
or, its initial condit
i
on is
[
(
0
)
(0
)]
qs
X
,
[(
0
)
]
qs
,
[(
0
)
]
q
X
is defined
as follo
ws:
1
[
(
0)
]
[
(
0
)
(
0)
]
[
(
0
)
]
M
i
qq
s
i
q
s
i
XX
(23)
Step 2: the
rand
om
sam
p
les
()
{(
)
,
1
,
,
1
,
}
s
i
ki
N
s
M
w
are
o
b
tained
by random
sampli
ng a
c
cordin
g to kno
w
n PDF
[(
)
(
)
]
qk
s
k
w
of th
e system n
o
ise vector.
Step 3: one
step p
r
e
c
a
s
ting sample
s
()
(1
)
s
i
k
X
are o
b
taine
d
by the syst
em state
equatio
n,
()
(1
)
s
i
k
X
are
defined a
s
follows:
()
()
(
1
)
[
()
,
(
)
,
()
]
ss
ii
i
kk
s
k
k
Xf
X
w
(24)
Step 4: the sample
s
(1
)
i
k
X
are o
b
tained a
s
fol
l
ows:
1
(1
)
(
1
)
[
(
)
]
M
s
k
ii
j
kk
f
s
k
j
X
XZ
(25)
Step 5: the n
o
rmali
z
e
d
we
ighted facto
r
s
{(
0
)
,
1
,
}
i
iN
X
are ca
cula
ted by equation
(24), namely
PDFs
of probability factors
(1
)
i
qk
and the sy
stem stru
cture
1
((
1
)
)
k
fs
k
Z
are defin
ed a
s
follows:
1
1
1
1
11
[
(
1)
,
(
1)
,
]
(1
)
[
(1
)
]
[
(
1)
,
(
1)
,
]
M
ik
j
k
ii
MN
ik
ji
ks
k
j
qk
f
k
ks
k
j
AX
z
XZ
A
Xz
(26
)
1
1
1
1
11
[
(
1)
,
(
1)
,
]
((
1
)
)
[(
1
)
,
(
1
)
,
]
N
ik
k
i
MN
ik
ji
ks
k
fs
k
ks
k
j
AX
z
Z
A
Xz
(27
)
Whe
r
e,
1
1
11
[
(
1)
,
(
1)
,
]
[
(
1)
,
(
)
,
(
1
)
,
(
)
,
]
MN
k
ik
i
j
k
lj
ks
k
k
k
s
ks
k
l
AX
z
A
X
x
z
Z
(28
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Anti-interfe
re
nce T
r
a
cki
ng
Method
s of Maneu
ve
ring T
a
rget for Stru
cture
…
(Jianf
eng Wu)
5311
1
1
[(
1
)
,
(
)
,
(
1
)
,
(
)
,
]
[(
1
)
,
(
1
)
]
[
(
1
)
(
)
,
(
)
]
[(
1
)
(
)
,
(
)
]
[
(
)
]
[(
)
]
k
ij
k
ki
i
j
kk
jj
kk
s
k
s
k
f
ks
k
f
k
k
s
k
qs
k
s
k
k
f
k
f
s
k
AX
X
z
Z
zX
X
X
XX
Z
Z
(29)
1
1
[(
1
)
,
(
1
)
]
{
[
(1
)
,
(1
)
,
(1
)
]
}
[
(1
)
(
1
)
]
(
1
)
ki
ki
fk
s
k
ks
k
k
q
k
s
k
d
k
zX
zh
X
v
v
v
(30)
Whe
r
e
()
is Dirac
func
tion.
Since the probability of each random sam
p
le
{(
1
)
,
1
,
}
i
ki
N
X
is equ
al, so
[(
1
)
(
)
,
(
)
]
1
ij
fk
k
s
k
XX
,
1
[(
)
]
k
j
fk
N
XZ
.
Step 6: the random
sam
p
les
{(
)
,
1
,
}
ui
i
N
are extra
c
ted by the
uniform di
stri
bution
(0
,1
)
, and new ra
ndom sample
s
(1
)
(
1
)
il
kk
XX
are o
b
taine
d
by resam
p
li
ng acco
rdin
g
to the probability factors
(1
)
i
qk
, so a
s
to
re
alize th
e up
d
a
te and t
r
an
smit process of the
rand
om
sam
p
les. Th
e ra
ndom
sam
p
l
e
s
{(
)
,
1
,
}
ui
i
N
satisfy the rel
a
tional
expre
ssi
on
as
follows
:
1
0
00
(
1
)
(
)
(
1)
,
(
1)
0
(
1
,
)
ll
jj
jj
qk
u
i
qk
q
k
l
N
(31
)
Step 7: the
e
s
timation val
ue an
d va
ria
n
ce
of
the sy
stem states
a
nd
stru
cture states
are
cal
c
ulate
d
se
parately a
s
follows:
1
1
(1
)
(
1
)
N
i
i
kk
N
mX
(32
)
1
ˆ
(1
)
a
r
g
m
a
x
[
(
1
)
]
k
sS
sk
f
s
k
Z
(33
)
1
1
(
1
)
[
(1
)
(
1
)
]
[
(1
)
(
1
)
]
N
T
ii
i
kk
k
k
k
N
Xm
X
m
(34
)
So much fo
r that, BSF algorithm of SRJSs ca
n be re
alize
d
by loo
p
runni
ng the
pro
c
e
ss
of step 2 to step 7.
4. Results a
nd Analy
s
is
Assu
me that
the targ
et mo
tion with
con
s
tant
a
c
celeration in two-d
i
mensi
onal
pl
ane, the
scanni
ng p
e
ri
od of
rad
a
r
a
nd IR
se
eker are
T
. Th
e consta
nt a
ccel
e
ration
(CA)
model
used f
o
r
the state equ
ation of the target motion
i
s
given by Equation (35
)
[16
,
17].
22
22
10
0
/
2
0
/
4
0
01
0
0
0
/
2
0
00
1
0
/
2
0
/
4
(1
)
(
)
(
)
00
0
1
0
0
/
2
00
00
1
0
1
0
00
00
0
1
0
1
TT
T
TT
TT
T
kk
k
TT
XX
w
(35)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5305 – 53
15
5312
Whe
r
e
12
T
ww
w
，
()
k
w
denote Gau
s
s random
seq
u
e
n
ce
with ze
ro
mean and v
a
rian
ce
Q
,
1
w
and
2
w
are in
depe
ndent
a
nd have
the
same
varia
n
ce
2
,
so
2
QI
,
and satisfy
[(
)
]
Ek
0
w
,
T
[(
)
(
)
]
=
kj
Ek
j
ww
Q
.
The state ve
ctor of the targ
et may be defined a
s
follows:
(
)
()
()
()
()
()
()
T
k
x
k
x
ky
ky
k
x
ky
k
X
(36
)
The first two
states
refe
r to the po
sition
and the velo
city along the
x
-ax
i
s, re
spe
c
tiv
e
ly
,
and th
e n
e
xt two
state
s
ref
e
r to
the
po
sition a
nd th
e v
e
locity al
ong
the
y
-ax
i
s,
re
spe
c
tiv
e
ly
，
an
d
the last two st
ates ref
e
r to the accel
e
rati
on alon
g the
x
-axi
s and
y
-
a
x
i
s,
re
spe
c
t
i
v
e
ly
.
Due to the i
n
fluen
ce of e
x
ternal environment
(su
c
h as radio i
n
terfere
n
ce), the ra
dar
measurement
erro
rs exi
s
t the dra
m
atic j
u
mp at r
and
o
m
times, nam
ely the pip interferen
ce sig
nal
exist in the obse
r
vation sy
stem, so the
observation e
quation i
s
as
follows:
11
1
(1
)
(
(
1
)
)
(
,
1
)
(
1
,
2
)
kk
s
k
s
S
Zh
X
v
(37)
Whe
r
e,
1
(1
)
(1
)
(1
)
rk
k
k
Z
(38
)
22
1
(1
)
(
1
)
((
1
)
)
(1
)
a
r
ct
an
(1
)
xk
y
k
k
yk
xk
hX
(39)
Here in
ord
e
r to di
sting
u
ish th
e st
ructure lab
e
l
r
of the sy
stem, the di
stan
ce
measurement
is ma
rked
r
,
1
(2
,
)
k
v
denote the
pip interfe
r
en
ce
sign
al. Since th
e infra
r
e
d
measurement
is
not affe
cte
d
by the
pip
i
n
terfer
en
ce
si
gnal, thu
s
th
e
mea
s
u
r
eme
n
t
noise i
s
zero
-
mean Ga
ussi
an distri
butio
n with a varia
n
ce of
2
()
Rk
.
The initial co
ndition
s of the syste
m
n
o
is
e, ob
se
rv
ation noi
se
and sy
stem
state are
descri
bed a
s
follows:
()
()
(
)
(
)
00
0
0
11
1
11
1
11
()
[
(
)
0
,
]
[,
]
(1
,
)
[
(
1
,
)
,
(1
,
)
]
(2
,
)
[
(
2
,
)
,
(2
,
)
]
(1
,
)
(
2
,
)
ss
s
s
kk
kk
k
kk
k
kk
0
0
wN
w
Q
XN
X
m
vN
v
R
vN
v
R
RR
(40
)
The tran
sition
proba
bility of the system state is described a
s
follows:
()
(1
,
)
(
,
1
,
)
(
)
(
1
,
2
)
sr
qk
k
q
s
k
r
k
q
s
s
(41)
Assu
me that
the pro
babil
i
ty of the pip
jamming signal
of
the observation noise
is
(2
)
1
qs
. The initial
condition
s
of the
simulatio
n
ca
n b
e
writte
n re
sp
ectively
as follo
ws:
T
=20
m
s,
λ
=0
.05,
R
1
(1,
k
)=diag[ 0.36
×1
0
4
(m)
2
, 3×10
-6
(rad
)
2
],
R
1
(2
,
k
)=diag[ 9.0
×
10
4
(m)
2
, 7.5
×
10
-
6
(rad
)
2
],
R
2
(
k
)=
1×
10
-6
(r
ad)
2
,
0
s
m
=[1000
0m, 300m/s, 400
0
m
, 150m/s, 5
m
/s
2
, 4m/s
2
]
T
,
2
=100.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Anti-interfe
re
nce T
r
a
cki
ng
Method
s of Maneu
ve
ring T
a
rget for Stru
cture
…
(Jianf
eng Wu)
5313
All error
curv
es applyin
g
NGAF algo
rit
h
m
are obtai
ned by Monte Carl
o simul
a
tion with
50 ru
ns, an
d
the simul
a
tio
n
re
sults a
r
e
sho
w
n in Fi
g
u
re 1
-
5. We can se
e that whether th
e target
positio
n e
s
timation, velocity estimation
and a
c
cele
ration e
s
timation a
r
e very
close to th
e a
c
tual
trajecto
ry, the root mea
n
squ
a
re e
r
ro
r
(RMSE)
of the positio
n est
i
mation of
x
-axis is basi
cally
maintaine
d
at
abo
ut 20m.
From
Figu
re
5, we
can
cle
a
rly see th
at t
he e
r
ror
of NGAF alg
o
rith
m is
much
small
e
r than EKF algorith
m
tha
t
without
co
nsid
erin
g the
pip interference sig
nal,
it
effectively overcome
s the target tra
c
king
difficultie
s
res
u
lt in
th
e
r
and
o
m
in
te
r
f
er
en
c
e
. O
b
viou
s
l
y,
unde
r the co
ndition of ra
ndom jum
p
and vari
able
in
noise ch
ara
c
teri
stics,
anti-interfe
r
ence
tracking
alg
o
rithm
ba
se
d on
NGAF
ca
n a
c
curately track
the man
euv
ering
target, its
perfo
rman
ce
s are sig
n
ifica
n
tly higher th
an EKF.
Figure 1. Target Traje
c
tory
Figure 2. RM
S Position Error
0
200
40
0
600
80
0
100
0
25
0
30
0
35
0
40
0
45
0
T
r
u
e
tr
aje
c
to
r
y
E
s
timat
e trajecto
r
y
Ti
me s
t
ep (k
)
Figure 3. RM
S Velocity Error
Figure 4. RM
S Accele
ratio
n
Erro
r
Figure 5. Position Average
Erro
r
The sim
u
latio
n
con
d
ition
s
as ab
ove me
ntions, the
si
mulation results co
mpa
r
in
g BSF
algorith
m
wit
h
NGAF alg
o
rithm are sho
w
n in Figu
re
6-8. 500 g
r
ou
ps of sam
p
le
s are u
s
e
d
in BSF
algorith
m
. Co
mpared with
NGAF alg
o
rit
h
m, we ca
n
see that the filtering
re
sults
of BSF algorithm
are m
o
re
clo
s
e to the tru
e
value from
the
simul
a
tion re
sult
s. F
u
rthe
rmo
r
e,
RMSE of BSF
algorith
m
is
much l
e
ss th
an NAG
F
alg
o
rithm, on
thi
s
a
c
count we
can
see th
at the perfo
rma
n
ce
s
of BSF algorithm is sig
n
ificantly supe
rio
r
to NGAF alg
o
rithm.
Figure 6. RM
S Position Error
Figure 7. RM
S Velocity Error
F
i
gure 8. RMS
Acceler
a
tion Er
ror
Ho
wever, BSF algorithm i
s
req
u
ire
d
to operatio
n for each sampl
e
in the sam
p
les set,
thus the
com
putation of B
S
F algorith
m
is mu
ch
la
rge
r
than
NGAF
algorith
m
, an
d its computa
t
ion
is m
u
ltiplied i
n
crea
sed
wit
h
the i
n
crea
se of the
nu
m
ber of
sampl
e
s. A
s
the
la
w of l
a
rge
nu
mber
can
be
se
en t
hat the tru
e
-v
alue
s of the
sample
s
se
t a
r
e clo
s
e
to rea
l
values wh
en
the num
be
r
o
f
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5305 – 53
15
5314
sampl
e
s te
n
d
s to infinity, therefo
r
e th
e numb
e
r
of sampl
e
s
ha
s a
certai
n i
n
fluen
ce o
n
the
perfo
rman
ce
s of BSF algorithm.
The comp
ari
s
on
re
sults
a
r
e sho
w
n in
Figur
e 9-11,
whi
c
h the n
u
m
ber
of sam
p
les i
s
N
=500 an
d
N
=100
0 re
sp
ectively. As can be
see
n
from the simulation re
sults, when t
h
e
sampli
ng nu
mber i
s
dou
bl
ed, the perfo
rmances
of
BSF algorithm
are alm
o
st no
thing to impro
v
e,
but the co
mp
utation is
sign
ificant
ly incre
a
se
d. Theref
ore, in
cre
a
si
n
g
the numb
e
r of sampl
e
s
may
not sig
n
ifica
n
t
ly improve the pe
rform
a
n
c
e
s
of
BSF
algorith
m
, the re
solvem
e
n
t method
sh
ould
con
s
id
er sele
cting
the app
rop
r
iate sam
p
ling
n
u
mb
e
r
s to si
gnifica
ntly redu
ce t
he computati
on
without mu
ch
loss of the p
e
rform
a
n
c
e
s
of BSF algorithm.
Figure 9. RM
S Position Error
Comp
ari
s
ion
Figure 10. RMS Velocity
Erro
r Com
p
a
r
ision
Figure 11. RMS Accele
rat
i
on
Erro
r Com
p
a
r
ision
5. Conclusio
n
and future
w
o
r
k
The
re
sults o
f
applying
NGAF an
d BS
F algo
rithm
on a
n
ti-inte
r
feren
c
e
target
tra
ckin
g
probl
em of SRJS
s dem
on
strate its
sta
b
ility and
rob
u
stne
ss. It is sho
w
n that
EKF perform
s
poorly in the
pre
s
en
ce of random inte
rf
eren
ce a
nd structu
r
e un
ce
rtainties. However, NGAF
and
BSF algo
rith
m are abl
e t
o
overco
me
these
difficu
lt
ies, a
nd p
r
ov
ide a
stabl
e
estimate
of the
states
，
so b
o
th them
ha
ve highe
r
pe
rforma
nces
i
n
a
c
cura
cy
and
stability than EKF.
The
comp
ari
s
o
n
result
s of NG
AF and BSF
algorith
m
d
e
mon
s
trate t
hat BSF alg
o
rithm h
a
s b
e
tter
adapta
b
ility than NGAF algorith
m
. Firstly, NGAF
algorithm i
s
lineari
z
ed b
y
Taylor se
ries
expan
sion m
e
thod, so it has the lo
ss
of inform
atio
n of higher o
r
de
r term
s. Seco
ndly, NG
AF
algorith
m
h
a
s the
defe
c
ts
b
y
Gau
ssi
an
a
pproxim
ate m
e
thod. Fi
nally
, NGAF
alg
o
ri
thm only
use
s
one sample
of the sampl
e
sets to b
e
the state
vari
able
s
, by con
t
rast, the sa
mple sets of BSF
algorith
m
carries a la
rge
a
m
ount of info
rmation,
so
it
can
more a
c
curately re
prese
n
t real
value
than
NGAF a
l
gorithm. Sin
c
e BSF
algo
rithm is
not st
rict limited
by
the sy
stem i
n
itial state
an
d
noise distri
bu
tion, so BSF algorithm m
a
y over
co
me
the defects
of NGAF alg
o
rithm, and i
t
s
perfo
rman
ce
s is
signifi
cantl
y
supe
rio
r
to
NGAF
algo
rithm. Ho
weve
r, the num
ber
of sam
p
le
s h
a
s
a certai
n infl
uen
ce
on
th
e a
c
curacy
a
nd
com
putati
on of
BSF al
gorithm. It i
s
noted
that t
h
e
accuracy a
n
d
comp
utation
are two
de
si
gner-cho
se
n perfo
rman
ce
s and furthe
r i
n
vestigatio
n on
the quantified
relation bet
ween tho
s
e pe
rforman
c
e
s
is
expecte
d in future.
Ackn
o
w
l
e
dg
ements
We
want to
thank t
he
helpful
com
m
ent
s and sug
g
e
s
tion
s from
the an
onymou
s
reviewers. T
h
is
wo
rk
wa
s sup
p
o
r
ted b
y
the Na
tu
re
Scientific fu
ndame
n
tal
Rese
arch P
r
og
ram
funded by Sh
aanxi Provin
cial Educat
io
n Dep
a
rtme
nt (No. 201
2JM
8
020).
Referen
ces
[1]
Simon D. Ka
l
m
an F
ilteri
ng
w
i
t
h
State Co
n
s
traint
s: A Surve
y
of Lin
ear a
nd No
nl
ine
a
r A
l
gorit
hms.
IET
Contro
l T
heory
and App
lic
atio
ns
. 2010; 4(
8): 130
3-13
18.
[2]
Z
hao S, Liu
F
.
State Esti
mation i
n
Non
-
line
a
r
Markov
Jump S
y
ste
m
s
w
i
th U
n
ce
rtain S
w
itch
in
g
Proba
bil
i
ties.
IET
Control T
h
e
o
ry and Ap
pl
ications
. 20
12; 6
(5): 641-65
0.
[3]
McGinnit
y
S, Ir
w
i
n GW
. Multiple Mo
del B
oot
strap F
ilter
for Maneuver
ing T
a
rget
T
r
ackin
g
.
IEEE
T
r
ansactio
n
s o
n
Aerosp
ace a
nd Electro
n
ic S
ystems
. 20
00; 36: 100
6-1
012.
[4]
Log
othetis
A,
Krishn
amurth
y V. Expectati
o
n
Ma
xi
m
i
zati
o
n
Al
gorithms
for MAP Estim
a
tion
of Jum
p
Markov Li
near
S
y
stems.
IEEE Transactions
on Signal Pr
oc
essing
. 199
9; 47(8): 213
9-2
1
5
6
.
Evaluation Warning : The document was created with Spire.PDF for Python.