TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.5, May 2014, pp
. 3338 ~ 33
4
7
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i5.4935
3338
Re
cei
v
ed O
c
t
ober 1
9
, 201
3; Revi
se
d Novem
b
e
r
30, 2013; Accept
ed De
cem
b
e
r
18, 2013
A Kind of
H
2
/
H
∞
Filtering Sheme on Deformation
Monitoring Data
Chen
gman Sha*, Yachun
Mao, Dongm
ei Yang
Schoo
l of Reso
urce & Civil E
n
gin
eeri
ng, Nort
heaster
n
Univ
e
r
sit
y
, She
n
y
an
g, Chin
a, 110
8
1
9
No. 11, La
ne 3
,
W
enhua Ro
a
d
, Hepi
ng Distr
ict, Shen
yan
g
Cit
y
,
Ch
in
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: shache
ngm
a
n
@mai
l.ne
u.ed
u.cn
A
b
st
r
a
ct
Based o
n
the r
e
searc
h
of t
he Kal
m
a
n
filter, a kind of H
2
/
H
∞
filter is put forw
ard w
h
ich de
p
ends o
n
the mode
l. H
2
filter assu
mes
that the nois
e
i
s
w
h
ite noise,
but do
es not r
equ
ire the stati
s
tical pro
perti
e
s
,
also ig
nor
es co
lor nois
e
. H
∞
fil
t
er only consi
d
ers the non-w
h
ite and
e
nergy-
limit
ed no
ise. It can ensure th
e
accuracy
of t
he filt
er in
w
o
rst frequ
ency
poi
nt, but
d
oesn
’
t
cons
id
e
r
the i
n
flu
enc
e of w
h
ite
no
ise.
Synthesi
z
i
n
g
t
he
adv
antag
es
an
d disa
dva
n
tages
of H
2
a
nd H
∞
fi
lter sc
he
me,
H
2
/
H
∞
hyb
r
id filt
er d
e
vi
de
the
nois
e
into the
w
h
ite noise a
n
d
non-w
h
ite n
o
i
se of li
mite
d e
nergy. Bae
d
o
n
nor
m an
alysi
s about the n
o
i
s
e
and syste
m
, th
e opti
m
i
z
a
t
io
n i
ndex J is
ado
pt
ed as th
e opt
i
m
i
z
at
io
n go
al,
and its p
h
ysic
a
l mean
in
g is gi
ven.
T
he hybri
d
filt
er is des
ign
e
d
by solvi
ng th
e corresp
on
din
g
Riccati E
q
u
a
tion, the s
i
mulati
on a
nd ac
tu
a
l
calcul
atio
n exa
m
p
l
e ar
e give
n
.
Ke
y
w
ords
: H
2
/
H
∞
filter, defor
mati
on
mo
nitor
i
ng,
estim
a
te error, filtering
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
No
wad
a
ys,
with the d
e
v
elopment of
eco
nomi
c
con
s
tru
c
tion,
high-ri
se b
u
ilding,the
proje
c
ts
of water
con
s
e
r
vancy & hyd
r
o
power, t
he la
rge b
r
id
ge p
r
oject a
nd o
p
en-pit mi
ning
are
becoming mo
re and mo
re.
It is the most comm
on ge
ologi
cal environmen,in
whi
c
h the huma
n
engin
eeri
ng
and th
e life
a
c
tivities filled.
Its defo
r
mati
on
can
cau
s
e the
deteri
o
ration
of hu
m
a
n
survival g
eol
ogical enviro
n
ment, even
bring
seve
re disa
ste
r
to the human.
In the relev
ant
geolo
g
ical
di
sa
sters, its d
a
mage
s i
s
o
n
ly next to
earthqu
ake. T
he a
c
curate
monitori
ng pl
ay an
importa
nt patr in land
slide
disa
ster a
n
a
l
ysis an
d pre
d
iction. So d
a
ta analysi
s
of deformatio
n
monitori
ng ha
ve becom
e a
n
importa
nt rese
arch di
re
ction.
In the pra
c
tice of foundati
on pit and
sl
ope
defo
r
mat
i
on monito
rin
g
, the accura
cy and
efficien
cy of monitori
ng
is g
r
eatly
impr
ove
d
wit
h
the imp
r
o
v
ement of
automation
and
measurement
preci
s
io
n of monitori
ng eq
uipment.
But some n
e
w p
r
oblem
s are p
u
t forwa
r
d to the
traditional
da
ta pro
c
e
s
sin
g
method
with the ha
rd
ware level e
nhan
ce
d [1-5]. The current
observation
d
a
ta filtering
method
s can
be divided
i
n
to seve
ral
categori
e
s [6
-7]. The traditi
onal
data filtering
method in
clu
de three
kind
s of met
hod:
One is rel
a
ted to gro
ss
error, aimin
g
at
jumping o
n
large
r
data p
r
oce
s
sing, su
ch a
s
Layda
Criteri
on, etc; Another i
s
to suppress the
observation noise,
su
ch as
the data point
pe
plce
d
with the ave
r
age val
ue of
few ne
ar p
o
i
nts
instea
d of itself; Another i
s
si
gnal filtering meth
o
d
. Fouri
e
r tran
sform of si
gn
al is a filteri
n
g
method. It co
nvert the sig
nal in time domain
into freque
ncy dom
ain, and elimi
nate the part
of
high fre
que
ncy part as
noi
se. In re
cent y
ears, so
me
of
the ne
w filtering metho
d
h
a
ve sp
run
g
u
p
.
They
can
be
divided i
n
to t
w
o
cate
gori
e
s. On
e
kind
depe
nd
s o
n
t
he m
odel
,an
d
the
othe
r o
n
e
doe
s not rely
on model.
Wavelet metho
d
[8-9], for ex
ample, is n
o
t depe
nd on th
e model, mai
n
ly
is time-freq
uen
cy deco
m
positio
n of
sign
als.
It se
t the thresho
l
d value an
d
eliminate
so
me
freque
ncy
pa
rts. Th
e Kalm
an Filte
r
met
hod i
s
dep
en
d on
the m
o
d
e
l [5], and
its filtering
effect is
excelle
nt. But it is neede
d
both to e
s
ta
blish th
e mot
i
on mod
e
l of
deform
a
tion
and move
me
n
t
system
an
d statistical ch
aracteri
stic of
n
o
ise
given.
T
he filter
in
g
m
e
thod whi
c
h
we have refe
rred
to is p
opul
ar
in currently. But they gen
erally a
s
sum
e
that the
ob
servatio
n n
o
i
s
e i
s
white
n
o
ise.
su
ch a
s
sump
tion in the p
r
actice of the
observation
i
s
not fully e
s
tablished, at l
east n
o
t accu
rat
e
enou
gh. A ki
nd of filter i
s
given i
n
th
is pa
per, the
y
are de
pen
dent on th
e
model.
H
∞
Filt
er
con
s
id
ers the
wo
rst
ca
se
about the
noi
se
com
pon
en
t, and minimi
zie m
a
ximum
gain of filteri
n
g
system fro
m
noise to esti
mated erro
r o
n
frequ
en
cy
domain. So It make
su
re th
e pre
c
i
s
ion in
the
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Kind of H
2
/H
∞
Filterin
g Shem
e on Def
o
rm
ation Mon
i
toring Data (Che
ngm
an Sha)
3339
worst
point
s.
But it is con
s
ervative b
e
ca
use
it d
o
e
s
n
o
t pay
attention to
the
othe
r
sign
als
out
side
of
the wo
rst point.
H
2
Filtering
sch
e
m
e
is u
s
ed
to
deal
with
white noi
se,
a
nd i
s
the
ov
erall
optimal filter whe
n
co
nsi
d
e
r
ing the hi
ghe
st pro
bab
ility at the most g
enerally situa
t
ion. But it does
not con
s
id
er
the worst case about noi
se outsid
e
of
white, so it cannot en
su
re
the pre
c
isio
n of
the individ
u
a
l
wo
rst poi
nt. Synthesi
z
in
g the
adva
n
t
ages an
d
d
i
sadva
n
tage
s of the
si
ng
le
optimizatio
n i
ndex of
H
2
fi
lter a
nd
H
∞
fi
lter, the
hybri
d
filter
sche
me i
s
p
r
op
osed. It uses t
he
hybrid optimi
z
ation in
dex J as th
e opti
m
ization
goal
base
d
on the
norm an
alysi
s
abo
ut the n
o
ise
sign
al and
m
odel. On th
e
con
d
ition of
the
H
∞
no
rm of the filter syste
m
le
ss than th
e gi
ve
n
value
γ
, the
H
2
norm of filter syste
m
is
overall optimi
z
ed. compa
r
eing
H
2
,
H
∞
and
H
2
/
H
∞
three
kind
s of filter sch
eme, the
advantag
e
s
and di
sa
dvantage
s of va
riou
s optio
n
s
is discu
ssed
adeq
uately.
2. The Filter Design o
f
H
H
/
2
De
formatio
n Sy
stem
2.1. Defo
rma
t
ion Sy
stem
Modeling an
d Filter Desi
gn
Con
s
id
erin
g the disto
r
tion
system
P(s)
is de
scribe
d by the followin
g
st
ate spa
c
e m
ode
l
[10].
11
0
1
22
1
2
()
,
(
0
)
()
x
Ax
B
t
x
x
zC
x
yC
x
D
t
&
,
(1)
Whe
r
e, state
variabl
e
x
is
a vector,
[,
,
,
,
,
]
T
xy
z
x
y
z
x
xx
x
x
x
x
&&
&
,
,,
x
yz
x
xx
res
p
ec
t the
relative co
ordinate
s
of x
、、
y
z
dire
ction in observ
a
tion point relative to th
e control poi
nts
r
e
spec
tively.
,,
x
yz
x
xx
&&
&
are the relative displa
ceme
nt of the ob
servati
on point in
x
、、
yz
dire
ction.
t
is the ob
servatio
n time, and
()
t
is the ob
serv
ation noi
se, whi
c
h is a
s
su
med to be
contai
ning
g
a
u
ssian white noise
and co
lored noi
se,
z
is estim
a
ted
deform
a
tion
vector.
y
is
obsrved di
spl
a
cem
ent valu
e. A is relate
d to mod
e
l system.
11
2
2
1
,C
,C
,
D
B
are
th
e co
nsta
nt of
the model
s.
0
x
Is the initial st
ate, which is
known. the
first observed v
a
lue will act as the initial
state.
De
signi
ng a dynamic filter
()
Ls
:
0
ˆˆ
ˆ
(
)
()
()
,
0
ˆ
ˆ
()
(
)
LL
LL
xt
A
x
t
B
y
t
x
zC
x
t
D
y
t
&
(
2
)
Make
sure
th
e differe
nt no
rm ind
e
x of
t
he tra
n
sfe
r
fu
nctionfrom from the
noi
se
(sy
s
tem
1
()
t
and observation noise
2
()
t
)to the
estimation e
rro
r
ˆ
zz
z
sat
i
sf
y
cert
a
i
n
requi
rem
ents.
Set
ˆ
[(
)
(
)
]
TT
x
xt
z
t
, then the dynamic e
q
uation of f
iltering error i
s
gi
ven as:
0
()
()
()
,
0
()
(
)
xt
A
x
t
B
t
x
zC
x
t
D
t
&%
%
%%
%
%
%
%
%
(3)
Whe
r
e
0
L
L
A
A
B
CA
%
,
L
B
B
B
D
%
,
12
f
L
CC
D
C
C
%
,
L
D
DD
%
.
2.2. Filtering D
y
namic S
y
stem No
rm
The no
rm de
finition of the following si
g
nal
and the
system tra
n
sfer function
on time
domain a
nd freque
ncy dom
ain are int
r
od
uce
d
.
The n
o
rm
def
inition of the
observation
n
o
ise
sig
nal
12
[(
)
(
)
]
tt
, filtering
estim
a
te
the error outp
u
t
z
on time domain are foll
owin
g:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3338 – 33
47
3340
If
)
(
sup
,
)
(
2
t
ess
dt
t
, the
2
L
of o
b
se
rvation
n
o
ise
sig
nal
()
s
are
defined a
s
:
2
2
()
,
td
t
(4)
su
p
(
)
es
s
t
)
(
)
(
sup
t
t
ess
T
,
(5)
If
2
z(
)
,
s
u
p
(
)
td
t
e
s
s
t
, the
2
L
and
L
n
o
rm of o
b
servation noi
se
sign
al
()
zs
are defin
ed a
s
:
2
2
()
,
zz
t
d
t
,
(6)
su
p
z
(
)
ze
s
s
t
sup
(
)
z
(
)
T
es
s
z
t
t
,
(
7
)
e
s
s
s
u
p
m
e
a
n
s
e
s
sure
d u
pper limit.
The no
rm d
e
finition of the ob
servati
on noi
se
、
Estimation e
r
ror outp
u
t
z
an
d the
trans
fer func
tion
G
of filtering syste
m
on
complex fre
q
uen
cy domai
n
s
are followi
ng:
if
2
(
s
)
,
sup
(
s)
ds
e
s
s
.The signal
doe
s not hav
e pulse, The
2
L
norm
of obse
r
vatio
n
noise
(s)
is de
fined as:
2
2
1
(s
)
2
ds
1
(s
)
(
s
)
2
T
ds
,
(8)
if
2
z(s
)
,
s
u
p
z(s
)
ds
e
s
s
.The signal
doe
s not hav
e pulse, The
2
L
norm
of
observation n
o
ise
z(
s
)
is define
d
as:
2
2
11
(s
)
(
s
)
(s
)
22
T
zz
d
s
z
z
d
s
,
(9)
the
H
n
o
r
m
o
f
G sy
stem i
n
the fo
rm o
f
transfe
r fun
c
tion in
freq
uen
cy dom
ai
n is
defined a
s
:
(s
)
s
u
p
(
)
GG
s
,
or
22
()
s
u
p
(
/
)
Gs
z
(10)
Whe
r
e
i
s
a
positive
kno
w
n n
u
mb
er,
s ia th
e com
p
lex freq
uen
cy,
σ
is th
e l
a
rge
s
t
sing
ular
value. The
2
H
n
o
r
m
o
f
sy
ste
m
’s in the form of transfe
r
function in fre
quen
cy doma
i
n is define
d
as:
2
1
(s
)
(
(s
)
*
(
s
)
2
T
Gt
r
a
c
e
G
G
d
s
,
(11)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Kind of H
2
/H
∞
Filterin
g Shem
e on Def
o
rm
ation Mon
i
toring Data (Che
ngm
an Sha)
3341
Whe
r
e
s
i
s
th
e co
mplex fre
quen
cy, t
r
a
c
e
is the
matri
x
’s
tra
c
e. If the de
scription
from the
stat
e of
the system e
quation,
2
H
norm of system G can b
e
exp
r
esse
d as:
)
(
)
(
'
1
1
2
LC
C
trace
s
G
,
Whe
r
e
L
i
s
t
h
e
s
o
l
u
t
i
o
n
o
f
Lyapnov e
q
u
a
t
i
o
n
a
s
f
o
l
l
o
w
s
:
0
'
1
1
B
B
LA
AL
. The definition of
sign
al
syste
m
in time
do
main a
nd fre
quen
cy
do
m
a
in i
s
eq
uival
ent, they can
be
conve
r
t
each
other by F
o
u
r
i
e
r
Tran
sfo
r
m.
2.3.
2
H
Filter Design
For th
e filter
descri
bed
by
the
formula (2),
if
it can
make
2
H
norm
2
G
o
f
tr
an
s
f
er
function
G
fro
m
the
system
noise
()
t
to es
timate the
error
ˆ
zz
z
to get the m
i
nimum n
o
rm
index, the filt
er d
e
scribe
d
by the fo
rm
ula (2) is its
2
H
filter.
Want
tp get th
e
2
H
filter (2
),
followin
g
two
Hamilton m
a
trix (12-13
) in the first pla
c
e
are given:
T
T
T
T
T
T
C
D
B
A
C
D
D
C
B
B
C
D
B
A
H
)
(
)
(
:
1
12
2
1
*
12
*
12
1
2
2
1
12
2
2
,
(12)
)
(
)
(
)
(
:
2
21
1
1
*
21
*
21
1
2
2
2
21
1
2
C
D
B
A
B
D
D
B
C
C
C
D
B
A
J
T
T
T
T
T
T
,
(13)
If the filter so
lution exist
s
,
and m
a
trix
2
:
H
an
d
2
J
have
no e
i
genvalue
on
the imagi
nary axis,
,
XY
are the two
positive se
midefinite sol
u
ti
on of the Ricatti equati
on co
rrespon
ded to
Hamilton m
a
trix (12),
(
13
), the
2
H
optimized
filter is given by (14) [10]:
0
ˆ
2
2
2
F
L
A
K
.
(14)
Whe
r
e
2
2
2
2
2
ˆ
C
L
F
B
A
A
,
)
(
:
2
2
1
12
2
X
B
C
D
F
T
T
)
(
2
2
21
1
2
T
T
C
Y
D
B
L
,
I
D
D
*
12
12
,
I
D
D
*
21
21
,
namely
*
21
*
12
,
D
D
a
s
t
h
e
ortho
gon
a
l
array of
21
12
,
D
D
.
Based o
n
2
H
n
o
rm definition
of the system
G
, as the input noise si
gnal is a
s
su
med to
be ga
ussia
n
white n
o
ise p
r
ocess to
2
H
filter,It is
not
hard to d
educ
e
d the integral term in
2
H
norm d
e
finition is the gai
n function a
r
ray about freq
uen
cy (formu
la 11). The i
n
tegral
re
sult to
freque
ncy
ref
l
ects the
gain
on th
e frequ
ency
domai
n. So p
h
ysi
c
al
meanin
g
of t
he
system’
s
2
H
norm i
s
that i
t
reflect
s
the
con
d
ition to
suppr
ess n
o
ise of filtering
sys
tem when t
he noi
se
sig
n
a
l
probability is
uniform
on
the whole frequency domai
n
. Accordingl
y, it is not hard to known i
t
s
geomet
ric si
g
n
ifican
ce
is the a
r
ea
surround
ed
by
la
rge
s
t
singul
a
r
value
curve
and
fre
quen
cy
axis.
For
2
H
filtere
d
system
expre
s
sed
by
the
form of t
r
an
sfer fun
c
tion
(
)
()
()
zs
G
s
s
,
Assu
me th
at the n
o
ise
si
gnal
s i
s
g
a
u
ssi
an
whit
e
noise p
r
o
c
e
s
s. With
out lo
sing
ge
nerali
t
y,
assume
as m
ean value 0,
variance 1, then
i
T
i
=
I
(unit matrix),
22
2
zG
G
. S
o
optimizin
g th
e
2
z
mean
s o
p
timizing
the
G
, The
optimi
z
a
t
ion go
al is o
n
ly relate
d to
the sy
stem
G
itself. Thus the
2
H
filter is to redu
ce the
estimation e
r
ror
ˆ
zz
z
in the frequen
cy doma
i
n
(or time
) on the wh
ole, as
far as p
o
ssibl
e
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3338 – 33
47
3342
2.4.
H
Filter Design
For th
e filter
descri
bed
by
the
form
ula (2),
if
it can make
H
norm
G
of tran
sfer
function
G
f
r
om t
h
e
sy
st
e
m
noi
se
()
t
to es
timate the error
ˆ
zz
z
to get th
e minim
u
m, t
he
filter describe
d
by the form
ula (2
) is it
s
H
filter. Want t
o
get the
H
filter (2), follo
wi
ng two
Hamilton m
a
trix in the first place are giv
en:
T
T
T
T
T
T
T
C
D
B
A
C
D
D
C
B
B
B
B
C
D
B
A
H
)
(
)
(
:
1
12
2
1
*
12
*
12
1
2
2
1
1
2
1
12
2
(15)
)
(
)
(
)
(
:
2
21
1
1
*
21
*
21
1
2
2
1
1
2
2
21
1
C
D
B
A
B
D
D
B
C
C
C
C
C
D
B
A
J
T
T
T
T
T
T
T
(16)
If the filter solution exists,
and matrix
H
an
d
J
have no ei
genvalu
e
on the imagin
a
ry
axis, an
d the
pola
r
radi
us
2
,
,
XY
are the
two po
sitive
se
midefinite
sol
u
tion of th
e
Ricatti equation co
rre
sp
on
ded to Hamil
t
on matrix (1
5-16
), the
H
op
timized filter is given by
(17) [10]:
0
ˆ
F
L
Z
A
K
(17)
Whe
r
e
2
2
1
1
2
ˆ
C
L
Z
F
B
X
B
B
A
A
T
,
)
(
:
2
1
12
X
B
C
D
F
T
T
,
)
(
2
21
1
T
T
C
Y
D
B
L
,
1
2
)
(
X
Y
I
Z
,
I
D
D
*
12
12
,
I
D
D
*
21
21
,
namely
*
21
*
12
,
D
D
a
s
t
h
e
orthog
on
al array of
21
12
,
D
D
H
Filter ca
n also be got by solving a
set
of linear m
a
trix inequalit
y (LMI) for the
following:
0
2
I
D
D
P
B
PB
C
C
A
P
P
A
T
T
T
,
Whe
r
e P is a
s
ymmetri
c
po
sitive definite matrix, 0 mea
n
s the ne
gati
v
e semi defini
t
e.
It also can b
e
see
n
from the definition
of
H
norm
s
G
of filter system. Its physi
cal
meanin
g
is the bigg
est g
a
in of filter syste
m (the
worst freq
ue
ncy point), a
nd its geom
etric
meanin
g
is th
e pea
k of the large
s
t sin
gul
ar value curv
e.
On the frequ
ency d
o
main
, only wh
en f
r
equ
en
cy
eq
uals
a
certai
n value, the
gain o
f
f
ilt
er sy
st
e
m
G f
r
o
m
to es
timated
error
z
a
c
q
u
i
r
e
to ma
x
i
mu
m. It i
s
H
norm
s
G
of filter
sy
st
em
.
Con
s
ide
r
ing th
e
freque
ncy p
o
int whi
c
h i
s
on the maxi
mum gai
n system, the
H
filter minimi
zed the
su
pre
m
um that fro
m
t
he filter
noise inp
u
t to e
s
t
i
m
a
t
e
d
e
r
r
o
r
z
(en
e
r
g
y
amplificatio
n
)
, which fre
quen
cy i
s
al
so
call
ed
a
s
the wo
rst
noise ca
se.
The
H
filte
r
consider the worst possi
ble frequency point (l
ow probability) only in the
whol
e frequency
domain. Wh
en
index
G
is smalle
r, it mean
s the infl
uen
ce de
gre
e
to estimat
e
error i
s
smalle
r o
n
t
h
is frequ
en
cy point.
H
filter con
s
ide
r
colo
red
noi
se, and it
gu
arante
e
the
es
timate error
z
minimizatio
n
on
the
wo
rst freq
uen
cy
point, but it
wa
s n
o
t be
consi
dered
on
other fre
que
n
c
y point wh
e
n
L(
)
s
is be de
sig
ned. The
r
efore it c
onsi
der t
hat the noise rea
c
h the
worst-ca
se in
a limited ran
ge of ene
rgy values. Th
e filtered wavefo
rm also re
pre
s
ent
s the wo
rst-
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Kind of H
2
/H
∞
Filterin
g Shem
e on Def
o
rm
ation Mon
i
toring Data (Che
ngm
an Sha)
3343
ca
se n
o
ise
waveform, so its filtering i
s
conser
vative a
nd reli
able. It's impo
rtant t
o
note that
H
filter is not d
e
sig
ned fo
r
white n
o
ise, and the
i
n
flu
ence of white noi
se o
n
the si
gnal
ca
n be
treated throu
gh other m
e
a
n
s such as
2
H
fil
t
er.
2.5.
2
H
/
H
Filter Design
For the
H
2
filter, assum
e
that the noi
se
signal i
s
Ga
ussian
white
noise(
∈
B
S
). Th
e
optimizatio
n goal is only
asso
ciated
with the syst
e
m
itself and is overall o
p
timization. Fo
r a
colo
red
noi
se
(incl
udin
g
the wo
rst
-
case
that of
small
prob
ability), It is not optimal and
ca
n
not
assure the
H
norm le
ss than
a given valu
e
(suboptimal). Besi
de, the
stability of the filtering
perfo
rman
ce i
ndex
z
is not consi
dered.
H
filtering me
thod avoid
e
d
the dra
w
ba
cks of the
2
H
filtering method in statisti
c
pro
c
e
s
s of n
o
ise
si
gnal.
Con
s
id
erin
g t
he
worst
situ
ation that
col
o
red
noi
se
si
gnal i
s
boun
ded
,
prop
ose a filtering m
e
thod
to minimize
2
z
. However, this
method is
n
o
t optimize
d
for commo
n
white
noi
se
signal
(hig
h p
r
obability), a
n
d
in m
o
st
sit
uation
s
it
sh
ows a
con
s
e
r
vative and
b
ad
effec
t.
The trad
eoff
2
H
/
H
filtering method ove
r
com
e
this si
ded
n
e
ss. It divided the noi
se
sign
al
()
t
(inclu
ding system
noise
1
and observed noi
se
2
) into a
determi
nisti
c
part
belon
gs to a
boun
ded
set
(col
ored noi
se) an
d a ra
nd
om part d
e
scribed
by the random va
riab
le
(Gau
ssian
white noise). Reco
rde
d
a
s
：
01
ww
,
0
w
∈
B
S
,
1
w
∈
P
)
。
Th
us, the distortio
n
sy
st
em
P(s)
(form
u
la 1) can be
descri
bed by
the follo
win
g
state sp
ace model (fo
r
mul
a
18).
00
1
1
1
22
0
0
2
1
1
xA
x
B
w
B
w
zC
x
y
Cx
D
w
D
w
&
(18)
The tran
sfe
r
functio
n
form can b
e
de
scri
bed by
：
01
0
1
[
(
)
(
)][
(
)
(
)
]
T
zG
G
s
G
s
s
s
.
(19)
To ta
ke
P
z
-no
r
m a
s
o
p
timization in
dex, it ca
n
be
se
en
from
the
defi
n
ition of
P-n
o
r
m that
the
physi
cal me
a
n
ing of
P
z
-norm
is the total e
nergy in
stea
d
of
the total power which differs
from
2
L
-no
r
m. It’s e
a
s
y to un
de
rst
and that
wh
en
P
z
is optimal
the
ǁ
2
z
will be optimal
either.
Thus, the foll
owin
g optimization index was ad
opted:
22
2
1
su
p
(
)
pp
w
Jz
,where.
1
G
.
(20)
It
can be prov
ed
that
2
2
JG
,w
h
en
0
is orth
ogo
na
l to
1
,
0
∈
B
S
,
1
∈
P
,
G
<
. Thus, the p
h
ysical meani
ng of hybrid fi
lter
ing p
e
rfo
r
mance indi
ca
tor J is to mi
n
i
mize
2
G
in
con
d
ition of
ǁ
G
ǁ
∞
<
. For
syst
ems
expressed by th
e
stat
e Equatio
n
(18
)
,
whe
n
the colored
noise si
gnal
satisfie
s th
e
con
d
ition
1
∈
P
and
0
is
orth
ogon
al to
1
, a stable
optim
ized
filter can b
e
g
o
t to minimize the index J
in con
d
ition o
f
ǁ
G
ǁ
∞
<
. The hy
brid filtering issue will be
turned into
standa
rd
H
2
issue wh
en
→
or stand
ard
H
issue
whe
n
0
=0.
These hyb
r
id
filtering
met
hod
s
will hav
e so
lution
s
whe
n
the
foll
owin
g
conditi
ons are
sat
i
sf
ie
d:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3338 – 33
47
3344
(1)
Hamilton
matrix corre
s
po
ndin
g
to
the system
doe
sn’t have
eigenvalue
s on the
imagina
ry axis and the
corresp
ondi
ng Ri
ccati Equatio
ns definite.
(2) T
here is
certain
,,
L
YP
to satisfy the follows couplin
g Ri
ccati equatio
n and Lya
pno
v
equatio
ns:
T
T
T
D
B
PX
PC
D
B
LR
Y
21
1
2
2
20
0
0
(
0
)
21
1
2
1
2
T
D
PYB
PYLR
(21)
Y
LD
B
LD
B
Y
Y
A
YA
T
T
ml
ml
)
)(
(
21
1
21
1
2
0
F
F
T
(22)
0
Y
and
Y
LD
B
LD
B
A
T
ml
)
)(
(
21
1
21
1
2
(23)
T
T
ml
T
ml
Y
LD
B
LD
B
A
P
P
Y
LD
B
LD
B
A
}
)
)(
(
{
}
)
)(
(
{
21
1
21
1
2
21
1
21
1
2
0
)
)(
(
20
0
20
0
T
LD
B
LD
B
(24)
Whe
n
these above conditi
ons a
r
e satisf
ied, the hy
bri
d
optimizatio
n filter (linea
r fraction
al
transfo
rmatio
n form) i
s
given by:
0
|
|
:
)
(
2
F
L
F
B
A
s
K
ml
Whe
r
e,
T
T
D
D
R
D
D
R
21
21
1
20
20
0
,
,
)
(
1
21
2
2
1
1
2
X
B
D
C
L
X
B
B
A
A
T
T
ml
m
)
(
2
1
12
X
B
C
D
F
T
T
.
The
equatio
ns
above
ca
n be
solved
thro
ugh
the
followi
ng
st
eps of the
algorith
m
prog
ram
m
ing
in matlab en
vironme
n
t:
(1) Initiali
ze t
he
> 0,
0
L
= 0; To plu
g
i
L
in equatio
n (2
2) the
n
calculate the
i
Y
;
Conditions to plug
i
Y
in (9
), i
f
satisfy the
condition
s,
we
ca
n exe
c
ute
the nex
t
step,
if doe
s
not,
do the agai
n;
(2)
Usi
ng
i
L
、
i
Y
,
plus them int
o
the equatio
n (24
)
for gett
i
ng
i
P
;
(3) Plu
s
ing
i
Y
、
i
P
into the equa
tion (21
)
for g
e
tting
1
i
L
;
(4) If
1
ii
LL
≤
(requ
ired a
c
cura
cy), the prog
ram is over,
i
L
,
i
Y
,
i
P
are answer,
otherwise
1
ii
LL
, repeat
step
(2), In the a
c
t
ual ca
lculatio
n, If no conv
erge
nce of
1
ii
L
L
iterative calculation, or (2
3) d
oes not
satisfy the co
n
d
itions, it is likely the diffe
ren
c
e i
s
too
big
betwe
en the i
n
itial value a
nd the a
c
tual,
we
shoul
d a
d
just the initi
a
l value, and
value sh
oul
d
be gra
dually redu
ced.
3. Filter Effe
ct An
aly
s
is
T
he effect
o
f
three
kin
d
s
of filteri
n
g
schem
e a
r
e analy
s
ed
with a
kn
own mod
e
l
example
s
. As sho
w
n in
Figure 1, X, Y, Z resp
e
c
tively represe
n
t the displa
ceme
nt valu
e of
observation
p
o
ints. T
he
ab
sci
ssa i
s
th
e
observation
time
sequ
en
ce
, assumi
ng th
e ob
se
rvation
of
100 d
a
ys, sa
mpled o
n
ce
a day. The t
o
tal displa
ce
ment of X, Y, Z is 0.8m
m
,
0.7mm, 1.6
m
m
respe
c
tively,
and the defo
r
mation is a
s
sumed to b
e
uniforma
n
d
linear. The noise sign
al is
comp
osed
of two p
a
rts.
T
hey ar
e ga
ussian
col
o
red
noise (Fi
g
u
r
e
2(e
))
and
wh
ite noise (Fig
ure
2(d
))
re
sp
ecti
vely. The ma
ximal displa
cement of
the
colo
red
noi
se
is
0.5mm.
T
he true valu
e
of
the displa
ce
ment and co
mbined
with noise rep
r
e
s
ent observed
values (a
s shown in Figu
re 3
contai
ns n
o
ise in the ob
served cu
rve). T
houg
ht thr
ee
different ki
nd
s of filtering schem
e, Figu
re 3
sho
w
s the cu
rves filtere
d
. Table 1 sho
w
s the thr
ee types of pe
rformance i
ndi
ca
tors of the no
ise
before a
nd after filtering. The noise
value in cu
rves after filtering i
s
got according to
the
differen
c
e bet
wee
n
the wav
e
form data af
ter filtering an
d the true value.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Kind of H
2
/H
∞
Filterin
g Shem
e on Def
o
rm
ation Mon
i
toring Data (Che
ngm
an Sha)
3345
(a) T
he true v
a
lue of X dire
ction di
spla
ce
ment, (b) the
true value of Y directio
n di
spla
cem
ent, (c)
the true value
of Z direction
displa
cem
e
n
t, (d) the colo
red noi
se, (e) white gau
ss noise
Figure 2. The
Time-di
s
pla
c
e Curve
s
of T
r
ue Di
spl
a
ce and Noise on
Known Mo
de
l
Table 1. Noi
s
e Perfo
r
ma
n
c
e Indi
cators
before a
nd af
ter Filtering
Before
filter
rmm)
H2
filtering(mm)
HINF
filtering(mm)
Mixed
F
ilter
(
mm)
The noise
standard dev
iation
X 0.1484
0.0440
0.0479
0.0446
Y 0.1484
0.0440
0.0479
0.0447
Z 0.1484
0.0455
0.05
0.0465
The Noise
mean value
X 0.005
0.0115
0.0065
0.009
Y 0.005
0.0115
0.0065
0.009
Z 0.005
0.0115
0.0065
0.009
The noise
max
i
mum
value
X 0.5
0.1471
0.0987
0.1195
Y 0.5
0.1471
0.0988
0.1195
Z 0.5
0.1471
0.11
0.1195
The effe
ct o
f
each
kind
of filter i
s
obviou
s
thro
ugh th
e Ta
b
l
e 1
and
Fi
gure
2,
Deformation
curve
smo
o
ths si
gnifican
t
ly after f
ilte
r
ing, an
d largely in coi
n
cide
nt with the
displ
a
cement
deform
a
tion
law. But com
pare
d
with th
e ideal di
spl
a
ceme
nt true
value, som
e
gap
also
exist. Th
is i
s
no
rmal
becau
se
we
can
not
sep
a
r
ate the
noi
se from
the
si
gnal
com
p
let
e
ly.
The n
o
ise m
ean i
s
n
o
t eq
ual to
ze
ro b
e
ca
use of
co
ntaining th
e
colore
d n
o
ise. After filtering
by
0
10
20
30
40
50
60
70
80
90
100
0
0.
5
1
X/
m
m
0
10
20
30
40
50
60
70
80
90
100
0
0.
5
1
Y/
m
m
0
10
20
30
40
50
60
70
80
90
100
-2
-1
0
Z/
m
m
0
10
20
30
40
50
60
70
80
90
100
0
0.
5
1
Co
l
o
r
e
d
No
i
s
e
/
m
m
0
10
20
30
40
50
60
70
80
90
100
-0
.
5
0
0.
5
sa
mp
l
e
d
t
i
me
s
e
r
i
e
s
W
h
it
w
G
a
u
s
s
N
o
is
e
/
m
m
(a
)
(d
)
(e
)
(c
)
(b
)
Fi
g
ure 1.
Sketch of Filter
S
y
st
em
z
y
P
(
s
)
G
(
s
)
ˆ
z
L
(
s
)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3338 – 33
47
3346
the
2
H
filter
sch
e
me,
the noi
se stan
dard deviation
i
s
t
he mini
mum,
the
curve
is smo
o
th, an
d
filtering to
white noi
se
is the m
o
st
effective.
But t
he n
o
ise of
averag
e a
n
d
maximum
is the
bigge
st and t
he re
sult of filtering colo
re
d noise
is ba
d, the supp
ression effe
ct to the colored
noise at the 42 point i
s
the
wo
rst. After filtering by
the
H
filter sch
e
me, the noi
se sta
nda
rd
deviation is t
he high
est, the effect of filteri
ng the
white noise is
not good, b
u
t
the mean a
n
d
maximum of noise is the minimum, the result of
sup
p
re
ssi
ng and
filtering the colo
red n
o
ise
is
good. Th
e su
ppre
s
sion
effect to the col
o
red
noi
se at
the 42 poi
nt is the be
st, bu
t it is at the cost
the filtering effect of other point. Th
e hybr
id scheme, not o
n
ly consi
d
e
r
s the effecti
v
e
sup
p
re
ssion
to the colo
re
d noi
se, b
u
t also
con
s
id
ers to filter t
he white n
o
i
s
e
as
mu
ch
as
possibl
e.The
r
efore, in the
actual
work
can be
cho
o
se
d acco
rdi
ng to tolera
n
c
e of col
o
re
d
noise, so it is a kind of effe
ctive filtering scheme.
Figure 3. The
Time-di
s
pla
c
e Curve
s
of
Observed a
n
d
Filtered Val
ue on Kno
w
n
Model
4. Calculatio
n Example
Figure 4. The
Time-di
s
pla
c
e Curve
s
of
Observ
ed a
n
d
Filtered Val
ue on a Strip
Mine in Liao
n
i
ng
P
r
ov
inc
e
,
(a)
X
displa
ce/
m
m,
(b) Y
displ
a
ce/
mm,
(
c
)
Z displa
ce
0
10
20
30
40
50
60
70
80
90
100
-0.
5
0
0.
5
1
1.
5
X/
m
m
0
10
20
30
40
50
60
70
80
90
100
-0.
5
0
0.
5
1
Y/
m
m
0
10
20
30
40
50
60
70
80
90
100
-2
-1.
5
-1
-0.
5
0
0.
5
Sa
m
p
le
d
t
i
m
e
s
e
r
i
e
s
Z /
m
m
i
deal
c
u
rv
e w
i
t
hout
n
o
i
s
e
obs
erv
e
d
c
u
rv
e w
i
t
h
boi
s
e
H
i
nf
f
i
l
t
eri
ng c
u
rv
e
H
2
f
i
l
t
eri
ng c
u
re
hy
bri
d
f
i
l
t
eri
ng c
u
rv
e
z
-2
mm
0
mm
0
mm
2
mm
0
mm
2
mm
Sa
m
p
led ti
m
e
se
rie
s
x
y
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Kind of H
2
/H
∞
Filterin
g Shem
e on Def
o
rm
ation Mon
i
toring Data (Che
ngm
an Sha)
3347
Figure 3 is
a
deform
a
tion
observation
waveform dia
g
ram from a
monitori
ng po
int on the
slop
e of an open
-pit iron
ore in Lia
oni
ng Provin
ce
, Chin
a. The h
o
rizontal
axis represents t
he
observation ti
me and x,y, z rep
r
e
s
ent th
e displa
ce
me
nt comp
one
nt of the ob
serv
ation poi
nt fro
m
the co
ntrol
p
o
int in the th
ree coo
r
din
a
te
re
spe
c
ti
vely. The
solid line is th
e o
r
igin
al ob
se
rvatio
nal
data. Accordi
ng to the d
a
ta analy
s
is
a
nd colored n
o
ise tol
e
ra
nce, the
H
2
/
H
∞
hybrid filteri
n
g
method
was
applie
d to filter the ori
g
inal
obse
r
va
tion
data and the
filtered dat
a curve
is sho
w
n
as
the dotted lin
e in Figu
re
4. From th
e an
alysis
of
defo
r
mation
cu
rve
s
befo
r
e a
nd
after filtering,
it
can
be
se
en
that amplitud
e of ob
se
rvat
ion noi
se
exceed th
e defo
r
mation true v
a
lue
s
. In oth
e
r
words, the d
e
formatio
n value is sm
all while the o
b
servation noi
sy is large. Th
us, the filterin
g is
very importan
t. A good result has bee
n achi
eved afte
r usin
g the
H
2
/
H
∞
hybrid
filtering meth
od
as
we
can
see from th
e
Figure 4.
Th
e
H
2
/
H
∞
hyb
r
id filteri
ng
m
e
thod i
s
not
only theo
reti
cally
rigo
rou
s
but a
l
so practi
cabl
e.
5. Conclusio
n
A kind of
H
2
/
H
∞
filter sche
me is pro
p
o
s
ed, with
re
sp
ect to the high-p
r
e
c
isi
on o
b
se
rved
data by
mea
s
ureme
n
t rob
o
t. Base
d o
n
the
previo
u
s
wo
rk,
H
2
a
nd
H
∞
filterin
g sch
e
me
are
prop
osed, an
d the solving
method of
filter is given. T
he assum
p
tio
n
that noise
consi
s
ts of whi
t
e
gau
ss
noi
se
and col
o
r noi
se of
limited ener
gy is p
r
o
posed.
Referring to th
e
H
2
and H
∞
ffiltering
algorith
m
respectively, an
d
ba
sed
on
the
analy
s
is of
th
e no
rm
of
sig
nals an
d
syst
ems, th
e hyb
r
id
perfo
rman
ce
index J is giv
en and an
alysed. Fo
r a given
value, by solving the corre
s
p
ondin
g
riccati e
quati
on to
get a
r
ith
m
etic to
mini
mize
i
ndex
J.
Thro
ugh
theo
retical
an
alysi
s
a
nd
nume
r
i
c
al
example
s
, th
e advanta
g
e
s
and di
sa
dva
n
tage
s of
H
2
,
H
∞
and
hybri
ed filter a
r
e
discusse
d. It is
an
effective met
hod to
deal
with the noi
se
contai
ni
ng
no
n-white n
o
ise
,
unde
r the
premise
of limiting
the impa
ct of the non
-white
noise
com
p
o
nent, as fa
r a
s
po
ssi
ble to redu
ce the i
n
fluen
ce of whit
e
noise.
Referen
ces
[1]
José F
,
R
o
sa
n
a
R,
Da
nie
l
C.
Detecti
o
n
of
Dis
pl
acem
ents
on
T
enerife
Islan
d
, Ca
nar
ies,
Usi
ng
Ra
dar
Interferometr
y
.
Geophys
i
cal J
ourn
a
l Intern
ati
ona
l.
200
5; 16
0(1): 33-4
5
.
[2]
Ach F
,
Schm
ied
e
l A, K
a
e
m
ling
A, Hi
g
h
Po
w
e
r Infr
ared
Las
er B
eam Mo
nitor
i
n
g
b
y
Optic
a
l
Measur
ement
of Mirror Surface Deformati
on
.
Laser Physics
Letters.
2005; 2(5):
267-
27
1.
[3]
Xi
ufen
g H, Def
o
rmatio
n
Mo
ni
to
ri
ng
Me
th
od
an
d
i
t
s Ap
pl
i
c
a
t
io
n
,
Beij
in
g: Sci
ence P
ubl
ishi
n
g
Comp
an
y.
200
7.
[4]
Jon M, Ian N,
Gu
y
C P
e
irs
on. Pavem
ent
Defo
rmatio
n
Monitori
ng in a
Rol
lin
g
L
o
a
d
Facilit
y,
T
he
Photogr
a
m
met
r
ic Record
. 20
01; 17(9
7
): 07-
24.
[5]
Che
n
L,
Lil
ong
L, do
ng
yi
n
C. Ada
p
tive K
a
l
m
an
F
ilter
ing
Method
is Us
e
d
for D
e
format
i
on M
onit
o
rin
g
Data Processing,
Surveyin
g Engi
neer
in
g.
20
08; 17(1): 4
8
-5
4.
[6]
Ale
x
a
ndra S, Anthon
y O. Ba
yesi
an Infere
nce for
Non-st
ation
a
r
y
S
pati
a
l Cov
a
ria
n
ce
Structure vi
a
Spatia
l Defor
m
ations.
Jo
urn
a
l of the R
o
y
a
l Statistica
l S
o
ciety
:
Seri
es B
(Statistical Method
olo
g
y)
,
200
3; 65(3): 74
3-75
8.
[7]
Kääb W
,
Ha
eb
erli G, Hilm
ar
G. Anal
ysin
g the Cr
eep
of M
ounta
i
n P
e
rma
frost Using
Hi
gh Prec
isio
n
Aerial
Phot
ogr
ammetr
y
:
25
Years of Mo
nit
o
rin
g
Grube
n
Rock Glac
ier,
S
w
i
ss Al
ps,
Permafrost and
Perigl
aci
a
l Pro
c
esses
. 199
7; 8(4): 409-
42
6.
[8]
Yachu
n
M, W
e
i
w
ei J, C
h
e
n
g
m
an S, End
e
W
,
Dongmei
Y
.
Slope
Defor
m
ation
in a Gr
e
y
F
o
r
e
casti
n
g
Method Bas
ed
on W
a
vel
e
t Anal
ysis.
Min
e
ral Engi
neer
in
g
. 2010; 8(6): 1
7
-2
0.
[9]
Yachu
n
M, C
h
engm
an S,
En
de W
.
T
he Re
search
of W
a
v
e
let
Den
o
isi
n
g
Use
d
to th
e
Deformati
o
n
Monitori
ng D
a
ta.
Metal Mine
.
201
0; 45(9): 13
9-14
2.
[10]
Kemin Z
,
John
D. Essentials o
f
Robus
t Contr
o
l, Ne
w
Jers
e
y
: Prentice Hal
l
. 199
8.
Evaluation Warning : The document was created with Spire.PDF for Python.