Ta
.
p ch´ı Tin ho
.
c v`a Diˆe
`
u khiˆe
’
n ho
.
c, T.21, S.3 (2005), 244—247
NH
ˆ
A
.
N DA
.
NG M
`
U CHU
ˆ
O
˜
I HAMMERSTEIN B
ˆ
A
.
C HAI
TR
ˆ
A
`
N THI
.
HO
`
ANG OANH
1
, D
ˆ
O
`
NG S
˜
I THI
ˆ
EN CH
ˆ
AU
2
1
Tru
.
`o
.
ng
Da
.
i ho
.
c Cˆong nghiˆe
.
p, Tp Hˆo
`
Ch´ı Minh
2
Tru
.
`o
.
ng
Da
.
i ho
.
c B´an cˆong Tˆon D´u
.
c Th˘a
´
ng, Tp Hˆo
`
Ch´ı Minh
Abstract. In this paper, a method of blind identification of second order Hammerstein series is
considered. This method is developed on the combination of stochastic approximation and Tixonop
method.
T´om t˘a
´
t. Du
.
.
a trˆen su
.
.
kˆe
´
t ho
.
.
p gi˜u
.
a hai phu
.
o
.
ng ph´ap chı
’
nh h´oa Tixonop v`a l´y thuyˆe
´
t xˆa
´
p xı
’
ngˆa
˜
u
nhiˆen, b`ai b´ao
dˆe
`
cˆa
.
p dˆe
´
n mˆo
.
t phu
.
o
.
ng ph´ap nhˆa
.
n da
.
ng chuˆo
˜
i Hammerstein bˆa
.
c hai.
GI
´
O
.
I THI
ˆ
E
.
U
C´ac mˆo h`ınh Hammerstein
du
.
o
.
.
c ´u
.
ng du
.
ng
dˆe
’
mˆo ta
’
hˆe
.
thˆo
´
ng phi tuyˆe
´
n d˜a v`a dang du
.
o
.
.
c
´u
.
ng du
.
ng nhiˆe
`
u trong c´ac qu´a tr`ınh sinh ho
.
c, h´oa ho
.
c, viˆe
˜
n thˆong,
diˆe
`
u khiˆe
’
n v`a xu
.
’
l´y t´ın
hiˆe
.
u [1, 2, 3]. Ta x´et chuˆo
˜
i Hammerstein bˆa
.
c hai sau
dˆay:
y(n) =
k
+
1
k=k
−
1
h
k
(n)x(n − k) +
k
+
2
k=k
−
2
h
kk
(n)x
2
(n − k); (1)
h
1
(0) = 1; k
1
, k
2
l`a bˆa
.
c cu
’
a hˆe
.
thˆo
´
ng;
x(n)
l`a t´ın hiˆe
.
u dˆa
`
u v`ao d`u
.
ng c´o trung b`ınh b˘a
`
ng khˆong
da
.
ng Gauss.
B`ai to´an nhˆa
.
n da
.
ng m`u
du
.
o
.
.
c
d˘a
.
t ra l`a du
.
.
a trˆen c´ac thˆong tin
dˆa
`
u ra
y(n)
v`a dˆa
`
u v`ao
x(n)
, h˜ay x´ac di
.
nh c´ac gi´a tri
.
h
k
(n)
v`a
h
kk
(n)
.
Du
.
a
y(n) = X
T
(n)h(n),
o
.
’
dˆay ta k´y hiˆe
.
u:
h(n) = (h
k
−
1
h
k
+
1
.
.
.
h
k
−
2
k
−
2
h
k
+
2
k
+
2
)
T
,
X(n) = (x(n − k
−
1
) x(n − k
+
1
)
.
.
.
x
2
(n − k
−
2
) x
2
(n − k
+
2
))
T
. (2)
X´et mˆo h`ınh:
ˆy(n) =
k
+
1
k=k
−
1
ˆ
h
k
(n)x(n − k) +
k
+
2
k=k
−
2
ˆ
h
kk
(n)x
2
(n − k), (3)
ˆy(n) = X
T
(n)
ˆ
h(n), (4)
o
.
’
dˆay ta k´y hiˆe
.
u
ˆ
h(n) = (
ˆ
h
k
−
1
ˆ
h
k
+
1
.
.
.
ˆ
h
k
−
2
k
−
2
.
.
.
ˆ
h
k
+
2
k
+
2
)
T
.
Lˆa
.
p tiˆeu chuˆa
’
n d´anh gi´a tˆo
´
i u
.
u:
NH
ˆ
A
.
N DA
.
NG M
`
U CHU
ˆ
O
˜
I HAMMERSTEIN B
ˆ
A
.
C HAI
245
E{e
2
(n)} → min
ˆ
h
,
o
.
’
dˆay sai sˆo
´
c´o da
.
ng:
e(n) = y(n) − ˆy(n) = X
T
(n)(h(n) −
ˆ
h(n)). (6)
T`u
.
diˆe
`
u kiˆe
.
n tˆo
´
i thiˆe
’
u h´oa theo tiˆeu chuˆa
’
n d´anh gi´a (5) ta thu du
.
o
.
.
c:
ˆ
h
k
(n + 1) = γ
ˆ
h
k
(n) + F
k
(
ˆ
h
k
(n)
ˆ
h
k
(n − 1)) − η(n)E{e(n)x(n − k)},
k = k
−
1
, , 0, 1, 2, , k
+
1
, 0 < γ 1, 0 < F
k
<< 1, (7)
ˆ
h
kk
(n + 1) = γ
ˆ
h
kk
(n) + F
kk
(
ˆ
h
kk
(n) −
ˆ
h
kk
(n − 1)) − η(n)E{e(n)x
2
(n − k)},
k = k
−
2
, , 0, 1, 2, , k
+
2
. (8)
Trong (7) v`a (8) c´ac bu
.
´o
.
c l˘a
.
p
η(n)
du
.
o
.
.
c cho
.
n sao cho tho
’
a m˜an c´ac
diˆe
`
u kiˆe
.
n hˆo
.
i tu
.
cu
’
a
Robbin—Monro [4] du
.
.
a theo l´y thuyˆe
´
t xˆa
´
p xı
’
ngˆa
˜
u nhiˆen:
0 < η(n) → 0 khi n → ∞;
η(n − 1) − η(n)
η(n)
→ 0 khi n → ∞, (9)
∞
n=1
η(n) = ∞;
∞
n=1
η
p
(n) < ∞; p ≥ 2. (10)
C´o thˆe
’
du
.
a c´ac thuˆa
.
t to´an (7) v`a (8) vˆe
`
da
.
ng kh´ac sau
dˆay:
ˆ
h
k
(n + 1) =
I − η(n)
x(n − k)x
T
(n − k)
x(n − k)
2
2
ˆ
h
k
(n) + F
k
(
ˆ
h
k
(n) −
ˆ
h
k
(n − 1))
+ η(n)y(n)
x(n − k)
x(n − k)
2
2
,
ˆ
h
kk
(n + 1) =
I − η(n)
x(n − k)x
T
(n − k)
x(n − k)
2
2
ˆ
h
kk
(n) + F
k
(
ˆ
h
kk
(n) −
ˆ
h
kk
(n − 1))
+ η(n)y(n)
x(n − k)
x(n − k)
2
2
.
Ta x´et hˆe
.
thˆo
´
ng dˆo
.
ng phi tuyˆe
´
n c´o dˆa
`
u ra du
.
o
.
.
c mˆo ta
’
bo
.
’
i phu
.
o
.
ng tr`ınh:
y(n) =
N
a
i=0
a
i
(n)x(n − i) +
N
d
i=0
N
d
i=0
d
ii
(n)x
2
(n − i) +
N
b
i=1
b
i
(n)y(n − i)
+
N
c
i=1
N
c
i=1
c
ii
(n)y
2
(n − i). (11)
246
TR
ˆ
A
`
N THI
.
HO
`
ANG OANH, D
ˆ
O
`
NG S
˜
I THI
ˆ
EN CH
ˆ
AU
Ta c´o h`am quan s´at
z(n)
:
z(n) = y(n) + v(n), (12)
o
.
’
dˆay
v(n)
l`a nhiˆe
˜
u quan s´at da
.
ng Gauss
E{v(n)} = 0, E{v
2
(n)} = σ
2
v
< ∞, (13)
E{.}
l`a k`y vo
.
ng to´an ho
.
c.
Diˆe
`
u kh´ac biˆe
.
t o
.
’
dˆay v´o
.
i c´ac t`ai liˆe
.
u
d˜a cˆong bˆo
´
[1, 2, 3] ta gia
’
thiˆe
´
t c´ac thˆong sˆo
´
a
i
(n), d
ii
(n), b
i
(n), c
ii
(n)
biˆe
´
n dˆo
.
ng theo th`o
.
i gian.
O
.
’
dˆay, b`ai to´an nhˆa
.
n da
.
ng th´ıch nghi khˆong d`u
.
ng
du
.
o
.
.
c du
.
.
a v`ao c´ac quan s´at
dˆa
`
u ra
z(n)
v`a dˆa
`
u v`ao
x(n)
dˆe
’
d´anh gi´a c´ac thˆong sˆo
´
hˆe
.
thˆo
´
ng dˆo
.
ng
ˆa
i
(n),
ˆ
d
ii
(n),
ˆ
b
i
(n), ˆc
ii
(n).
Gia
’
thiˆe
´
t r˘a
`
ng ta c´o thˆe
’
du
.
a hˆe
.
thˆo
´
ng
dˆo
.
ng phi tuyˆe
´
n (11) vˆe
`
da
.
ng vecto
.
sau
dˆay:
z(n, θ) = φ
T
(n)θ(n) + v(n) = θ
T
(n)φ(n) + v(n). (14)
B˘a
`
ng c´ach du
.
a v`ao vecto
.
thˆong sˆo
´
θ(n)
θ(n) = (a
i
(n) (i = 0, , N
a
); d
ii
(n) (i = 0, , N
d
); b
i
(n) (i = 0, , N
b
); c
ii
(n) (i = 0, , N
c
))
T
,
v`a vecto
.
quan s´at
φ(n)
φ(n) = (x(n − i) (i = 0, , N
a
); x
2
(n − i) (i = 0, , N
d
);
y(n − i) (i = 0, , N
b
); y
2
(n − i) (i = 0, , N
c
))
T
.
Dˆo
`
ng ´y v´o
.
i c´ac t´ac gia
’
Erik Weyer v`a M.C. Campi [5] ta
du
.
a v`ao c´ac tiˆeu chuˆa
’
n
d´anh gi´a
tˆo
´
i u
.
u:
V (
ˆ
θ) = E{ε
2
(n,
ˆ
θ)}, (15)
o
.
’
dˆay
ε(n, θ) = z(n) − ˆy(n,
ˆ
θ), (16)
ˆy(n,
ˆ
θ) = φ
T
(n)
ˆ
θ(n − 1). (17)
Su
.
’
du
.
ng phu
.
o
.
ng ph´ap chuˆa
’
n b`ınh phu
.
o
.
ng tˆo
´
i thiˆe
’
u du
.
.
a theo ´y tu
.
o
.
’
ng cu
’
a Alimed 2003
d˜a du
.
o
.
.
c ca
’
i biˆen, ta thu
du
.
o
.
.
c:
ˆ
θ(n + 1) =
ˆ
θ(n) + F (n)(
ˆ
θ(n) −
ˆ
θ(n − 1)) + µ(n)ε(n,
ˆ
θ)
φ(n)
φ(n)
2
2
. (18)
T`u
.
diˆe
`
u kiˆe
.
n tˆo
´
i thiˆe
’
u h´oa (17), ta thu du
.
o
.
.
c l`o
.
i gia
’
i
d´anh gi´a tˆo
´
i u
.
u vecto
.
θ :
θ
opt
= R
−1
f,
o
.
’
dˆay
R = E{φ(n)φ
T
(n)}; f = E{φ(n)z(n)}. (19)
Du
.
.
a theo ´y tu
.
o
.
’
ng cu
’
a c´ac t´ac gia
’
Erik Weyer v`a M.C. Campi ta lˆa
.
p tiˆeu chuˆa
’
n tˆo
´
i thiˆe
’
u
h´oa phiˆe
´
m h`am Tixonop:
NH
ˆ
A
.
N DA
.
NG M
`
U CHU
ˆ
O
˜
I HAMMERSTEIN B
ˆ
A
.
C HAI
247
V
N
(
ˆ
θ) = lim
N→∞
1
2N
N
n=1
ε
2
(n,
ˆ
θ) +
α
2
ˆ
θ
2
→ min
ˆ
θ
, (20)
N > N
a
+ N
b
+ N
c
+ N
d
, 0 < α.
Thˆong sˆo
´
b´e Tixonop c´o thˆe
’
cho
.
n:
α
opt
= min{σ
2
v
,
1
n
). (21)
T`u
.
diˆe
`
u kiˆe
.
n tˆo
´
i thiˆe
’
u h´oa tiˆeu chuˆa
’
n d´anh gi´a tˆo
´
i u
.
u (12) kˆe
´
t ho
.
.
p v´o
.
i c´ach lu
.
.
a cho
.
n
thˆong sˆo
´
b´e Tixonop theo (21) ta thu
du
.
o
.
.
c
d´anh gi´a b`ınh phu
.
o
.
ng tˆo
´
i thiˆe
’
u:
ˆ
θ
Nα
= R
−1
Nα
f
N
, (22)
o
.
’
dˆay
R
Nα
= lim
N→∞
1
N
N
n=1
φ(n)φ
T
(n) + α
opt
I, (23)
I
l`a ma trˆa
.
n do
.
n vi
.
N × N,
f
N
= lim
N→∞
1
N
N
n=1
φ(n)z(n). (24)
Kˆe
´
t luˆa
.
n. Viˆe
.
c t`ım mˆo h`ınh gˆa
`
n d´ung xˆa
´
p xı
’
c´ac hˆe
.
phi tuyˆe
´
n l`a mˆo
.
t vˆa
´
n dˆe
`
rˆa
´
t quan tro
.
ng
du
.
o
.
.
c nhiˆe
`
u nh`a nghiˆen c´u
.
u quan tˆam. Du
.
.
a theo ´y tu
.
o
.
’
ng cu
’
a c´ac t´ac gia
’
Erik Weyer v`a M.C.
Campi c`ung v´o
.
i viˆe
.
c su
.
’
du
.
ng phiˆe
´
m h`am chı
’
nh h´oa Tixonop ch´ung ta
du
.
a ra
du
.
o
.
.
c c´ac thuˆa
.
t
to´an tˆo
´
i u
.
u bˆe
`
n v˜u
.
ng
dˆe
’
nhˆa
.
n da
.
ng chuˆo
˜
i Hammerstein bˆa
.
c hai. C´ac kˆe
´
t qua
’
thu du
.
o
.
.
c s˜e
du
.
o
.
.
c ´ap du
.
ng trong qu´a tr`ınh cˆong nghˆe
.
h´oa ho
.
c, sinh ho
.
c v`a viˆe
˜
n thˆong.
T
`
AI LI
ˆ
E
.
U THAM KHA
’
O
[1] P. Koukoulas, N. Kalouptsidis, Blind identification of second order Hammerstein series,
Signal Processing 83 (2003) 213—234.
[2] N. Kalouptsidis, P. Koukoulas, Blind identification of Bilinear Systems, IEEE Transac-
tions on Signal Processing 51 (2) (2003) 484—499.
[3] Jean-Marc Le Caillec, Rene Garello, Time series nonlinearity modeling: A Giannakis
formula type approach, Signal Processing 83 (2003) 1759—1788.
[4] H. Robbins, S. Monro, Stochastic approximation method, Ann Math, Startist 22 (1951)
400—407.
[5] Erik Weyer, M. C. Campi, Non-asymptotic confidence ellipsoids for the least-square esti-
mate, Automatica 38 (2002) 1539—1547.
Nhˆa
.
n b`ai ng`ay 1 - 6 - 2005
. (2005), 244—247
NH
ˆ
A
.
N DA
.
NG M
`
U CHU
ˆ
O
˜
I HAMMERSTEIN B
ˆ
A
.
C HAI
TR
ˆ
A
`
N THI
.
HO
`
ANG OANH
1
, D
ˆ
O
`
NG S
˜
I THI
ˆ
EN CH
ˆ
AU
2
1
Tru
.
`o
.
ng
Da
.
i. Tixonop
method.
T´om t˘a
´
t. Du
.
.
a trˆen su
.
.
kˆe
´
t ho
.
.
p gi˜u
.
a hai phu
.
o
.
ng ph´ap chı
’
nh h´oa Tixonop v`a l´y thuyˆe
´
t xˆa
´
p xı
’
ngˆa
˜
u
nhiˆen,