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and Equation (1) can be written as
A
(
z
)
y
(
t
)=
B
(
z
)((
m
1
−
m
2
)
u
(
t
)
h
(
u
(
t
))
+
1
2
(
m
1
+
m
2
)
u
(
t
)) +
v
(
t
)
.
(3)
From (3), we can see that the output
y
(
t
) of the nonlinear block can be written
as an analytic function of the input.
3 The Estimation Algorithms
Define the parameter vector
θ
and the information vector
ϕ
(
t
)as
θ
:= [
b
1
(
m
1
−
m
2
)
,b
2
(
m
1
−
m
2
)
,b
3
(
m
1
−
m
2
)
,
···
,
m
2
)
,
1
2
b
1
(
m
1
+
m
2
)
,
1
b
n
(
m
1
−
2
b
2
(
m
1
+
m
2
)
,
1
2
b
3
(
m
1
+
m
2
)
,
,
1
···
2
b
n
(
m
1
+
m
2
)
,
3
n
,
a
1
,a
2
,a
3
,
···
,a
n
]
T
∈
R
ϕ
(
t
):=[
u
(
t
−
1)
h
(
t
−
1)
,u
(
t
−
2)
h
(
t
−
2)
,
u
(
t
−
3)
h
(
t
−
3)
,
···
,u
(
t
−
n
)
h
(
t
−
n
)
,
u
(
t
−
1)
,u
(
t
−
2)
,u
(
t
−
3)
,
···
,
u
(
t
−
n
)
,
−
y
(
t
−
1)
,
−
y
(
t
−
2)
,
···
,
3
n
,
−
y
(
t
−
n
)]
T
∈
R
gets
y
(
t
)=
ϕ
T
(
t
)
θ
+
v
(
t
)
.
(4)
If
has been estimated, none of the identification schemes can distinguish
b
i
,i
=
1
,
2
,
3
,
θ
. Therefore, to get a unique
parameterization, in this paper, we adopt the assumption that the first coecient
b
1
equals 1, i.e.,
b
1
=1.
The parameter vector
···
,n
and
m
i
,i
=1
,
2 from the estimated
θ
θ
and the information vector
ϕ
(
t
) be defined as
θ
:= [(
m
1
−
m
2
)
,b
2
(
m
1
−
m
2
)
,b
3
(
m
1
−
m
2
)
,
m
2
)
,
1
···
,b
n
(
m
1
−
2
(
m
1
+
m
2
)
,
1
2
b
2
(
m
1
+
m
2
)
,
1
2
b
3
(
m
1
+
m
2
)
,
,
1
···
2
b
n
(
m
1
+
m
2
)
,a
1
,
a
2
,a
3
,
3
n
,
···
,a
n
]
T
∈
R
(5)
ϕ
(
t
):=[
u
(
t
−
1)
h
(
t
−
1)
,u
(
t
−
2)
h
(
t
−
2)
,
u
(
t
−
3)
h
(
t
−
3)
,
···
,u
(
t
−
n
)
h
(
t
−
n
)
,
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