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model, a stochastic gradient algorithm (SG) is proposed to estimate the unknown
parameters of the systems.
Briefly, the paper is organized as follows. Section 2 describes the piece-wise
linearities and derives an identification model. Section 3 studies estimation algo-
rithms for the identification model. Section 4 provides an illustrative example.
Finally, concluding remarks are given in Section 5.
2 The Piece-Wise Linearities
Consider a Hammerstein system
A
(
z
)
y
(
t
)=
B
(
z
)
f
(
u
(
t
)) +
v
(
t
)
,
(1)
where
y
(
t
) is the system output,
u
(
t
) is the system input, and
v
(
t
) is a stochastic
white noise with zero mean, and
A
(
z
)and
B
(
z
) are polynomials in the unit
backward shift operator [
z
−
1
y
(
t
)=
y
(
t
−
1)] and
A
(
z
):=1+
a
1
z
−
1
+
a
2
z
−
2
+
+
a
n
z
−n
,
B
(
z
):=
b
1
z
−
1
+
b
2
z
−
2
+
b
3
z
−
3
+
···
+
b
n
z
−n
.
···
The nonlinear input
f
(
u
(
t
)) is a piece-wise linearity which is shown in Figure 1
and can be expressed as
f
(
u
(
t
)) =
m
1
u
(
t
)
,u
(
t
)
0
,
m
2
u
(
t
)
,u
(
t
)
<
0
,
≥
where
m
1
and
m
2
are the corresponding segment slopes.
Define a switching function,
h
(
t
):=
h
[
u
(
t
)] =
2
,u
(
t
)
≥
0
,
1
2
,u
(
t
)
<
0
.
−
Then the output
y
(
t
) can be written as
m
2
)
u
(
t
)
h
(
u
(
t
)) +
1
f
(
u
(
t
)) = (
m
1
−
2
(
m
1
+
m
2
)
u
(
t
)
,
(2)
f
(
u
)
m
1
u
m
2
Fig. 1.
The piece-wise linearity
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