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Considering an nonlinear plant (
P
) type
⊧
⊨
x
1
=
˙
x
2
˃
+
i
=
˃
1
˄
+
j
=
˄
w
1
j
2
(
i
x
2
=
˙
f
2
p
(
x
1
,
x
2
)
+
u
=
1
(
x
1
)w
x
2
)
f
2
P
(
i
,
j
)
+
u
⊩
if a controller
˃
+
1
˄
+
1
j
i
=
(
x
1
,
x
2
)
=
w
1
(
x
1
)w
2
(
x
2
)
(
,
)
u
C
C
i
j
i
=
˃
j
=
˄
is considered, the closed-loop
CP
will be described as
˙
x
1
=
x
2
x
2
=
˙
f
2
cp
(
x
1
,
x
2
)
=
f
2
p
(
x
1
,
x
2
)
+
(
x
1
,
x
2
)
C
In order to make the closed-loop behavior of the system be approximated to the
performance of a desired plant
DP
(
f
2
cp
=
f
2
dp
), the controller can be directly
computed as,
=
(
x
1
,
x
2
)
=
f
2
dp
(
x
1
,
x
2
)
−
f
2
p
(
x
1
,
x
2
).
u
C
Example 8.3.6
Considering the so called “
Van der Pol
” oscillator, which is a non-
linear plant expressed as,
x
1
=
˙
x
2
x
1
)
x
2
=−
˙
x
1
+
(
1
−
x
2
+
u
If the operational region of the systems is selected as
−
4
x
1
4 and
−
4
x
2
4. Thus, the partition of the whole region into piecewise regions is done
through
d
1
=
. The piecewise bilinear model representation
of the
f
2
p
is the following (Table
8.1
).
d
2
=
(
−
4
,
−
2
,
0
,
2
,
4
)
Table 8.1
PB representation
of
f
2
p
x
1
\
x
2
d
2
=−
4
d
2
=−
2
d
2
=
0
d
2
=
2
d
2
=
4
d
1
=−
−
−
4
64
34
4
26
56
d
1
=−
2
14
8
2
−
4
−
10
d
1
=
0
−
4
−
2
0
2
4
d
1
=
2
10
4
−
2
−
8
−
14
d
1
=
4
56
26
−
4
−
34
−
64
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