Environmental Engineering Reference
In-Depth Information
The state variables are taken as the currents of the three inductors and the voltage of the
capacitor (
u
o
, since
S
c
is closed). The external input variables (disturbances and references)
are
i
d
,
u
g
and
u
ref
and the control input is
u
. Thus,
⎡
⎤
⎡
⎤
i
d
u
g
u
ref
-----
u
i
1
i
2
i
3
u
o
u
⎣
⎦
,
⎣
⎦
.
x
=
=
The state equations of the plant are
B
2
]
u
x
=
Ax
+
[
B
1
(4.1)
with
⎡
⎣
⎤
⎦
R
f
r
f
r
f
−
0
0
−
(
R
f
+
r
f
)
L
f
(
R
f
+
r
f
)
L
f
R
g
r
g
(
R
g
+
r
g
(
R
g
+
0
−
0
−
r
g
)
L
g
r
g
)
L
g
=
,
A
Rr
r
0
0
−
(
R
+
r
)
L
(
R
+
r
)
L
1
R
1
C
r
f
r
g
(
R
g
+
r
1
1
R
g
+
−
−
r
+
r
f
+
(
R
f
+
r
f
)
C
r
g
)
C
(
R
+
r
)
C
+
R
f
+
r
g
⎡
⎣
⎤
⎦
r
f
0
0
0
(
R
f
+
r
f
)
L
f
r
g
(
R
g
+
0
0
0
r
g
)
L
g
=
[
B
1
B
2
]
=
.
B
0
0
0
0
1
C
1
(
R
g
+
1
−
0
r
g
)
C
(
R
f
+
r
f
)
C
The output signals from the plant are the tracking error
e
=
u
ref
−
u
o
and the current
i
o
,so
[
e
o
]
T
. The output equations are
that
y
=
C
1
C
2
x
D
11
u
D
12
y
=
+
(4.2)
D
21
D
22
with
⎡
⎣
⎤
⎦
,
0
1
-----------------------------------------------------
0
0
0
−
C
1
C
2
C
=
=
r
g
R
g
+
r
1
1
R
g
+
−
r
+
+
+
r
g
R
r
R
r
g
⎡
⎤
D
11
0
0
1
0
D
12
⎣
1
R
g
+
⎦
.
D
=
=
1
−
00
D
21
D
22
r
g
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