Environmental Engineering Reference
In-Depth Information
After some mathematical manipulations, the problem of finding an anti-windup
compensator is reduced to a state feedback F satisfying the following optimization
problem with LMI constraints:
minimize m
ð
Q
;
U
;
L
Þ
,
subject to
2
3
B
u
U
QF
T
QC
y
þ
L
T
D
yu
ð
AQ
þ
B
u
L
Þþð
I
Þ
0
4
5
I
2U
I
UD
yu
\0
;
I
I
mI
0
I
I
I
mI
Q
¼
Q
T
[ 0
;
where I are inferred by symmetry. The state feedback gain is then obtained as
F
¼
LQ
1
(see [
16
] for more details).
4.6 Results
The system behavior was evaluated by simulation on the 5 MW NREL wind tur-
bine benchmark [
6
]. Simulations were performed in the FAST/Simulink
r
=
Matlab
r
environment. A more complete 16 degrees-of-freedom model available in
FAST [
7
] was used as a way to assess the robustness of the proposed control scheme
against unmodeled dynamics. The wind turbine data are given in Table
4.1
,
whereas the limit values of the operating locus are listed in Table
4.2
.
The pitch controller was designed according to the control setup in Fig.
4.11
with
W
e
ð
s
Þ¼
M
ð
s
Þ
W
e
ð
s
Þ¼
1
s
k
e
;
W
u
¼
k
u
s
=
0
:
1x
u
þ
1
s
=
x
u
þ
1
where k
e
¼
0
:
3, x
u
¼
50 and k
u
¼
0
:
25. The frequency response of the weights
W
e
, W
D
and W
u
are shown in Fig.
4.14
. Remember that W
u
must be more
demanding than W
u
and W
D
at every frequency. So, as can be seen in Fig.
4.14
,it
suffices to choose W
u
¼
W
u
.
The
1
-norm of the closed loop transfer function T
zw
resulted in 0.977. In
particular, the norm of the transfer function from X
N
to the control signal b, i.e.,
jj
K
ð
I
þ
KG
Þ
1
jj
1
¼
0
:
972
:
As the norm is lower than 1, stability against covered modeling errors is
guaranteed.
