Environmental Engineering Reference
In-Depth Information
r
e
u
G
K
(
s
)
(
s
)
Fig. 4.9 Closed loop system with uncertain plant
augmented plant (Eq.
4.8
) by selecting the performance output z and the distur-
bance w. More details about this process will be given in the next following
sections.
One of the most common uses of H
?
-synthesis is the design of robust con-
trollers. The dynamics of a system can be described by a model, but this
description is always an approximation. There always exists discrepancy between
system behavior and response predicted by the model, caused for example for
changes in some parameters or unmodeled phenomena. To consider this situation,
the system is described as a family of models in the form
G
ð
s
Þ¼
G
o
ð
s
Þþ
W
D
ð
s
Þ
D
;
k
D
k
1
\1
;
ð
4
:
10
Þ
where G
o
is the nominal model and W
D
is a filter describing the modeling error at
different frequencies. D is an unknown LTI system with
1
-norm bounded by 1. It
can be proved that the closed loop comprising system G
ð
s
Þ
and controller
K(s) (Fig.
4.9
) is stable for all
k
D
k
1
\1 if the
1
-norm of the transfer function
from r to u is lower than 1, i.e.,
k
K
ð
s
Þð
I
þ
G
ð
s
Þ
K
ð
s
ÞÞ
1
k
1
\1
:
ð
4
:
11
Þ
When a controller K(s) ensures stability for all plants in the set in Eq.
4.10
,itis
said that the closed loop is robustly stable and Eq.
4.11
is the robust stability
condition associated to the uncertainty representation in Eq.
4.10
. Notice that there
are other uncertainty representations, but Eq.
4.10
is one of the most commonly
used (for more details see for example, [
10
]).
4.5 Wind Turbine Control Design
In this section, we present an H
?
pitch controller. As said before, an anti-windup
controller was added to avoid undesirable behavior caused by pitch saturation in
the transition from region 1 to region 3. The controller is designed following a
two-step process. First, the H
?
pitch controller is designed disregarding the pitch
saturation. Then, the anti-windup controller is designed for a suitable performance
in the transition region.
