Environmental Engineering Reference
In-Depth Information
C
t
is given from the state-space system. Now, when the output gain matrix is
required, a solution to (
9
) needs to be found such that the product YX
1
factors as
YX
1
¼
KC
t
:
ð
6
:
10
Þ
To solve this, [
20
] suggests the following change of variables
X
¼
QX
Q
Q
T
þ
RX
R
R
T
;
ð
6
:
11
Þ
Y
¼
Y
R
R
T
;
ð
6
:
12
Þ
where X
Q
and X
R
are symmetric matrices with dimensions
R
ð
n
m
Þð
n
m
Þ
and
R
m
m
, respectively, and Y
R
has dimension
R
p
m
. The matrix Q is the null-space
of C
t
and R can be calculated as follows:
1
þ
QL
;
R
¼
C
t
C
t
C
t
ð
6
:
13
Þ
where L is an arbitrary matrix with dimensions
R
ð
n
m
Þ
m
.
In order to obtain a diagonal structure on the gain matrix K, simply force a
diagonal structure on X
R
and Y
R
, while X
Q
is a full matrix.
X
R
¼
diag X
R1
;
X
R2
;
X
R3
f
g
;
ð
6
:
14
Þ
Y
R
¼
diag Y
R1
;
Y
R2
;
Y
R3
f
g
:
ð
6
:
15
Þ
In order to solve the
LMI
s (
9
), first define m
¼
c
2
. Then, maximize m and
solve the
LMI
s in terms of X
Q
;
X
R
;
Y
R
. Once X and Y from (
11
,
12
) are cal-
culated it is possible to find the gain matrix K
¼
Y
R
X
R
, satisfying YX
1
¼
KC.
Additional information and proofs about this can be found in the aforementioned
references.
6.4 Simulation Results
In this section, the proposed control design methodology is applied to the wind
turbine system. The simulation is divided into two parts, one dealing with the
linear model and one dealing with the nonlinear model. First, both the full
information gain and the constrained gain are tested with the linear model. Second,
the constrained gain controller is tested with the nonlinear model. The figures
contain plots for simulations done with the constrained controller and with the
baseline controller. The baseline is intended as a reference plot and is included in
the FAST package. But first of all, suitable matrix values for B
1
, C
z
and D
z
need to
be found.
