Environmental Engineering Reference
In-Depth Information
There are several solutions for the c-suboptimal H
?
synthesis problem, but
the most popular nowadays is the formulation as an optimization problem with
linear matrix inequalities (LMIs) constraints [
11
]. Considering a controller with a
state-space realization
the controller matrices can be found by solving the following optimization
problem:
minimize c
ð
R
;
S
;
B
k
;
C
k
;
D
k
Þ
,
subject to
2
3
AR
þ
B
u
C
k
þð
I
Þ
I
I
4
5
\0
;
ð
B
w
þ
B
y
D
k
D
yw
Þ
T
cI
n
w
I
C
z
R
þ
D
zu
C
k
D
zw
þ
D
zu
D
k
D
yw
cI
n
z
2
3
ð
SA
þ
B
k
C
y
Þþð
I
Þ
I
I
4
5
\0
;
ð
SB
w
þ
B
k
D
yw
Þ
T
cI
n
w
I
C
z
þ
D
zu
D
k
C
y
D
zw
þ
D
zu
D
k
D
yw
cI
n
z
[ 0
;
RI
IS
with ''['' and ''\'' denoting positive and negative definite matrices, respectively,
and I represents the matrices needed to obtain a symmetric matrix.
After finding the positive definite matrices R and S and matrices B
k
, C
k
and D
k
,
the controller matrices can be computed from
A
k
¼ð
A
þ
B
u
D
k
C
y
Þ
T
þ
SB
w
þ
B
k
D
yw
C
z
þ
D
zu
D
k
C
y
"
#
1
"
#
ð
D
zw
þ
D
zu
D
k
D
yw
Þ
T
ð
B
w
þ
B
y
D
k
D
yw
Þ
T
C
z
R
þ
D
zu
C
k
cI
D
zw
þ
D
zu
D
k
D
yw
cI
B
k
¼
N
1
ð
B
c
SB
u
D
c
Þ;
C
k
¼ð
C
k
D
k
C
y
R
Þ
M
T
;
with MN
T
¼
I
RS.
The optimization problem involved in the H
?
-synthesis can be effectively
solved with available software such as Sedumi [
12
] and YALMIP [
13
]. It is also
available as a command in the Robust Control Toolbox for Matlab. Therefore, the
design process of an H
?
optimal control requires only to put the control specifi-
cations in terms of the minimization of the norm in Eq.
4.9
, i.e., to construct the