Image Processing Reference
In-Depth Information
6. Generalized KYP lemma for finite frequency design problems
In this section, we reduce the problems given in the previous section to semidefinite
programming. As in the
H
∞
optimization, we first formulate the problems in state-space
representation, and then derive semidefinite programming via
generalized KYP lemma
(Iwasaki & Hara, 2005).
6.1 State-space representation
As in the
H
∞
optimization in Section 4, we employ state-space representation. Let
T
(
)=
z
(
)
−
(
)
(
)=
(
)
(
)
−
P
z
Q
z
for the approximation problemor
T
z
P
z
Q
z
1 for the inversion problem.
(
)
(
)=
(
)+
(
)
(
)
Then
T
z
can be described by
T
z
T
1
z
Q
z
T
2
z
as in (2). Then our problems are
described by the following min-max optimization:
e
j
ω
)+
e
j
ω
)
e
j
ω
)
Ω
(
T
1
+
QT
2
) =
T
1
(
(
T
2
(
min
V
min
max
ω∈
Ω
Q
.
(8)
(
)
∈F
N
(
)
∈F
N
Q
z
Q
z
Let
{
A
i
,
B
i
,
C
i
,
D
i
}
,
i
=
1, 2 be state-space matrices of
T
i
(
z
)
. By using the same technique as in
Section 4, we can obtain a state-space representation of
T
(
z
)
as
A
B
T
(
z
)=
(
z
)
,
(9)
C
(
α
N
:0
)
D
(
α
0
)
α
N
:0
=[
α
N
,...,
α
0
]
where
is the coefficient vector of the FIR filter to be designed as defined in
(4).
6.2 Semidefinite programming by generalized KYP lemma
The optimization in (8) can be equivalently described by the following problem:
minimize
γ
subject to
Q
(
z
)
∈F
N
and
≤ γ
e
j
ω
)+
e
j
ω
)
e
j
ω
)
T
1
(
(
T
2
(
max
ω∈
Ω
Q
(10)
To describe this optimization in semidefinite programming, we adopt the following lemma
(Iwasaki & Hara, 2005):
Lemma 2
(Generalized KYP Lemma)
.
Suppose
A B
C D
(
)=
(
)
T
z
z
is stable, and the state-space representation
{
A
,
B
,
C
,
D
}
of T
(
z
)
is minimal. Let
Ω
be a closed interval
[
ω
1
,
ω
2
]
⊂
[
0,
π
]
.Let
γ
>
0
. Then the following are equivalent conditions:
≤ γ
e
j
ω
)
Ω
(
)=
(
1. V
T
max
T
.
ω∈
[
ω
1
,
ω
2
]
2. There exist symmetric matrices Y
>
0
and X such that
⎡
⎣
⎤
⎦
≤
C
M
1
(
X
,
Y
)
M
2
(
X
,
Y
)
)
M
3
(
2
M
2
(
)
0,
X
,
Y
X
,
γ
D
−
C
D
1