Image Processing Reference
In-Depth Information
4. KYP lemma for
H
∞
design problems
In this section, we show that the
H
∞
design problems given in the previous section
are efficiently solved via
semidefinite programming
(Boyd & Vandenberghe, 2004). For this
purpose, we first formulate the problems in state-space representation reviewed in Section
2. Then we bring in
Kalman-Yakubovich-Popov
(KYP) lemma (Anderson, 1967; Rantzer, 1996;
Tuqan & Vaidyanathan, 1998) to reduce the problems into semidefinite programming.
4.1 State-space representation
The transfer functions
(
(
)
−
(
))
(
)
(
(
)
(
)
−
)
(
)
P
z
Q
z
W
z
and
Q
z
P
z
1
W
z
in
Problems 1
and
2
,
respectively, can be described in a form of
T
(
z
)=
T
1
(
z
)+
Q
(
z
)
T
2
(
z
)
,
(2)
where
T
1
(
z
)=
P
(
z
)
W
(
z
)
,
T
2
(
z
)=
−
W
(
z
)
,
for
Problem 1
and
T
1
(
z
)=
−
W
(
z
)
,
T
2
(
z
)=
P
(
z
)
W
(
z
)
,
for
Problem 2
. Therefore, our problems are described by the following min-max optimization:
e
j
ω
)+
e
j
ω
)
e
j
ω
)
min
)
∈F
N
T
1
+
QT
2
∞
=
min
max
ω∈
[
T
1
(
Q
(
T
2
(
,
(3)
Q
(
z
Q
(
z
)
∈F
N
0,
π
]
F
where
N
is the set of
N
-th order FIR filters, that is,
Q
.
N
i
=
0
α
i
z
−
i
,
α
i
∈
R
F
=
(
)
(
)=
N
:
z
:
Q
z
To reduce the problem of minimizing (3) to semidefinite programming, we use state-space
representations for
T
1
(
z
)
and
T
2
(
z
)
in (2). Let
{
A
i
,
B
i
,
C
i
,
D
i
}
(
i
=
1, 2
)
are state-space matrices
of
T
i
(
z
)
in (2), that is,
:
A
i
B
i
C
i
D
i
A
i
)
−
1
B
i
+
T
i
(
z
)=
C
i
(
zI
−
D
i
=
(
z
)
,
i
=
1, 2.
(
)
Also, a state-space representation of an FIR filter
Q
z
is given by
⎡
⎣
⎤
⎦
0 1 0 ... 0 0
001
.
.
.
. .
00
.
.
.
.
.
.
0 .
. .
.
.
.
010
0 0 ... 0 0 1
α
N
α
N
−
1
...
:
A
q
B
q
α
N
:1
α
0
N
n
=
0
α
n
z
−
n
(
)=
=
(
)=
(
)
Q
z
z
z
,
(4)
α
2
α
1
α
0
=
α
N
α
N
−
1
...
α
1
.
where
α
N
:1
: