Digital Signal Processing Reference
In-Depth Information
respectively. Substituting Eqs. (6) into Eq. (5), the L
2
-sensitivity can be rewritten as
∂
H
(
z
)
2
2
∂
H
(
z
)
2
2
∂
H
(
z
)
2
2
S
(
A
,
B
,
C
)=
+
+
∂
A
∂
B
∂
C
2
2
2
G
T
(
z
)
F
T
(
z
)
G
T
(
z
)
F
T
(
z
)
=
2
+
2
+
2
.
(9)
We can express the L
2
-sensitivity S
(
A
,
B
,
C
)
by using complex integral as [1]
I
1
2
π
j
|z|=1
F
(
z
)
G
(
z
)(
F
(
z
)
G
(
z
))
†
dz
S
(
A
,
B
,
C
)=
tr
z
I
I
1
2
π
j
|
z
|=
1
G
†
(z)
G
(z)
dz
1
2
π
j
|
z
|=
1
F
(z)
F
†
(z)
dz
+tr
+ tr
.
(10)
z
z
Applying Parseval's relation to Eq. (10), Hinamoto et al. expressed the L
2
-sensitivity in terms
of the general Gramians such as [2]
∞
i=1
tr(
W
i
)tr(
K
i
).
S(
A
,
B
,
C
) = tr(
W
0
)tr(
K
0
) + tr(
W
0
) + tr(
K
0
) + 2
(11)
The general controllability Gramian
K
i
and the general observability Gramian
W
i
in Eq. (11)
are defined as the solutions to the following Lyapunov equations:
K
i
=
AK
i
A
T
+
1
2
A
i
BB
T
+
BB
T
(
A
T
)
i
(12)
W
i
=
A
T
W
i
A
+
1
2
C
T
CA
i
+ (
A
T
)
i
C
T
C
(13)
for i
=
0, 1, 2,
· · ·
, respectively. The general controllability and observability Gramians are
natural expansions of the controllability and observability Gramians, respectively. Letting
i = 0 in Eqs. (12) and (13), we have the Lyapunov equations for the controllability Gramian
K
0
and the observability Gramian
W
0
as follows:
K
0
=
AK
0
A
T
+
BB
T
(14)
W
0
=
A
T
W
0
A
+
C
T
C
.
(15)
The controllability Gramian
K
0
and the observability Gramian
W
0
are positive definite
symmetric, and the eigenvalues
θ
i
(
i
=
1,
· · ·
, N
)
of the matrix product
K
0
W
0
are all positive.
