Digital Signal Processing Reference
In-Depth Information
Balanced realization
A
b
B
b
C
b
D
b
R
T
B
2
R
A
opt
B
opt
C
opt
D
opt
R
T
U
1
U
M
A
opt
B
opt
A
opt
B
opt
C
opt
A
opt
B
opt
C
opt
D
opt
C
opt
D
opt
D
opt
1
M
Limit Cycle Free
Minimum L
2
-sensitivity realizations without L
2
-scaling constraints
FIGURE6.
Synthesisoftheminimum L
2
-sensitivityrealizationwhichdoesnotgeneratelimitcycles.
W
(opt)
0
K
(opt)
0
=
B
opt
B
opt
.
(106)
Eq.
(106) is equivalent to a sufficient condition for the absence of limit cycles proposed in
Therefore, the minimum L
2
-sensitivity realization
(
A
opt
,
B
opt
,
C
opt
,
D
opt
)
does not
Ref.
[6].
generate limit cycles.
✷
Theorem 5 shows that we can synthesize the minimum
L
2
-sensitivity realization without
limit
cycles
by
choosing
appropriate
orthogonal
matrix
U
.
Fig.
6
shows
the
synthesis
procedure of the minimum
L
2
-sensitivity realization which does not generate limit cycles.
The
coefficient
matrices
of
the
minimum
L
2
-sensitivity
realization
without
limit
cycles
(
A
opt
,
B
opt
,
C
opt
, D
opt
)
are given by
2
4
3
5
B
−
2
opt
B
−
2
1
2
A
opt
B
opt
opt
RA
b
R
T
B
opt
RB
b
=
.
(107)
D
opt
1
2
opt
C
opt
C
b
R
T
B
D
5.2. Numerical examples
We present numerical examples to demonstrate the validity of our proposed method. We
synthesize the minimum L
2
-sensitivity realizations of second-order and fourth-order digital
filters which do not generate limit cycles.
