National Instruments NI MATRIXx Xmath User Manual
Page 83
Chapter 4
Frequency-Weighted Error Reduction
© National Instruments Corporation
4-13
3.
Compute weighted Hankel Singular Values
σ
i
(described in more
detail later). If the order of C
r
(s) is not specified a priori, it must be
input at this time. Certain values may be flagged as unacceptable for
various reasons. In particular nscr cannot be chosen so that
σ
nscr
=
σ
nscr + 1
.
4.
Execute reduction step on stable part of C(s), based on a modification
of
redschur( )
to accommodate frequency weighting, and yielding
stable part of C
r
(s).
5.
Compute C
r
(s) by adding unstable part of C(s) to stable part of C
r
(s).
6.
Check closed-loop stability with C
r
(s) introduced in place of C(s),
at least in case C(s) is a compensator.
More details of steps 3 and 4, will be given for the case when there is an
input weight only. The case when there is an output weight only is almost
the same, and the case when both weights are present is very similar, refer
to [Enn84a] for a treatment. Let
be a stable transfer function matrix to be reduced and its stable weight.
Thus, W(s) may be P(I + CP)
–1
, corresponding to
"input stab"
, and will
thus have been calculated in step 2; or it maybe an independently specified
stable V(s). Then
The controllability grammian P satisfying
is written as
C s
( )
D
c
C
c
sI A
c
–
(
)
1
–
B
c
+
=
W
S
s
( )
D
w
C
w
sI A
w
–
(
)
1
–
B
w
+
=
C
s
s
( )W s
( )
D
c
D
w
C
c
D
c
C
w
sI A
c
–
B
c
C
w
0
sI A
w
–
1
–
B
c
D
w
B
w
+
=
P
A
c
′
0
C
w
′
B
C
′
A
w
′
A
c
B
c
C
w
0
A
w
P
B
c
D
w
B
w
D
w
′
B
c
′
B
w
′
+
+
0
=
P
P
cc
P
cw
P
cw
′
P
ww
=