National Instruments NI MATRIXx Xmath User Manual
Page 89
Chapter 4
Frequency-Weighted Error Reduction
© National Instruments Corporation
4-19
6.
Check the stability of the closed-loop system with C
r
(s). When the
type="left perf"
is specified, one works with
(4-11)
which is formed from the numerator and denominator of the MFD
in Equation 4-5. The grammian equations (Equation 4-8 and
Equation 4-9) are replaced by
redschur( )
-type calculations are used to reduce E(s) and Equation 4-10
again yields the reduced-order controller. Notice that the HSVs obtained
from Equation 4-10 or the left MFD (Equation 4-5) of C(s) will in general
be quite different from those coming from the right MFD (Equation 4-6). It
may be possible to reduce much more with the left MFD than with the right
MFD (or vice-versa) before closed-loop stability is lost.
As noted in the
fracred( )
input listing,
type="left stab"
and
"right stab"
focus on a stability robustness measure, in conjunction
with Equation 4-5 and Equation 4-6, respectively. Leaving aside for the
moment the explanation, the key differences in the algorithm computations
lie solely in the calculation of the grammians P and Q. For
type="left
stab"
, these are given by
and for
"right stab"
,
(4-12)
(4-13)
E s
( )
K
R
sI A K
E
C
+
–
(
)
1
–
B K
E
=
P A K
E
C
–
(
)′
A K
E
C
–
(
)P
+
BB
′
–
K
E
K
E
′
–
=
Q A K
E
C
–
(
)
A K
E
C
–
(
)′Q
+
K
R
′
K
R
–
=
P A BK
R
–
(
)′
A BK
R
–
(
)P
+
BB
′
–
=
Q A K
E
C
–
(
)
A K
E
C
–
(
)′Q
+
K
R
′
K
R
–
=
P A BK
R
–
(
)′
A BK
R
–
(
)P
+
K
E
K
E
′
–
=
Q A K
E
C
–
(
)
A K
E
C
–
(
)′Q
+
C
–
′C
=