National Instruments Xmath - page 89

Chapter 4 Frequency-Weighted Error Reduction
© National Instruments Corporation 4-19 Xmath Model Reduction Module
6. Check the stability of the closed-loop system with Cr(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 Equat ion 4-5. The grammian equations (Equation 4-8 and
Equation4-9) 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 (Equation4-5) 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 Equation4-6, 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)
Es() KRsI A KEC+–()
1–BK
E
=
PA K
EC–()′AK EC–()P+BB′–KEKE
′
–=
QA K
EC–()AK EC–()′Q+KR
′KR
–=
PA BK
R
–()′ABK
R
–()P+BB′–=
QA K
EC–()AK EC–()′Q+KR
′KR
–=
PA BK
R
–()′ABK
R
–()P+KEKE′–=
QA K
EC–()AK EC–()′Q+C–′C=