User's Manual

Chapter 4 Frequency-Weighted Error Reduction
Xmath Model Reduction Module 4-20 ni.com
Additional Background
A discussion of the stability robustness measure can be found in [AnM89]
and [LAL90]. The idea can be understood with reference to the transfer
functions E(s) and E
r
(s) used in discussing type="right perf". It is
possible to argue (through block diagram manipulation) that
• C(s) stabilizes P(s) when E(s) stabilizes (as a series compensator) with
unity negative feedback .
• E
r
(s) also will stabilize [P(s)I], and then C
r
(s) will stabilize P(s),
provided
(4-14)
Accordingly, it makes sense to try to reduce E by frequency-weighted
balanced truncation. When this is done, the controllability grammian for
E(s) remains unaltered, while the observability grammian is altered. (Hence
Equation 4-5, at least with Q
yy
= I, and Equation 4-12 are the same while
Equation 4-6 and Equation 4-13 are quite different.) The calculations
leading to Equation 4-13 are set out in [LAL90].
The argument for
type="left perf" is dual. Another insight into
Equation 4-14 is provided by relations set out in [NJB84]. There, it is
established (in a somewhat broader context) that
The left matrix is the weighting matrix in Equation 4-14; the right matrix is
the numerator of C(jω) stacked on the denominator, or alternatively
E(jω) +
This formula then suggests the desirability of retaining the weight in the
approximation of E(jω) by E
r
(jω).
P
ˆ
s()
Ps()I
=
CjωIA– K
E
C+()
1–
BICjωIA– K
E
C+()
1–
K
E
–
Ejω()E
e
jω()–[]
∞
1<
CjωIA– K
E
C+()
1–
BICjωIA– K
E
C+()
1–
K
E
–{}
K
r
sI A BK
R
+–()
1–
K
E
ICjωIA– BK
R
+()
1–
K
E
+
× I=
0
I