User's Manual

Chapter 4 Frequency-Weighted Error Reduction
© National Instruments Corporation 4-11 Xmath Model Reduction Module
• Reduce the order of a transfer function matrix C(s) through
frequency-weighted balanced truncation, a stable frequency weight
V(s) being prescribed.
The syntax is more accented towards the first use. For the second use,
the user should set S =0, NS = 0. This results in (automatically)
SCLR = NSCLR = 0. The user will also select the
type="input
spec".
Let C
r
(s) be the reduced order approximation of C(s) which is being
sought. Its order is either specified in advance, or the user responds to
a prompt after presentation of the weighted Hankel singular values.
Then the different types concentrate on (approximately) minimizing
certain error measures, through frequency weighted balanced
truncation. These are shown in Table 4-1.
These error measures have certain interpretations, as shown in Table 4-2.
In case C(s) is not a compensator in a closed-loop and the error measure
is of interest, you can work with
type="input spec" and C', V' in lieu
of C and V.
There is no restriction on the stability of C(s) [or indeed of P(s)] in the
algorithm, though if C(s) is a controller the closed-loop must be stabilizing.
Also, V(s) must be stable. Hence all weights (on the left or right of
C(jω)–C
r
(jω) in the error measures) will be stable. The algorithm,
however, treats unstable C(s) in a special way, by reducing only the stable
part of C(s) (under additive decomposition) and copying the unstable part
into C
r
(s).
Table 4-1. Types versus Error Measures
Type Error Measure
"input stab"
"output stab"
"match"
"match spec"
"input spec"
CC
r
–[]PI CP+[]
1–
∞
IPC+[]
1–
PC C
r
–[]
∞
IPC+[]
1–
PC C
r
–[]IPC+[]
1–
∞
IPC+[]
1–
PC C
r
–[]IPC+[]
1–
V
∞
CC
r
–[]V
∞
Vjω()Cjω()C
r
jω()–[]
∞