National Instruments NI MATRIXx Xmath - Manual (Page 82)

Chapter 4 Frequency-Weighted Error Reduction
Xmath Model Reduction Module 4-12 ni.com
This rather crude approach to the handling of the unstable part of a
controller is avoided in
fracred( ) , which provides an alternative to
wtbalance( ) for controller reduction, at least for an important family
of controllers.
Algorithm
The major steps of the algorithm are as follows:
1. Check dimension, syntax, stability of SysV , closed-loop stability, and
decomposition of C(s) into the sum of a stable part (poles in Re[s] < 0)
and unstable part (poles in Re[s]
≥ 0);
stable( ) is used for this
purpose.
2. Compute input (right) weight and/or output (left) weight as appropriate
for the specified type.
Table 4-2. Error Measure Interpretation for wtbalance
Type Error Measure Interpretations
"input stab" A stability robustness argument, based on breaking the loop at the controller
output, indicates that if C is stabilizing for P and the error measure is less
than 1, then Cr is stabilizing for P. The smaller the error measure is, the
greater the stability robustness.
"output stab" A similar stability robustness argument, but based on breaking the loop
at the controller input, indicates that if C is stabilizing for P and the error
measure is less that 1, then C
r
is stabilizing for P. The smaller the error
measure is, the greater the stability robustness.
"match" If T = PC(I + PC) –1 and T
r
= PC
r
(I + PC
r
) –1 are the two closed-loop transfer
function matrices, then T – T
r
to first order in C – C
r
is given by
(I + PC) –1 P[C
r
– C][I + PC] –1 , so that the error measure looks at matching
of the closed-loop transfer function matrix.
"match spec" It may be important to match closed-loop transfer function matrices more
at certain frequencies than others; frequency weighting is achieved by
introducing V(s). Frequencies corresponding to larger values of |V(j
ω)| or
V(j
ω)V*(jω) will be the frequencies at which T(jω) and T
r
(j
ω) should have
smaller error.
"input spec" This is the one error measure that is not associated with a plant, or
closed-loop of some kind. It simply allows the user to emphasize certain
frequencies in the reduction procedures.