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
ofco ntrollers.
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
atthe controller input , indicates that if C is stabilizing for P and the error
measure is less that 1, then Cr is stabilizing for P. The smaller the error
measure is, the greater the stability robustness.
"match" If T = PC(I+PC)–1 and Tr = PCr(I+PCr)–1 are the two closed-loop transfer
function matrices, then T–Tr to first order in C–Cr is given by
(I+PC)–1P[Cr–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
atcertain frequencies than others; frequency weighting is achieved by
introducing V(s). Frequencies corresponding to larger values of |V(jω)| or
V(jω)V*(jω) will be the frequencies at which T(jω) and Tr(jω) 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.