Chapter 4 Frequency-Weighted Error Reduction
Xmath Model Reduction Module 4-10 ni.com
(Here, the Wi and Vi are submatrices of W,V.) Evidently,
Some manipulation shows that trying to preserve these identities after
approximation of DL, NL or NR, DR suggests use of the error measures
and . For further details, refer to [AnM89] and
[LAL90].
In all four fracred( ) options, it is possible to construct (weighted)
Hankel singular values, and to use them as a guide to the likely quality of
approximation. The patterns tend to be different for the four options.
The fracred( ) options are normally different in outcome from the
wtbalance( ) options. However, if the controller has been designed
bythe loop transfer recovery method and is stable, then one of the
fracred( ) options is essentially the same as one of the wtbalance( )
options, refer to [LiA90].
More precisely, if the LTR design is performed with input noise or process
noise weighting tending to infinity, reduction with fracred( ) and
type="left stab", which uses the error measure , leads to
effectively the same reduction as wtbalance () with the type="input
stab". If the LTR design is performed with state or output weighting
tending to infinity (in the index determining the state feedback law),
reduction with fracred( ) and type="right stab" using the error
measure leads to effectively the same reduction as
wtbalance( ) with type="output stab".
wtbalance( )[SysCR,SysCLR,HSV]
=
wtbalance(Sys,SysC,type,{nscr,SysV})
The wtbalance( ) function calculates a frequency weighted balanced
truncation of a system.
wtbalance( ) has two separate uses:
• Reduce the order of a controller C(s) located in a stable closed-loop,
with the plant P(s) known. Frequency-weighted balanced truncation is
used, with the weights involving P(s) and being calculated in a
predominantly standard way.
DLNLVI=and WNR
DR
I=
WG G
r
–()
∞HH
r
–()V∞
HH
r
–()V
WG G
r
–()
∞