Wtbalance( ), Wtbalance( ) -10 – National Instruments NI MATRIXx Xmath User Manual
Page 80
Chapter 4
Frequency-Weighted Error Reduction
4-10
ni.com
(Here, the W
i
and V
i
are submatrices of W,V.) Evidently,
Some manipulation shows that trying to preserve these identities after
approximation of D
L
, N
L
or N
R
, D
R
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
by the 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.
D
L
N
L
V
I
=
and
W
N
R
D
R
I
=
W G G
r
–
(
)
∞
H H
r
–
(
)V
∞
H H
r
–
(
)V
W G G
r
–
(
)
∞