Chapter 3 Multiplicative Error Reduction
© National Instruments Corporation 3-19 Xmath Model Reduction Module
• and stand in the same relation as W(s) and G(s), that is:
–
– With , there holds
or
– With there holds
or
–
– is the stable strictly proper part of .
• The Hankel singular values of (and ) are the first as–r Hankel
singular values of F,
• has the same zeros in Re[s]>0 as G(s).
These properties mean that one is immediately positioned to repeat the
reduction procedure on , with almost all needed quantities being on
hand.
W
ˆs() G
ˆs
W
ˆ′s–()W
ˆs() G
ˆs()G
ˆ′s–()=
P
ˆA
ˆ′FA
ˆFP
ˆ
+B
ˆFB
ˆ′F
–=
BW
ˆP
ˆCG
ˆ
′BG
ˆDG
ˆ
′
+=
B
ˆFD′V1C′+P
ˆDC
ˆFB′WU1Σ1
+()′B
ˆFIv
nsT′–()D′+=
Q
ˆA
ˆFA
ˆF
′Q
ˆ
+C
ˆ′–FC
ˆF
=
CW
ˆDG
ˆ
1–CG
ˆB′W
ˆQ
ˆ
–()=
Iv
nsT′–()Iv
nsT–()
1–C
ˆFDI v
nsT–()[]
1–
=
DC
ˆFB′WU1Σ1B
ˆFD′V1C′+[]′Q
ˆ
–()+{}
DW
ˆD′G
ˆ
=
F
ˆW
ˆ1–s–()()G
ˆs()
F
ˆpF
ˆ
P
ˆΣ1
1–U1
′QV1V1
′QU1Σ1
1–
==
Q
ˆV1
′PU1Σ1Σ1U1
′PV1
==
G
ˆs
G
ˆs