Chapter 3 Multiplicative Error Reduction
Xmath Model Reduction Module 3-18 ni.com
Note The expression is the strictly proper part of . The matrix
is all pass; this property is not always secured in the multivariable case
when ophank( ) is used to find a Hankel norm approximation of F(s).
5. The algorithm constructs and , which satisfy,
and,
through the state variable formulas
and:
Continue the reduction procedure, starting with , , and
repeating the process till Gr of the desired degree nsr is obtained.
Forexample, in the second iteration, is given by:
(3-4)
Consequences of Step 5 and Justification of Step 6A number of properties are true:
• is of order ns–r, with:
(3-5)
F
ˆps() F
ˆs()
vns
1–Fs() F
ˆs()–[]
G
ˆW
ˆ
G
ˆs() Gs() W′s–()Fs() F
ˆs()–[]–=
W
ˆs() Iv
nsT′–()Iv
nsT–()
1–
=
Ws() Fs() F
ˆs()–[]G′–s–()+{}
G
ˆs() DI v
nsT–()()DC
ˆFBW
′UΣ1
+[]sI A
ˆF
–()
1–B
ˆF
=()
W
ˆs() Iv
nsT′–()D′Iv
nsT′–()Iv
nsT–()
1–
+=
C
ˆFsI A
ˆF
–()
1–B
ˆFD′V1
′C′+[]
G
ˆW
ˆF
ˆ
G
ˆs()
^
G
ˆs() G
ˆs() W
ˆ′��s–()F
ˆps() F
ˆs()–[]+=
^^
G
ˆs()
G1–GG
ˆ
–()
∞vns
=