User's Manual

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 G
r
of the desired degree nsr is obtained.
For example, in the second iteration, is given by:
(3-4)
Consequences of Step 5 and Justification of Step 6
A number of properties are true:
• is of order ns – r, with:
(3-5)
F
ˆ
p
s() F
ˆ
s()
v
ns
1–
Fs() F
ˆ
s()–[]
G
ˆ
W
ˆ
G
ˆ
s() Gs() W′ s–()Fs() F
ˆ
s()–[]–=
W
ˆ
s() Iv
ns
T′–()Iv
ns
T–()
1–
=
Ws() Fs() F
ˆ
s()–[]G′– s–()+{}
G
ˆ
s() DI v
ns
T–()()DC
ˆ
F
B
W
′
UΣ
1
+[]sI A
ˆ
F
–()
1–
B
ˆ
F
=()
W
ˆ
s() Iv
ns
T′–()D′ Iv
ns
T′–()Iv
ns
T–()
1–
+=
C
ˆ
F
sI A
ˆ
F
–()
1–
B
ˆ
F
D′ V
1
′
C′+[]
G
ˆ
W
ˆ
F
ˆ
G
ˆ
s()
^
G
ˆ
s() G
ˆ
s() W
ˆ
′– s–()F
ˆ
p
s() F
ˆ
s()–[]+=
^^
G
ˆ
s()
G
1–
GG
ˆ
–()
∞
v
ns
=