User's Manual

Chapter 3 Multiplicative Error Reduction
© National Instruments Corporation 3-17 Xmath Model Reduction Module
singular values of F(s) larger than 1– ε (refer to steps 1 through 3 of the
Restrictions section). The maximum order permitted is the number of
nonzero eigenvalues of W
c
W
o
larger than ε.
4. Let r be the multiplicity of ν
ns
. The algorithm approximates
by a transfer function matrix of order ns – r, using Hankel norm
approximation. The procedure is slightly different from that used in
ophank( ).
Construct an SVD of :
with Σ
1
of dimension (ns – r) × (ns – r) and nonsingular. Also, obtain
an orthogonal matrix T, satisfying:
where and are the last r rows of and , the state variable
matrices appearing in a balanced realization of . It is
possible to calculate T without evaluating , as it turns out (refer
to [AnJ]), and the algorithm does this. Now with
there holds:
Fs() C
w
sI A–()
1–
B=
F
ˆ
s()
QP v
ns
2
I–
QP v
NS
2
I– U
Σ
1
0
00
= V′ U
1
U
2
[]
Σ
1
0
00
V
1
′
V
2
′
=
B
2
C′
w2
T+ 0=
B
2
C′
w2
B C
w
′
C′
w
sI A–()
1–
B
B
B C
w
F
ˆ
s() D
ˆ
F
C
ˆ
F
sI A
ˆ
F
–()
1–
B
ˆ
F
+=
F
ˆ
p
s() C
ˆ
F
sI A
ˆ
F
–()B
ˆ
F
=
A
ˆ
F
Σ
1
1–
U
1
′
v
ns
2
A′ QAP v
ns
C
w
′
TB′–+[]V
1
=
B
ˆ
F
Σ
1
1–
U
1
′
QB v
ns
C
w
′T
+[]=
C
ˆ
F
C
w
Pv
ns
TB′+()V′=
D
ˆ
F
v–
ns
T=