National Instruments NI MATRIXx Xmath - Manual (Page 63)

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:
F s ( ) C
w
sI A
–
( ) 1 – B
=
Fˆ s ( )
QP v
ns
2
I –
QP v
NS
2
I – U Σ 1
0
0 0
= V ′ U
1
U
2
[ ] Σ 1
0
0 0
V
1
′
V
2
′
=
B
2
C ′
w2
T + 0 =
B
2
C ′
w2
B C
w
′
C ′
w
s I A –
( ) 1 –
B
B B C
w
Fˆ s ( ) Dˆ
F
Cˆ
F
sI Aˆ
F –
( )
1 –
Bˆ
F + =
Fˆ
p
s ( ) Cˆ
F
sI Aˆ
F –
( )Bˆ
F =
Aˆ
F
Σ
1
1 – U
1
′
v
ns
2
A ′ QAP v
ns
C
w
′
TB ′
– +
[ ]V
1
=
Bˆ
F
Σ
1
1 – U
1
′
QB v
ns
C
w
′T
+
[ ]
=
Cˆ
F
C
w
P v
ns
TB ′
+
( )V′
=
Dˆ
F
v
–
ns
T
=