National Instruments Xmath - 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 WcWo larger than ε.
4. Let r be the multiplicity of νns. 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 Σ1 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() CwsI A–()
1–B=
F
ˆs()
QP vns
2I–
QP vNS
2I–UΣ10
00
=V′U1U2
[]
Σ10
00
V1
′
V2
′
=
B2C′w2T+0=
B2C′w2B Cw
′
C′wsI A–()
1–B
BB Cw
F
ˆs() D ˆFC
ˆFsI A ˆF
–()
1–B
ˆF
+=
F
ˆps() C
ˆFsI A ˆF
–()B
ˆF
=
A
ˆFΣ1
1–U1 ′vns 2A′QAP vnsCw ′TB′–+[]V1
=
B
ˆFΣ1
1–U1 ′QB vnsCw ′T
+[]=
C
ˆFCwPv
nsTB′+()V′=
D
ˆFv–nsT=