User's Manual

Chapter 3 Multiplicative Error Reduction
© National Instruments Corporation 3-7 Xmath Model Reduction Module
strictly proper stable part of θ(s), as the square roots of the eigenvalues
of PQ. Call these quantities ν
i
. The Schur decompositions are,
where V
A
, V
D
are orthogonal and S
asc
, S
des
are upper triangular.
4. Define submatrices as follows, assuming the dimension of the reduced
order system
nsr is known:
Determine a singular value decomposition,
and then define transformation matrices:
The reduced order system G
r
is:
where step 4 is identical with that used in
redschur( ), except
the matrices P, Q which determine V
A
, V
D
and so forth, are the
controllability and observability grammians of C
W
(sI – A)
–1
B rather
than of C(sI – A)
–1
B, the controllability grammian of G(s) and the
observability grammian of W(s).
The error formula [WaS90] is:
(3-2)
All ν
i
obey ν
i
≤ 1. One can only eliminate ν
i
where ν
i
< 1. Hence, if nsr is
chosen so that ν
nsr + 1
= 1, the algorithm produces an error message. The
algorithm also checks that
nsr does not exceed the dimension of a minimal
V
A
′
PQV
A
S
asc
= V
D
′
PQV
D
S
des
=
V
lbig
V
A
0
I
nsr
= V
rbig
V
D
I
nsr
0
=
U
ebig
S
ebig
V
ebig
V
lbig
′
V
rbig
=
S
lbig
V
lbig
U
ebig
S
ebig
12⁄–
=
S
rbig
V
rbig
V
ebig
S
ebig
12⁄–
=
A
R
S
lbig
′
AS
rbig
=
A
R
CS
rbig
=
B
R
S
lbig
′
B=
D
R
D=A
R
CS
rbig
=
G
1–
GG
r
–()
∞
2
v
i
1 v
i
–
-------------
∑
≤