National Instruments NI MATRIXx Xmath User Manual
Page 53
Chapter 3
Multiplicative Error Reduction
© National Instruments Corporation
3-7
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
1 2
⁄
–
=
S
rbig
V
rbig
V
ebig
S
ebig
1 2
⁄
–
=
A
R
S
lbig
′
AS
rbig
=
A
R
CS
rbig
=
B
R
S
lbig
′
B
=
D
R
D
=
A
R
CS
rbig
=
G
1
–
G G
r
–
(
)
∞
2
v
i
1 v
i
–
-------------
∑
≤