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
–
-------------
∑
≤
