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Chapter 2 Additive Error Reduction
Xmath Model Reduction Module 2-18 ni.com
being approximated by a stable G
r
(s) with the actual error (as opposed to
just the error bound) satisfying:
Note G
r
is optimal, that is, there is no other G
r
achieving a lower bound.
Onepass Algorithm
The first steps of the algorithm are to obtain the Hankel singular values of
G(s) (by using
hankelsv( )) and identify their multiplicities. (Stability of
G(s) is checked in this process.) If the user has specified
nsr and this does
not coincide with one of 0,n
1
,n
2
, ... an error message is obtained; generally,
all the σ
i
are different, so the occurrence of error messages will be rare.
The next step of the algorithm is to calculate the sum G(s)=G
r
(s)+G
u
(s),
following [SCL90]. (A separate function
ophred( ) is called for this
purpose.) The controllability and observability grammians P and Q are
found in the usual way.
AP + PA′ = –BB′
QA + A′Q = –C′C
and then a singular value decomposition is obtained of the
matrix :
There are precisely n
i
– n
i –1
zero singular values, this being the multiplicity
of σ
n
i
. Next, the following definitions are made:
Gs() G
r
s()–
∞
σ
ns
=
QP σ
n
i
2
I–
U
1
U
2
S
B
0
00
V
1
′
V
2
′
QP σ
n
i
2
I–=
A
11
A
12
A
21
A
22
U
1
′
U
2
′
= σ
n
i
2
A′ QAP+()V
1
V
2
()
B
1
B
2
U
1
′
U
2
′
QB=
C
1
C
2
[]CP V
1
V
2
[]=