Chapter 2 Additive Error Reduction
Xmath Model Reduction Module 2-18 ni.com
being approximated by a stable Gr(s) with the actual error (as opposed to
just the error bound) satisfying:
Note Gr is optimal, that is, there is no other Gr 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,n1,n2, ... an error message is obtained; generally,
all the σi are different, so the occurrence of error messages will be rare.
Thenext step of the algorithm is to calculate the sum G(s)=Gr(s)+Gu(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 ni–ni–1
zero singular values, this being the multiplicity
of σni. Next, the following definitions are made:
Gs() Grs()–∞σns
=
QP σni
2I–
U1U2
SB0
00
V1′
V2′QP σni
2I–=
A11 A12
A21 A22
U1′
U2′
=σni
2A′QAP+()V1V2
()
B1
B2
U1′
U2′QB=
C1C2
[]CP V1V2
[]=