Filter
Top Products
Chapter 2 Additive Error Reduction
Xmath Model Reduction Module 2-20 ni.com
to choose the D matrix of G
r
(s), by splitting between G
r
(s) and G
u
(s).
This is done by using a separate function
ophiter( ). Suppose G
u
(s) is
the unstable output of
stable( ), and let K(s)=G
u
(–s). By applying the
multipass Hankel reduction algorithm, described further below, K(s) is
reduced to the constant K
0
(the approximation), which satisfies,
that is, if it is larger than,
then one chooses:
This ensures satisfaction of the error bound for G – G
r
given previously,
because:
Multipass Algorithm
We now explain the multipass algorithm. For simplicity in first explaining
the idea, suppose that the Hankel singular values at every stage or pass are
distinct.
1. Find a stable order ns – 1 approximation G
n–1
(s) of G(s) with:
(This can be achieved by the algorithm already given, and there is no
unstable part of the approximation.)
D
˜
Ks() K
0
–
∞
σ
1
K() ... σσ
n
s
n
i
–
K()++≤
σ≤
n
i
1+
G() ... σ
n
s
G()++
G
u
s–()K
0
–
∞
σ
k
G()
kn
i
1+=
n
s
∑
≤
G
r
G
˜
r
K
0
+=
G
u
G
˜
u
K
0
+=
GG
r
–
∞
GG
˜
r
G
˜
u
–– G
˜
u
K
0
–()+
∞
=
GG
˜
r
G
˜
u
––
∞
= KK
0
–
∞
+
σ
n
i
G() σ
n
i
1+
G() ... σ
n
s
G()+++≤
Gjω()G
ns 1–
jω–
∞
σ
ns
G()=