Chapter 2 Additive Error Reduction
Xmath Model Reduction Module 2-20 ni.com
to choose the D matrix of Gr(s), by splitting between Gr(s) and Gu(s).
This is done by using a separate function ophiter( ). Suppose Gu(s) is
the unstable output of stable( ), and let K(s)=Gu(–s). By applying the
multipass Hankel reduction algorithm, described further below, K(s) is
reduced to the constant K0 (the approximation), which satisfies,
that is, if it is larger than,
then one chooses:
This ensures satisfaction of the error bound for G–Gr 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 Gn–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() K0
–∞σ1K() ... σσ
nsni
–K()++≤
σ≤ ni1+G()... σnsG()++
Gus–()K0
–∞σkG()
kn
i1+=
ns
∑
≤
GrG
˜rK0
+=
GuG
˜uK0
+=
GG
r
–∞GG
˜rG
˜u
–– G
˜uK0
–()+∞
=
GG
˜rG
˜u
–– ∞
=KK
0
–∞
+
σniG() σ
ni1+G() ... σnsG()+++≤
Gjω()Gns 1–jω–∞σns G()=