User's Manual

Chapter 2 Additive Error Reduction
© National Instruments Corporation 2-7 Xmath Model Reduction Module
function matrix. Consider the way the associated impulse response maps
inputs defined over (–∞,0] in L
2
into outputs, and focus on the output over
[0,∞). Define the input as u(t) for t <0, and set v(t)=u(–t). Define the
output as y(t) for t > 0. Then the mapping is
if G(s)=C(sI-A)
–1
B. The norm of the associated operator is the Hankel
norm of G. A key result is that if σ
1
≥σ
2
≥ ···, are the Hankel singular
values of G(s), then .
To avoid minor confusion, suppose that all Hankel singular values of G are
distinct. Then consider approximating G by some stable of prescribed
degree k much that is minimized. It turns out that
and there is an algorithm available for obtaining . Further, the
optimum which is minimizing does a reasonable job
of minimizing , because it can be shown that
where n =deg G, with this bound subject to the proviso that G and are
allowed to be nonzero and different at s = ∞.
The bound on is one half that applying for balanced truncation.
However,
• It is actual error that is important in practice (not bounds).
• The Hankel norm approximation does not give zero error at ω = ∞
or at ω = 0. Balanced realization truncation gives zero error at ω = ∞,
and singular perturbation of a balanced realization gives zero error
at ω =0.
There is one further connection between optimum Hankel norm
approximation and L
∞
error. If one seeks to approximate G by a sum + F,
with stable and of degree k and with F unstable, then:
yt() CexpAt r+()Bv r()dr
0
∞
∫
=
G
H
G
H
σ
1
=
G
ˆ
GG
ˆ
–
H
inf
G
ˆ
of degree k
GG
ˆ
–
H
σ
k 1+
G()=
G
ˆ
G
ˆ
GG
ˆ
–
H
GG
ˆ
–
∞
GG
ˆ
–
∞
σ
j
jk1+=
∑
≤
G
ˆ
GG
ˆ
–
G
ˆ
G
ˆ
inf
G
ˆ
of degree k and F unstable
GG
ˆ
– F–
∞
σ
k 1+
G()=