National Instruments Xmath

Chapter 2 Additive Error Reduction

function matrix. Consider the way the associated impulse response maps

inputs defined over (–∞,0] in L2 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)–1B. The norm of the associated operator is the Hankel

norm of G. A key result is that if σ1≥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

ofminimizing , 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=∞

orat ω= 0. Balanced realization truncation gives zero error at ω=∞,

andsingular perturbation of a balanced realization gives zero error

atω=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

∞

∫

GHσ1

–H

infG

ˆofdegree k GG

–Hσk1+G()=

ˆGG

–H

–∞

–∞σj

jk1+=

∑

≤

–

infG

ˆofdegree k and F unstable GG

–F–∞σk1+G()=