National Instruments NI MATRIXx Xmath User Manual
Page 30
Chapter 2
Additive Error Reduction
© National Instruments Corporation
2-7
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:
y t
( )
CexpA t r
+
(
)Bv r
( )dr
0
∞
∫
=
G
H
G
H
σ
1
=
Gˆ
G Gˆ
–
H
inf
Gˆ of degree k
G Gˆ
–
H
σ
k 1
+
G
( )
=
Gˆ
Gˆ
G Gˆ
–
H
G Gˆ
–
∞
G Gˆ
–
∞
σ
j
j
k 1
+
=
∑
≤
Gˆ
G Gˆ
–
Gˆ
Gˆ
inf
Gˆ of degree k and F unstable
G Gˆ
–
F
–
∞
σ
k 1
+
G
( )
=