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
( )
C
exp
A t r
+
(
)
Bv r
( )
dr
0
∞
∫
=
G
H
G
H
σ
1
=
G
ˆ
G G
ˆ
–
H
inf
G
ˆ
of degree k
G G
ˆ
–
H
σ
k
1
+
G
( )
=
G
ˆ
G
ˆ
G G
ˆ
–
H
G G
ˆ
–
∞
G G
ˆ
–
∞
σ
j
j
k
1
+
=
∑
≤
G
ˆ
G G
ˆ
–
G
ˆ
G
ˆ
inf
G
ˆ
of degree k and F unstable
G G
ˆ
–
F
–
∞
σ
k
1
+
G
( )
=
