Chapter 2
Additive Error Reduction
©
National Instruments Corporation
2-17
Thus, the penalty for not being allowed to include
G
u
in the approximation
is an increase in the error bound, by
σ
n
i
+ 1
+ ... +
σ
ns
. A number of
theoretical developments hinge on bounding the Hankel singular values of
G
r
(
s
) and
G
u
(–
s
) in terms of those of
G
(
s
). With
G
r
(
s
) of order
n
i
– 1
, there
holds:
The transfer function matrix
G
u
(
s
), being unstable, does not have Hankel
singular values; however,
G
u
(–
s
) (which is stable) does have Hankel
singular values. They satisfy:
In most cases, the Hankel singular values of
G
(
s
) are distinct. If,
accordingly,
then
G
r
has degree (
i
– 1),
G
u
has degree
ns
–
i
and
(2-4)
Observe that the bound (Equation 2-3 or Equation 2-4), which is not
necessarily obtained, is one half that applying for both balanced truncation
(as implemented by
balmoore( )
or, effectively, by
redschur( )
); it
also is one half that obtained when applying
mreduce
to a balanced
realization. In general, the D matrices of
G
and
G
r
are different, that is,
G
(
∞
)
≠
G
r
(
∞
) (in contrast to
balmoore( )
and
redschur( )
). Similarly,
G
(0)
≠
G
r
(0) in general (in contrast to the result when
mreduce
is applied
to a balanced realization). The price paid for obtaining a smaller error
bound overall through Hankel norm reduction is that one no longer
(normally) secures zero error at
ω
=
∞
or
ω
= 0.
Two special cases should be noted. If
nsr
= 0 then
G
r
(
s
) is a constant only.
This constant can be added onto
G
u
(
s
), so that
G
(
s
) is then being
approximated by a totally unstable transfer function matrix, with error
σ
1
;
this type of approximation is known as Nehari approximation. The second
special case arises when
nsr
=
n
m
– 1
(or
NS
– 1 if the smallest Hankel
singular value has multiplicity 1). In this case,
G
u
(
s
) becomes a constant,
which can then be lumped in with
G
r
(
s
), so that
G
(
s
), of degree
NS
, is then
σ
k
G
r
) σ
k
G
( )
k
≤
(
1 2
…
n
i
1
–
, , ,
=
σ
k
G
u
s
–
( )
[
] σ
n
i
k
+
G
( )
≤
G Gr
–
G
u
–
∞
σ
i
=
G G
r
–
∞
σ
i
σ
i
1
+
...
σ
ns
+
+
+
=
