Chapter 2
Additive Error Reduction
2-22
ni.com
We use
sysZ
to denote G(z) and define:
bilinsys=makepoly([-1,a]/makepoly([1,a])
as the mapping from the z-domain to the s-domain. The specification is
reversed because this function uses backward polynomial rotation. Hankel
norm reduction is then applied to H(s), to generate, a stable reduced order
approximation H
r
(s) and unstable H
u
(s) such that:
Here, the
s
ni
are the Hankel singular values of both G(z) and H(s); they are
the same:
We then implement the s-domain equivalent with:
sysS=subsys(sysZ,bilinsys)
There is no simple rule for choosing
α
; the choice
α
= 1 is probably as good
as any. The orders of
G
r
and
G
u
are the same as those of
H
r
and
H
u
,
respectively. The error formulas are as follows:
Impulse Response Error
If
G
r
is determined by the first (single-pass) algorithm, the impulse
response error (for
t
> 0) between the impulse responses of
G
and
G
r
can
be bounded. As shown in Corollary 9.9 of [Glo84], if
G
r
is of degree
i
– 1
and the multiplicity of the
i
th larger singular value
σ
i
of
G
is
r
, then:
H H
r
H
u
–
–
σ
i
=
H H
r
–
σ
i
σ
n
i
1
+
...
σ
ns
+
+
+
=
G
r
z
( )
H
r
α
z
1
–
z
1
+
-----------
⎝
⎠
⎛
⎞
=
G
u
z
( )
H
u
α
z
1
–
z
1
+
-----------
⎝
⎠
⎛
⎞
=
G e
j
ω
(
)
G
r
e
j
ω
(
)
–
G
u
e
j
ω
(
)
–
∞
σ
n
i
=
G e
j
ω
(
)
G
r
e
j
ω
(
)
–
∞
σ
n
i
σ
n
i
1
+
...
σ
ns
+
+
≤
σ
j
G G
r
–
[
] σ
i
G
for
j
≤
1 2 ... 2
i
2
r
+
–
, , ,
=
σ
j i
–
1
+
G
( )
for
j
≤
2
i
1
–
r
...,
ns i
1
–
+
,
+
=
