Chapter 3
Multiplicative Error Reduction
3-8
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state-variable representation of
G
. In this case, the user is effectively asking
for
G
r
=
G
. When the phase matrix has repeated Hankel singular values,
they must all be included or all excluded from the model, that is,
ν
nsr
=
ν
nsr + 1
is not permitted; the algorithm checks for this.
The number of
ν
i
equal to 1 is the number of zeros in
Re
[
s
]>0 of
G
(
s
), and
as mentioned already, these zeros remain as zeros of
G
r
(
s
).
If
error
is specified, then the error bound formula (Equation 3-2) in
conjunction with the
ν
i
values from step 3 is used to define
nsr
for step 4.
For nonsquare
G
with more columns than rows, the error formula is:
If the user is presented with the
ν
i
, the error formula provides a basis for
intelligently choosing
nsr
. However, the error bound is not guaranteed to
be tight, except when
nsr
=
ns
– 1.
Securing Zero Error at DC
The error
G
–1
(
G
–
G
r
) as a function of frequency is always zero at
ω
=
∞
.
When the algorithm is being used to approximate a high order plant by a
low order plant, it may be preferable to secure zero error at
ω
= 0. A method
for doing this is discussed in [GrA90]; for our purposes:
1.
We need a bilinear transformation of sys = 1/z. Given
G
(
s
) we generate
H
(
s
) through:
bilinsys=makepoly([b3,b4]/makepoly([b1,b2])
sys=subsys(sys,bilinsys)
2.
Reduce with the previous algorithm:
[sr,nsr,hsv] = bst(sys)
3.
Use the bilinear transformation s = 1/z again:
[sr1,nsr1] = bilinear(sr,nsr,[0,1,1,0])
The
ν
i
are the same for
G
(
s
) and
H
(
s
) =
G
(
s
–1
). The error bound formula is
the same;
H
is stable and
H
(
j
ω
)
H
'(–
j
ω
) of full rank for all
ω
including
ω
=
∞
if and only if
G
has the same property; right half plane zeros of
G
are
still preserved by the algorithm. The error
G
–1
(
G
–
G
r
), though now zero at
ω
= 0, is in general nonzero at
ω
=
∞
.
G G
r
–
(
)
*
G
*
G
(
)
1
–
G G
r
–
(
)
∞
1 2
⁄
2
v
i
1
v
i
–
-------------
i
nsr
1
+
=
ns
∑
≤
