Chapter 3
Multiplicative Error Reduction
3-12
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Any zero (or rank reduction) on the
j
ω
-axis of
G
(
s
) becomes a zero (or rank
reduction) in
Re
[
s
] > 0 of
, and if
G
(
s
) has a zero (or rank reduction)
at infinity, this is shifted to a zero (or rank reduction) of
at the point
b
–1
, (in
Re
[
s
] > 0). If all poles of
G
(
s
) are inside the circle of diameter
[–
b
–1
+
j
0,
a
+
j
0], all poles of
will be in
Re
[
s
] < 0, and if
G
(
s
) has no
zero (or rank reduction) on this circle,
will have no zero (or rank
reduction) on the
j
ω
-axis, including
ω
=
∞
.
If
G
(
s
) is nonsingular for almost all values of
s
, it will be nonsingular or
have no zero or rank reduction on the circle of diameter [–
b
–1
+
j
0, –
a
+
j
0]
for almost all choices of
a
,
b
. If
a
and
b
are chosen small enough,
G
(
s
) will
have all its poles inside this circle and no zero or rank reduction on it, while
then will have all poles in
Re
[
s
] < 0 and no zero or rank reduction on
the
j
ω
-axis, including
s
=
∞
. The steps of the algorithm, when
G
(
s
) has a
zero on the
j
ω
-axis or at
s
=
∞
, are as follows:
1.
For small
a
,
b
with 0 <
a
<
b
–1
, form
as shown for
gtildesys
.
2.
Reduce
to
, this being possible because
is stable and
has full rank on
s
=
j
ω
, including
ω
=
∞.
3.
Form
as shown for
gsys
.
The error
will be overbounded by the error
, and
G
r
will contain the same zeros in
Re
[
s
]
≥
0 as
G
.
If there is no zero (or rank reduction) of
G
(
s
) at the origin, one can take
a
= 0 and
b
–1
= bandwidth over which a good approximation of
G
(
s
) is
needed, and at the very least
b
–1
sufficiently large that the poles of
G
(
s
)
lie in the circle of diameter [–
b
–1
+
j
0, –
a
+
j
0]. If there is a zero or rank
reduction at the origin, one can replace
a
= 0 by
a
=
b
. It is possible to take
b
too small, or, if there is a zero at the origin, to take
a
too small. The user
will be presented with an error message that there is a
j
ω
-axis zero and/or
that the Riccati equation solution may be in error. The basic explanation is
that as
b
→
0, and thus
a
→
0, the zeros of
approach those of
G
(
s
).
Thus, for sufficiently small
b
, one or more zeros of
may be identified
as lying on the imaginary axis. The remedy is to increase
a
and/or
b
above
the desirable values.
The procedure for handling
j
ω
-axis zeros or zeros at infinity will be
deficient if the number of such zeros is the same as the order of
G
(
s
)—for
example, if
G
(
s
) = 1/
d
(
s
), for some stable
d
(
s
). In this case, it is possible
G
˜
s
( )
G
˜
s
( )
G
˜
s
( )
G
˜
s
( )
G
˜
s
( )
G
˜
s
( )
G
s a
–
bs
–
1
+
-------------------
⎝
⎠
⎛
⎞
=
G
˜
s
( )
G
˜
r
s
( )
G
˜
s
( )
G
r
s
( )
G
˜
r
s a
+
bs
1
+
---------------
⎝
⎠
⎛
⎞
=
G
1
–
G G
r
–
(
)
∞
G
˜
1
–
G
˜
G
˜
r
–
(
)
∞
G
˜
s
( )
G
˜
s
( )
