Chapter 3
Multiplicative Error Reduction
©
National Instruments Corporation
3-23
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. In these
cases an error message results, saying 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 previous 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
again with a bilinear transformation to secure multiplicative
approximations over a limited frequency band. Suppose that
Create a system that corresponds to
with:
gtildesys=subs(gsys,(makep([-eps,1])/makep([1,-]))
bilinsys=makep([eps,1])/makep([1,0])
sys=subsys(sys,bilinsys)
Under this transformation:
•
Values of
G
(
s
) along the
j
ω
-axis correspond to values of
around
a circle in the left half plane on diameter (–
ε
–1
+
j
0, 0).
•
Values of
along the
j
ω
-axis correspond to values of
G
(
s
) around
a circle in the right half plane on diameter (0,
ε
–1
+
j
0).
G
1
–
G G
r
–
(
)
∞
G
˜
1
–
G
˜
G
˜
r
–
(
)
∞
G
˜
s
( )
G
˜
s
( )
G
˜
s
( )
G
s
ε
s
1
+
--------------
⎝
⎠
⎛
⎞
=
G
˜
s
( )
G
˜
s
( )
G
˜
s
( )
