National Instruments NI MATRIXx Xmath User Manual
Page 69
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
+ j0, –a + j0]. 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
+ j0, 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
+ j0).
G
1
–
G G
r
–
(
)
∞
G˜
1
–
G˜ G˜
r
–
(
)
∞
G˜ s
( )
G˜ s
( )
G˜ s
( )
G
s
εs 1
+
--------------
⎝
⎠
⎛
⎞
=
G˜ s
( )
G˜ s
( )
G˜ s
( )