National Instruments NI MATRIXx Xmath User Manual
Page 59
Chapter 3
Multiplicative Error Reduction
© National Instruments Corporation
3-13
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).
Multiplicative approximation of
(along the j
ω-axis) corresponds to
multiplicative approximation of G(s) around a circle in the right half plane,
touching the j
ω-axis at the origin. For those points on the jω-axis near the
circle, there will be good multiplicative approximation of G( j
ω). If it is
desired to have a good approximation of G(s) over an interval [–j
Ω, jΩ],
then a choice such as
ε
–1
= 5
Ω or 10 Ω needs to be made. Reduction then
proceeds as follows:
1.
Form .
2.
Reduce
through
bst( )
.
3.
Form
with:
gsys=subsys(gtildesys(gtildesys,
makep([-eps,-1])/makep[-1,-0]))
Notice that the number of zeros of G(s) in the circle of diameter
sets a lower bound on the degree of G
r
(s)—for such zeros become right half
plane zeros of
, and must be preserved by
bst( )
. Obviously, zeros at
s =
∞ are never in this circle, so a procedure for reducing G(s) = 1/d(s) is
available.
G˜ s
( )
G
s
εs 1
+
--------------
⎝
⎠
⎛
⎞
=
G˜ s
( )
G˜ s
( )
G˜ s
( )
G˜ s
( )
G˜ s
( )
G˜ s
( )
G
r
s
( )
G˜
r
s 1
εs
–
(
)
⁄
(
)
–
=
0
ε
1
–
,
j0
+
(
)s
G˜ s
( )