National Instruments NI MATRIXx Xmath User Manual
Page 68
Chapter 3
Multiplicative Error Reduction
3-22
ni.com
The values of G(s) along the j
ω-axis are the same as the values of
around a circle with diameter defined by [a – j0, b
–1
+ j0] on the positive
real axis (refer to Figure 3-2). Also, the values of
along the j
ω-axis
are the same as the values of G(s) around a circle with diameter defined by
[–b
–1
+ j0, –a + j0].
We can implement an arbitrary bilinear transform using the
subsys( )
function, which substitutes a given transfer function for the s- or z-domain
operator, as previously shown.
To implement
use:
gtildesys=subsys(gsys,makep([-b,1]/makep([1,-a])
To implement
use:
gsys=subsys(gtildesys,makep([b,1]/makep([1,a])
Note
The systems substituted in the previous calls to subsys invert the function
specification because these functions use backward polynomial rotation.
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
, again in Re[s] > 0. If all poles of G(s) are inside the circle of diameter
[–b
–1
+ j0, a + j0], 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
+ j0, – a + j0]
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
.
G˜ s
( )
G˜ s
( )
G˜ s
( )
G
s a
–
bs
–
1
+
-------------------
⎝
⎠
⎛
⎞
=
G s
( )
G˜
s a
+
bs 1
+
---------------
⎝
⎠
⎛
⎞
=
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
+
---------------
⎝
⎠
⎛
⎞
=