Chapter 3 Multiplicative Error Reduction
Xmath Model Reduction Module 3-22 ni.com
The values of G(s) along the jω-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 Figure3-2). Also, the values of along the jω-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 jω-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 jω-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 jω-axis, including s=∞.
The steps of the algorithm, when G(s) has a zero on the jω-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=jω, including ω=∞.
3. Form as shown for gsys.
G
˜s()
G
˜s()
G
˜s() Gsa–
bs–1+
-------------------
=
Gs() G
˜sa+
bs 1+
---------------
=
G
˜s()
G
˜s()
G
˜s()
G
˜s()
G
˜s()
G
˜s() Gsa–
bs–1+
-------------------
=
G
˜s() G
˜rs() G
˜s()
Grs() G
˜r
sa+
bs 1+
---------------
=