National Instruments NI MATRIXx Xmath - Manual (Page 70)

Chapter 3 Multiplicative Error Reduction
Xmath Model Reduction Module 3-24 ni.com
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 a good
approximation of G(s) over an interval [–j
Ω, jΩ] it is desired, then
ε –1 = 5
Ω or 10 Ω are good choices. 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 (0,
ε –1 + j0)
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( ) . Zeros at s =
∞ are
never in this circle, so a procedure for reducing G(s) = 1/d(s) is available.
There is one potential source of failure of the algorithm. Because G(s) is
stable,
certainly will be, as its poles will be in the left half plane circle
on diameter . If acquires a pole outside this circle
(but still in the left half plane of course)—and this appears possible in
principle—G
r
(s) will then acquire a pole in Re [s] >0. Should this difficulty
be encountered, a smaller value of
ε should be used.
Related Functions
singriccati() , ophank() , bst() , hankelsv()
G˜ s ( )
G˜ s ( )
G˜ s ( )
G
r
s ( ) G˜
r
s 1 εs
–
( ) ⁄ ( )
– =
G˜ s ( )
G˜ s ( )
ε
–
1 – j0 0 ,
=
( ) G˜
r
s ( )