National Instruments NI MATRIXx Xmath - Manual (Page 67)

Chapter 3 Multiplicative Error Reduction
© National Instruments Corporation 3-21 Xmath Model Reduction Module
For mulhank( ) , this translates for a scalar system into
and
The bounds are double for bst( ) .
The error as a function of frequency is always zero at
ω = ∞ for
bst( )
(or at
ω = 0 if a transformation s → s –1 is used), whereas no such particular
property of the error holds for mulhank( ) .
Imaginary Axis Zeros (Including Zeros at
∞
)
When G(j
ω) is singular (or zero) on the jω axis or at ∞, reduction can be
handled in the same manner as explained for bst( ) .
The key is to use a bilinear transformation [Saf87]. Consider the bilinear
map defined by
where 0 < a < b –1 and mapping G(s) into through
86.9 v
i
dB 20log
10
< Gˆ
nsr
G ⁄
i nsr 1 + =
ns
∑
–
8.69 v
i
i nsr 1 + =
ns
∑
< dB
phase error v
i
radians
i nsr 1 + =
ns
∑
<
s
z a –
bz
–
1
+
------------------- =
z
s a +
bs 1 +
--------------- =
G˜ s ( )
G˜ s ( ) G
s a –
bs – 1 +
-------------------
⎝ ⎠
⎛ ⎞
=
G s ( ) G˜
s a
+
bs 1
+
---------------
⎝ ⎠
⎛ ⎞
=