Imaginary axis zeros (including zeros at ∞), Imaginary axis zeros (including zeros at – National Instruments NI MATRIXx Xmath User Manual
Page 56
Chapter 3
Multiplicative Error Reduction
3-10
ni.com
which also can be relevant in finding a reduced order model of a plant.
The procedure requires G again to be nonsingular at
ω = ∞, and to have no
j
ω-axis poles. It is as follows:
1.
Form H = G
–1
. If G is described by state-variable matrices A, B, C, D,
then H is described by A – BD
–1
C, BD
–1
, –D
–1
C, D
–1
. H is square,
stable, and of full rank on the j
ω-axis.
2.
Form H
r
of the desired order to minimize approximately:
3.
Set G
r
= H
–1
r
.
Observe that
The reduced order G
r
will have the same poles in Re[s] > 0 as G, and
be minimum phase.
Imaginary Axis Zeros (Including Zeros at
∞
)
We shall now explain how to handle the reduction of G(s) which has a rank
drop at s =
∞ or on the jω-axis. 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:
H
1
–
H H
r
–
(
)
∞
H
1
–
H H
r
–
(
)
I H
1
–
H
r
–
=
I GG
r
1
–
–
=
G
r
G
–
(
)G
r
1
–
=
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
+
---------------
⎝
⎠
⎛
⎞
=