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Chapter 3 Multiplicative Error Reduction
Xmath Model Reduction Module 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
jω-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 jω-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–
HH
r
–()
∞
H
1–
HH
r
–()IH
1–
H
r
–=
IGG
r
1–
–=
G
r
G–()G
r
1–
=
s
za–
bz– 1+
-------------------=
z
sa+
bs 1+
---------------=
G
˜
s()
G
˜
s() G
sa–
bs– 1+
-------------------
=
Gs() G
˜
sa+
bs 1+
---------------
=