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.
Theprocedure 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–1C, BD–1, –D–1C, D–1. H is square,
stable, and of full rank on the jω-axis.
2. Form Hr of the desired order to minimize approximately:
3. Set Gr = H–1r .
Observe that
The reduced order Gr 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:
H1–HH
r
–()
∞
H1–HH
r
–()IH
1–Hr
–=
IGG
r
1–
–=
GrG–()Gr
1–
=
sza–
bz–1+
-------------------=
zsa+
bs 1+
---------------=
G
˜s()
G
˜s() Gsa–
bs–1+
-------------------
=
Gs() G
˜sa+
bs 1+
---------------
=