National Instruments NI MATRIXx Xmath - Manual (Page 56)

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
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 +
---------------
⎝ ⎠
⎛ ⎞
=