National Instruments NI MATRIXx Xmath - Manual (Page 25)

Chapter 2 Additive Error Reduction
Xmath Model Reduction Module 2-2 ni.com
Truncation of Balanced Realizations
A group of functions can be used to achieve a reduction through truncation
of a balanced realization. This means that if the original system is
(2-1)
and the realization is internally balanced, then a truncation is provided by
The functions in question are:
• balmoore( )
• balance( ) (refer to the Xmath Help)
• truncate( )
• redschur( )
One only can speak of internally balanced realizations for systems which
are stable; if the aim is to reduce a transfer function matrix G(s) which
contains unstable poles, one must additively decompose it into a stable part
and unstable part, reduce the stable part, and then add the unstable part back
in. The function
stable( ) , described in Chapter 5, Utilities , can be used
to decompose G(s). Thus:
G(s) = G
s
(s) + G
u
(s)(G
s
(s) stable, G
u
(s) unstable)
G
sr
(s) = found by algorithm (reduction of G
s
(s))
G
r
(s) = G
sr
(s) + G
u
(s) (reduction of G(s))
x·
1
x·
2
A
11
A
12
A
21
A
22
x
1
x
2
B
1
B
2
u
+ =
y C
1
C
2
x D
u
+ =
x·
1
A
11
x
1
B
1
u + =
y C
1
x
1
Du + =