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Chapter 6 State-Space Design
© National Instruments Corporation 6-35 Xmath Control Design Module
and σ
1
2
through σ
n
2
are the singular values of the matrix H satisfying
Σ
2
= H'H. They are termed the Hankel singular values. The σ
k
2
terms are
ordered so that σ
1
2
≥ σ
2
2
≥ … ≥ σ
n
2
≥ 0.
The balanced system essentially gives the best compromise between how
well conditioned the system is with regard to controllability and
observability.
For model reduction problems, consider the balanced model partition as:
with
The essence of a balanced model reduction is that if σ
2
k
>> σ
2
k+1
,
the input/output behavior of the states in x
2
is much less important than
that of the states in x
1
. Eliminating the part of the model corresponding to
x
2
will result in a reduced-order model which retains the most important
input-output characteristics of the original system.
balance( )
[SysB,HSV,T] = balance(Sys)
The balance( ) function performs input/output balancing on a linear
system, returning the system transformed to a balanced form as
SysB. HSV
contains the second-order modes of the balanced system, or the singular
values of
H, where H is as defined previously.
x
·
1
x
·
2
=
A
11
A
12
A
21
A
22
x
1
x
2
B
1
B
2
u+
yC
1
C
2
[]
x
1
s
2
Du+=
Σ
2
Σ
1
2
0
0 Σ
2
2
=
Σ
1
2
diagonal σ
1
2
....σ
k
2
()=
Σ
2
2
diagonal σ
k 1+
2
....σ
n
2
()=