Chapter 6 State-Space Design
© National Instruments Corporation 6-35 Xmath Control Design Module
and σ12 through σn2 are the singular values of the matrix H satisfying
Σ2= H'H. They are termed the Hankel singular values. The σk2 terms are
ordered so that σ12 ≥ σ22 ≥ … ≥ σn2 ≥ 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 σ2k >> σ2k+1
,
the input/output behavior of the states in x2 is much less important than
that of the states in x1. Eliminating the part of the model corresponding to
x2 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
=A11 A12
A21 A22
x1
x2
B1
B2
u+
yC
1C2
[]
x1
s2
Du+=
Σ2Σ1
20
0Σ2
2
=
Σ1
2diagonal σ1
2....σk
2
()=
Σ2
2diagonal σk1+
2....σn
2
()=