User's Manual

Chapter 4 Frequency-Weighted Error Reduction
Xmath Model Reduction Module 4-14 ni.com
and the observability grammian Q, defined in the obvious way, is written as
It is trivial to verify that so that Q
cc
is the
observability gramian of C
s
(s) alone, as well as a submatrix of Q.
The weighted Hankel singular values of C
s
(s) are the square roots of the
eigenvalues of P
cc
Q
cc
. They differ from the usual or unweighted Hankel
singular values because P
cc
is not the controllability gramian of C
s
(s) but
rather a weighted controllability gramian. The usual controllability
gramian can be regarded as when C
s
(s) is excited by white noise.
The weighted controllability gramian is still , but now C
s
(s) is
excited by colored noise, that is, the output of the shaping filter W(s), which
is excited by white noise.
Small weighted Hankel singular values are a pointer to the possibility
of eliminating states from C
s
(s) without incurring a large error in
. No error bound formula is known, however.
The actual reduction procedure is virtually the same as that of
redschur( ), except that P
cc
is used. Thus Schur decompositions of
P
cc
Q
cc
are formed with the eigenvalues in ascending and descending order
The maximum order permitted is the number of nonzero eigenvalues of
P
cc
Q
cc
that are larger than ε.
The matrices V
A
, V
D
are orthogonal and S
asc
and S
des
are upper triangular.
Next, submatrices are obtained as follows:
and then a singular value decomposition is formed:
Q
Q
cc
Q
cw
Q
cw
′
Q
ww
=
Q
cc
A
c
A
c
′
Q
cc
+ C
c
′
– C
c
=
Ex
c
x
c
′
[]
Ex
c
x
c
′
[]
Cjω()C
r
jω()–[]Wjω()
∞
V
A
′
P
cc
Q
cc
V
A
S
asc
=
V
D
′
P
cc
Q
cc
V
D
S
des
=
V
lbig
V
A
0
I
nscr
= V
rbig
V
D
I
nscr
0
=
U
ebig
S
ebig
V
ebig
V
lbig
′
V
rbig
=