National Instruments NI MATRIXx Xmath User Manual
Page 84
Chapter 4
Frequency-Weighted Error Reduction
4-14
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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
=
E x
c
x
c
′
[
]
E x
c
x
c
′
[
]
C j
ω
( ) C
r
j
ω
( )
–
[
]W jω
( )
∞
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
=