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 Qcc is the
observability gramian of Cs(s) alone, as well as a submatrix of Q.
The weighted Hankel singular values of Cs(s) are the square roots of the
eigenvalues of PccQcc. They differ from the usual or unweighted Hankel
singular values because Pcc is not the controllability gramian of Cs(s) but
rather a weighted controllability gramian. The usual controllability
gramian can be regarded as when Cs(s) is excited by white noise.
The weighted controllability gramian is still , but now Cs(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 Cs(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 Pcc is used. Thus Schur decompositions of
PccQcc are formed with the eigenvalues in ascending and descending order
The maximum order permitted is the number of nonzero eigenvalues of
PccQcc that are larger than ε.
The matrices VA, VD are orthogonal and Sasc and Sdes are upper triangular.
Next, submatrices are obtained as follows:
and then a singular value decomposition is formed:
QQcc Qcw
Qcw
′Qww
=
QccAcAc
′Qcc
+Cc
′
–Cc
=
Ex
cxc
′
[]
Ex
cxc
′
[]
Cjω()Crjω()–[]Wjω()
∞
VA
′PccQccVASasc
=
VD
′PccQccVDSdes
=
Vlbig VA
0
Inscr
=Vrbig VD
Inscr
0
=
UebigSebigVebig Vlbig
′Vrbig
=