Chapter 2 Additive Error Reduction
© National Instruments Corporation 2-13 Xmath Model Reduction Module
Next, Schur decompositions of WcWo are formed with the eigenvalues of
WcWo in ascending and descending order. These eigenvalues are the square
of the Hankel singular values of Sys, and if Sys is nonminimal, some can
be zero.
The matrices VA, VD are orthogonal and Sasc, Sdes are upper triangular. Next,
submatrices are obtained as follows:
and then a singular value decomposition is found:
From these quantities, the transformation matrices used for calculating
SysR are defined:
and the reduced order system is:
An error bound is available. In the continuous-time case it is as follows. Let
G(jω) and GR(jω) be the transfer function matrices of Sys and SysR,
respectively.
For the continuous case:
V′AWcWoVASasc
=
V′DWcWoVDSdes
=
Vlbig VA
0
Insr
=Vrbig VD
Insr
0
=
UebigSebigVebig V′lbigVrbig
=
Slbig VlbigUebigSebig
12⁄
=
Srbig VrbigVebigSebig
12⁄
=
ARSlbig
′ASrbig
=
ARCSrbig
=
BRSlbig
′B=
D
Gjω()GRjω()–∞2σnsr 1+σnsr 2+... σns
+++()≤