Chapter 2 Additive Error Reduction
© National Instruments Corporation 2-19 Xmath Model Reduction Module
and finally:
These four matrices are the constituents of the system matrix of ,
where:
Digression:
This choice is related to the ideas of [Glo84] in the following way;
in[ Glo84], the complete set is identified of satisfying
with having a stable part of order ni–1
. The set is parameterized in
terms of a stable transfer function matrix K(s) which has to satisfy
with C2, B2 being two matrices appearing in the course of the algorithm
of [Glo84], and enjoying the property . The particular
choice
in the algorithm of [Glo84] and flagged in corollary 7.3 of [Glo84] is
equivalent to the previous construction, in the sense of yielding the
same , though the actual formulas used here and in [Glo84] for the
construction procedure are quite different. In a number of situations,
including the case of scalar (SISO)G(s), this is the only choice.
The next step of the algorithm is to call stable( ), to separate into
its stable and unstable parts, call them and , stable( ) will
always assign the matrix to , and the final step of the algorithm is
A
˜SB
1–A11 A12–A22
#A21
–()=
B
˜SB
1–B1A12–A22
#B2
–()=
C
˜C1C2A22
#A21
#
–=
D
˜DC
2A22
#B2
–=
G
˜s()
G
˜s() Grs() Gus()+=
G
˜s()
Gjω()G
˜jω()–∞σni
=
G
˜
C2Ks()B2
′
+0=
IK′jω–()Kjω()–0forallω≤
C2
′C2B2B2
′
=
Ks() C2C′2C2
()
#B2
–=
G
˜s
G
˜s()
G
˜s() G
˜us()
D
˜G
˜rs()