National Instruments NI MATRIXx Xmath User Manual
Page 42
Chapter 2
Additive Error Reduction
© National Instruments Corporation
2-19
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 n
i – 1
. The set is parameterized in
terms of a stable transfer function matrix K(s) which has to satisfy
with C
2
, B
2
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˜
S
B
1
–
A
11
A
12
–
A
22
#
A
21
–
(
)
=
B˜
S
B
1
–
B
1
A
12
–
A
22
#
B
2
–
(
)
=
C˜
C
1
C
2
A
22
#
A
21
#
–
=
D˜
D C
2
A
22
#
B
2
–
=
G˜ s
( )
G˜ s
( )
G
r
s
( ) G
u
s
( )
+
=
G˜ s
( )
G j
ω
( ) G˜ jω
( )
–
∞
σ
n
i
=
G˜
C
2
K s
( )B
2
′
+
0
=
I K
′ jω
–
(
)K jω
( )
–
0 for all
ω
≤
C
2
′
C
2
B
2
B
2
′
=
K s
( )
C
2
C
′
2
C
2
(
)
#
B
2
–
=
G˜
s
G˜ s
( )
G˜ s
( )
G˜
u
s
( )
D˜
G˜
r
s
( )