Restrictions, Algorithm, Restrictions -4 algorithm -4 – National Instruments NI MATRIXx Xmath User Manual
Page 50
Chapter 3
Multiplicative Error Reduction
3-4
ni.com
The objective of the algorithm is to approximate a high-order stable transfer
function matrix G(s) by a lower-order G
r
(s) with either
inv(g)(g-gr)
or
(g-gr)inv(g)
minimized, under the condition that G
r
is stable and of the
prescribed order.
Restrictions
This function has the following restrictions:
•
The user must ensure that the input system is stable and nonsingular at
s = infinity.
•
The algorithm may be problematic if the input system has a zero on the
j
ω-axis.
•
Only continuous systems are accepted; for discrete systems use
makecontinuous( )
before calling
bst( )
, then discretize the
result.
Sys=bst(makecontinuous(SysD));
SysD=discretize(Sys);
Algorithm
The modifications described in this section allow you to circumvent the
previous restrictions.
The objective of the algorithm is to approximate a high order stable transfer
function matrix G(s) by a lower order G
r
(s) with, in the square G(s) case,
either
or
(approximately) minimized,
under the constraint that G
r
is stable and of prescribed order
nsr
. In case
G is not square but has full row rank, the algorithm seeks to minimize:
Recall that
so that when
,
When G is not square but has full column rank, the algorithm seeks to
minimize:
G G
r
–
(
)G
1
–
∞
G
1
–
G G
r
–
(
)
∞
G G
r
–
(
)
*
GG
*
(
)
1
–
G G
r
–
(
)
∞
X
*
s
( )
X
′ s
–
( )
=
s
j
ω
=
X
*
j
ω
( )
X
*
j
ω
( )
=
G G
r
–
(
) G
*
G
(
)
1
–
G G
r
–
(
)
*
∞