User's Manual

Chapter 4 Frequency-Weighted Error Reduction
Xmath Model Reduction Module 4-16 ni.com
3. Only continuous systems are accepted; for discrete systems use
makecontinuous( ) before calling bst( ), then discretize the
result.
Sys=fracred(makecontinuous(SysD));
SysD=discretize(Sys);
Defining and Reducing a Controller
Suppose P(s) = C(sI – A)
–1
B and A – BK
R
and A – K
E
C are stable (where
K
R
is a stabilizing state feedback gain and K
E
a stabilizing observer gain).
A controller for the plant P(s) can be defined by
(with u the plant input and y the plant output). The associated series
compensator under unity negative feedback is
and this may be written as a left or right MFD as follows:
(4-5)
(4-6)
The reduction procedures
"right perf" and "left perf" have similar
rationales. We shall describe
"right perf", refer to [AnM89] and
[LiA86]. The first rationale involves observing that to reduce C(s), one
might as well reduce its numerator and denominator simultaneously, and
then form a new fraction C
r
(s) of lower order than C(s).
This amounts to reducing
(4-7)
x
ˆ
·
Ax
ˆ
Bu K
E
Cx
ˆ
y–()–+=
uK
R
x
ˆ
–=
Cs() K
R
sI A BK
R
K
E
C++–()
1–
K
E
=
Cs() IK
R
sI A K
E
C+–()
1–
B+[]
1–
K
R
sI A K
E
C+–()
1–
K
E
=
Cs() K
R
sI A BK
R
+–()
1–
K
E
ICsIABK
R
+–()
1–
K
E
+[]
1–
=
Es()
K
R
C
sI A BK
R
+–()
1–
K
E
=