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Chapter 4
Frequency-Weighted Error Reduction
4-16
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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
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
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)
xˆ
·
Axˆ Bu K
E
Cxˆ y
–
(
)
–
+
=
u
K
R
xˆ
–
=
C s
( )
K
R
sI A BK
R
K
E
C
+
+
–
(
)
1
–
K
E
=
C s
( )
I K
R
sI A K
E
C
+
–
(
)
1
–
B
+
[
]
1
–
K
R
sI A K
E
C
+
–
(
)
1
–
K
E
=
C s
( )
K
R
sI A BK
R
+
–
(
)
1
–
K
E
I C sI A BK
R
+
–
(
)
1
–
K
E
+
[
]
1
–
=
E s
( )
K
R
C
sI A BK
R
+
–
(
)
1
–
K
E
=