Hardware Reference
In-Depth Information
a
b
a
a
p
3
p
4
s
3
s
4
b
b
a
a
a
b
b
p
0
p
1
p
2
s
0
s
1
s
2
a
,
b
b
c
d
a
b
c
4
c
4
c
0
c
1
b
a
b
a
b
c
0
c
1
c
2
b
Fig. 15.2
Illustration of Example
15.11
.(
a
)Plant
P
;(
b
) Specification
S
;(
c
) S
ma
llest controller
Pref
C
m
D
Pref
.S /
;(
d
) A controller
C
i
such that
C
m
C
i
C
M
where
C
M
D
.P
[
S/
is the
largest controller shown in Fig.
15.3
In summary, we notice the following facts to compare the two approaches:
1. In the supervisory control approach, (a) necessary and sufficient conditions for
the existence of a controller are established, and (b) a single (smallest) controller
Pref
is derived.
2. In the language approach, (a) necessary and sufficient conditions for the existence
of a controller are established, and (b) the set of all controller
s
is derived,
including the smallest controller
Pref
.S /
.S /
and the largest one
P
\
.P
[
S/
Pref
.
3. Both approaches have the same complexity.
15.2.3
Progressive Solutions Under Full Controllability
Definition 15.2.1.
A state
s
of an automaton
G
is
co-accessible
if there is a path
in the state transition diagram of
to an accepting or final state (in the
supervisory control literature, usually an accepting state is called a
marked
state).
An automaton is
trim
if every state is reachable from the initial state and every
state is co-accessible (i.e., from any state an accepting state can be reached).
G
from
s
Unless otherwise stated, we assume that
are trim automata.
If the languages of the plant and of the specification are not prefix-closed, then
the automaton of the intersection
P
,
S
and
C
P
\
C
is not necessarily a trim automaton, and thus
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