Hardware Reference
In-Depth Information
Pref
Pref
Given that
˛
2
cProg
..P
[
S/
/
(because
C
i
cProg
..P
[
S/
/
)
Pref
that is progressive by construction,
P
\
cProg
..P
[
S/
/
reaches under
˛
a co-accessible state. So by defin
iti
on of co-accessible state, there is a string
ˇ
Pref
such that
˛ˇ
leads
P
\
cProg
..P
[
S/
/
to an accepting state, i.e.,
˛ˇ
2
Pref
P
\
cProg
..P
[
S/
/
D
S
.
Finally, from
˛ˇ
2
P
,
˛ˇ
2
S
,and
P
\
C
i
D
S
(
C
i
is a solution), it follows that
˛ˇ
2
C
i
, contradicting that
P
\
C
i
under
˛
reaches a state that is not co-accessible.
t
15.3
Supervisory Control with Partial Controllability
The set
˙
of actions is partitioned into two sets,
˙
D
˙
c
[
˙
u
c
:theset
˙
c
˙
u
c
of uncontrollable actions. When an
uncontrollable action occurs in the plant the controller cannot disable this action.
of controllable actions and the set
15.3.1
Supervisory Control Approach
Definition 15.3.1.
Let
S
and
P
D
Pref
.P /
be languages over the event alphabet
˙
,con
˙
u
c
˙
.
S
is said to be
controllable
with respect to
P
and
˙
u
c
,ifforall
strings
s
2
Pref
.S /
and for all events
2
˙
u
c
it is
s
2
P
)
s
2
Pref
.S /;
(controllability condition) equivalent to
Pref
.S /˙
u
c
\
P
Pref
.S /:
When the controllability condition is valid,
Pref
.S /
is a controller that solves the
supervision problem under partial controllability.
15.3.2
Equation Solving Approach
When modeling the problem of partial controllability by solving language equa-
tions, one should characterize all the solutions that do not block the uncontrollable
actions, i.e., to determine necessary and sufficient conditions for a prefix-closed
language
C
over
˙
to be a controller under partial controllability.
Definition 15.3.2.
A prefix-closed language
C
over
˙
is a
controller
for
S
, with
respect to a plant
P
and a set of uncontrollable events
˙
u
c
,if
P
\
C
D
S
,andfor
each state
.p; c/
2
P
\
C
and each
2
˙
u
c
, the following implication holds:
if
is an event defined at state
p
then
is an event defined at state
s
.
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