The underlying difference between the controlling strategies obtained
by the different models is related to the methods' capabilities for
handling sequential specifications. Because we were only able to
obtain the safety controller with TTM's and Petri Nets, these
solutions still allow the pushers to move back and forth without
accomplishing the desired task (bringing the workpiece from position 1
to position 3). Notice that the NCES sequential controller consists of
a single sequence of interim places to follow. This same path is also
allowed by the R&W supervisor as well as several others. Depending
on our objectives for the system, the restriction of the NCES
controller may be undesirable. However, for purposes of implementing
controller code (in a PLC, for example), a separate entity would be
needed to ``prune'' the R&W supervisor to get a specific sequence of
events to be realized by the system. The policies used by this
entity in pruning the supervisor would depend,, again, on the goals for
system behavior.
The R&W theory guarantees that the supervisor is maximally
permissive; that is, it only restricts those events which are not
legal with respect to the specifications and no others. The safety
controller based on Petri Nets is also maximally permissive. With
TTM's and NCES's, however, it is not generally the case that the
safety controllers are maximally permissive. The reason is
that these algorithms use a backward search in the structures that
generate the global states, instead of the ``reachability graph''
itself which is equivalent to R&W's finite state machine.
In terms of the synthesis procedures, it is well known that the R&W algorithm is difficult to implement due to the state explosion problem. (The shuffled plant can grow exponentially in the number of states.) Although explicit representation of all states is what makes the R&W theory complete in terms of its capabilities, it is also what makes the procedure impractical for large, complex systems. With Petri Nets and their compact matrix representation, monitor controllers are efficiently obtained because they require only several matrix operations. Both TTM's and NCES avoid a full search of the state space, thus providing more efficient solutions as well. Furthermore, with NCES it is possible to automatically generate control code for PLC's.