Supervisory Control - The Unknown Component Problem: Theory and Applications - page 235

Fig. 15.1 Supervisory
control topology

M P

U V

S

M C

M S

Even though the theory of supervisory control holds also for discrete event
systems whose number of states is not finite, for reasons of implementability we
restrict ourselves to regular languages and the finite automata that generate them.
In this chapter we will denote with the same symbol a regular language and
its related automaton, unless it is required to distinguish them. When we denote
operations over languages we mean that they are performed on their representations
as automata.

15.2.1 Special Case: S and P Prefix-Closed

Suppose for the sake of simplicity that S and P are prefix-closed languages.

15.2.1.1 Supervisory Control Approach

Given the plant and the specification as prefix-closed languages P and S over the

alphabet ˙ , we would like to derive a prefix-closed language C of the controller

P \ C D S S P

such that . A controller language C exists if and only if .

S P Proposition 15.1. A controller exists if and only if . If so, the prefix-closed

language S can be chosen as the language of the controller.

S 6 P S \ P Proposition 15.2. If , i.e., a controller does not exist, then is the

supremal controllable sublanguage of S , i.e., the largest language contained in S

for which a controller exists.