On the Existence of Finite State Supervisors for

Given two preex closed languages K, L , where K L represents the desired closed-loop behavior and L is the open-loop behavior, there exists a nite-state supervisor that enforces K in the closed-loop if and only if there is a regular, preex-closed language M , such that (i) M u \ L M, and (ii) M \ L = K (cf. 9] for an equivalent deenition). In this paper we show that this is equivalent (i) the controllability of supfP K L j pr(P) = Pg with respect to , and (ii) the regularity of supfP K L j pr(P) = Pg, where L =-L and pr() is the set of preexes of strings in the language argument. We use this property to investigate the issue of deciding the existence of a nite-state supervisor for diierent representations. We also present some properties of the language supfP K L j pr(P) = Pg along with implications to the synthesis of solutions to the supervisory control problem with the fewest states.