On the existence of finite state supervisors for arbitrary supervisory control problems

Given two prefix closed languages K, L /spl sube/ /spl Sigma/*, where K /spl sube/ L represents the desired closed-loop behavior and L is the open-loop behavior, there exists a finite-state supervisor that enforces K in the closed loop if and only if there is a regular, prefix-closed language M /spl sube/ /spl Sigma/*, such that: 1) M/spl Sigma//sub u//spl cup/L/spl sube/M, and 2) M/spl cup/L=K. In this paper, we show that this is equivalent to: 1) the controllability of sup/spl lcub/P/spl sube/K/spl cup/L/spl macr//spl verbar/pr(P)=P/spl rcub/ with respect to /spl Sigma/*; and 2) the regularity of sup/spl lcub/P/spl sube/K/spl cup/L/spl macr//spl verbar/pr(P)=P/spl rcub/, where L/spl macr/=/spl Sigma/*/spl minus/L:and pr(/spl middot/) is the set of prefixes of strings in the language argument. We use this property to investigate the issue of deciding the existence of a finite-state supervisor for different representations. We also present some properties of the language sup/spl lcub/P/spl sube/K/spl cup/L/spl macr//spl verbar/pr(P)=P/spl rcub/, along with implications to the synthesis of solutions to the supervisory control problem with the fewest states.<
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