On the existence of finite state supervisors for arbitrarysupervisory control problems

摘要

Given two prefix closed languages K, L ? Σ*, where K ? 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 ? Σ*, such that: 1) MΣu∪L?M, and 2) M∪L=K. In this paper, we show that this is equivalent to: 1) the controllability of sup{P?K∪Lˉ|pr(P)=P} with respect to Σ*; and 2) the regularity of sup{P?K∪Lˉ|pr(P)=P}, where Lˉ=Σ*-L:and pr(·) 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{P?K∪Lˉ|pr(P)=P}, along with implications to the synthesis of solutions to the supervisory control problem with the fewest states