Prefix-closed Languages Research Articles

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. >

A note on deciding the controllability of a language K with respect to a language L

It has been shown by P.J. Ramadge and W.M. Wonham (1987) if G is a plant automaton and K contained in L(G) is a prefix-closed language, there exists a supervisor Theta that is complete with respect to G so that L( Theta /G)=K intersection L(G)=K if and only if K is controllable with respect to the plant language L(D). It is shown that if L(G) and K are represented as free-labeled Petri nets, then the controllability of K with respect to L(G) is detectable. This result is a direct consequence of the decidability of the Petri net and reachability problem. In effect, a modeling framework capable of finitely representing a class of infinite-state systems is identified, and controllability is decidable within this framework.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On a weaker notion of controllability of a language K with respect to a language L

A necessary and sufficient condition for a prefix-closed language K contained in Sigma * to be controllable with respect to another prefix-closed language L contained in Sigma * is that K contained in L. A weaker notion of controllability where it is not required that K contained in L is considered here. If L is the prefix-closed language generated by a plant automaton G, then essentially there exists a supervisor Theta that is complete with respect to G such that L( Theta mod G)=K intersection L if and only if K is weakly controllable with respect to L. For an arbitrary modeling formalism it is shown that the inclusion problem is reducible to the problem of deciding the weaker notion of controllability. Therefore, removing the requirement that K contained in L from the original definition of controllability does not help the situation from a decidability viewpoint. This observation is then used to identify modeling formalisms that are not viable for supervisory control of the untimed behaviors of discrete-event dynamic systems. >