On supervisory control of sequential behaviors

The authors address the supervisory synthesis problem of controlling the sequential behaviors of discrete-event dynamical systems (DEDSs) under complete and partial information through the use of synchronous composition of the plants and the supervisors. The authors present the notion of complete languages, discuss some of its algebraic properties, and show its close relation to ω-languages. The authors prove that the supremal (closed) complete and controllable sublanguage of a given language exists, and present an algorithm to compute it. They present a closed-form expression for the supremal ω-controllable sublanguage of a given ω-language in terms of the supremal (closed) complete and controllable sublanguage. This closed-form expression suggests that certain operations on a given ω-language can alternatively be achieved by performing certain other similar operations on its prefix (which is a finite language) and then taking the limit (to obtain the desired ω-language). A necessary and sufficient condition for the existence of a supervisor in case of partial observation is presented in terms of ω-observability. Notion of ω-normality is also introduced, and a closed-form expression for the supremal ω-normal sublanguage, in terms of the supremal closed, complete, and normal sublanguage, is presented