On ω-controllability and ω-normality of deds

In this paper, we address the supervisory synthesis problem for controlling the sequential behaviors of Discrete Event Dynamical Systems (DEDS´s) under complete as well as partial information through the use of synchronous composition of the plants and the supervisors [7], [12]. We present the notion of complete languages and show its close relation to ω-languages. We prove that the supremal (closed,) complete and controllable sublanguage of a given language exists and present an algorithm to compute it. We 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 sublanguge. The notions of ω-observability and ω-normality are introduced. A necessary and sufficient condition for the existence of a supervisor in case of partial observation is presented in terms of ω-observability. A closed form expression for the supremal ω-normal sublanguage in terms of the supremal closed, complete and normal sublanguage is also presented.