Quantitatively Nonblocking Supervisory Control of Discrete-Event Systems

In this paper, we propose two new nonblocking properties of automata as quantitative measures of maximal distances to marker states. The first property, called {\em quantitative nonblockingness}, captures the practical requirement that at least one of the marker states (representing e.g. task completion) be reachable within a prescribed number of steps from any non-marker state and following any trajectory of the system. The second property, called {\em heterogeneously quantitative nonblockingness}, distinguishes individual marker states and requires that each marker state be reached within a given bounded number of steps from any other state and following any trajectory of the system. Accordingly, we formulate two new problems of quantitatively nonblocking supervisory control and heterogeneously quantitatively nonblocking supervisory control, and characterize their solvabilities in terms of new concepts of quantitative language completability and heterogeneously quantitative language completability, respectively. It is proved that there exists the unique supremal (heterogeneously) quantitatively completable sublanguage of a given language, and we develop effective algorithms to compute the supremal sublanguages. Finally, combining with the algorithm of computing the supremal controllable sublanguage, we design algorithms to compute the maximally permissive solutions to the formulated (heterogeneously) quantitatively nonblocking supervisory control problems.