Quantitatively nonblocking supervisory control of discrete-event systems

In this paper, we propose a new property of quantitative nonblockingness of an automaton with respect to a given cover on its set of marker states. This property quantifies the standard nonblocking property by capturing the practical requirement that every subset (i.e. cell) of marker states can be reached within a prescribed number of steps from any reachable state and following any trajectory of the system. Accordingly, we formulate a new problem of quantitatively nonblocking supervisory control, and characterize its solvability in terms of a new concept of quantitative language completability. It is proven that there exists the unique supremal quantitatively completable sublanguage of a given language, and we develop an effective algorithm to compute the supremal sublanguage. Finally, combining with the algorithm of computing the supremal controllable sublanguage, we design an algorithm to compute the maximally permissive solution to the formulated quantitatively nonblocking supervisory control problem.