Control of stochastic discrete event systems modeled by probabilistic languages

In Garg et al. (1999) and Garg (1992) the formalism of probabilistic languages for modeling the stochastic qualitative behavior of discrete event systems (DESs) was introduced. In this paper, we study their supervisory control where the control is exercised by dynamically disabling certain controllable events thereby nulling the occurrence probabilities of disabled events, and increasing the occurrence probabilities of enabled events proportionately. This is a special case of "probabilistic supervision" introduced in Lawford and Wonham (1993). The control objective is to design a supervisor such that the controlled system never executes any illegal traces (their occurrence probability is zero), and legal traces occur with minimum prespecified occurrence probabilities. In other words, the probabilistic language of the controlled system lies within a prespecified range, where the upper bound is a "nonprobabilistic language" representing a legality constraint. We provide a condition for the existence of a supervisor. We also present an algorithm to test this existence condition when the probabilistic languages are regular (so that they admit probabilistic automata representation with finitely many states). Next, we give a technique to compute a maximally permissive supervisor online.