Using L∞ Norm to Measure Controllability of Probabilistic Discrete Event Systems

A discrete-event system is a dynamic system in which a state change takes place according to the events.Discrete-event systems are usually modeled by various automata as well as Petri nets. In this thesis, discrete-event systems will be modeled by Büchi automata. A supervisor for a discrete-event system is a controller that can enable or disable each event in each state of the system. Some typical controllability issues of discrete–event systems include: Is a discrete-event system controllable by a supervisor? If the controlled discrete-event system is not controllable, does there exist a smaller discrete-event system which is controllable when supervised by the controller? Moreover, how to find the new discrete-event system? On the other hand, does there exist a larger discrete-event system which is controllable when supervised by the controller? How to find the new larger discrete-event system?
In the real life many systems are of probabilistic nature. In this thesis, we discuss some supervisory control problems of probabilistic discrete-event systems which are modeled by Büchi automata. In the first part, we restrict the behavior of a discrete-event system to be of finite length. Further, we prove the necessary and sufficient conditions for the supervisory control problem on probabilistic discrete-event systems. In the second part of the thesis, we discuss the infinite behaviors of probabilistic discrete-event systems. Then, we extend the controller to probabilistic controller and also prove the problems on infinite behavior.
The contributions of this thesis include: we have used the L∞-norm to measure the distance between two probabilistic languages of finite length and infinite length. Moreover, we have given the computation of probabilistic languages of infinite length. We have presented how to find the supremal controllable sublanguage and the infimal closed controllable superlanguage if given a language of finite length or infinite length. We have also used the probabilistic supervisor to reduce the L∞ distance between two probabilistic languages.