Two new optimal models for controlling discrete event systems

Supervisory control belongs essentially to the logic level for control problems in discrete event systems (DESs) and its corresponding control task is hard. This is unlike many practical optimal control problems which belong to the performance level and whose control tasks are soft. In this paper, we present two new optimal control problems of DESs: one with cost functions for choosing control inputs, and the other for occurring events. Their performance measures are to minimize the maximal discounted total cost among all possible strings that the system generates. Since this is a nonlinear optimization problem, we model such systems by using Markov decision processes. We then present the optimality equations for both control problems and obtain their optimal solutions. When the cost functions are stationary, we show that both the optimality equations and their solutions are also stationary. We then use these equations and solutions to describe and solve uniformly the basic synthesizing problems in the two branches of the supervisory control area: those being the event feedback control and the state feedback control. Moreover, we show that the control invariant languages and the control invariant predicates with their permissive supervisors and state feedbacks not only have meanings in supervisory control of DESs, but are also the optimal solutions for some optimal control problems. This shows a link existing between the logic level and the performance level for the control of discrete event systems. Finally, a numerical example is given to illustrate some results for supervisory control of a DES.