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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.