Two new optimal models for controlling discrete event systems
Qiying Hu, Wuyi Yue · Journal of Industrial and Management Optimization · 2005
Supervisory control belongs essentially to the logiclevel for control problems in discrete event systems (DESs) and itscorresponding control task is hard. This is unlike many practical optimalcontrol problems which belong to the performance level and whose controltasks are soft. In this paper, we present two new optimal control problemsof DESs: one with cost functions for choosing control inputs, and the otherfor occurring events. Their performance measures are to minimize the maximaldiscounted total cost among all possible strings that the system generates.Since this is a nonlinear optimization problem, we model such systems byusing Markov decision processes. We then present the optimality equationsfor both control problems and obtain their optimal solutions. When the costfunctions are stationary, we show that both the optimality equations andtheir solutions are also stationary. We then use these equations andsolutions to describe and solve uniformly the basic synthesizing problems inthe two branches of the supervisory control area: those being the eventfeedback control and the state feedback control. Moreover, we show that thecontrol invariant languages and the control invariant predicates with theirpermissive supervisors and state feedbacks not only have meanings insupervisory control of DESs, but are also the optimal solutions for someoptimal control problems. This shows a link existing between the logic leveland the performance level for the control of discrete event systems.Finally, a numerical example is given to illustrate some results forsupervisory control of a DES.