Fuzzy Discrete-Event Systems Under Fuzzy Observability and a Test Algorithm

In order to more effectively cope with the real-world problems of vagueness, impreciseness, and subjectivity, fuzzy discrete-event systems (FDESs) were proposed recently. Notably, FDESs have been applied to biomedical control for human immunodeficiency virus (HIV)/acquired immune deficiency syndrome (AIDS) treatment planning and sensory information processing for robotic control. Qiu independently developed supervisory control theory of FDESs. We note that the controllability of events in Qiu's work is fuzzy, but the observability of events is crisp, and the observability of events in Cao and Ying's work is also crisp, although the controllability is not completely crisp since the controllable events can be disabled with any degrees. Motivated by the necessity to consider the situation that the events may be observed or controlled with some membership degrees, in this paper, we establish the supervisory control theory of FDESs with partial observations in which both the observability and controllability of events are fuzzy instead. We formalize the notions of fuzzy controllability condition and fuzzy observability condition. In addition, controllability and observability theorem of FDESs is set up in a more generic framework. In particular, we present a detailed computing flow to verify whether the controllability and observability conditions hold. Thus, this result can decide the existence of supervisors. Also, we use this computing method to check the existence of supervisors in the controllability and observability theorem of classical discrete-event systems (DESs), which is a new method and different from classical case. A number of examples are elaborated on to illustrate the presented results.