  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
HIV/AIDS treatment planning and sensory information processing for robotic
control. Qiu, Cao and Ying 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. And 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.
