  It is well known that type-1 fuzzy sets (T1 FSs) have limited capabilities to
handle some data uncertainties directly, and type-2 fuzzy sets (T2 FSs) can
cover the shortcoming of T1 FSs to a certain extent. Fuzzy discrete event
systems (FDESs) were proposed based on T1 FSs theory. Hence, FDES may not be a
satisfactory model to characterize some high-uncertainty systems. In this
paper, we propose a new model, called as bi-fuzzy discrete event systems
(BFDESs), by combining classical DESs theory and T2 FSs theory. Then, we
consider the supervisory control problem of BFDESs. The bi-fuzzy
controllability theorem and nonblocking bi-fuzzy controllability theorem are
demonstrated. Also, an algorithm for checking the bi-fuzzy controllability
condition is presented. In addition, two controllable approximations to an
uncontrollable language are investigated in detail. An illustrative example is
provided to show the applicability and the advantages of BFDESs model.
