Bifuzzy Discrete Event Systems and Their Supervisory Control Theory

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 bifuzzy discrete event systems (BFDESs), by combining classical DESs theory and T2 FSs theory. Then, we consider the supervisory control problem of BFDESs. The bifuzzy controllability theorem and nonblocking bifuzzy controllability theorem are demonstrated. In addition, an algorithm for checking the bifuzzy 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 the BFDES model.