Stochastic Fuzzy Discrete Event Systems and Their Model Identification

We introduce a new class of fuzzy discrete event systems (FDESs) called stochastic FDESs (SFDESs), which is significantly different from the probabilistic FDESs (PFDESs) in the literature. It offers an effective modeling framework for applications that are unsuitable for the PFDES framework. An SFDES is comprised of multiple fuzzy automata that occur randomly one at time with different occurrence probabilities. It uses either the max-product fuzzy inference or the max–min fuzzy inference. This article focuses on single-event SFDES—each of the fuzzy automata of such an SFDES has one event. Assuming nothing is known about an SFDES, we develop an innovative technique capable of determining number of fuzzy automata and their event transition matrices as well as estimating their occurrence probabilities. The technique, called prerequired-pre-event-state-based technique, creates and uses merely
$N$
particular pre-event state vectors of dimension
$N$
to identify event transition matrices of
$M$
fuzzy automata, involving a total of
$MN^{2}$
unknown parameters. One necessary and sufficient condition and three sufficient conditions are established for the identification of SFDES with different settings. The technique does not have any adjustable parameter or hyperparameter to set. A numerical example is provided to concretely illustrate the technique.