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On modeling of fuzzy hybrid systems

A hybrid system is a system containing a mixture of discrete event components and continuous variable components. The existing hybrid system modeling methods are effective to handle crisp cases but can be difficult to represent deterministic uncertainties and subjectivity inherited in many real-world applications. We generalize the crisp hybrid system framework to a fuzzy hybrid system framework by using fuzzy set theory; the latter contains the former as a special case. We utilize fuzzy sets, type-1 and type-2, to capture and represent uncertainties in the hybrid system's states and variables. We develop algorithms to calculate fuzzy states and their transitions and propose a parallel composition method for modeling a complex fuzzy hybrid system through composing its components. This new formal, mathematical framework, capable of modeling a hybrid system with fuzzy states and various types of continuous dynamic processes, regardless whether they are available explicitly or implicitly e.g., fuzzy systems and neural networks, establishes a basis for systematic study of the fuzzy hybrid systems. It can also be employed for computer simulation investigation, analogous to the discrete event simulation methodology. An example fuzzy hybrid system involving fuzzy differential equations as continuous variable component is provided to illustrate the new theory.