The challenges of modeling using fuzzy standard interval arithmetic: A case study in electrical engineering

Abstract

In this work, the aim is to shed light on the challenges in deriving a mathematical model of dynamical systems when fuzzy standard interval arithmetic (FSIA) serves as a mathematical tool. The challenges in question are investigated through two approaches called the direct and devious approach. Specifically, in the direct approach, the challenges lie in the high complexities of deriving the fuzzy model while maintaining conformity with the laws of physics. Additionally, it is possible that the resulting fuzzy model may not have a solution. Concerning the devious approach, the primary challenge is the potential violation of the physics laws governing the system, which means that the validity of the fuzzy model is not guaranteed. Moreover, in both cases, the UBM phenomenon poses another challenge, preventing the attainment of a unique fuzzy model. As a result, it is demonstrated that FSIA and any related concepts, such as the strongly generalized Hukuhara derivative (SGH-derivative), generalized Hukuhara derivative (gH-derivative), generalized derivative, etc., mainly suffer from the catastrophe of physics laws violation (CPLV), which can be considered the most significant drawback. Furthermore, it is explained that the reason for a fuzzy differential equation under concepts such as the SGH-derivative or gH-derivative having multiple solutions is due to the CPLV. To clarify the CPLV, a simple electrical circuit with uncertain elements is examined as a case study.