Polynomial chaos expansion with fuzzy and random uncertainties in dynamical systems

This paper proposes a surrogate model which is able to deal with mixed uncertain dynamical systems: some uncertain parameters are modelled by random variables whereas others are represented by fuzzy variables. Polynomial chaos expansions (PCE) were developed for uncertainty propagation through random systems. However, fuzzy variables may also be described with polynomial chaos: in particular, Legendre polynomials are well adapted to fuzzy variables. Hence we propose a polynomial chaos expansion, which is able to describe an uncertain dynamical system with both random and fuzzy variables. The method is applied to simulate an uncertain bar and the results are compared to MCS results. Thus the PCE is successfully applied to dynamical systems with both fuzzy and random uncertainties. This study also highlights the issue of the description of the outputs: which quantities should be calculated to represent the behaviour of the uncertain outputs? In the case studies, the fuzzy mean and the fuzzy standard deviation are used to describe the output properties.