Extending Polynomial Chaos to Include Interval Analysis

Polynomial chaos theory (PCT) has been proven to be an efficient and effective way to represent and propagate uncertainty through system models and algorithms in general. In particular, PCT is a computationally efficient way to analyze and solve dynamic models under uncertainty. This paper presents a new way to use a polynomial expansion to incorporate uncertainties that are not expressed in terms of a probability density function (pdf). This paper presents the formalization of the process and some simple applications. The authors show that, within the framework introduced in this paper, it is possible to incorporate interval analysis. The long-term goal of this paper is to support the claim that the proposed framework can extract and represent uncertain behaviors in a form more general than previously used for these engineering problems. The proposed approach is first applied to an algebraic model and then to a differential equation model. The results thus obtained are analyzed in two different perspectives: 1) interpreting the PCT expansion as a fully probabilistic method and 2) in the framework of possibility theory. The conclusions in these two cases are compared and discussed.