Analytical Standard Uncertainty Evaluation Using Mellin Transform

Abstract

Uncertainty evaluation plays an important role in ensuring that a designed system can indeed achieve its desired performance. There are three standard methods to evaluate the propagation of uncertainty: 1) analytic linear approximation; 2) Monte Carlo (MC) simulation; and 3) analytical methods using mathematical representation of the probability density function (pdf). The analytic linear approximation method is inaccurate for highly nonlinear systems, which limits its application. The MC simulation approach is the most widely used technique, as it is accurate, versatile, and applicable to highly nonlinear systems. However, it does not define the uncertainty of the output in terms of those of its inputs. Therefore, designers who use this method need to resimulate their systems repeatedly for different combinations of input parameters. The most accurate solution can be attained using the analytical method based on pdf. However, it is unfortunately too complex to employ. This paper introduces the use of an analytical standard uncertainty evaluation (ASUE) toolbox that automatically performs the analytical method for multivariate polynomial systems. The backbone of the toolbox is a proposed ASUE framework. This framework enables the analytical process to be automated by replacing the complex mathematical steps in the analytical method with a Mellin transform lookup table and a set of algebraic operations. The ASUE toolbox was specifically designed for engineers and designers and is, therefore, simple to use. It provides the exact solution obtainable using the MC simulation, but with an additional output uncertainty expression as a function of its input parameters. This paper goes on to show how this expression can be used to prevent overdesign and/or suboptimal design solutions. The ASUE framework and toolbox substantially extend current analytical techniques to a much wider range of applications.