Arbitrary Polynomial Chaos for Uncertainty Propagation of Correlated Random Variables in Dynamic Systems

Dynamic simulation of stochastic systems requires uncertainty propagation. Traditional sample-based uncertainty propagation methods are often computationally intractable for online optimization-based estimation and control applications. Generalized polynomial chaos (gPC) is an efficient uncertainty propagation method that has been used for solving various nonlinear estimation and optimal control problems. However, gPC requires knowledge of the exact probability distribution of the random variables, and does not explicitly account for correlations between these random variables. This paper demonstrates the use of arbitrary polynomial chaos (aPC) for propagation of correlated multivariate random variables. aPC constructs orthogonal polynomial basis functions from only the raw moments of the random variables. Thus, aPC can be used for propagation of uncertainties with arbitrary probability distributions, even if their functional forms are unknown. The main contributions of this paper consist of presenting an algorithm for generating a set of orthogonal polynomial basis functions for correlated multivariate random variables and applying the Galerkin projection to compute closed-form expressions for the dynamics of the aPC expansion coefficients. An algorithm is also presented for efficient calculation of inner products between polynomial basis functions needed in the Galerkin projection. The error convergence properties of aPC are investigated and compared to that of gPC and Monte Carlo using a dynamic simulation case study. (C) 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.