Distributional uncertainty analysis using power series and polynomial chaos expansions

This paper provides an overview of computationally efficient approaches for quantifying the influence of parameter uncertainties on the states and outputs of nonlinear dynamical systems with finite-time control trajectories, focusing primarily on computing probability distributions. The advantages and disadvantages of various uncertainty analysis approaches, which use approximate representations of the full nonlinear model using power series or polynomial chaos expansions, are discussed in terms of computational cost and accuracy in computing the shape and tails of the state and output distributions. Application of the uncertainty analysis methods to a simulation study is used to provide advice as to which uncertainty analysis methods to select for a particular application. In particular, the results indicate that first-order series analysis can be accurate enough for the design of real-time robust feedback controllers for batch processes, although it is cautioned that the accuracy of such analysis should be confirmed a posteriori using a more accurate uncertainty analysis method. The polynomial chaos expansion is well suited to robust design and control when the objectives are strongly dependent on the shape or tails of the distributions of product quality or economic objectives.