Numerical Methods for Uncertainty Analysis in Dynamical Systems

The current methods for uncertainty analysis in dynamical systems are restricted in terms of computational cost and evaluation domain since they either use grid points or work only along trajectories. To break through these problems we present a new method: the Rothe & maximum-entropy method which follows the steps below.

A deterministic dynamical system with initial value uncertainties can be analyzed via the uncertainty propagation which is based on the Liouville equation in the form of the first-order linear partial differential equation. On this equation we conduct a semi-discretization in time via A-stable rational approximations of consistency order k and this yields the stationary spatial problem. This spatial problem now can be solved by the spatial discretization scheme: we propose the maximum-entropy approximation which provides unbiased interpolations even with fewer numbers of scattered points. Through these steps we finally obtain a system of linear equations for the evolution of the probability density function ut, which can be easily solved in several ways.

This method can provide more efficiency in terms of computation time thanks to using fewer numbers of scattered points instead of grid points. Also, it enables the constant tracking of probability density functions in a specific fixed domain of interest and this is especially effective for switched systems.