State uncertainty propagation in the presence of parametric uncertainty and additive white noise

We present a new approach to describe the evolution of uncertainty in linear dynamic models with parametric and initial condition uncertainties, and driven by additive white Gaussian stochastic forcing. This is based on the polynomial chaos (PC) series expansion of second order random processes, which has been used in several domains to solve stochastic systems with parametric and initial condition uncertainties. The PC solution is found to be an accurate approximation to ground truth, established by Monte Carlo simulation, while offering an efficient computational approach for large systems with a relatively small number of uncertainties. However, when the dynamic system includes an additive stochastic forcing term varying with time, the computational cost of using the PC expansions for the stochastic forcing terms is expensive and increases exponentially with the increase in the number of time steps, due to the increase in the stochastic dimensionality. In this work, an alternative approach is proposed for uncertainty evolution in linear uncertain models driven by white noise. The uncertainty in the model states due to additive white Gaussian noise can be described by the mean and covariance of the states. This is combined with the PC based approach to propagate the uncertainty due to Gaussian stochastic forcing and model parameter uncertainties which can be non-Gaussian.