Uncertainty Quantification for Stochastic Nonlinear Systems using Perron-Frobenius Operator and Karhunen-Loève Expansion

Abstract: In this paper, a methodology for propagation of uncertainty in stochastic nonlinear dynamical systems is investigated. The process noise is approximated using Karhunen- Lo`eve (KL) expansion. Perron-Frobenius (PF) operator is used to predict the evolution of uncertainty. A multivariate Kolmogorov-Smirnov test is used to verify the proposed framework. The method is applied to predict uncertainty evolution in a Duffing oscillator and a Vanderpol oscillator. It is observed that the solution of the approximated stochastic dynamics converges to the true solution in distribution. Finally, the proposed methodology is combined with Bayesian inference to estimate states of a nonlinear dynamical system, and its performance is compared with particle filter. The proposed estimator was found to be computationally superior than the particle filter.