Uncertainty Propagation in Dynamical Systems Using Koopman Eigenfunctions

Abstract

In many complex dynamical system analyses, it is essential to understand the dynamic behavior of the states as accurately as possible. Considering the uncertain environment, it is important to be able to predict how the uncertainties in the inputs will propagate in the system dynamics and will affect the system’s performance. Such information provides an important analysis for systems’ operation and stability analysis. This paper proposes an approach for uncertainty propagation analysis using the Koopman operator theory for uncertain inputs for the dynamical systems. The uncertain input can be characterized by the probability distribution function (PDF). For linear dynamical systems, uncertainty propagation analysis is obtained using an analytical expression for the first and second moment, i.e., mean and variance. This paper extends the same concept to linear dynamical systems using the Koopman operator theory, which involves the computation of the Koopman eigenfunctions. The efficacy of the proposed approach is demonstrated using linear quarter car dynamics simulations showing the mean and variance propagation of the states. A comparison is provided between our proposed approach with the Monte Carlo simulations for computing mean and variance propagation for bench-marking the efficacy of the approach.