Conjugate Unscented Transform-Based Approach to Solve the Fokker-Planck-Kolmogorov Equation

摘要

This paper presents a new approach to solve the Fokker-Planck-Kolmogorov Equation (FPKE). The main purpose of this research is to provide a methodology to efficiently determine the time-varying state probability density function (PDF) for a general nonlinear system. Collocation-based methods often rely on high order cubature points, and require a large number of basis functions to accurately determine the solution of a partial differential equation. The recently developed Conjugate Unscented Transform is therefore proposed to provide a minimal set of collocation points. In conjunction with this minimal cubature point method, by employing an l_1 norm minimization to optimally select the appropriate basis functions from the larger complete dictionary of polynomial basis functions, a minimal polynomial expression for the state PDF is obtained at each time step.