Stochastic Reachability Analysis using Sparse-Collocation Method

A computationally efficient approach is presented to compute the reachability set for a nonlinear system. A reachability set is defined as the computation of states a system can reach given the bounds on system inputs, parameters, and initial conditions. The main idea of the developed approach is to represent the reachability set as the probability density function (pdf) and find the evolution of the state pdf. A non-product sampling method known as Conjugate Unscented Transformation (CUT), in conjunction with the sparse approximation method, is used to find the time evolution of system state pdf. The CUT method helps alleviate the curse of dimensionality, which occurs as the number of collocation points increases with the increase in uncertain variables. Furthermore, the sparse approximation methods help in finding a parsimonious representation of state pdf from an over-complete dictionary of basis functions. Finally, two numerical examples are presented to show the efficacy of the developed approach. Conventional Monte Carlo simulations are used to assess the performance of the developed approach.