Hybrid solution of nonlinear stochastic optimal control problems using Legendre Pseudospectral and generalized Polynomial Chaos algorithms

Abstract

A novel hybrid technique is discussed to numerically solve nonlinear stochastic optimal control (OC) problems with numerous potential applications. The hybrid technique combines a Legendre Pseudospectral Method (LPM) with a generalized Polynomial Chaos (gPC) method, which are highly accurate numerical methods for solving deterministic OC problems and stochastic differential equations (SDE), respectively. The hybrid algorithm first selects samples from the random space using collocation nodes, inserts those sample values into the differential equations of the OC problem, and then uses a pseudospectral-based deterministic solver to generate solutions for each of the resulting deterministic problems. The set of deterministic solutions (i.e., ensemble of random realizations) are then used to construct a polynomial representation of the solution to the stochastic OC problem as a function of the random inputs using a gPC method (i.e., a high-order stochastic collocation method). The hybrid technique is used to solve a nonlinear stochastic OC problem to demonstrate its utility. The algorithm is a highly accurate technique for solving nonlinear OC problems with uncertain parameters. The hybrid technique will allow the user to analyze the solution to an OC problem and understand how uncertainty on the parameters such as the states, initial conditions, and boundary conditions can affect the solution (i.e., uncertainty quantification). The algorithm may also be useful for near real-time OC since a new trajectory can be followed if the uncertainty can be estimated.