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A numerical algorithm combining the Gauss Pseudospectral Method (GPM) with a Generalized PolynomialChaos (gPC) method to solve nonlinear stochastic optimal control problems with constraint uncertainties is presented. The GPM and gPC have been shown to be spectrally accurate numerical methods for solving deterministic optimal control problems and stochastic differential equations, respectively. The gPC uses collocation nodes to sample the random space, which are then inserted into the differential equations and solved using standard solvers to generate a set of deterministic solutions used to characterize the distribution of the solution by constructing a polynomial representation of the output as a function of uncertain parameters. The proposed algorithm investigates using GPM optimization software in place of deterministic differential equation solvers traditionally used in the gPC, providing minimum cost deterministic solutions that meet path, control, and boundary constraints. A trajectory optimization problem is considered where the objectives are to find the path through a two-dimensional space that minimizes the probability a vehicle will be 'killed' by lethal threats whose locations are uncertain and to characterize the effects those uncertainties have on the solution by estimating the statistical properties.

This paper presents an optimal sliding mode control design method for a nonlinear system with a "cascade" or two-loop structure. The outer loop is designed using a State Dependent Riccati Equation optimal controller. This outer loop controller generates an optimal sliding surface which is followed by a sliding mode controller in the inner loop. The control design method is demonstrated on a nonlinear model of an F-16 aircraft. Simulation results show excellent command tracking and robustness, with substantial errors in aerodynamic coefficients having little effect on command tracking or stability. The results presented in this paper indicate that the combination of an SDRE controller in the outer loop and a sliding mode controller in the inner loop has a great deal of promise for nonlinear robust control.