On Optimal Control Based on Parametric Gradient Approximations for Nonlinear Systems with Stochastic Parameters

This paper presents a design method for a suboptimal feedback controller to minimize the expectation of a cost function for uncertain nonlinear systems. The uncertainty is described by time-invariant stochastic parameters, which cause difficulties in solving the optimal control problem. The conventional notion of the principle of optimality cannot be applied to solve this problem. Furthermore, the optimal input and the expected cost cannot be described explicitly because of the nonlinearity of the system. These difficulties are overcome by combining a parametric approximation with a gradient-based optimization method. This approach enables us to obtain the gradient of an approximated cost function in an explicit form. A Monte Carlo (MC) approximation is employed to calculate the expectation of the derived gradient with respect to the stochastic parameters. This expected gradient is used to optimize the parameter of the controller.