Stochastic Optimal Control based on Value-Function Approximation using Sinc Interpolation

An efficient approach for solving stochastic optimal control problems is to employ dynamic programming (DP). For continuous-valued nonlinear systems, the corresponding DP recursion generally cannot be solved in closed form. Thus, a typical approach is to discretize the DP value functions in order to be able to carry out the calculation. Especially for multidimensional systems, either a large number of discretization points is necessary or the quality of approximation degrades. This problem can be alleviated by interpolating the discretized value function. In this paper, we present an approach based on optimal low-pass interpolation employing sinc functions (sine cardinal). For the important case of systems with Gaussian mixture noise (including the special case of Gaussian noise), we show how the calculations required for this approach, especially the nontrivial calculation of an expected value of a Gaussian mixture random variable transformed by a sinc function, can be carried out analytically. We illustrate the effectiveness of the proposed interpolation scheme by an example from the field of Stochastic Nonlinear Model Predictive Control (SNMPC).