Open-loop feedback control of nonlinear stochastic systems based on deterministic dirac mixture densities

The main problem of stochastic nonlinear model predictive control (SNMPC) is that the equations for state prediction and calculation of the expected reward are in general not solvable in closed form. A popular approach is to approximate the occurring continuous probability density functions by a discrete density representation, which allows an analytical solution of the SNMPC equations. In this paper, we propose to draw the samples not randomly as in Monte Carlo based methods, but systematically by minimizing a distance measure. In doing so, fewer components are generally required to represent the underlying probability density while achieving the same approximation quality. Especially if the evaluation of the expected reward is computationally expensive, this property affects the complexity of computation significantly. By means of a path planning problem, we have substantiated this statement with several simulation runs.