  Model predictive control is an advanced control approach for multivariable
systems with constraints, which is reliant on an accurate dynamic model. Most
real dynamic models are however affected by uncertainties, which can lead to
closed-loop performance deterioration and constraint violations. In this paper
we introduce a new algorithm to explicitly consider time-invariant stochastic
uncertainties in optimal control problems. The difficulty of propagating
stochastic variables through nonlinear functions is dealt with by combining
Gaussian processes with polynomial chaos expansions. The main novelty in this
paper is to use this combination in an efficient fashion to obtain mean and
variance estimates of nonlinear transformations. Using this algorithm, it is
shown how to formulate both chance-constraints and a probabilistic objective
for the optimal control problem. On a batch reactor case study we firstly
verify the ability of the new approach to accurately approximate the
probability distributions required. Secondly, a tractable stochastic nonlinear
model predictive control approach is formulated with an economic objective to
demonstrate the closed-loop performance of the method via Monte Carlo
simulations.
