Combining Gaussian processes and polynomial chaos expansions for stochastic nonlinear model predictive control

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.