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Model predictive control is an advanced control approach for multivariablesystems with constraints, which is reliant on an accurate dynamic model. Mostreal dynamic models are however affected by uncertainties, which can lead toclosed-loop performance deterioration and constraint violations. In this paperwe introduce a new algorithm to explicitly consider time-invariant stochasticuncertainties in optimal control problems. The difficulty of propagatingstochastic variables through nonlinear functions is dealt with by combiningGaussian processes with polynomial chaos expansions. The main novelty in thispaper is to use this combination in an efficient fashion to obtain mean andvariance estimates of nonlinear transformations. Using this algorithm, it isshown how to formulate both chance-constraints and a probabilistic objectivefor the optimal control problem. On a batch reactor case study we firstlyverify the ability of the new approach to accurately approximate theprobability distributions required. Secondly, a tractable stochastic nonlinearmodel predictive control approach is formulated with an economic objective todemonstrate the closed-loop performance of the method via Monte Carlosimulations.

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