Batch processes play a vital role in the chemical industry, but are difficult to control due to highly nonlinear behaviour and unsteady state operation. Nonlinear model predictive control (NMPC) is therefore one of the few promising approaches. Batch process models are however often affected by uncertainties, which can lower the performance and cause constraint violations. In this paper we propose a shrinking horizon NMPC algorithm accounting for these uncertainties to optimize a probabilistic objective subject to chance constraints. At each sampling time only noisy output measurements are observed. Polynomial chaos expansions (PCE) are used to express the probability distributions of the uncertainties, which are updated at each sampling time using a PCE state estimator and exploited in the NMPC formulation. The approach considers feedback by using time-invariant linear feedback gains, which alleviates the conservativeness of the approach. The NMPC scheme is verified on a polymerization semi-batch reactor case study. (C) 2019 Elsevier Ltd. All rights reserved.