This paper proposes a new predictive controller approach for nonlinear process based on a reduced complexity homogeneous, quadratic discrete-time Volterra model called quadratic S-PARAFAC Volterra model. The proposed model is yielded by using the symmetry property of the Volterra kernels and their tensor decomposition using the PARAFAC technique that provides a parametric reduction compared to the conventional Volterra model. This property allows synthesising a new nonlinear-model-based predictive control (NMBPC). We develop the general form of a new predictor, and therefore, we propose an optimisation algorithm formulated as a quadratic programming under linear and nonlinear constraints. The performances of the proposed quadratic S-PARAFAC Volterra model and the developed NMBPC algorithm are illustrated on a numerical simulation and validated on a benchmark as a continuous stirred-tank reactor system. Moreover, the efficiency of the proposed quadratic S-PARAFAC Volterra model and the NMBPC approach are validated on an experimental communicating two-tank system.