In this paper, we investigate the use of Model Predictive Control (MPC) applications for quasi-Linear Parameter Varying (qLPV) systems subject to faults along the input channels. We propose a Fault Tolerant Control (FTC) mechanism based on a robust state-feedback MPC synthesis, considering polytopic inclusions. In order to alleviate the numerical burden of the robust min-max procedure, we use small prediction horizons, in such a way that the solution becomes viable for real-time systems. The FTC system is able to tolerate time-varying saturation of the actuator, which may happen due to malfunctions. Recursive feasibility and poly-quadratic stability guarantees are ensured through the synthesis of adequate terminal ingredients. Accordingly, we present a catalogue of three different LMI remedies, considering: (a) parameterindependent ingredients, (b) a parameter-dependent terms and (c) a parameter-dependent maps that take into account bounded rates of parameter variation. An autonomous driving car example is used to illustrate the performances of the proposed technique, which is compared to other MPCs from the literature. The proposed FTC method is able to ensure good performances, obtained with reduced computational demand.