A robust Model Predictive Control algorithm is proposed for learning-based control with model represented by an affine combination of basis functions. The online optimization is formulated as a sequence of convex programming problems derived by linearizing concave components of the dynamic model. A tube-based approach ensures satisfaction of constraints on control variables and model states while avoiding conservative bounds on linearization errors. The linear dependence of the model on unknown parameters is exploited to allow safe online parameter adaptation. The resulting algorithm is recursively feasible and provides closed loop stability and performance guarantees. Numerical examples are provided to illustrate the approach.