This work addresses the design of a path tracking controller for autonomous vehicles. It reformulates the Reference Aware MPC in order to guarantee closed-loop stability, while maintaining a safe and comfortable ride, and minimizing wear and tear of vehicle components. Stability is proved via Lyapunov techniques. Furthermore, to adapt the response of the controller online while in operation, a novel model for the nonlinear curvature response of the vehicle is proposed. This model is estimated online by means of Kalman filtering. Both the proposed controller and curvature response model are evaluated with simulations and through experiments on a Scania construction truck, where the advantages to the previous state-of-the-art are highlighted and discussed.