This paper presents a path following controller for autonomous vehicles, making use of the linear time-varying model predictive control (LTV-MPC) framework. The controller takes into consideration control input rates and accelerations, not only to account for limitations in the steering dynamics, but also to provide a safe and comfortable ride while minimizing wear and tear of the vehicle components. Furthermore, it introduces a method to handle model references generated by motion planning algorithms that can consider different vehicle models from the controller. The proposed controller is verified by simulations and through experiments in a Scania construction truck, and is shown to have better performance than the state-of-the-art smooth and accurate MPC.