Model predictive control (MPC) frameworks have been effective in collision avoidance, stabilization, and path tracking for automated vehicles in real-time. These MPC formulations use a variety of vehicle models that capture specific aspects of vehicle handling, focusing either on low-speed scenarios or highly dynamic maneuvers. However, these models individually are unable to handle all operating regions with the same performance. This work introduces a novel linearization of a brush tire model that is affine, timevarying, and effective at any speed. We incorporate this tire model into a convex MPC formulation for lateral control on a low friction surface. Experimental results on an automated Volkswagen Golf GTI demonstrate effective steering control for path tracking from a standstill up to the limits of tire-road friction.