This paper shows the design of a lane keeping steering control, that considers the nonlinear behavior of the tire-road lateral forces, by parameterizing its dynamics with respect to the yaw rate, as it is a low cost measurable variable. The proposed control strategy uses a piecewise linear (PWL) proportional double-integral derivative (PIIDi) control based on the lateral offset, measured by the vision system. The switches of the PWL controller are triggered by the yaw rate to take into account the nonlinear behavior of the tire forces. The stability is proved by a piecewise quadratic Lyapunov function. Simulation tests, including disturbances rejections and step references, are carried out on a standard CarSim D-Class vehicle model to explore the robustness with respect to unmodelled dynamics. The simulation results confirm that the proposed PWL control can greatly improve the vehicle stability and it is advantageous in very demanding maneuvers.