This paper presents a control architecture based on a linear MPC formulation that addresses the lane keeping and obstacle avoidance problems for a passenger car driving on low curvature roads. The proposed control design decouples the longitudinal and lateral dynamics in two successive stages. First, plausible braking or throttle profiles are defined over the prediction horizon. Then, based on these profiles, linear time-varying models of the vehicle lateral dynamics are derived and used to formulate the associated linear MPC problems. The solutions of the optimization problems are used to determine for every time step, the optimal braking or throttle command and the corresponding steering angle command. Simulations show the ability of the controller to overcome multiple obstacles and keep the lane. Experimental results on an autonomous passenger vehicle driving on slippery roads show the effectiveness of the approach.