Model predictive control (MPC) is the mainstream method in the motion control of autonomous vehicles. However, due to the complex and changeable driving environment, the perturbation of vehicle parameters will cause the steady-state error problem, which will lead to the degradation of controller performance. In this paper, the offset-free MPC control method is proposed to solve the steady-state error problem systematically. The core idea of this method is to model the model mismatch, control input offset, and external disturbances as disturbance terms, then use filters to observe these disturbances and finally eliminate the influence of these disturbances on the steady-state error in the MPC solution stage. This paper uses the Kalman filter as an observer, which is integrated into our latest designed MPC solver. Based on state-of-the-art sparse quadratic programming (QP) solver operator splitting solver for quadratic programs (OSQP), an offset free model predictive control (OF-MPC) framework based on disturbance observation and MPC is formed. The proposed OF-MPC solver can efficiently deal with common model mismatch problems such as tire stiffness mismatch, steering angle offset, lateral slope disturbance, and so on. This framework is very efficient and completes all calculations in less than 7 ms when the horizon length is 50. The efficiency and robustness of the algorithm are verified on our newly designed robot operating system (ROS)-Unreal4-carsim real-time cosimulation platform and real vehicle experiments.