In this work, we present a novel approach based on linear matrix inequalities to design a linear-time varying model predictive controller for a nonlinear system with guaranteed stability. The proposed method utilizes a multi-model description to model the nonlinear system where the dynamics is represented by a group of linear-time invariant plants, which makes the resulting optimization problem easy to solve in real-time. In addition, we apply the control invariant set designed as the final stage constraint to bound the additive disturbance introduced to the plant by other subsystems interfacing with the controller. We show that the persistent feasibility is ensured with the presence of such constraint on the disturbance of the specified kind. The proposed method is then put into the context of emergency lane change for steering control of automated vehicles and its performance is verified via numerical evaluation.