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We describe the application of model predictive control (MPC) using state mapping to the automatic landing system of a spaceplane. The controller has to be designed to enable the vehicle to follow reference signals with respect to trajectory and velocity, which are expressed as functions of the downrange. To track the reference signals, a control input is generally designed so that one or more ideal transient responses are realized. We also have developed a flight control system based on this concept. However, it was not satisfactory for use as the controller of an actual vehicle due to the lack of consideration of the input and state constraints. A controller that considers these constraints can be designed by applying MPC. On the other hand, it is difficult to predefine the transient responses necessary to track the reference signals, which vary with the current downrange. In this paper, a new approach using state mapping and feedback linearization is proposed to improve the control performance of tracking the reference signals. State mapping is derived to describe a system in which the reference signals can be handled as a constant set point, and the described system is linearized using state feedback. The proposed system is applied to the Japanese automatic landing flight experiment (ALFLEX) vehicle. A numerical simulation is performed to verify the validity of the proposed method based on MPC.