A composite adaptive model predictive control (MPC) scheme is proposed to mitigate model mismatches due to parameter uncertainties and disturbances in controller designs for DC-DC boost converters. First, a model-based feedforward compensator based on a cascade proportional-integral (PI) MPC controller was developed as a current reference to improve the dynamic response performance. Then, an observer was employed to construct an accurate model and thereby avoid overcompensation and poor prediction due to model mismatch. Finally, a finite control set MPC was implemented to select a suitable switching state with which to realise output voltage regulation. Compared with a traditional PI MPC, the proposed controller has a rapid dynamic response, and compared with an unknown offset free MPC, the proposed controller is well suited to a simplified model. The proposed methods and benefits were validated by simulations in MATLAB/Simulink and by experiments on an NI Compact RIO rapid control prototype test bench.