Vertical Takeoff and Landing (VTOL) Unmanned Aerial Vehicles (UAVs) possess several beneficial attributes, including requiring minimal space to takeoff, hover, and land. The tailsitter is a type of VTOL airframe that combines the benefits of VTOL capability with the ability to achieve efficient horizontal flight. One type of tailsitter, the Quadrotor Biplane (QRBP), can transition the vehicle from hover as a quadrotor to horizontal flight as a biplane. The vehicle used in this thesis is a QRBP designed with special considerations for fully autonomous operation in an outdoor environment in the presence of model uncertainties. QRBPs undergo a rotation of 90° about its pitch axis during transition from vertical to horizontal flight that induces strong aerodynamic forces that are difficult to model, thus necessitating the use of a robust control method to overcome the resulting uncertainties in the model. A feedback-linearizing controller augmented with an H-Infinity robust control is developed to regulate the altitude and pitch angle of the vehicle for the whole flight regime, including the ascent, transition forward, and landing. The performance of the proposed control design is demonstrated through numerical simulations in MATLAB and outdoor flight tests. The H-Infinity controller successfully tracks the prescribed trajectory, demonstrating its value as a computationally inexpensive, robust control technique for QRBP tailsitter UAVs.