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. / Master of Science / Vertical Takeoff and Landing (VTOL) Unmanned Aerial Vehicles (UAVs) are a special type of UAV that can takeoff, hover, and land vertically, which lends several benefits. VTOL aircraft have recently gained popularity due to their potential to serve as fast and efficient payload delivery vehicles for e-commerce. One type of VTOL aircraft, the Quadrotor Biplane (QRBP) combines the ability of a quadrotor aircraft to hover, with the efficient horizontal flight of a biplane. Such a vehicle is able to takeoff and land in confined spaces, and also travel large distances on a single battery. However, the takeoff maneuver of a QRBP involves pitching from vertical to horizontal flight, which causes the vehicle to experience strong aerodynamic effects that are difficult to accurately model. Thus, to autonomously perform this unique maneuver, a robust control technique is necessary. A robust UAV controller is one that functions even when there is a degree of uncertainty in the predicted behavior of the vehicle, such as differences between estimated and actual vehicle parameters, or the presence of external disturbances such as wind. Therefore, a robust controller known as H-Infinity is developed to regulate the altitude and pitch angle of the QRBP as it takes off, transitions to forward flight, flies as a biplane, transitions back to vertical flight, and lands. The performance of the proposed control design is validated using numerical simulations performed in MATLAB, and flight tests. The H-Infinity controller successfully tracks the prescribed trajectory, demonstrating its value as a reliable, computationally inexpensive, robust control technique for QRBP UAVs.