In this paper, a novel adaptive sliding-mode control algorithm is proposed for the attitude control of quadrotor unmanned aerial vehicles (UAVs) under the delta operator framework. First, the delta operator technique is used to discretize the attitude control systems of a quadrotor UAV. Then, based on the linear matrix inequality technique, a linear sliding surface is designed to ensure the asymptotical stability of the quadrotor UAV attitude control system during the sliding motion process. Second, by the estimated external disturbance using a radical basis function (RBF) neural network, an adaptive sliding-mode attitude controller is designed such that the states of the quadrotor UAV attitude systems can be driven towards the desired sliding surface, and thus the attitude control objective of the qudarotor UAV is achieved. Compared with the traditional adaptive sliding-mode control algorithm, the proposed adaptive sliding-mode control algorithm can effectively realize the attitude control of a quadrotor UAV subject to strong disturbances and couplings. Finally, comparisons of the simulation results verify the effectiveness and superiority of the control algorithm proposed in this paper.