In this article, we introduce an adaptive strictly negative-imaginary (SNI) autopilot for a low-cost quadrotor aerial vehicle, specifically designed to achieve high precision hovering and perform accurate trajectory tracking under time-varying dynamic load (i.e., displacement, velocity, and acceleration). Leveraging the learning ability of an artificial neural network, our adaptive SNI controller is robustly designed to overcome uncertainties in flight environments such as variations in the centre-of-gravity, modeling errors, and unpredictable wind gusts. The efficacy of the proposed adaptive control system is investigated under extensive flight tests in addition to numerous computer simulations and rigorous comparison with other control techniques, namely, fixed-gain SNI, fuzzy-SNI, and conventional PID controllers. We also conduct a stability analysis of the proposed control system using the SNI theorem.