This paper proposes a quadrotor system control scheme using an intelligent–proportional–integral–differential control (I-PID)-based controller augmented with a radial basis neural network (RBF neural network) and the proposed adaptive robust term. The I-PID controller, similar to the widely utilized PID controller in quadrotor systems, demonstrates notable robustness. To enhance this robustness further, the time-delay estimation error was compensated with an RBF neural network. Additionally, an adaptive robust term was proposed to address the shortcomings of the neural network system, thereby constructing a more robust controller. This supplementary control input integrated an adaptation term to address significant signal changes and was amalgamated with a reverse saturation filter to remove unnecessary control input during a steady state. The adaptive law of the proposed controller was designed based on Lyapunov stability to satisfy control system stability. To verify the control system, simulations were conducted on a quadrotor system maneuvering along a spiral path in a disturbed environment. The simulation results demonstrate that the proposed controller achieves high tracking performance across all six axes. Therefore, the controller proposed in this paper can be configured similarly to the previous PID controller and shows satisfactory performance.