## 1 Introduction
[26] designed a novel weight update law by utilizing adaptive technology, which removed the PE condition. Unfortunately, most of the existing methods still rely on real data. Thus, how to effectively avoid the use of PE conditions is still an open problem. Meanwhile, uncertainties can seriously degrade the performance of a quadrotor UAV flight control system. Uncertainties usually have time-varying characteristics, and the ADP technique lacks the ability to tackle time-varying signals [26; 27]. Thus, it alone cannot be utilized to achieve high-quality control of quadrotor UAVs. Therefore, integrating the uncertainty attenuation technique with the ADP technique is of great significance in solving the control problems of quadrotor UAVs. In recent years, due to their strong fitting ability, neural networks (NNs) have provided some promising approaches to the uncertainty attenuation problem in practical systems such as quadrotor UAVs [28; 29; 30; 31; 32]. [PERSON] et al. [30] proposed an NN-based adaptive control scheme for the unknown and continuous dynamics of quadrotor UAVs, which cannot achieve fast convergence characteristics. To improve the convergence speed of the control system, some scholars have integrated finite-time control techniques with NN-based techniques. [PERSON] et al. [31] proposed a control scheme based on a finite-time multivariate NN interference observer for helicopter systems. [PERSON] et al. [32] proposed an adaptive NN-learning-based control scheme with finite-time convergence characteristics for the quadrotor UAV fault-tolerant control problem. However, the performance of the finite-time control techniques depends heavily on the initial state errors or observation errors, which limits the scope of their application in quadrotor UAVs. Therefore, combining NN-based techniques with fixed-time techniques for application in a quadrotor UAV control system requires further research. In addition, NN units commonly directly approximate the uncertainties in the above-mentioned studies, which leads to a large computational load and parameter explosion. Motivated by the aforementioned significant research, this paper investigated the control scheme of a quadrotor UAV under external disturbances and parameter uncertainties. The investigation focused on tracking control by integrating the NN-based observer technique with the ADP-based control technique, ensuring that the resulting control laws demonstrate both robust and approximately optimal characteristics. The main contributions are outlined below:  
1. Two fixed-time NN-based observers (FTNNOBs) were developed to compensate the control inputs of quadrotor UAV, which can estimate external disturbances and parameter uncertainties in a fixed time. Different from the traditional NN approximators designed in [33; 34; 35], the FTNNOBs only need to adjust the scalar parameters rather than the weight vectors or matrices. They provided a simple structure and inexpensive computation. 2. A novel ADP-based robust control scheme is proposed by combining the ADP technique with the estimated information from FTNNOBs, which improves the tracking control accuracy and optimizes the control cost consumption. Different from the existing weight update laws of AC NNs [16; 25], two novel weight update laws were introduced in this work. They not only have a simple structure, but are also independent of the PE condition. Moreover, two auxiliary terms related to the system states were introduced to improve the data utilization efficiency of the AC NNs and make the training results more effective. The rest of the paper is organized as follows. Section 2 introduces the dynamics model of a quadrotor UAV and derives the position and attitude error systems. Section 3 describes the design process of the ADP-based robust control scheme and the stability analysis of the overall closed-loop system. Section 4 introduces the simulation results. Section 5 presents the conclusions of this paper. **Notations 1. **\(\|\cdot\|\)_is utilized for the Euclidean norm of vectors and matrices.