## 1 Introduction  
In the field of industrial control, it is often necessary to estimate the system state on-line and control it according to the estimated state, however, the existence of faults will make the control more difficult. Usually, fault detection, isolation and estimation are three major tasks of fault diagnosis. Compared with fault detection and isolation, fault estimation is more difficult and more significant. Because fault estimation is often the key link of fault-tolerant control and fault regulation system, the quality of fault estimation directly affects the control accuracy and the control performance of the whole system. At present, researchers have done a lot of research on fault diagnosis methods, including data-based fault diagnosis algorithm and model-based fault diagnosis algorithm [1]. 1. Fault diagnosis algorithm based on data The data-based fault diagnosis algorithm mainly uses statistical methods to obtain the characteristics of the data, and constructs the detection threshold through statistics to diagnose the fault. Furthermore, some intelligent algorithms, such as support vector machine (SVM), K-nearest neighbor (KNN) and so on, are also used in fault diagnosis [2, 3]. However, most of the data-based fault diagnosis methods are used for fault detection and identification, and it is difficult to estimate the fault amplitude. Model based fault diagnosis algorithm The model-based fault diagnosis algorithm usually needs to accurately model the diagnosis object [4], and obtain the system state by comparing the reference model output with the actual output, so as to judge whether there is a fault in the system [5]. Although this method has high accuracy and reliability, it often takes a lot of energy to accurately model the diagnosis object. Due to the unmanned aerial vehicle (UAV) control system requires high reliability, it needs more accurate fault diagnosis algorithm [6]. The data-based fault diagnosis algorithm is not suitable for UAV control system because of its low accuracy and inability to estimate fault effectively [7]. On the contrary, a large number of scholars have conducted in-depth research on UAV control system and established perfect mathematical and physical models [8]. Therefore, the model-based fault diagnosis algorithm can provide highly reliable fault diagnosis results for UAV control system. A large number of scholars have studied the model-based UAV fault diagnosis method. The proportional integral (PI) control algorithm is the most widely used control algorithm. It mainly considers the historical information, current information and future information of dynamic system output error to construct state observer to realize the state and fault monitoring. In References [9; 10], the method of fault detection observer was first proposed. Its basic idea is to detect the fault of linear system by designing fault detection observer. In Reference [11], a set of structured residuals is proposed, in which each residuals is only sensitive to some faults but not to others. In Reference [12], the observer based method is deeply studied, and an improved PI observer based on adaptive gain scheduling method and its design method are proposed. In addition, in References [13; 14], the parameter estimation problem of PI observer was considered, and the optimal planning algorithm for determining the optimal parameters was designed. In the simulation, the overshoot of fault estimation process was obviously reduced, and the robustness of PI control system was effectively improved. In References [15; 16; 17], a fault estimation method based on PI observer and considering imprecise scheduling parameters is proposed for a class of nonlinear systems described by linear variable parameter mathematical model. Most UAV fault diagnosis algorithms are based on PI control algorithm. In References [18; 19], a widely applicable fault estimation observer was designed by the optimization technology for the UAV system with an actuator fault and sensor fault.