# Sliding Mode Thau Observer for Actuator Fault Diagnosis of Quadcopter UAVs
## 1 Introduction  
Quadcopter unmanned aerial vehicles (UAVs) have been used in a variety of applications, due to their numerous advantages, such as small size, agility, low cost, mechanical simplicity, and indoor and outdoor operability, which have led to their increased popularity compared to other UAV systems. As a result, they have been investigated and tested in a range of environments and applications which include target tracking [1; 2], fault detection and fault-tolerant control [3; 4], and formation flight [5; 6]. Particularly the topic of fault-tolerant control (FTC) has received a large amount of attention in the community, which led to quadcopter UAVs that are less error-prone and, thus, more reliable during flight. In general, there are two types of FTC: passive and active. Several studies investigated passive FTCs [7; 8], which have the advantage that they do not require any fault diagnosis scheme, but the resulting disadvantage is that they have a lower fault tolerance [9]. To overcome this limitation, active FTCs have been introduced to improve said fault tolerance. Fault diagnosis (FD) is the essential requirement for active FTCs to determine the location and magnitude of faults. Through FD, active FTCs can be designed to compensate the effect of faults and, thus, improve flight control and stability, which makes FD the main task of active FTCs. The FD approach has been studied by numerous authors. [PERSON] et al. [10; 11] investigated a model-based fault diagnosis which can be used to monitor sensor faults and detect actuator faults. In this method, residuals are used to distinguish between system and observer outputs, but these methods are inaccurate and unsuitable for quantifying the magnitude of a fault. [PERSON] andZhang [12; 13] proposed a method for fault estimation based on a Kalman filter, but their approach is of insufficient robustness with regard to disturbances if the transfer matrices are inaccurate. Several effective approaches, such as sliding mode observer [14; 15], neural network [16; 17], and adaptive observer [18; 19], have been investigated, but none of these approaches focused on a real quadcopter UAV. Moreover, recent studies [20; 21] used fuzzy methods for fault diagnosis problems, but these approaches do not focus on real quadcopters, and may be overly complex to implement in a flight controller. Only few studies focused on the problem of fault diagnosis in a real quadcopter UAV, verified through real flight data. While [4] used an actuator fault estimation with an adaptive observer based on \(H_{\infty}\), and demonstrated the effectiveness of the proposed scheme, this method may not be sufficiently robust to external disturbances because the underlying mathematical model neglects both nonlinear terms and external disturbances. The most recent application of an adaptive Thau observer (ATO) for actuator fault diagnosis was proposed in [22]. While this approach is capable of handling model uncertainties in the nonlinear quadcopter model, it is rather complex and time-consuming because it uses system identification to find the drag terms, and the filter to eliminate sensor noise. In the present study, we try to overcome these limitations by combining a sliding mode observer based on Walcott-Zak observer design [23], with ATO to handle the actuator fault diagnosis. This method is capable of accounting for time-varying actuator faults. We then derive the Lyapunov stability and other conditions to obtain the desired matrices and associated parameters. Finally, a straightforward method based on linear matrix inequalities (LMI) is proposed to allow relaxing the derived conditions, which is a useful feature in flight controllers. Unlike previous methods, our approach is simple and not overly time-consuming, which makes it amenable for use in real quadcopters. Moreover, our method can handle uncertainties of magnitudes that are unknown, a priori, through an adaptive law approach.