# Failure Identification Method and LMI-Based Control System for Small Spacecraft
## 1 Introduction
In this way, it is possible to design a robust controller that also achieves high performance when failure occurs. However, this design process makes the controller extremely sensitive, making actuators generally work at their maximum or minimum torque. This limitation makes it difficult to design an effective control allocation algorithm to further improve controller performance, as control allocation algorithms are the main solution for achieving an effective failure rejection, as seen in [19; 20; 21; 22]. Instead, a control correction approach was proposed that considers the recovery of disturbances (identified as external disturbances and failure) that are acting on the spacecraft dynamics. First, it is necessary to identify when a failure occurs. For this purpose, nonlinear observer-based methods have been considered. Failure identification is generally seen as a signal estimation problem. Lately, methods based on observers have been under investigation for failure identification and detection [23]. In particular, sliding mode observers (SMOs) [24; 25; 26; 27] are usually developed as failure estimation methods and are able to identify where a failure occurs. Considering this approach, it is possible to identify the failing actuator system and take advantage of this information to evaluate the effects of the failure and disturbance on the spacecraft dynamics. This study proposed the combination of two SMOs: one for the identification of the failing actuator system, and the second for the estimation of disturbances acting on the spacecraft. Generally, the Kalman filter is the typical algorithm that is used to estimate unknown variables [28; 29]. However, in this case, system nonlinearities make it difficult to design an effective corrective strategy based on the Kalman filter technique. An observer-based algorithm better fits the problem formulation and, for this reason, it is preferred to classical approaches. Finally, a robust controller was combined with an active fault detection and correction method, which was based on the partial recovery of the estimated disturbances. The performance of the H\({}_{\infty}\) controller has been evaluated and compared while considering different designing approaches and while including the failure correction algorithm based on failure and disturbance estimation. The effectiveness of the proposed approach was verified by considering extensive simulations with varying initial conditions and external disturbances. The work proposed in this study was based on the thesis of [30]; this study's novelty and extension of [30] are mainly related to the design of a sliding mode observer for the identification of the failure. Moreover, additional mission scenarios are included to show the effectiveness of the combination of an observer and robust controller. The novelties of this study are: (i) the design of a fault-tolerant H\({}_{\infty}\) controller for small spacecraft using the LMI design method to achieve robustness to system uncertainties that are related to the presence of MEDs as actuators, and failures; (ii) the detection of a failure, the identification of the failing actuator, and the estimation of disturbances acting on a spacecraft with failure through the use of observer-based algorithms; and (iii) the combination of the proposed control strategy with a correction approach that is based on the observer's output. This paper is organized as follows. The satellite attitude dynamics are presented in Section 2, including the model of the actuator. Section 3 describes the design process of the \(H_{\infty}\) controller; in Section 4, the sliding mode observer is introduced as a failure detection method, including a failure correction approach that was employed to improve controller performance. The simulation results are shown in Section 5. Finally, some concluding remarks are listed in Section 6.