# Failure Identification Method and LMI-Based Control System for Small Spacecraft
## 6 Conclusions  
In this study, a fault-tolerant robust controller design was addressed for small spacecraft attitude control. Robust controllers are adopted to deal with bounded system uncertainties. In particular, an H\({}_{\infty}\) controller was designed as a spacecraft attitude controller and started from the LMI formulation to achieve robustness for bounded plant uncertainties. A spacecraft equipped with four reaction wheels was considered. Failures were modeled and included in the design process as plant uncertainties. The H\({}_{\infty}\) suboptimal problem was then solved for the definition of the feedback controller gain. However, despite the controller being robust to the failure when it occurs, the system results are more sensitive to perturbations. For this reason, an active strategy based on failure identification and control correction was included. First, an observer was designed to detect the failure and identify the failing reaction wheel. Then, considering the presence of the failure, the disturbances acting on the spacecraft dynamics were estimated by means of a second observer. The disturbances take into account external disturbances, failure effects, and sensor uncertainties. The controller input was then coupled and corrected with the estimated disturbances, improving controller performance. By means of extensive simulation, the controller performance was investigated by considering random initial conditions and external disturbances.  
The simulation results verify the controller robustness, and the proposed control correction shows a significant performance improvement in terms of the amount of time needed to accomplish the maneuver. The accuracy is compromised by the high sensor uncertainties; it could be improved by implementing a navigation algorithm, but that is not the main purpose of this work. However, the controller robustness to sensor uncertainties is verified, which was also verified while considering the controller reliability in an Earth observation mission scenario. The proposed study addresses the design of a simple and low computational cost technique for failure-tolerant attitude control. However, some aspects can be further improved in future studies. The desired controller is able to satisfy the requirements, but this design method makes the tuning process not trivial. Different approaches may be investigated to improve controller tuning according to system requirements. Moreover, failure identification is based on reaction wheel data and can be further improved. More sophisticated techniques may be considered (i.e., machine learning and artificial intelligence) to detect and identify the failure. Finally, robustness to multiple failures may be addressed, evaluating the proposed design approach or possible action plan to avoid complete mission failure.  
Figure 5: Augmented controller simulation campaign with the random initial condition, external disturbances, and failure. Each colored line refers to a different simulation parameters and initial condition: attitude time variation of spacecraft stabilization (**left**); Earth pointing mission scenario (**right**).  
**Author Contributions:** Conceptualization, E.C. and H.P.; methodology, N.C. and D.R.; software, N.C. and D.R.; validation, N.C. and D.R.; data curation, all authors; writing--original draft preparation, D.R.; writing--review and editing, all authors; supervision, E.C. and H.P.; project administration, E.C. and H.P.; funding acquisition, E.C. and H.P. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data used in the current study are available from the corresponding authors upon request.  
**Conflicts of Interest:** The authors declare no conflict of interest.