# Failure Identification Method and LMI-Based Control System for Small Spacecraft
## 1 Introduction  
In the last few years, technological improvements and the resulting miniaturization of processors and electronics have given small satellites new capabilities and better performance. The increased usage of small satellites has led to bigger attention from customers for any kind of failure that can affect the spacecraft at any moment. Particularly, spacecraft attitude, communication, or scientific observation impose strict requirements in terms of attitude stability and accuracy. The attitude accuracy is usually a function of the mission scenario and spacecraft characteristics; some mathematics related to the accuracy evaluation for different missions can be found in [1] and in [2] for a real-time implementation. A pointing stability index, for example, can be evaluated during the most critical maneuver phases, as detailed in [3]. Moreover, the reduced mass causes the spacecraft to be more sensitive to external perturbations, and momentum exchange devices (MEDs) have more limitations due to their smaller dimensions. For this reason, any kind of failure in the actuation system may critically affect the success of the mission. Generally, over-actuated systems guarantee redundancy in the actuators to withstand failures, but the controller design must also allow for this eventuality. In this context, robust control methods for small satellites have been attracting the attention of the space systems community, which needs effective approaches for withstanding failures. Moreover, robust controllers are generally adopted to deal with bounded plantuncertainties, which guarantees the required performance and stability. A first definition of robustness, although not so rigorous, can be the capability of the control system to work well under sets of parameters that are different from the nominal one. For example, these parameters can be uncertainties within the system that are not known but are bounded. Failures can be considered as a bounded uncertainty of the system, and the controller can be designed to guarantee a robust performance. Robust controllers can be designed to act as passive fault-tolerant algorithms, as in [4; 5]; however, they have limited fault tolerance capabilities, as can be seen in [6; 7]. Despite the robustness to failure being verified, some perturbation may eventually lead to instability or loss of performance. This is directly related to the loss of control effectiveness. An active approach would result in a more effective performance, but it requires a fault detection system for the complete awareness of the system in real time, as seen in [8; 9] for aerial multirotors. Identifying the failure may allow for the reallocation of the control to between the remaining actuators, which would limit the control action loss. Among robust control strategies, the H\({}_{\infty}\) control method is well established and studied. Referring to this controller makes the designer aware of its closed-loop system properties and performance. Recent studies on the H\({}_{\infty}\) controller show the introduction of the suboptimal problem [10; 11; 12]. As conducted in [13; 14], the suboptimal problem is brought back to a linear matrix inequalities (LMIs) problem. Solving the LMI problem using numerical methods allows the design of the controller to follow a theoretical approach; in this way, the controller is already designed to satisfy the system requirements. The LMI design approach is under investigation for different control strategies [15; 16; 17; 18]. With respect to previous work, the current study proposed the inclusion of failure and uncertainties directly into the design process. In this study, the solution of the suboptimal problem in the LMI formulation was addressed to include plant uncertainties related to parameter variation in the design process of the H\({}_{\infty}\) controller. Failures were modeled as unknown disturbances acting on the systems and were included as uncertainties in the design process.