# Failure Identification Method and LMI-Based Control System for Small Spacecraft
## 5 Simulations Results  
In this section, extensive simulations are carried out to evaluate the controller's performance. The LMI-based robust design was proposed to achieve fault-tolerant controllers for space applications. In particular, small spacecraft equipped with reaction wheels are considered for their sensitivity to failure and external perturbation. Considering the model of the spacecraft attitude dynamics as expressed by Equation (2), the capability of the controller to stabilize the spacecraft with failure and perform a pointing maneuver was investigated. The attitude controller was defined as a state-feedback controller, taking into account both the rotation and angular velocity of the spacecraft. A sliding mode observer was considered to detect and identify the failure (Equation (9)). Then, considering the failure, a secondary observer, Equation (11), was designed to estimate the disturbances acting on the spacecraft. Disturbances estimation takes into account both the failure recovery and external disturbances, and it was coupled with the controller input to improve its effectiveness. In order to consider a detailed model, sensors noise that was based on real sensors was also included. The attitude measurement noise was modeled according to [39], where two star-tracker sensors were combined with an IMU, providing the attitude with a 10 arcsec accuracy at 10 Hz. The angular velocity was measured at 50 Hz while considering a gyroscope model based on stim380h[40] (noise power spectral density \(0.8462\cdot 10^{-9}\) (rad/s)\({}^{2}\)/Hz). For the first observer, the reaction wheels' angular momentum was assumed to be measured at 10 Hz with an accuracy of \(10^{-5}\) kg m\({}^{2}\)s\({}^{-1}\). First, the H\({}_{\infty}\) state-feedback controllers were compared while considering the formulations from Equations (5) and (6) and with the inclusion of the proposed disturbances correction algorithm Equation (12). In Figure 3, the spacecraft stabilization from random initial conditions is considered, and a comparison between the design approach is made. The inclusion of the failure in the evaluation of the H\({}_{\infty}\) state-feedback controller was mandatory to achieve the desired robustness to the failure of one reaction wheel. In the right plot from Figure 3, with the controller in the state matrix (only A uncertainties), the closed-loop system does not converge to the desired value. In Figure 3 and 4, a comparison is also made between the H\({}_{\infty}\) state-feedback controller with and without the correction algorithm. The proposed approach significantly reduces the time required to accomplish the stabilization, with comparable accuracy for both cases. The simulation results proved that including failures as uncertainties in the controller design allowed us to design a robust, failure-tolerant controller. The proposed corrective algorithm does not affect the system's capabilities to withstand failures, but it is helpful in the stabilization process by reducing the maneuvering time. Finally, a simulation campaign was carried out to point out the effectiveness of the augmented controller. The simulation parameters are reported in Table 1. The considered spacecraft follows in the small spacecraft category, with a mass of 200 kg. The initial conditions and external disturbances are changed in each simulation while considering random values. In the left plot of Figure 5, the attitude time variation for the spacecraft stabilization of different initial conditions is shown, considering random external disturbances and failure cases. Moreover, to test the reliability and robustness of the controller, a real mission scenario was considered. The Earth observation consisted of an 86,400 s mission, where the attitude was kept in the same configuration for most of the time, and the actual maneuver lasted for 659 s. In the right plot from Figure 5, only the part concerning this maneuver is reported. The maneuver can be described in three phases.