## 1 Introduction
In [14], a reconfigurable spacecraft formation FTC approach was designed based on a super-twisting sliding mode observer, which had strong robustness to measurement noise and J2 orbital perturbation. The reconfigurable FTC approach in different failure modes could effectively maintain the spacecraft formation configuration. In [15], a decentralized FTC algorithm was proposed for the three-inline array tethered spacecraft formation system. In [16], two new adaptive FTC algorithms were developed to estimate the effectiveness of the actuator, spacecraft masses, and the upper bound of external disturbances for spacecraft formation control systems that are subject to time-varying communication delays. It is noteworthy that robust fault reconstruction observer-based reconfigurable spacecraft formation FTC has not been considered in the existing literature, although fault reconstruction and spacecraft formation FTC have been separately addressed. During the past decade, fault diagnosis and FTC based on iterative learning algorithms have been widely studied in time-delay systems, Takagi-Sugeno fuzzy systems, etc., [17; 18; 19], and applied in robotics and spacecraft systems [6; 7]. Recently, iterative learning algorithms have been attracting much attention from aerospace scholars mainly because they can be applied to robust fault reconstruction and reconfigurable FTC for spacecraft systems. In [20], an ILO was designed to estimate the torque deviation for rigid spacecraft attitude stabilization in the presence of external disturbances and actuator faults. In [21], a novel ILO that could simultaneously reconstruct actuator faults and disturbances was developed for the attitude control of spacecrafts against actuator faults. In [22], two new non-linear ILO were suggested in order to robustly reconstruct effectiveness factors and bias faults of reaction wheels for spacecraft attitude control systems (ACSs) subject to space disturbance torques. In [23], a new fault diagnosis observer was proposed based on the iterative learning methodology to reconstruct actuator failures in real time for spacecraft ACSs in the presence of actuator failures, external disturbances, and actuator saturation. In [24], a novel Barrier Function-based ILO was designed to reconstruct the lumped disturbance, including multiple disturbances and failure torque, for spacecraft attitude stabilization. However, so far few research results on iterative learning algorithms have been focused on spacecraft formation flying systems. In addition, in comparison with the existing integrator-based adaptive algorithm, the algebraic iterative learning algorithm required less computation and no existence of derivatives at some time instants [21; 22; 25]. Therefore, the iterative learning algorithm was easier to implement in practice, especially when considering the limited storage and computing power for on-board computers in the spacecraft formation flying system. In light of these, there was a strong incentive for us to investigate iterative learning algorithm-based fault reconstruction and reconfigurable FTC for spacecraft formation flying systems. Based on the above discussions, this paper investigates the issues of robust thruster fault reconstruction and reconfigurable fault tolerant control for a spacecraft formation flying system based on the iterative learning algorithms. Considering space perturbations and thruster faults, the relative orbital dynamics of spacecraft formation control systems are established. Considering the drawback in robustness of the existing ILOs [21, 22, 25] and the robustness of fault reconstruction against space perturbations in spacecraft formation flying systems, a novel ILO was proposed to robustly reconstruct thruster bias faults based on sliding mode methodology, and partial ILO-gain matrices could be conveniently solved by using LMI optimization techniques.