## 2 Problem Formulation  
In this paper, it is assumed that the chief spacecraft runs in a circular orbit and formation configuration is short distance. As shown in Figure 1, the considered spacecraft formation flying system comprised a chief spacecraft and a deputy spacecraft. \(r\) refers to the position vector of the chief spacecraft with respect to the earth-centered inertial (ECI) frame O-XYZ, which is attached to the center of the earth. The relative motion model for spacecraft formation flying was formulated in the local vertical and local horizontal (LVLH) coordinate frame that was fixed at the center of the chief spacecraft. In this frame, the \(x\), \(y\), and \(z\) components are referred to as radial direction, tangential direction, and normal direction, respectively.  
Considering space perturbation, the linear C-W equation [9; 26] describes the relative dynamic motion between the deputy spacecraft and the chief spacecraft and is as follows:  
\[\left\{\begin{array}{l}\ddot{x}=2n\dot{y}+3n^{2}x+u_{x}+d_{x}\\ \ddot{y}=-2n\dot{x}+u_{y}+d_{y}\\ \ddot{z}=-n^{2}z+u_{z}+d_{z}\end{array}\right. \tag{1}\]  
where \(x,y,z\) are the components of relative position in each axis, \(n\) is the constant angular velocity of the chief spacecraft around the earth, and \(u_{i},i=x,y,z\) are the components of orbit control force provided by thrusters along each axis of the deputy spacecraft. \(d_{i},i=x,y,z\) are the components of space perturbation along each axis.  
We define system state as \(\dot{\mathbf{q}}(t)=[x,y,z,\dot{x},\dot{y},\dot{z}]^{T}\), and the relative motion equation in (1) can be written into the following state-space model.  
\[\left\{\begin{array}{l}\dot{\mathbf{q}}=\mathbf{A}\mathbf{q}+\mathbf{B}\mathbf{u}+\mathbf{E}\mathbf{d}\\ \mathbf{y}=\mathbf{C}\mathbf{q}\end{array}\right. \tag{2}\]  
where the control input vector \(\mathbf{u}(t)=[u_{x},\,u_{y},\,u_{z}]^{T}\), space perturbation \(\mathbf{d}(t)=[d_{x},\,d_{y},\,d_{z}]^{T}\), and coefficient matrices are denoted by the following.  
\[\mathbf{A}=\left[\begin{array}{cc}\mathbf{0}_{3}&I_{3}\\ \mathbf{A}_{1}&A_{2}\end{array}\right],\;B=E=\left[\begin{array}{cc}\mathbf{0}_{3}\\ I_{3}\end{array}\right],\;\mathbf{C}=\mathbf{I}_{6},\;\text{and}\;\mathbf{A}_{1}=\left[ \begin{array}{ccc}3n^{2}&0&0\\ 0&0&0\\ 0&0&-n^{2}\end{array}\right],\;\mathbf{A}_{2}=\left[\begin{array}{ccc}0&2n&0\\ -2n&0&0\\ 0&0&0\end{array}\right].\]  
Considering the thruster bias fault, a fault-free spacecraft formation system (2) can be rewritten into the following form.  
\[\left\{\begin{array}{l}\dot{\mathbf{q}}=\mathbf{A}\mathbf{q}+\mathbf{B}\mathbf{u}+\mathbf{E}\mathbf{d}(t )+\mathbf{H}\mathbf{F}(t)\\ \mathbf{y}=\mathbf{C}\mathbf{q}\end{array}\right. \tag{3}\]  
where \(\mathbf{F}(t)\) represents control force results from thruster faults and the fault distribution matrix \(\mathbf{H}=\mathbf{B}\).  
**Assumption 1**.: _In this study, the space perturbation \(\mathbf{d}(t)\) is assumed to be bounded, and it satisfies the following:_  
\[\left\|\mathbf{d}(t)\right\|\leq d_{\max} \tag{4}\]  
_where \(d_{\max}\) is a positive constant._  
The main objective of this work was to develop robust thruster bias fault reconstruction and a learning output-feedback fault-tolerant tracking control approach for spacecraft  
Figure 1: ECI and LVLH coordinate frames for spacecraft formation flying.  
formation flying systems based on iterative learning algorithms. For reconstructing thruster bias fault robustly, a novel ILO was explored for the spacecraft formation dynamics (3). Based on the faults reconstructed using the designed ILO, an LOF\({}^{2}\)TC approach was developed for the purpose of accurate spacecraft formation configuration maintenance.