## 3 Iterative Learning Observer-Based Thruster Fault Reconstruction Approach
### Stability Analysis of the Iterative Learning Observer
\right]<0 \tag{27}\]  
Therefore, the solution of (8) and (9) was converted into a problem of finding a global solution of the following minimization problem:  
Minimize \(\lambda\), subject to (8) and (27). To solve the above minimization problem, a sufficiently small positive scalar \(\lambda\) could be chosen. Then matrices \(\mathbf{P}\), \(\mathbf{L}_{1}\), and \(\mathbf{L}_{2}\) were obtained by solving (8) and (27) using the Matlab LMI toolbox such that matrix \(\mathbf{H}^{T}\mathbf{P}\) could be approximate to \(\mathbf{L}_{2}\mathbf{C}\) with satisfactory precision. Moreover, the above minimization problem could also be solved by using the Solvers minx in the Matlab LMI toolbox. Finally, gain matrix \(\mathbf{Q}_{1}\) could be easily computed through solving (10). **Remark 2**. : _To guarantee time-varying actuator fault reconstruction with high accuracy, parameters in ILO (5) and Theorem 1 should be chosen appropriately such that the terms \(\eta^{2}/\gamma_{1}\) and \(\Delta f(t)\) in \(\Delta_{f}\) were sufficiently small. First, to guarantee that parameter \(\eta^{2}/\gamma_{1}\) was small, parameter \(\mu\) should be designed as small as possible in Theorem 1. In addition, to guarantee that parameter \(\Delta f(t)\) was small, the learning interval \(\tau\) should be selected to be sufficiently small and the gain matrix \(\mathbf{Q}_{1}\), which was close to \(\mathbf{I}_{3}\), should be designed to satisfy (10) in Theorem 1. To achieve this, it could be noted from (10) that \(\gamma_{1}\) should be selected as a sufficiently small number. However, it also made the term \(\eta^{2}/\gamma_{1}\) large. It could be noted from \(\Delta_{f}\) that \(\gamma_{1}\) had a much greater impact on \(\Delta\mathbf{f}^{T}(t)\Delta f(t)\) than \(\eta^{2}/\gamma_{1}\). Therefore, a small parameter \(\gamma_{1}\) should be chosen for fault-reconstructing performance._Furthermore, based on Theorem 1, the design procedures of the proposed ILO are given as follows:  
1. Select appropriate positive scalars \(\mu\) and \(\beta\), then compute \(\eta=1+\mu\) and \(Q_{2}=1/\beta\eta\). 2. Select a small positive scalar \(\gamma_{1}\), then compute \(\mathbf{Q}_{1}\) through solving (10). 3. Choose a sufficiently small positive scalar \(\lambda\), then matrices \(\mathbf{P}\), \(\mathbf{Y}\), and \(\mathbf{L}_{2}\) can be obtained readily by using the Matlab/LMI toolbox to solve inequalities (8) and (27). 4. Compute the gain matrix \(\mathbf{L}_{1}=\mathbf{P}^{-1}\mathbf{Y}\) by using the matrices obtained in step (3). 5. Choose the appropriate learning interval \(\tau\) and establish the learning observer in the form of (5) according to the above gain matrices and parameters. **Remark 3**. : _Compared with the existing ILOs proposed in [21], a term \(\gamma\text{sgn}(\mathbf{y}(t)-\hat{\mathbf{y}}(t))\) in (5) was employed for the robustness of ILO with respect to the external space perturbations. To achieve this point, the condition (11) that required that relation between the parameter \(\gamma\) and space perturbations was obtained in Theorem 1. Obviously, the designed ILO (5) could clearly reconstruct thruster faults in the presence of space perturbations, unlike the ILO that reconstructed the lumped disturbances, including external disturbances, and actuator faults in [21]._  
Based on Theorem 1, the following corollary could be obtained for the considered spacecraft formation control system, subject to constant thruster bias faults. **Corollary 1**. : _Suppose that \(\mathbf{Q}_{1}=\mathbf{I}_{3}\), if there exist a positive definite symmetric matrices \(\mathbf{P}\in\mathbb{R}^{6\times 6}\), a positive scalar \(\gamma\), and matrices \(\mathbf{L}_{2}\in\mathbb{R}^{3\times 6}\), \(\mathbf{Y}\in\mathbb{R}^{6\times 6}\) such that (8), (9), and (11) hold, then the ILO (5) can reconstruct constant thruster bias faults asymptotically, and the observer gain matrix \(\mathbf{L}_{1}\) can be obtained by \(\mathbf{L}_{1}=\mathbf{P}^{-1}\mathbf{Y}\)._  
**Proof of Corollary 1.