## 3 Iterative Learning Observer-Based Thruster Fault Reconstruction Approach
### Stability Analysis of the Iterative Learning Observer
\begin{array}{l}\beta\eta\mathbf{e}_{F}^{T}(t)\mathbf{e}_{F}(t)\leq\beta(\eta+ \gamma_{1})\mathbf{e}_{F}^{T}(t-\tau)\mathbf{Q}_{1}{}^{T}\mathbf{Q}_{1}\mathbf{e}_{F}(t-\tau) +\beta(\eta+\frac{\eta^{2}}{\gamma_{1}})\Delta\mathbf{\hat{f}}^{T}(t)\Delta\mathbf{f} (t)\\ +\beta\eta[\Delta\mathbf{\hat{f}}^{T}(t)\Delta\mathbf{\hat{f}}(t)\ -2\Delta\mathbf{\hat{f}}^{T}(t)\Delta\mathbf{f}(t) \ -2\Delta\mathbf{\hat{f}}^{T}(t)\mathbf{Q}_{1}\mathbf{e}_{F}(t-\tau)]\end{array}  \tag{21}\]  
Further, with the aid of (9), it is easily obtained from (16) that  
\[2\mathbf{e}_{q}^{T}\mathbf{P}\mathbf{H}=\frac{2}{Q_{2}}\Delta\mathbf{\hat{f}}^{T}(t) \tag{22}\]  
Defining \(\frac{1}{Q_{2}}=\beta\eta\) and noting (22), it follows that  
\[ \begin{array}{l}2\mathbf{e}_{q}^{T}\mathbf{P}\mathbf{H}[\mathbf{Q}_{1}\mathbf{e}_{F}(t-\tau)+ \Delta\mathbf{f}(t)-\Delta\mathbf{\hat{f}}(t)]\\ =\frac{2}{Q_{2}}\Delta\mathbf{\hat{f}}^{T}(t)[\mathbf{Q}_{1}\mathbf{e}_{F}(t-\tau)+\Delta \mathbf{f}(t)-\Delta\mathbf{\hat{f}}(t)]\\ =2\beta\eta\Delta\mathbf{\hat{f}}^{T}(t)[\mathbf{Q}_{1}\mathbf{e}_{F}(t-\tau)+\Delta\mathbf{f}( t)-\Delta\mathbf{\hat{f}}(t)]\\ =\beta\eta[2\Delta\mathbf{\hat{f}}^{T}(t)\mathbf{Q}_{1}\mathbf{e}_{F}(t-\tau)+2\Delta\mathbf{ \hat{f}}^{T}(t)\Delta\mathbf{f}(t)-2\Delta\mathbf{\hat{f}}^{T}(t)\Delta\mathbf{\hat{f}}(t)] \end{array}  \tag{23}\]Then, substituting (21) and (23) into (18) yields  
\[ \begin{array}{l}\dot{V}(t)\leq e_{q}^{T}(t)[(\mathbf{A}-\mathbf{L}_{1}\mathbf{C})^{T}\mathbf{P}+ \mathbf{P}(\mathbf{A}-\mathbf{L}_{1}\mathbf{C})]\mathbf{e}_{q}(t)-2\gamma e_{q}^{T}\mathbf{P}\mathbf{\mathrm{ sgn}}(\mathbf{e}_{q})+2\mathbf{e}_{q}^{T}\mathbf{P}\mathbf{\mathrm{{E}}}\mathbf{d}(t)\\ \quad-\beta\mu\mathbf{e}_{F}^{T}(t)\mathbf{e}_{F}(t)+\beta\mathbf{e}_{F}^{T}(t-\tau)[(\eta+ \gamma_{1})\mathbf{Q}_{1}{}^{2}-\mathbf{I}_{3}]\mathbf{e}_{F}(t-\tau)\\ \quad+\beta(\eta+\frac{\eta^{2}}{\gamma_{1}})\Delta\mathbf{f}^{T}(t)\Delta\mathbf{f}(t) \\ \quad=\mathbf{e}_{q}^{T}(t)[(\mathbf{A}-\mathbf{L}_{1}\mathbf{C})^{T}\mathbf{P}+\mathbf{P}(\mathbf{A}-\mathbf{ L}_{1}\mathbf{C})]\mathbf{e}_{q}(t)-2\mathbf{e}_{q}^{T}\mathbf{P}[\gamma\mathrm{sgn}(\mathbf{e}_{q})- \mathbf{\mathrm{{E}}}\mathbf{d}(t)]\\ \quad-\beta\mu\mathbf{e}_{F}^{T}(t)\mathbf{e}_{F}(t)+\beta\mathbf{e}_{F}^{T}(t-\tau)[(\eta +\gamma_{1})\mathbf{Q}_{1}{}^{2}-\mathbf{I}_{3}]\mathbf{e}_{F}(t-\tau)\\ \quad+\beta(\eta+\frac{\eta^{2}}{\gamma_{1}})\Delta\mathbf{f}^{T}(t)\Delta\mathbf{f}(t )\end{array}  \tag{24}\]  
According to (12), it follows that  
\[\|\Delta\mathbf{f}(t)\|\leq\|\mathbf{\mathrm{{F}}}(t)\|+\|\mathbf{\mathrm{{Q}}}_{1}\|\!\cdot \!\|\mathbf{\mathrm{{F}}}(t-\tau)\| \tag{25}\]  
then, \(\Delta\mathbf{f}(t)\) is bounded. If (8), (10), and (11) hold, we have  
\[ \begin{array}{l}\dot{V}(t) \leq-\lambda_{\min}(-\mathbf{\Phi})\mathbf{e}_{q}^{T}(t)\mathbf{e}_{q}(t)- \beta\mu\mathbf{e}_{F}^{T}(t)\mathbf{e}_{F}(t)+\beta(\eta+\frac{\eta^{2}}{\gamma_{1}} )\Delta\mathbf{f}^{T}(t)\Delta\mathbf{f}(t)\\ \leq-\delta\|\left[\begin{array}{c}\mathbf{e}_{q}(t)\\ \mathbf{e}_{F}(t)\end{array}\right]\|_{2}^{1}+\Delta_{f}\end{array}  \tag{26}\]  
where \(\delta=\min(\lambda_{\min}(-\mathbf{\Phi}),\beta\mu)\) with \(\mathbf{\Phi}=(\mathbf{A}-\mathbf{L}_{1}\mathbf{C})^{T}\mathbf{P}+\mathbf{P}(\mathbf{A}-\mathbf{L}_{1}\mathbf{C})\) and \(\Delta_{f}=\beta(\eta+\frac{\eta^{2}}{\gamma_{1}})\Delta\mathbf{f}^{T}(t)\Delta f (t)\). Therefore, state-estimating and fault-reconstructing errors are uniformly ultimately bounded. This completes the proof. \(\square\)  
**Remark 1**. : _It should be noted that (8), expressed in terms of LMI formation, could be conveniently solved using standard LMI tools. However, Theorem 1 included a linear matrix in Equation (9) such that solving (8) and (9) using the LMI toolbox of Matlab was not an easy task. To solve this difficult problem, equation (9) could be transformed into the following inequality using the Schur complement lemma [28]._  
\[\left[ \begin{array}{cc}-\lambda\mathbf{I}_{3}&\mathbf{H}^{T}\mathbf{P}-\mathbf{L}_{2}\mathbf{C }\\ \ast&-\mathbf{I}_{6}\end{array}