## 3 Iterative Learning Observer-Based Thruster Fault Reconstruction Approach
### Stability Analysis of the Learning Output-Feedback Fault-Tolerant Controller  
To guarantee the stability and convergence of the formation configuration tracking error dynamics system (29) under the controller (30), the stability proof of the closed-loop formation configuration tracking error dynamics (33) is provided in detail. **Theorem 2**. : _For the formation configuration tracking error dynamics system (29) under controller (30), if there exists positive definite symmetric matrices \(\mathbf{P}_{1}\in\mathbb{R}^{6\times 6}\), \(\mathbf{Q}\in\mathbb{R}^{6\times 6}\), \(\mathbf{G}_{3}\in\mathbb{R}^{3\times 3}\), matrix \(\mathbf{K}\in\mathbb{R}^{3\times 6}\), and a scalar \(G_{1}\) such that the following relations hold,_  
\[\mathbf{P}_{1}\mathbf{A}+\mathbf{A}^{T}\mathbf{P}_{1}-\mathbf{P}_{1}\mathbf{B}\mathbf{B}^{T}\mathbf{P}_{1}=-\bm {Q} \tag{34}\]  
\[\mathbf{K}\mathbf{C}=\mathbf{B}^{T}\mathbf{P}_{1} \tag{35}\]\[(1+\mu_{1}+\gamma_{3})G_{1}^{2}-1<0 \tag{36}\] \[(1+\mu_{1}+\gamma_{2})G_{3}^{T}G_{3}-\textbf{I}_{3}<0 \tag{37}\]  
_where \(G_{2}=1/l(1+\mu_{1})\) and \(G_{4}=-1/\beta_{1}(1+\mu_{1})\) with \(\mu_{1}>0\), \(\gamma_{2}>0\), \(\gamma_{3}>0\), the closed-loop formation configuration tracking error dynamics (33) is uniformly ultimately bounded._  
**Proof of Theorem 2. ** Define  
\[\left\{ \begin{array}{l}\rho_{1}(t)=\textbf{{h}}(t)-G_{3}\textbf{{h}}(t-\tau) \\ \rho_{2}(t)=v(t)-G_{3}v(t-\tau)\\ \rho_{3}(t)=\Sigma(t)-G_{1}\Sigma(t-\tau)\\ \rho_{4}(t)=\pi(t)-G_{1}\pi(t-\tau)\end{array} \right. \tag{38}\]  
where \(\textbf{{h}}(t)\) and \(\Sigma(t)\) denote adaptive learning terms of \(v(t)\) and \(\pi(t)\), respectively. Then substituting (31) and (32) into (38) yields  
\[\left\{ \begin{array}{l}\rho_{2}(t)=G_{4}\textbf{{K}}\textbf{{C}}\textbf{{e}} _{q}(t)\\ \rho_{4}(t)=G_{2}\textbf{{e}}_{q}^{T}(\textbf{{K}}\textbf{{C}})^{T}(\textbf{{K }}\textbf{{C}})\textbf{{e}}_{q}(t)\end{array} \right. \tag{39}\]  
Let \(e_{\Gamma}(t)=\Sigma(t)-\pi(t)\), it follows from (38) that  
\[\left\{ \begin{array}{l}\textbf{{e}}_{h}(t)=\rho_{1}(t)-\rho_{2}(t)+\textbf{ {G}}_{3}\textbf{{e}}_{h}(t-\tau)\\ \epsilon_{\Gamma}(t)=\rho_{3}(t)-\rho_{4}(t)+G_{1}\epsilon_{\Gamma}(t-\tau) \end{array} \right. \tag{40}\]  
therefore, the new form of \(\pi(t)\) can be described by  
\[ \begin{array}{l}\pi(t)=\Sigma(t)-\epsilon_{\Gamma}(t)\\ =\Sigma(t)-[\rho_{3}(t)-\rho_{4}(t)+G_{1}\epsilon_{\Gamma}(t-\tau)]\end{array}  \tag{41}\]  
Substituting (41) into (33) leads to  
\[ \begin{array}{l}\dot{\textbf{{e}}}_{q}(t)=(\textbf{{A}}-\textbf{{B}}\pi \textbf{{K}}\textbf{{C}})\textbf{{e}}_{q}(t)+\textbf{{B}}[G_{1}\epsilon_{ \Gamma}(t-\tau)+\rho_{3}(t)-\rho_{4}(t)]\textbf{{K}}\textbf{{C}}\textbf{{e}}_ {q}(t)\\ -\textbf{{B}}[\textbf{{G}}_{3}\textbf{{e}}_{h}(t-\tau)+\rho_{1}(t)-\rho_{2}(t)] \end{array}  \tag{42}\]  
Considering the following Lyapunov-Krasovski function candidate  
\[V(t)=\textbf{{e}}_{q}^{T}\textbf{{P}}_{1}\textbf{{e}}_{q}+\beta_{1}\int_{t- \tau}^{t}\textbf{{e}}_{h}^{T}(s)\textbf{{e}}_{h}(s)ds+l\int_{t-\tau}^{t} \textbf{{e}}_{\Gamma}^{T}(s)\epsilon_{\Gamma}(s)ds \tag{43}\]  
where \(\beta_{1}>0\) and \(l>0\), the derivative of the Lyapunov candidate \(\dot{V}(t)\) can be derived as  
\[\begin{array}{l}\dot{V}(t)=\textbf{{e}}_{q}^{T}(\textbf{{P}}_{1}\textbf{{A}} +\textbf{{A}}^{T}\textbf{{P}}_{1}-2\Sigma\textbf{{P}}_{1}\textbf{{B}}\textbf{{ K}}\textbf{{C}})\textbf{{e}}_{q}+2\textbf{{e}}_{q}^{T}\textbf{{P}}_{1}\textbf{{B}}[G_{1} \epsilon_{\Gamma}(t-\tau)+\rho_{3}(t)-\rho_{4}(t)]\textbf{{K}}\textbf{{C}} \textbf{{e}}_{q}\\ -2\textbf{{e}}_{q}^{T}\textbf{{P}}_{1}\textbf{{B}}[G_{3}\textbf{{e}}_{h}(t- \tau)+\rho_{1}(t)-\rho_{2}(t)]+\beta_{1}[-\mu_{1}\textbf{{e}}_{h}^{T}(t) \textbf{{e}}_{h}(t)+(1+\mu_{1})\textbf{{e}}_{h}^{T}(t)\textbf{{e}}_{h}(t)\\ -\textbf{{e}}_{h}^{T}(t-\tau)\textbf{{e}}_{h}(t-\tau)]+l[-\mu_{1}\textbf{{e}}_{ \Gamma}  
In this work, it was assumed that an abrupt constant fault occurred in the \(x\)-axis thruster, a slow time-varying fault occurred in the \(z\)-axis thruster, and the thruster in the \(y\)-axis was fault free.