## 3 Active Fault-Tolerant Control System Design
### Fault Diagnosis Scheme
 \tag{13}\]  
then the error dynamics can be obtained from Equations (7) and (9) as:  
\[\dot{e}_{x}(t)=(A-LC)e_{x}(t)+Be_{f}(t)+E_{d}d(t) \tag{14}\]  
Let us consider the following Lyapunov candidate:  
\[V(t)=e_{x}^{T}(t)Pe_{x}(t)+\rho^{-1}e_{f}^{T}(t)\Lambda^{-1}e_{f}(t) \tag{15}\]  
Its time derivative \(\dot{V}(t)\) is:  
\[ \begin{array}{ll}\dot{V}(t)&=\dot{e}_{x}^{T}(t)Pe_{x}(t)+e_{x}^{T}(t)P\dot{ e}_{x}(t)\\ &+2\rho^{-1}e_{f}^{T}(t)\Lambda^{-1}\dot{e}_{f}(t)\\ &=e_{x}^{T}(t)\Big{[}(A-LC)^{T}\!P+P(A-LC)\Big{]}\!e_{x}(t)\\ &+2\dot{e}_{x}^{T}(t)PBe_{f}(t)\ +2\dot{e}_{x}^{T}(t)PE_{d}d(t)\\ &+2\rho^{-1}e_{f}^{T}(t)\Lambda^{-1}(\dot{f}(t)-\hat{f}(t))\end{array}  \tag{16}\]  
Substituting Equations (11), (12), and (14) into Equation (16) we obtain:  
\[ \begin{array}{ll}\dot{V}(t)=&e_{x}^{T}(t)\Big{[}(A-LC)^{T}\!P+P(A-LC)\Big{]} \!e_{x}(t)\\ &+2e_{x}^{T}PE_{d}(t)\ +2\rho^{-1}e_{f}^{T}(t)\Lambda^{-1}f(t)\\ &-2e_{f}^{T}R_{2}C(A-LC)e_{x}-2e_{f}^{T}R_{2}CBe_{f}\\ &-2\dot{e}_{f}^{T}R_{2}CE_{d}d(t)\end{array}  \tag{17}\]  
From Lemma 1, Assumption 1, and Theorem 1, we obtain:  
\[ \begin{array}{ll}2\rho^{-1}e_{f}^{T}(t)\Lambda^{-1}\dot{f}(t)\\ \leq\rho^{-1}(e_{f}^{T}(t)Ge_{f}^{T}(t)\ +\dot{f}^{T}(t)\Lambda^{-1}G^{-1} \Lambda^{-1}\dot{f}(t))\\ \leq\rho^{-1}(e_{f}^{T}(t)Ge_{f}^{T}(t)\ +f_{1}^{2}\lambda_{\max}(\Lambda^{-1}G^{-1} \Lambda^{-1}))\end{array}  \tag{18}\]  
where \(\lambda_{\max}\) is the maximum eigenvalue of the associated matrix. The \(H_{\infty}\) tracking index function is chosen to ensure that the adaptive observer is robust against the external disturbance as:  
\[J=\int\limits_{0}^{\infty}\Big{[}e_{y}^{T}(t)e_{y}(t)-\varepsilon^{2}d^{T}(t )d(t)\Big{]}\!\mathrm{d}t \tag{19}\]With the initial conditions \((V(0)=0,\ V(\infty)>0)\), we obtain:  
\[ \begin{array}{ll}J&=\int\limits_{0}^{\infty}\Bigl{[}e_{y}^{T}(t)e_{y}(t)- \varepsilon^{2}d^{T}(t)d(t)+\dot{V}(t)\Bigr{]}\mathrm{d}t\ +V(0)-V(\infty)\\ &\leq\int\limits_{0}^{\infty}\Bigl{[}e_{x}^{T}(t)C^{T}Ce_{x}(t)- \varepsilon^{2}d^{T}(t)d(t)+\dot{V}(t)\Bigr{]}\mathrm{d}t\end{array}  \tag{20}\]  
From Equation (19), the integral term can be described as:  
\[ \begin{array}{ll}N&=e_{x}^{T}\Bigl{(}(A-LC)^{T}P+P(A-LC)+C^{T}C\Bigr{)}e_{x} \\ &+2e_{x}^{T}PE_{d}d(t)\ +\rho^{-1}e_{f}^{T}(t)Ge_{f}(t)\\ &+\rho^{-1}f_{1}^{2}\lambda_{\max}(\Lambda^{-1}G^{-1}\Lambda^{-1})\ -2e_{f}^{T}R_{2}C(A-LC)e_{x}\\ &-2e_{f}^{T}R_{2}CE_{f}-2e_{f}^{T}R_{2}CE_{d}d(t)-\varepsilon^{2}d^{T}(t)d(t) \\ =\left[\begin{array}{c}e_{x}(t)\\ e_{f}(t)\\ d(t)\end{array}\right]^{T}\left[\begin{array}{ccc}M_{11}&M_{12}&M_{13}\\ *&M_{22}&M_{23}\\ *&*&M_{33}\end{array}\right]\left[\begin{array}{c}e_{x}(t)\\ e_{f}(t)\\ d(t)\end{array}\right]\end{array}  \tag{21}\]  
where  
\[ \begin{array}{ll}M_{11}=(A-LC)^{T}P+P(A-LC)+C^{T}C\\ M_{12}=(A-LC)^{T}C^{T}R_{2}^{T}\\ M_{13}=PE_{d}\\ M_{22}=\rho^{-1}G-2R_{2}CB\\ M_{23}=-R_{2}CE_{d}\\ M_{33}=-\varepsilon^{2}l_{p}\\ \eta=\rho^{-1}f_{1}^{2}\lambda_{\max}(\Lambda^{-1}G^{-1}\Lambda^{-1})\end{array} \]  
If the following inequality holds:  
\[\left[ \begin{array}{ccc}M_{11}&M_{12}&M_{13}\\ *&M_{22}&M_{23}\\ *&*&M_{33}\end{array} \right]<0 \tag{22}\]  
when we have \(N<nI_{n}\) and thus can obtain \(J<\eta\). **Remark 1**. : _To solve Equations (10) and (11) simultaneously, the linear matrix inequalities (LMI) technique was applied to relax these conditions. Equation (11) was modified according to previous study [5]._  
\[\left[ \begin{array}{cc}\eta_{1}I&B^{T}P-R_{1}C\\ (B^{T}P-R_{1}C)^{T}&\eta_{1}I\end{array} \right]>0 \tag{23}\]