## 3 Design of Fault Detection and Estimation Observer
### Design of Fault Estimation Observer  
After fault detection, an adaptive fault observer is designed to estimate the total fault error of the system actuator, as shown in Equations (17) and (18). \[ \begin{array}{ll}\dot{\hat{x}}_{e}(t)&=A\hat{x}_{e}(t)+A_{m}\hat{x}_{e}(t-d(t ))+B(I+F(t))u_{c}+\hat{f}+L(y(t)-\hat{y}_{e}(t))\\ &+H(y(t-d(t))-\hat{y}_{e}(t-d(t)))\\ &\hat{y}_{e}(t)=C\hat{x}_{e}(t)\end{array}  \tag{17}\]  
The adaptive rate \(\hat{f}\) is designed as:  
\[\dot{\hat{f}}(t)=\Lambda\Big{(}K_{1}\tilde{y}+K_{2}\dot{\hat{y}}\Big{)} \tag{18}\]  
where \(\Lambda=\Lambda^{T}>0\), \(K_{1}\), \(K_{2}\) are the gain matrix. Let \(\tilde{y}=y-\hat{y}_{e}\), \(\tilde{f}=f-\hat{f}\), \(\widetilde{x}_{e}=x-\hat{x}_{e}\), combining with Equations (17) and (18), Equation (20) is obtained. \[\dot{\widetilde{x}}_{e}=(A-LC)\widetilde{x}_{e}+(A_{m}-HC)\widetilde{x}_{e}(t -d(t))+\tilde{f}+Dv(t) \tag{21}\]  
**Theorem 2**. : _Considering Assumptions 2 and 3, there are matrices \(P_{1}\), \(P_{2}\) which satisfy the relation in Equation (21) and \(P_{1}=P_{1}^{T}>0\), \(P_{2}=P_{2}^{T}>0\)._  
\[\left\{ \begin{array}{cc}P_{1}^{T}=K_{1}C\\ P_{2}\Lambda=I\end{array} \right. \tag{22}\]  
_There exist observation matrices \(L\), \(H\) such as:_  
\[ \begin{bmatrix}-2K_{2}C+2\beta^{-1}M&-2K_{2}CD&-2K_{2}C(A-LC)&2K_{2}C(A_{m}- HC)\\ *&\overline{d}^{2}I&D^{TP}P_{1}^{T}&0\\ *&*&\Omega_{1}&0\\ *&*&*&\Omega_{2}\end{bmatrix} <0 \tag{23}\]  
_where \(\Omega_{1}=(A-LC)^{T}P_{1}+P_{1}(A-LC)\), \(\Omega_{2}=(A_{m}-HC)^{T}P_{1}+P_{1}(A_{m}-HC)\), which makes the system state estimation error stable under the designed adaptive law._  
**Proof of Theorem 2. ** By Considering the Lyapunov candidate as:  
\[V_{1}=\widetilde{x}_{e}^{T}P_{1}\widetilde{x}_{e}+\widetilde{f}^{T}P_{2} \widetilde{f}+\int_{t-d(t)}^{t}\widetilde{x}_{e}^{T}P_{3}\widetilde{x}_{e}dt \tag{24}\]  
Its derivative is  
\[\dot{V}_{1}=\dot{\widetilde{x}}_{e}^{T}P_{1}\widetilde{x}_{e}+\widetilde{x}_{e }^{T}P_{1}\dot{\widetilde{x}}_{e}+2\widetilde{f}^{T}P_{2}\dot{\widetilde{f}}+ \widetilde{x}_{e}^{T}P_{3}\widetilde{x}_{e}-\Big{(}1-\dot{d}(t)\Big{)} \widetilde{x}_{e}^{T}P_{3}\widetilde{x}_{e} \tag{25}\]Combining with Equations (19)-(21), it is further simplified to obtain  
\[ \begin{array}{ll}\dot{V}_{1}&=\dot{\tilde{x}}_{e}^{T}\Big{[}\big{(}A-LC\big{)}^ {T}P_{1}+P_{1}(A-LC)\Big{]}\tilde{x}_{e}+2\tilde{x}_{e}P_{1}\tilde{f}+2\tilde{x }_{e}^{T}P_{1}Dv(t)\\ &+\dot{\tilde{x}}_{e}^{T}\big{(}t-d(t)\big{)}\Big{[}\big{(}A_{m}-HC\big{)}^{T}P _{1}+P_{1}(A_{m}-HC\big{)}\Big{]}\tilde{x}_{e}(t-d(t))\\ &+2\tilde{f}^{T}P_{2}\bigg{(}\dot{f}-\tilde{f}\bigg{)}+\tilde{x}_{e}^{T}P_{3} \tilde{x}_{e}-(1-d(t))\tilde{x}_{e}^{T}P_{3}\tilde{x}_{e}\\ &=\dot{\tilde{x}}_{e}^{T}\Big{[}\big{(}A-LC\big{)}^{T}P_{1}+P_{1}(A-LC)\Big{]} \tilde{x}_{e}+2\tilde{x}_{e}^{T}P_{1}Dv(t)+2\tilde{f}^{T}P_{2}\dot{f}\\ &+\dot{\tilde{x}}_{e}^{T}\big{(}t-d(t)\big{]}\big{(}A_{m}-HC\big{)}^{T}P_{1}+P_ {1}(A_{m}-HC)\Big{]}\tilde{x}_{e}(t-d(t))\\ &-2\tilde{f}^{T}K_{2}C(A-LC)\tilde{x}_{e}-2\tilde{f}^{T}K_{2}C\tilde{f}+ \tilde{x}_{e}^{T}P_{3}\tilde{x}_{e}-(1-d(t))\tilde{x}_{e}^{T}P_{3}\tilde{x}_{e }\\ &-2f^{T}K_{2}CDv(t)-2f^{T}K_{2}C(A_{m}-HC)\tilde{x}_{e}(t-d(t))\end{array}  \tag{25}\]  
According to Lemma 1, there exist matrix \(M=M^{T}\), constant \(\beta>0\), which satisfy:  
\[ \begin{array}{ll}2\tilde{f}^{T}P_{2}\dot{f}&\leq 2\beta^{-1}\tilde{f}^{T}Mf +2\beta\tilde{f}^{T}\big{(}{P_{2}}^{T}M^{-1}P_{2}\big{)}\dot{f}\\ &\leq 2\beta^{-1}\tilde{f}^{T}Mf+2\mu^{2}\lambda_{max}({P_{2}}^{T}M^{-1}P_ {2})\end{array}