## 4 Sensor Fault Isolation Scheme
The fault estimation observer gain matrix is \(\bar{L}\left(\theta_{k}\right)=\sum\limits_{i=1}^{4}\eta_{k,i}\left(\theta_{k} \right)\bar{L}_{i}\) with \(\bar{L}_{i}=\mathcal{P}_{1}{}^{-1}\bar{\mathcal{Y}}_{v,i}\)._  
Proof of Theorem 2.: According to the polytopic decompostion, parameter-dependent matrix \(\bar{A}\left(\theta_{k}\right)\) and \(L\left(\bar{\theta}_{k}\right)\) are as follows:  
\[ \begin{array}{c}\bar{A}\left(\theta_{k}\right)=\sum\limits_{i=0}^{4}\eta_{k, i}\bar{A}_{v,i},\\ \bar{L}\left(\theta_{k}\right)=\sum\limits_{i=0}^{4}\eta_{k,i}\bar{L}_{v,i}. \end{array}  \tag{38}\]  
Consider the following Lyapunov function  
\[V\left(k\right)=\xi^{T}\left(k\right)\mathcal{P}_{1}\xi\left(k\right). \tag{39}\]  
The cost function \(J_{\infty}\) is defined as  
\[J_{\infty}=\sum\limits_{k=0}^{\infty}\left[\Delta V\left(k\right)+\frac{1}{ \gamma_{1}}e_{f}^{T}\left(k\right)e_{f}\left(k\right)-\gamma_{1}\bar{d}^{T} \left(k\right)\bar{d}\left(k\right)\right]<0. \tag{40}\]By substituting \(\Delta V\left(k\right)\) and \(e_{f}^{T}\left(k\right)e_{f}\left(k\right)\) into Equation (40), we have  
\[ \begin{split} J_{\infty}=&\left[\begin{array}{c} \xi\left(k\right)\\ \bar{d}\left(k\right)\end{array}\right]^{T}\left[\begin{array}{cc}\bar{A}^{T} \left(\theta_{k}\right)\mathcal{P}_{1}-\bar{C}^{T}\bar{L}^{T}\left(\theta_{k} \right)\mathcal{P}_{1}\\ \bar{B}_{d}^{T}\mathcal{P}_{1}\end{array}\right]\mathcal{P}_{1}^{-1}\left[ \begin{array}{cc}\mathcal{P}_{1}\bar{A}\left(\theta_{k}\right)-\mathcal{P}_ {1}\bar{L}\left(\theta_{k}\right)\mathcal{C}&\mathcal{P}_{1}\bar{B}_{d}\end{array} \right]\left[\begin{array}{c}\xi\left(k\right)\\ \bar{d}\left(k\right)\end{array}\right]\\ &+\left[\begin{array}{c}\xi\left(k\right)\\ \bar{d}\left(k\right)\end{array}\right]^{T}\left[\begin{array}{cc}\frac{1}{ \tau_{1}}C_{e}^{T}C_{e}-\mathcal{P}_{1}&0\\ 0&-\gamma_{1}I\end{array}\right]\left[\begin{array}{c}\xi\left(k\right)\\ \bar{d}\left(k\right)\end{array}\right].\end{split}  \tag{41}\]  
Furthermore, Equation (41) can be rewritten as  
\[\left[ \begin{array}{cc}\bar{A}^{T}\left(\theta_{k}\right)\mathcal{P}_{1}- \bar{C}^{T}\bar{L}^{T}\left(\theta_{k}\right)\mathcal{P}_{1}\\ \bar{B}_{d}^{T}\mathcal{P}_{1}\end{array} \right]\mathcal{P}_{1}^{-1}\left[ \begin{array}{cc}\mathcal{P}_{1}\bar{A}\left(\theta_{k}\right)-\mathcal{P}_ {1}\bar{L}\left(\theta_{k}\right)\bar{C}&\mathcal{P}_{1}\bar{B}_{d}\end{array}  \right]+\left[ \begin{array}{cc}\frac{1}{\tau_{1}}C_{e}^{T}C_{e}-\mathcal{P}_ {1}&0\\ 0&-\gamma_{1}I\end{array} \right]<0. \tag{42}\]  
Note that Equation (37) is a sufficient condition of (42). Thus, if Equation (37) holds, the estimation error system satisfies \(\mathcal{H}_{\infty}\) performance \(\left\|e_{f}\left(k\right)\right\|_{2}^{2}\leq\gamma_{1}^{2}\|\bar{d}\left(k \right)\|_{2}^{2}\). The proof is completed. This method presents a biased estimation of sensor fault \(\hat{f}\left(k\right)\) with an upper bound \(\gamma_{1}^{2}\|\bar{d}\left(k\right)\|_{2}^{2}\) due to the disturbances \(\bar{d}\left(k\right)\). However, it has the ability to locate the fault sensor phase by the tuned isolation threshold for each estimation \(\hat{f}_{a}\left(k\right)\), \(\hat{f}_{b}\left(k\right)\) and \(\hat{f}_{c}\left(k\right)\). **Remark 3**. : _Before designing the proposed residual generator and fault estimator, it is necessary to check observability of the pair \(\left(A\left(\theta_{k}\right),\mathcal{C}\right)\) and \(\left(\bar{A}\left(\theta_{k}\right),\bar{C}\right)\). This paper checks this property by analyzing the observability of each pair on the vertices._