## 4 Sensor Fault Isolation Scheme  
This section deals with the fault isolation problem of a three-phase current sensor. Since it is difficult to isolate the sensor fault by direct residual analysis, a robust fault estimation based method is presented to generate distinguished residual sensitive to each phase current sensor fault. First, a parameter-dependent fault estimation observer is constructed for system (7)  
\[\hat{x}\left(k+1\right) =A\left(\theta_{k}\right)\hat{x}\left(k\right)+B_{u}u\left(k \right)+L\left(\theta_{k}\right)\left(y\left(k\right)-\hat{y}\left(k\right) \right), \tag{33}\] \[\hat{y}\left(k\right) =C\hat{x}\left(k\right)+F_{f}\hat{f}\left(k\right),\] \[\hat{f}\left(k+1\right) =\hat{f}\left(k\right)+\Gamma\left(\theta_{k}\right)\left(y\left( k\right)-\hat{y}\left(k\right)\right),\]where \(\hat{x}\left(k\right)\in\mathcal{R}^{n_{x}},\hat{y}\left(k\right)\in\mathcal{R}^{n_ {y}}\) and \(\hat{f}\left(k\right)\in\mathcal{R}^{n_{f}}\) are observer state, observer output and estimate of sensor faults. \(L\left(\theta_{k}\right)\) and \(\Gamma\left(\theta_{k}\right)\) are the gain matrices. Suppose that  
\[\left[ \begin{array}{c}L\left(\theta_{k}\right)\\ \Gamma\left(\theta_{k}\right)\end{array} \right]=\sum\limits_{i=1}^{4}\eta_{k, i}\left[ \begin{array}{c}L_{i}\\ \Gamma_{i}\end{array} \right]. \tag{34}\]  
Let \(e_{x}\left(k\right)=x\left(k\right)-\hat{x}\left(k\right)\) and \(e_{f}\left(k\right)=f\left(k\right)-\hat{f}\left(k\right)\), the estimation error dynamics is expressed as  
\[ \begin{array}{c}e_{x}\left(k+1\right)=\left(A\left(\theta_{k}\right)-L \left(\theta_{k}\right)C\right)e_{x}\left(k\right)-L\left(\theta_{k}\right)F_ {f}e_{f}\left(k\right)+B_{d}d\left(k\right),\\ e_{f}\left(k+1\right)=-\Gamma\left(\theta_{k}\right)Ce_{x}\left(k\right)+ \left(I-\Gamma\left(\theta_{k}\right)F_{f}\right)e_{f}\left(k\right)+f\left(k +1\right)-f\left(k\right).\end{array}  \tag{35}\]  
Choose \(\xi\left(k\right)=\left[e_{x}^{T}\left(k\right),e_{f}^{T}\left(k\right) \right]^{T}\) and \(\Delta f\left(k\right)=f\left(k+1\right)-f\left(k\right),\)  
\[ \begin{array}{c}\xi\left(k+1\right)=\left(\bar{A}\left(\theta_{k}\right)- \bar{L}\left(\theta_{k}\right)\bar{C}\right)\xi\left(k\right)+\bar{B}_{d} \bar{d}\left(k\right),\\ e_{f}\left(k\right)=\bar{C}_{e}\xi\left(k\right),\end{array}  \tag{36}\]  
with \(\bar{d}\left(k\right)=\left[ \begin{array}{c}d\left(k\right)\\ \Delta f\left(k\right)\end{array} \right]\), \(\bar{A}\left(\theta_{k}\right)=\left[ \begin{array}{cc}A\left(\theta_{k} \right)&0_{n_{x}\times n_{f}}\\ 0_{n_{f}\times n_{x}}&{}_{n_{f}}\end{array} \right]\), \(\bar{L}\left(\theta_{k}\right)=\left[ \begin{array}{c}L\left(\theta_{k} \right)\\ \Gamma\left(\theta_{k}\right)\end{array} \right]\),  
\(\bar{B}_{d}=\left[ \begin{array}{cc}B_{d}&0_{n_{x}\times n_{f}}\\ 0_{n_{f}\times n_{d}}&{}_{1n_{f}\times n_{f}}\end{array} \right]\), \(\bar{C}=\left[ \begin{array}{cc}C&F_{f}\end{array} \right]\), \(\bar{C}_{e}=\left[ \begin{array}{cc}0_{n_{f}\times n_{x}}&{}_{1n_{f}\times n _{f}}\end{array} \right]\). **Theorem 2**. : _If there exists a symmetric positive definite matrix \(\mathcal{P}_{1}\), real matrices \(\bar{\mathcal{Y}}_{v,i}\) with appropriate dimensions, \(i=1,\cdots,4\), positive scalar \(\gamma_{1}\), such that the following linear matrix inequality holds_  
\[\left[ \begin{array}{ccc}-\mathcal{P}_{1}&0_{\left(n_{x}+n_{f}\right)\times n _{f}}&\mathcal{P}_{1}\bar{A}_{v,i}-\bar{\mathcal{Y}}_{v,i}\bar{C}&\mathcal{P} \bar{B}_{d}\\ *&-\gamma_{1}{}_{n_{f}\times n_{f}}&\bar{C}_{e}&0_{n_{f}\times\left(n_{d}+n_{f} \right)}\\ *&*&-\mathcal{P}_{1}&0_{\left(n_{x}+n_{f}\right)\times\left(n_{d}+n_{f} \right)}\\ *&*&*&-\gamma_{1}{}_{\left(n_{d}+n_{f}\right)\times\left(n_{d}+n_{f}\right)} \end{array} \right]<0, \tag{37}\]  
_then fault estimaiton error dynamics (36) satisfies \(\mathcal{H}_{\infty}\) performance \(\left\|e_{f}\left(k\right)\right\|_{2}^{2}\leq\gamma_{1}^{2}\|\bar{d}\left(k \right)\|_{2}^{2}\).