## 3 Current Sensor Fault Detection
### Parameter-Dependent Observer Design  
For system (14), a parameter-dependent observer based residual generator is designed to detect sensor fault  
\[\hat{x}\left(k+1\right) =A\left(\theta_{k}\right)\hat{x}\left(k\right)+B_{u}u\left(k \right)+L\left(\theta_{k}\right)r\left(k\right), \tag{20}\] \[\hat{y}\left(k\right) =C\hat{x}\left(k\right),\] \[r\left(k\right) =y\left(k\right)-\hat{y}\left(k\right),\]in which \(\hat{x}\left(k\right)\) is state estimation, \(r\left(k\right)\) is the desired residual to current sensor fault. \(L\left(\theta_{k}\right)\) denotes the observer gain. By defining the state estimation error \(e\left(k\right)=x\left(k\right)-\hat{x}\left(k\right)\), the error dynamics is obtained by substituting Equation (20) into system Equation (7)  
\[e\left(k+1\right) =\left(A\left(\theta_{k}\right)-L\left(\theta_{k}\right)C\right) e\left(k\right)+B_{d}d\left(k\right)-L\left(\theta_{k}\right)F_{f}f\left(k \right), \tag{21}\] \[r\left(k\right) =Ce\left(k\right)+F_{f}f\left(k\right).\]  
For the fault-free case \(f\left(k\right)=0\), the error dynamics (21) become  
\[e\left(k+1\right) =\left(A\left(\theta_{k}\right)-L\left(\theta_{k}\right)C\right) e\left(k\right)+B_{d}d\left(k\right), \tag{22}\] \[r\left(k\right) =Ce\left(k\right).\]  
The following theorem provides a method to determine the gain matrix \(L\left(\theta_{k}\right)\) and to guarantee the stability and convergency of the proposed residual generator. **Theorem 1**. : _For the system (14) and residual generator (22), suppose that there exists a scalar \(\gamma>0\), positive definite matrix \(P=P^{T}\) and real matrices \(U_{i}\), for \(i=1,\cdots,4\) such that_  
\[\left[ \begin{array}{cccc}-P&0_{n_{x}\times n_{x}}&PA_{v,j}-U_{i}C&PB_{d}\\ \ast&-\gamma I_{n_{x}\times n_{x}}&C&0_{n_{x}\times n_{d}}\\ \ast&\ast&-P&0_{n_{x}\times n_{d}}\\ \ast&\ast&\ast&-\gamma I_{n_{d}\times n_{d}}\end{array} \right]<0. \tag{23}\]  
_Then, the residual generator is a asymptotically stable and the following holds_  
\[\sum_{k=0}^{\infty}r^{T}\left(k\right)r\left(k\right)=\gamma^{2}\sum_{k=0}^{ \infty}d^{T}\left(k\right)d\left(k\right)+\gamma V\left(0\right) \tag{24}\]  
_and parameter-dependent observer gain is given for \(i=1,\cdots,4\)_  
\[L\left(\theta_{k}\right) =\sum_{i=1}^{4}\eta_{k,i}L_{i}, \tag{25}\] \[L_{i} =P^{-1}U_{i}.\]  
**Remark 2**. : _In this paper, only \(A\left(\theta_{k}\right)\) in system (6) is parameter-dependent while \(B_{u}\), \(B_{d}\), \(C\) and \(F_{f}\) remain constant. Otherwise, varying matrices \(C\) and \(F_{f}\) may lead to a bilinear matrix inequality (BMI) of Equation (23). Further procedures are required to deal with such BMIs._  
Proof of Theorem 1.: This proof contains two parts: one is to prove the stability of the residual generator and the other is to calculate the upper bound of residuals in Equation (24). First, assume that Equation (23) holds. By substituting Equation (25) into Equation (23),  
\[\left[ \begin{array}{cccc}-P&0&P\left(A_{v,i}-L_{i}C\right)&PB_{d}\\ \ast&-\gamma I&C&0\\ \ast&\ast&-P&0\\ \ast&\ast&\ast&-\gamma I\end{array} \right]<0 \tag{26}\]  
for all \(i=1,\cdots,N\), by multiplying Equation (26) with \(\eta_{k,i}\) and sum to obtain  
\[\left[ \begin{array}{cccc}-P&0&P\left(A\left(\theta_{k}\right)-L\left(\theta _{k}\right)C\right)&PB_{d}\\ \ast&-\gamma I&C&0\\ \ast&\ast&-P&0\\ \ast&\ast&\ast&-\gamma I\end{array} \right]<0. \tag{27}\]Letting \(A\left(\rho\right)=A\left(\theta_{k}\right)-L\left(\theta_{k}\right)C\), \(B\left(\rho\right)=B_{d}\), \(C\left(\rho\right)=C\) and \(D\left(\rho\right)=0\), Equation (27) implies Equation (18). According to Lemma 1, residual generator (22) is asymptotically stable. Second, consider the following Lyapunov function  
\[V\left(k\right)=e^{T}\left(k\right)Pe\left(k\right), \tag{28}\]  
where \(P\) is a positive definite matrix and \(\Delta V\left(k\right)=V\left(k+1\right)-V\left(k\right)\).