## 4 Robust Fault-Tolerant Predictive Control
\begin{array}{cccccccc}-\Xi_{1}&*&*&*&*&*&*\\ R^{\frac{1}{2}}Y_{r}&-\gamma I&*&*&*&*&*\\ Q^{\frac{1}{2}}X_{r}&0_{n}&-\gamma I&*&*&*&*&*\\ 0&0_{n}&0_{n}&-\frac{\epsilon^{2}I}{\gamma}&*&*&*&*\\ 0&0_{n}&0_{n}&0_{n}&-\Xi_{2}&*&*&*\\ -R^{\frac{1}{2}}Y_{r}E_{1}&0_{n}&0_{n}&0_{n}&0_{n}&-\gamma I&*&*\\ A_{r}X_{r}+B_{r}Y_{r}&-B_{r}Y_{r}E_{1}&D_{r}X_{r}&0_{n}&0_{n}&0_{n}&-X_{j}&*\\ 0_{n}&T_{1}Z_{r}\mathcal{A}_{r}-H_{r}E_{2}&Z_{r}T_{1}D_{r}&0_{n}&0_{n}&0_{n}&- W_{j}\end{array} \right]<0 \tag{31}\]  
\[r,j=1,\ldots,\ N\]  
where \(\Xi_{1}=-M_{r}+X_{r}^{T}+X_{r}\), \(\Xi_{2}=-N_{r}+X_{r}^{T}+X_{r}\) and the matrices \(M_{r}\) and \(N_{r}\) are positives. Furthermore, since \(X>0\), we have  
\[(X_{r}^{T}-M_{r})^{T}X^{-1}(X_{r}-M_{r})\geq 0\Rightarrow-M_{r}+X_{r}^{T}+X_{r }\leq X_{r}^{T}M_{r}^{-1}X_{r} \tag{32}\]  
Consequently,  
\[\Rightarrow-N_{r}+X_{r}^{T}+X_{r}\leq X_{r}^{T}N_{r}^{-1}X_{r} \tag{33}\]By taking into account Equations (32) and (33), we can state that (17) implies  
\[r,j=1,\ldots,\ N\]  
where \(\Gamma(\theta_{k})=\sum\limits_{r=1}^{N}\rho_{r}(\theta_{k})\Gamma_{r}\) with \(\Gamma_{r}=\gamma M_{r}^{-1}\), and \(\Pi(\theta_{k})=\sum\limits_{r=1}^{N}\rho_{r}(\theta_{k})\Pi_{r}\) with \(\Pi_{r}=\gamma N_{r}^{-1}\). Furthermore, \(\Gamma(\theta_{k+1})=\sum\limits_{j=1}^{N}\rho_{r}(\theta_{k+1})\Gamma_{j}\) and \(\Pi(\theta_{k+1})=\sum\limits_{j=1}^{N}\rho_{r}(\theta_{k+1})\Pi_{j}\), and by substituting \(Y_{r}=F_{r}X_{r}\) and \(H_{r}=Z_{r}L_{r}\) into Equation (29), pre-multiplying and post-multiplying by Diag \([X_{r}^{-\Gamma},\ I,\ I,\ I,\ N_{r}^{-1},\ I,\ I,\ I]\), and using the Schur complement, we see that this is equivalent to:  
\[\left[ \begin{array}{ccccc}-\Gamma(\theta_{k})+Q+F(\theta_{k})^{T}RF(\theta_{ k})&*&*&*&*\\ 0_{n}&-\Pi(\theta_{k})+(-F(\theta_{k})E_{1})^{T}R(-F(\theta_{k})E_{1})&*&*&*\\ 0_{n}&0_{n}&-\alpha^{2}I&*&*\\ A(\theta_{k})+B(\theta_{k})F&-B(\theta_{k})F(\theta_{k})E_{1}&D&-\Gamma(\theta_ {k+1})^{-1}&*\\ 0_{n}&T_{1}\mathcal{A}(\theta_{k})-L(\theta_{k})E_{2}&T_{1}D&0_{n}&\Pi(\theta_ {k})^{-1}\end{array} \right]<0 \tag{35}\]  
Again, applying the Schur complement to Equation (35):  
\[\left[ \begin{array}{ccccc}A(\theta_{k})+B(\theta_{k})F(\theta_{k})&0\\ -B(\theta_{k})F(\theta_{k})E_{1}&T_{1}\mathcal{A}-LE_{2}\\ D&T_{1}D\end{array} \right]^{T}\left[ \begin{array}{ccccc}\Gamma(\theta_{k+1 })&0\\ 0&\Pi(\theta_{k+1})\end{array} \right]\left[ \begin{array}{ccccc}A(\theta_{k} )+B(\theta_{k})F&-B(\theta_{k})F(\theta_{k})E_{1}&D\\ 0&T_{1}\mathcal{A}(\theta_{k})-L(\theta_{k})E_{2}&T_{1}D\end{array} \right]\]  
\[+\left[ \begin{array}{ccccc}-\Gamma(\theta_{k})+Q+F(\theta_{k})^{T}RF(\theta _{k})&*&*\\ 0_{n}&-\Pi(\theta_{k})+(-F(\theta_{k})E_{1})^{T}R(-F(\theta_{k})E_{1})&*\\ 0_{n}&0_{n}&-\alpha^{2}I\end{array} \right]<0 \tag{36}\]  
Then, multiplying the resulting inequality from the left by \([x(k)\ e(k)\ \omega(k)]\) and from the right by \([x(k)\ e(k)\ \omega(k)]\) and taking into account Equations (2) and (3), we have:  
\[V(k+i+1/k)-V(k+i/k)\leq-\left[\|x(k+i/k)\|_{Q}^{2}+\|u\left(k+\frac{i}{k} \right)\|_{R}^{2}\right]+\mu^{2}\|\omega(k)\|^{2}\]  
This will conclude the proof of Theorem 1. Next, a closed-loop system is implemented as a polytopic observer-based fault-tolerant predictive controller for LPV systems subject to sensor faults.