## 4 Robust Fault-Tolerant Predictive Control  
Next, a fault-tolerant predictive controller based on an observer is designed for LPV systems subject to disturbances and sensor faults. As the initial part of this work, an observer was constructed to estimate faulty signals of the system subject to an external disturbance. Next, based on the observer information a fault-tolerant predictive controller is proposed to guarantee stability and maintain control performance. Next, the control law in Equation (10) is considered for compensating for the impacts of the faults in the closed-loop system and that rejects the disturbance's influence. \[u(k)=F(\theta_{k})E_{1}\hat{\eta}(k) \tag{10}\]  
where \(F\) is a time-varying gain matrix that depends on \(\theta_{k}\), and can be defined as \(F(\theta_{k})=\sum\limits_{r=1}^{N}\rho_{r}(\theta_{k})F_{r}\). Replacing Equation (10) in Equation (4), we obtain:  
\[ \begin{array}{ll}x(k+1)&=A(\theta_{k})x(k)+B(\theta_{k})(F(\theta_{k})x(k)- F(\theta_{k})E_{1}e(k))+D(\theta_{k})\omega(k)\\ &=(A(\theta_{k})+B(\theta_{k})F(\theta_{k}))x(k)-B(\theta_{k})F(\theta_{k})E_{ 1}e(k)+D(\theta_{k})\omega(k)\end{array}  \tag{11}\]  
To obtain the predictive controller, an optimization problem that minimizes the following worst-case quadratic objective function in an infinite horizon is considered [30]:  
\[\underset{u_{\epsilon}(k)}{min}\ max\big{\rvert}_{\infty}(k) \tag{12}\]  
Subject to (6) and \(\left\lVert u(k)\right\rVert_{2}\leq u_{max}\), where  
\[J_{\infty}(k)=\sum_{i=0}^{\infty}\left\lVert x(k+i)\right\rVert_{Q}^{2}+ \left\lVert u(k+i)\right\rVert_{R}^{2}+\alpha^{2}\left\lVert\omega(k)\right\rVert ^{2} \tag{13}\]  
where \(Q\) and \(R\) are known positive-definite weighting matrices. Next, sufficient stability conditions, in terms of LMIs, are derived to ensure the stability of the overall closed-loop system in Equation (5). **Theorem 1**. : _Consider the faulty LPV system in Equation (5). The state-feedback controller given by Equation (4) will robustly stabilize the system in Equation (6) subjected to external disturbances and sensor faults if there exist symmetric positive definite matrices \(M_{r}\), \(N_{r}\), \(X_{r}\), \(Y_{r}\), \(Z_{r}\), and \(H_{r}\)\(r=1,\dots,\ N\), and a positive scalar \(\gamma\) that satisfies the following convex optimization problem:_  
\[\underset{\gamma,M_{r},\ N_{r},\ X_{r},\ Y_{r},\ Z_{r},\ H_{r}}{\min}\gamma \tag{14}\]  
_Subject to_  
\[\left[ \begin{array}{ccc}-1&*&*\\ \overline{x}(k)&-M_{r}&*\\ e(k)&0&-N_{r}\end{array} \right] \tag{15}\]  
by the Schur complement, we obtain  
\[\left[ \begin{array}{ccc}-1&*&*\\ x(k)&-\gamma\Gamma(\theta_{k})^{-1}&*\\ e(k)&0&-\gamma\Pi(\theta_{k})^{-1}\end{array} \right] \tag{24}\]  
Substituting \(\Gamma_{r}=\gamma M_{r}^{-1}\) and \(\Pi_{r}=\gamma N_{r}^{-1}\) into the above inequality, and by applying the congruence transformation to the resulting inequality with Diag [1, \(M_{r}^{-1}\), \(N_{r}^{-1}\)], we conclude that Equation (15) holds. Consider the input constraint in Equation (2)  
\[\left\|u(k+i/k)\right\|_{max}\triangleq\underset{i}{max}u_{i}(k+i/k) \tag{25}\]  
\[\underset{i>0}{max}\left\|u(k)\right\|_{max}=\underset{i>0}{max}\left\|F\hat{ x}(k)\right\|_{max} \tag{26}\]  
\[\underset{i>0}{max}\left\|u(k)\right\|_{max}\leq\left\|F(\frac{1}{\gamma} \Gamma_{r})^{\frac{1}{2}}\right\|_{2}^{2}\Leftrightarrow \tag{27}\]  
\[u_{max}^{2}\leq\Gamma_{r}^{\frac{1}{2}}F^{T}F_{r}\frac{1}{2} \tag{28}\]  
\[-u_{max}^{2}+(\frac{1}{\gamma}\Gamma_{r})^{\frac{1}{2}}F^{T}F(\frac{1}{\gamma }\Gamma_{r})^{\frac{1}{2}}\leq 0 \tag{29}\]  
Using the Schur complement, we obtain:  
\[\left[ \begin{array}{ccc}-u_{max}^{2}&F\\ F^{T}&-\frac{1}{\gamma}\Gamma_{r}\end{array} \right]\leq 0 \tag{30}\]  
Substituting \(\Gamma_{r}=\gamma M_{r}^{-1}\) and \(F_{r}=\gamma_{r}X_{r}^{-1}\) followed by multiplying the right by \(\left[ \begin{array}{cc}I&0\\ 0&X_{r}\end{array} \right]\) and the left by \(\left[ \begin{array}{cc}I&0\\ 0&X_{r}^{T}\end{array} \right]\), we obtain (16). Now, (17) implies that:  
\[\left[