## 2 Convex Polytopic Model  
Consider the following polytopic discrete-time model given by:  
\[\left\{\begin{array}{c}x(k+1)=A(\theta_{k})x(k)+B(\theta_{k})u(k)\\ y(k)=C(\theta_{k})x(k)\end{array}\right. \tag{1}\]  
where \(x(k)\in\mathbb{R}^{n_{x}}\) is the system state, \(y(k)\in\mathbb{R}^{n_{y}}\) is the system output vector, and \(u(k)\in\mathbb{R}^{n_{u}}\) is the system input vector. The system matrices can be written in the polytopic form as \(A(\theta_{k})=\sum_{\tau=1}^{N}\rho_{r}(\theta_{k})A_{r}\), \(B(\theta_{k})=\sum_{\tau=1}^{N}\rho_{r}(\theta_{k})B_{r}\), and \(C(\theta_{k})=\sum_{\tau=1}^{N}\rho_{r}(\theta_{k})C_{r}\), where \(\rho_{r}(\theta_{k})\geq 0\) and \(\sum_{\tau=1}^{N}\rho_{r}(\theta_{k})=1\).  
The parameters \(\theta_{k}\), \(k=1,2,\ldots,N\) are bounded in a predefined compact set \(\Theta\), i.e., \(\theta_{k}\in\Theta\;\forall\;k\in\mathbb{N}\).  
Consequently, the possible values of \(\theta_{k}\) are held within a polytope convex \(Co\{\rho_{1},\ldots,\rho_{r}\}\) of \(r=2^{m}\) vertices \(\{\theta_{1},\ldots,\theta_{r}\}\). The images of the matrix \(A(\rho)\) for each vertex \(\theta_{j}\) correspond to a set \(\{\rho_{1},\ldots,\rho_{r}\}\), while the components of the set \(\{\rho_{1},\ldots,\rho_{r}\}\) are known as the extreme of the polytope [15]. The system in Equation (1) can be represented by a convex interpolation of the vertices \(\Theta\) and each vertex \(\theta_{j}\) of the polytope, which is obvious because of the linear dependence on \(\theta_{k}\).  
Note that the control action is set in order to satisfy the constraints below:  
\[\left\|u(k)\right\|_{2}\leq u_{max} \tag{2}\]  
where \(u_{max}\) is the maximum value (upper limit) of the control input.  
Since the proposed polytopic discrete-time system is subject to both sensor faults and external disturbances, the real output signal measured by the sensor may be altered when a sensor fault occurs. The output of the system is then modeled as follows:  
\[y(k)=C(\theta_{k})x(k)+f_{s}(k) \tag{3}\]  
where \(f_{s}\in\mathbb{R}^{n_{y}}\) is the vector of additive sensor faults.  
Throughout this work, the following Lemmas and assumptions are needed to derive the main results.  
**Lemma 1** ([PERSON] complements lemma [29]).: _Given any real matrices \(S\), \(R\), and \(Z\) with \(S=S^{\mathsf{T}}\) and \(Z>0\), then_  
\[S-RZ^{-1}R^{T}<0\]  
If and only if  
\[\left[\begin{array}{cc}S&R^{T}\\ R&Z\end{array}\right]<0\]  
**Assumption 1**.: Sensor faults \(f_{s}(k)\) and the external disturbance \(\omega(k)\) are bounded, and there exists a positive scalar \(\delta\) such that \(\left\|\omega(k)\right\|<\delta\).  
**Remark 1**.: _Note that Assumption 1 ensures that the increments of faults and the disturbances between two sampling time instants are bounded._  
In this paper, only the sensor faults are considered and the model in Equation (1) can be rewritten as:  
\[\left\{\begin{array}{c}x(k+1)=A(\theta_{k})x(k)+B(\theta_{k})u(k)+D(\theta_{ k})\omega(k)\\ y_{f}(k)=Cx(k)+f_{s}(k)\end{array}\right. \tag{4}\]where \(\mathbf{u_{f}}\in\mathbb{R}^{\mathbf{n_{u}}}\), \(\mathbf{x}\in\mathbb{R}^{\mathbf{n_{x}}}\), and \(\mathbf{y_{f}}\in\mathbb{R}^{\mathbf{n_{y}}}\) denote the vector of the faulty control input, the vector of the faulty system state, and the faulty measurement output vector, respectively. Note that \(\mathbf{f_{s}}\in\mathbb{R}^{\mathbf{n_{y}}}\) is assumed to be an additive sensor fault.