## 3 Design of Observer-Based Control Law  
The main objective of this work is to propose a virtual observer to improve the estimations and provide better accuracy for both fault observation and the system's states.  
First, an augmented system, which is proposed based on the system in Equation (4) and exposed to an external disturbance, is proposed for designing an observer that can estimate system state vector \(x(k)\) and sensor faults \(f_{s}(k)\). The augmented system is defined as:  
\[\left\{\begin{array}{c}E_{1}\eta(k+1)=\mathcal{A}(\theta_{k})\eta(k)+B( \theta_{k})u(k)+D(\theta_{k})\omega(k)\\ y_{f}(k)=E_{2}\eta(k)\end{array}\right. \tag{5}\]  
where  
\[\eta=\left[\begin{array}{c}x(k)\\ f_{s}(k)\end{array}\right]\text{, }\mathcal{A}(\theta_{k})=\left[A(\theta_{k}) \,0\right]\text{, }E_{1}=\left[I\,0\right]\text{, }E_{2}=\left[C(\theta_{k})\ I\right]\]  
Multiplying \(T_{1}\) to both sides of (4), and by using fact (5), i.e., \(T_{1}E_{1}+T_{2}E_{2}=I\), where \(T_{1}=\left[\begin{array}{c}I_{n_{x}}\\ -C\end{array}\right]\) and \(T_{2}=\left[\begin{array}{c}0_{n_{x}\times n_{y}}\\ I_{n_{y}}\end{array}\right]\), we have:  
\[\eta(k+1)=T_{1}\mathcal{A}(\theta_{k})\eta(k)+T_{1}B(\theta_{k})u(k)+T_{1}D \omega(k)+T_{2}E_{2}\eta(k+1) \tag{6}\]  
Let us define the following virtual observer:  
\[\hat{\eta}(k+1)=T_{1}\mathcal{A}(\theta_{k})\hat{\eta}(k)+T_{1}B(\theta_{k})u (k)+T_{1}D(\theta_{k})\omega(k)+T_{2}E_{2}\eta(k+1)+L(\theta_{k})(y_{f}(k)-E _{2}\hat{\eta}(k)) \tag{7}\]  
where \(\hat{\eta}=\left[\begin{array}{c}\hat{x}(k)\\ \hat{f}_{s}(k)\end{array}\right]\) is the estimation of the system state \(x(k)\) and sensor fault signals \(f_{s}(k)\), and \(L(\theta_{k})\) is the observer gains, which will be obtained later.  
To compute these gains, we will consider an augmented system with estimation error \(e(k)\) of the observer in Equation (7) as \(e(k)=\eta(k)-\hat{\eta}(k)\). Subtracting Equation (7) from Equation (6), the error dynamic equation is obtained as  
\[\begin{array}{rl}e(k+1)&=(T_{1}\mathcal{A}-LE_{2})e+T_{1}D\omega(k)\\ &=(T_{1}\mathcal{A}(\theta_{k})-L(\theta_{k})E_{2})e+T_{1}D\omega(k)\end{array} \tag{8}\]  
**Remark 2**: _The virtual observer cannot be realized since it includes unreachable information, i.e., \(T_{2}E_{2}\eta(k+1)\). Consequently, a real observer based on the virtual observer's information is synthesized by eliminating the unknown terms._  
Therefore, we define the auxiliary variable \(\mathcal{V}(k)=\hat{\eta}(k)-T_{2}E_{2}\eta(k+1)\). The real observer for the system in Equation (4) can be obtained by:  
\[\mathcal{V}(k+1)=\overline{\mathcal{A}}(\theta_{k})\mathcal{V}(k)+\overline{ B}(\theta_{k})u(k)+\mathcal{L}(\theta_{k})\hat{\eta}(k)\]  
\[\hat{\eta}(k)=\mathcal{V}(k)+\overline{\mathcal{C}}(\theta_{k})y_{f}(k) \tag{9}\]  
where a new auxiliary variable, \(\mathcal{V}(k)\), is introduced along with the matrices \(\overline{\mathcal{A}}(\theta_{k})\), \(\overline{B}(\theta_{k})\), and \(\overline{\mathcal{C}}(\theta_{k})\), and the gain matrices \(\mathcal{L}(\theta_{k})\) are the observer matrices to be computed later as follows:\[\overline{\mathcal{A}}(\theta_{k})=T_{1}\mathcal{A}(\theta_{k})-L(\theta_{k})E_{2}, \ \overline{B}(\theta_{k})=T_{1}B(\theta_{k}),\ \overline{\mathcal{C}}(\theta_{k})=T_{2}E_{2},\ \text{and}\]  
\[\mathcal{L}(\theta_{k})=L(\theta_{k})+(T_{1}\mathcal{A}(\theta_{k})-L(\theta_{ k})E_{2})T_{2}\]  
where \(L(\theta_{k})\) is a time-varying observer gain matrix that depends on \(\theta_{k}\), which is defined as  
\[L(\theta_{k})=\sum_{r=1}^{N}\rho_{r}(\theta_{k})L_{r}\]