## 2 Problem Statement
### Polytropic Decomposition of the System Model  
An LPV model of PMSG with sensor faults is presented as follows with bounded varying parameters:  
\[x\left(k+1\right) =A\left(\theta_{k}\right)x\left(k\right)+B_{u}u\left(k\right)+B_{d }d\left(k\right), \tag{7}\] \[y\left(k\right) =Cx\left(k\right)+F_{f}f\left(k\right),\]  
where \(F_{f}\) is the fault distribution matrix. \(A\left(\theta_{k}\right)\) contains two time-varying terms \(\sin\left(\theta_{k}\right)\) and \(\cos\left(\theta_{k}\right)\), an auxiliary variable is defined \(\mu\left(\theta_{k}\right)=\left[\mu_{1}\left(\theta_{k}\right)\quad\mu_{2} \left(\theta_{k}\right)\right]^{T}\),  
\[\mu_{1}\left(\theta_{k}\right) =\sin\left(\theta_{k}\right), \tag{8}\] \[\mu_{2}\left(\theta_{k}\right) =\cos\left(\theta_{k}\right).\]  
It is obvious that \(A\left(\theta_{k}\right)\) depends affinely on the parameter \(\mu\left(\theta_{k}\right)\)  
\[A\left(\theta_{k}\right)=\mathcal{A}_{0}+\mu_{1}\left(\theta_{k}\right) \mathcal{A}_{1}+\mu_{2}\left(\theta_{k}\right)\mathcal{A}_{2}, \tag{9}\]  
where \(\mathcal{A}_{0}\), \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) are constant matrices. The time-varying parameter vector \(\mu\left(\theta_{k}\right)\) is determined by the rotor electrical angle \(\theta_{k}\). Moreover, \(\mu_{1}\left(\theta_{k}\right)\) and \(\mu_{2}\left(\theta_{k}\right)\) are trigonometric function and bounded by the lower and upper bounds  
\[\begin{array}{l}\mu_{1}\in\left[\begin{array}{cc}\mu_{1}&\bar{\mu}_{1}\\ \mu_{2}\in\left[\begin{array}{cc}\mu_{2}&\bar{\mu}_{2}\\ \end{array}\end{array}\right]\\ \mu(\theta_{k})^{T}\mu\left(\theta_{k}\right)=1\end{array} \tag{10}\]  
in which \(\underline{\mu}_{1}=\underline{\mu}_{2}=-1\) and \(\bar{\mu}_{1}=\bar{\mu}_{2}=1\). A convex polytope \(\Theta\) with four vertices \(\mu_{v,1}\), \(\mu_{v,2}\), \(\mu_{v,3}\), \(\mu_{v,4}\) is defined to ensure that the trajectory of parameter \(\mu\left(\theta_{k}\right)\) is enclosed:  
\[\begin{array}{l}\mu_{v,1}=\left[\begin{array}{c}\underline{\mu}_{1}\\ \underline{\mu}_{2}\end{array}\right]\quad\mu_{v,2}=\left[\begin{array}{c} \underline{\mu}_{1}\\ \bar{\mu}_{2}\end{array}\right]\\ \mu_{v,3}=\left[\begin{array}{c}\bar{\mu}_{1}\\ \underline{\mu}_{2}\end{array}\right]\quad\mu_{v,1}=\left[\begin{array}{c} \bar{\mu}_{1}\\ \bar{\mu}_{2}\end{array}\right].\end{array} \tag{11}\]  
Consequently, parameter \(\mu\left(\theta_{k}\right)\) can be expressed as a convex combination of the vertices with coordinates \(\eta_{k}=\left[\eta_{k,1}\quad\eta_{k,2}\quad\eta_{k,3}\quad\eta_{k,4}\right] ^{T},\)  
\[\begin{array}{l}\mu\left(\theta_{k}\right)=\left[\begin{array}{cc}\mu_{v,1 }&\mu_{v,2}&\mu_{v,3}&\mu_{v,A}\\ \eta_{k,1}+\eta_{k,2}+\eta_{k,3}+\eta_{k,4}=1,\end{array}\right]\eta_{k},\end{array} \tag{12}\]  
in which \(\forall i=1,\cdots,4\), \(\eta_{k,i}\geq 0\) and the parameter-dependent matrix \(A\left(\mu\right)\) is rewritten by a combination of coordinate vector \(\eta_{k}\)  
\[A\left(\mu\right)=\sum_{i=1}^{4}\eta_{k,i}A_{v,j}, \tag{13}\]  
in which \(A_{v,i}=A\left(\mu_{v,i}\right)\) with \(i=1,\cdots,4\). The system Equation (7) can be transformed into a polytopic form  
\[\begin{array}{l}x\left(k+1\right)=\sum_{i=1}^{4}\eta_{k,i}A_{v,j}x\left(k \right)+B_{u}u\left(k\right)+B_{d}d\left(k\right),\\ y\left(k\right)=Cx\left(k\right)+F_{f}f\left(k\right).\end{array} \tag{14}\]  
Since Equation (12) is an underdetermined equation, further constraints are required to solve this equation. In [20; 21; 22], a vertex expansion technique is presented to get a unique solution of \(\eta_{k}\). Furthermore, this work decomposes \(A\left(\theta_{k}\right)\) with this method.