## 6 Conclusions  
In this paper, a robust observer based sensor FDI scheme, targeting both additive and multiplicative faults, is presented for PMSG in WECSs. This method isolates multiple sensor faults via two procedures: robust residual generation based fault detection and fault estimation based isolation. The system is reformulated as a LPV model in the \(\alpha-\beta\) frame by introducing electromechanical dynamics of PMSG. The polytopic decomposition technique is applied to obtain the parameter-dependent form of the system model by defining a convex polytope with four vertices. Furthermore, the gain-scheduled residual generator and fault estimator are designed by \(\mathcal{H}_{\infty}\) synthesis in the form of LMIs. The proposed gain-scheduled FDI scheme is capable of online monitoring of three-phase currents and isolating multiple sensor faults under varying disturbances.  
The proposed scheme is implemented in a MATLAB/SIMULINK environment and multiple sensor faults are isolated correctly. Due to the lack of experimental validation, the corresponding issues are discussed in Section 5.4, of which the challenging issue is the influence of disturbances and harmonics on threshold selection and system dynamics. In a real power conversion system, the diagnosis thresholds in (32) need to be investigated further by defining the augmented disturbances including measurement noise and parameter uncertainties. However, it does not affect the theoretical results of Theorem 1 and Theorem 2 since the observability of the pair \(\left(A\left(\theta_{k}\right),C\right)\) and \(\left(\bar{A}\left(\theta_{k}\right),C\right)\) is independent of disturbances. Since further results is lacked about the influence of harmonics on system dynamics, it is difficult to quantify these effects on the system. The results in [20; 21; 22] indicate  
Figure 8: Simulation results on simultaneous isolation of three-phase sensor faults.  
that the gain-scheduled controller design based on the this LPV model suffers less from the harmonics. Observer design as the dual problem of controller design is less dependent on the harmonic problem.  
In addition, fault estimates are sensitive enough to distinguish each phase current sensor fault but cannot be applied directly for the purpose of fault compensation because of the unknown disturbances. The future work will be focused on the unbiased fault estimation and fault tolerant control for PMSG based WECS system on the basis of this LPV modeling technology.  
Methodology and Writing--Original Draft, Z.Y.; Writing--Review and Editing, Y.C. and H.Y.; Data Curation, S.T.  
This research was funded by the National Natural Science Foundation of China Grant No. 61633005 and 61773080, and Chongqing Nature Science Foundation of Fundamental Science and Frontier Technologies Grant No. cstc2015 cyjB0569, and the Scientific Reserve Talent Programs of Chongqing University Grant No. cqu2017 CDHB1B04.  
The authors would like to thank the editors and all of the anonymous reviewers for their valuable insights and comments.  
The authors declare no conflict of interest.  
The system matrices of Equation (6) are as follows:  
\[A\left(\theta_{k}\right)=\left[\begin{array}{ccc}1-\frac{R_{k}T_{k}}{L_{s}}& 0&\frac{n_{p}\psi T_{s}}{L_{s}}\sin\left(\theta_{k}\right)&0\\ 0&1-\frac{R_{k}T_{s}}{L_{s}}&-\frac{n_{p}\psi T_{s}}{L_{s}}\cos\left(\theta_{ k}\right)&0\\ -\frac{3n_{p}\psi T_{s}}{2}\sin\left(\theta_{k}\right)&\frac{3n_{p}\psi T_{s} }{2}\cos\left(\theta_{k}\right)&1-\frac{\Gamma T_{s}}{L}&0\\ 0&0&n_{p}T_{s}&1\end{array}\right],\]  
\[B_{u}=\left[\begin{array}{ccc}\frac{T_{s}}{L_{s}}&0\\ 0&\frac{T_{s}}{L_{s}}\\ 0&0\\ 0&0\end{array}\right];\quad B_{d}=\left[\begin{array}{ccc}0\\ 0\\ -\frac{T_{s}}{L}\\ 0\end{array}\right];\quad C=\left[\begin{array}{ccc}1&0&0&0\\ -\frac{1}{2}&\frac{\sqrt{3}}{2}&0&0\\ -\frac{1}{2}&-\frac{\sqrt{3}}{2}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right].\]