# Intelligent Fault Diagnosis Techniques Applied to an Offshore Wind Turbine System
## 3 Fault Diagnosis Techniques: Fuzzy Systems and Neural Networks
To this end, the design of these fault estimators is enhanced by the fault sensitivity analysis procedure reported in Section 3. The first method proposed in this paper for designing fault estimators relies on Takagi-Sugeno (TS) models [19]. This approach was formerly addressed in [14] for the approximation of nonlinear Multiple-Input Single-Output (MISO) dynamic systems with arbitrary accuracy. The general fault estimator \(\hat{f}\) has the form of Equation (6):  
\[\hat{f}=\frac{\sum_{i=1}^{n_{\text{C}}}\lambda_{i}(\mathbf{x})\,\left(\mathbf{a }_{i}^{T}\,\mathbf{x}+b_{i}\right)}{\sum_{i=1}^{n_{\text{C}}}\lambda_{i}( \mathbf{x})} \tag{6}\]  
The TS fuzzy model results are described as discrete-time linear AutoRegressive models with eXogenous input (ARX) of order \(o\), in which the regressor vector has the form of Equation (7):  
\[\mathbf{x}(k)=\big{[}\ldots,\,y_{l}(k-1),\,\ldots,\,y_{l}(k-o),\,\ldots\,u_{j} (k),\,\ldots,\,u_{j}(k-o),\,\ldots\big{]}^{T} \tag{7}\]  
where \(u_{l}(\cdot)\) and \(y_{j}(\cdot)\) are the components of the actual system input and output vectors \(\mathbf{u}(k)\) and \(\mathbf{y}(k)\) that are selected using the fault sensitivity analysis proposed in Section 3. The variable \(k\) represents the time step, with \(k=1,\,2,\,\ldots,\,N\). The parameters of the TS fuzzy model in Equation (6) are collected into the vector:  
\[\mathbf{a}_{i}=\big{[}\alpha_{1}^{(i)},\,\ldots,\,\alpha_{o}^{(i)},\,\delta_{1 }^{(i)},\,\ldots,\,\delta_{o}^{(i)}\big{]}^{T} \tag{8}\]  
where the \(\alpha_{j}^{(i)}\) coefficients refer to the output samples, while the \(\delta_{j}^{(i)}\) coefficients are associated with the input ones. This work proposes to solve the derivation of the TS models as a system identification problems from the noisy data of Equation (2). In particular, the design of the bank of fault estimators in Figure 3 requires the estimation of the consequent parameters \(\mathbf{a}_{i}\) and \(b_{i}\) of Equation (8). Note that the design method proposed in this work exploits the direct identification of the TS fuzzy models of Equation (6). In particular, the fuzzy model structure, i.e., the number of rules \(n_{\text{C}}\), the antecedents, and the fuzzy membership functions \(\lambda_{i}(\mathbf{x})\) in Equation (6), are derived by means of the Fuzzy Modeling and Identification (FMID) toolbox implemented in the MATLAB environment [14]. Moreover, the computation of the TS model parameters in Equation (8) was solved by the authors in [20] as an EIV estimation problem, as highlighted by the relations in Equation (2). On the other hand, the FMID toolbox uses the Gustafson-Kessel (GK) clustering method [14] to perform a partition of input-output data into a proper number \(n_{\text{C}}\) of regions (clusters), where the \(i^{text}\) model of Equation (6) is valid. This model is thus obtained after the selection of the model order \(o\) and the number of clusters \(n_{\text{C}}\). The FMID toolbox also determines the antecedent degrees of fulfillment \(\lambda_{i}(\mathbf{x})\) in Equation (6), which are derived with a curve fitting method [14]. This paper proposes a different data-driven approach that is based on NN, which is exploited to implement the scheme shown in Figure 3. According to this scheme, a bank of NN is used to reconstruct the faults affecting the system under diagnosis using a proper set of input and output measurements. The structure proposed in this work consists of a feedforward multilayer perceptron NN with three layers [21]. Moreover, this study suggests the use of a quasi-static NN, as it represents a suitable tool to predict dynamic relationships between the input-output measurements and the considered fault function \(f_{i}(k)\) with arbitrary accuracy [21].