# Robust Control Examples Applied to a Wind Turbine Simulated Model
## 3 Data-Driven and Model-Based Control Designs  
This section describes the two approaches considered in this paper for obtaining the control laws by using data-driven and model-based methodologies. Once a suitable mathematical description of the monitored process is provided, the derivation of the controller structure is sketched in Section 3.1 for the fuzzy approach, whilst Section 3.2 proposes a different method relying on an adaptive technique. The first method proposed in this paper for the derivation of the wind turbine controller is based on a fuzzy clustering technique to partition the available data into subsets characterized by linear behaviors. The integration between clusters and linear regression is exploited, thus allowing for the combination of fuzzy logic techniques with system identification methodologies. These tools are already available and implemented in the MATLAB(r) Fuzzy Modeling and IDentification (FMID) Toolbox recalled below [6]. This study proposes the use of TS fuzzy prototypes since they are able to model nonlinear dynamic systems with arbitrary accuracy [6]. The switching between the local affine submodels is achieved through a smooth function of the system state defined exploiting the fuzzy set theory and its tools. In more detail, the fuzzy estimation scheme relies on a two-step algorithm, in which, the working regions are first defined by exploiting the data fuzzy clustering tool, i.e., the Gustafson-Kessel (GK) method [6]. On the other hand, the second step performs the identification of the controller structure and its parameters using the estimation method proposed by the same authors in [7]. This estimation approach can be considered as a generalization of the general least-squares method for hybrid models. Under these assumptions, the TS fuzzy prototypes have the form of the model of Equation (14):  
\[y_{k+1}=\frac{\sum_{i=1}^{M}\mu_{i}\left(\mathbf{x}_{k}\right)\,y_{i}}{\sum_{i =1}^{M}\mu_{i}\left(\mathbf{x}_{k}\right)} \tag{14}\]  
where \(y_{i}=\mathbf{a}_{i}^{T}\,\mathbf{x}+b_{i}\), with \(\mathbf{a}_{i}\) the parameter vector (regressand), and \(b_{i}\) is the scalar offset. \(\mathbf{x}=\mathbf{x}_{k}\) represents the regressor vector, which contain delayed samples of the signals \(u_{k}\) and \(y_{k}\). Note that the discrete-time description of Equation (14), after the proper estimation of the parameter vector [6], will be used for reconstructing the sampled outputs \(P_{g}(t)\) and \(\omega_{g}(t)\) of the continuous-time nonlinear model of Equation (9) fed by the sampled input signals \(\beta(t)\) and \(\tau_{g}(t)\). The antecedent fuzzy sets \(\mu_{i}\) that determine the switching among the different submodels \(i\) in Equation (14) are estimated using the input-output data acquired from the wind turbine simulator, i.e., the input and output sampled signals \(\beta(t)\), \(\tau_{g}(t)\) and \(\omega_{g}(t)\) and \(P_{g}(t)\). These data are organized into proper clusters where affine relations hold, as described in [6]. The consequent parameters \(\mathbf{a}_{i}\) and \(b_{i}\) are also identified from these input-output data by means of the estimation methodology proposed in [7]. This identification scheme exploited for the estimation of the TS model parameters has been integrated into the FMID toolbox for MATLAB(r) by the authors. This approach is preferable when the TS model of Equation (14) is used as predictor, since it derives the consequent parameters via the so-called Frisch scheme, developed for the Errors-In-Variables (EIV) structures [7]. Once the description of the monitored process is obtained in the form of Equation (14), the data-driven approach for the design of the fuzzy controller proposed in this work is presented in Section 3.1. The second approach exploited for obtaining the mathematical description of the wind turbine system under investigation is based on a recursive methodology, which will be used for the design of the second control strategy presented in Section 3.2.