# Robust Control Examples Applied to a Wind Turbine Simulated Model
## 3 Data-Driven and Model-Based Control Designs
### Data-Driven Fuzzy Control Strategy  
This section describes the derivation of the fuzzy controller model. Once a reasonably accurate fuzzy description of the considered benchmark has been available, as described above, it is used off-line to directly estimate the nonlinear fuzzy controllers. As already remarked, this design procedure differs from the approach proposed in [29]. In fact, the control design proposed in this paper relies on the so-called inverse model principle, which is solved using the fuzzy identification approach recalled above. Note that as explained in the following, the fuzzy methodology is exploited twice. First, the fuzzy modeling and identification approach is used to derive a fuzzy representation of the process under investigation, by means of its sampled inputs and outputs. Second, by means of this fuzzy model, and in particular using its state, the design of the fuzzy controller can be achieved. The derivation of the fuzzy controller model is performed by employing the states of the process fuzzy model. However, the estimation of the parameters of the fuzzy controller are achieved by minimizing the error between the reference power \(P_{r}\) and the controlled power \(P_{g}(t)\), thus leading to maximize the wind turbine generated power. With reference to stable fuzzy systems, whose inverted dynamics are also stable, a nonlinear controller can be simply designed by inverting the fuzzy model itself. Moreover, when modeling errors and disturbances are not present, this controller is able to allow for exact tracking with zero steady-state errors. However, modeling errors and disturbance effects are always present in real conditions, which can be tackled by directly identifying the controller model (i.e., the inverse controlled model) using the FMID approach. Differently from [29], a robust control strategy is thus achieved by minimizing a cost function which includes the difference between the desired and controller outputs, and a penalty on the system stability. In general, a nonconvex optimization problem has to be solved, which hampers the direct application of the proposed approach. However, the optimization scheme described in [7] can be exploited, which is based on a parametrized search technique applied at a higher level to formulate the control objectives and constraints. Note that, as remarked in Section 2, the fuzzy approach proposed in this work is able to provide a high-fidelity description of the wind turbine behavior, which already includes uncertainty and disturbance, as described, e.g., in [22]. The fuzzy approach is thus used again to derive the formulation of the controller in the form of TS prototypes. The parameters of the controller fuzzy model are estimated by minimizing the difference between the monitored outputs and the reference ones, taking into account the disturbance and the uncertainty affecting the wind turbine process. Therefore, the approach proposed in this work is able to cope with external disturbance modeled in the wind turbine benchmark. In this way, the estimated controller based on the process inverse model and approximated via a fuzzy prototype is able to describe the complete behavior of the monitored plant in its different working conditions (i.e., partial and full load situations). In fact, the rule-based fuzzy inference system of Equation (14) has been derived for modeling the wind turbine dynamic process of Equation (9) in its equivalent discrete-time form of Equation (30):  
\[y_{k+1}=f\left(\mathbf{x}_{k},\,u_{k}\right) \tag{30}\]  
and, in particular, the TS fuzzy representation has the form of Equation (31):  
\[y_{k+1}=\frac{\sum_{i=1}^{M}\mu_{i}^{(m)}\left(\mathbf{x}_{k}^{(m)}\right)\, \left(\mathbf{a}_{i}^{(m)}\mathbf{x}_{k}^{(m)}+b_{i}^{(m)}\right)}{\sum_{i=1}^ {M}\mu_{i}^{(m)}\left(\mathbf{x}_{k}^{(m)}\right)} \tag{31}\]  
The current state \(\mathbf{x}_{k}=[y_{k},\,\ldots,\,y_{k-n+1},\,u_{k-1},\,\ldots,\,u_{k-n+1}]^{T}\) and the input \(u_{k}\) represent the inputs that drive the model of Equation (31).