# Robust Control Examples Applied to a Wind Turbine Simulated Model
## 3 Data-Driven and Model-Based Control Designs
### Data-Driven Fuzzy Control Strategy
Its output represents the prediction of the system output atthe next sample \(y_{k+1}\). The model of Equation (31) requires the estimated membership functions \(\mu_{i}^{(m)}\), the state \(\mathbf{x}^{(m)}\) and the parameters \(\mathbf{a}_{i}^{(m)}\), \(b_{i}^{(m)}\) of the controlled system, which are denoted by the superscript \((m)\). Therefore, the input \(u_{k}\) generated by the control law feeds the monitored process such that its output \(y_{k+1}\) asymptotically follows the desired (reference) output \(r_{k+1}\). This behavior is obtained by using the inverse model principle, represented by the expression of Equation (32):  
\[u_{k+1}=f^{-1}\left(\mathbf{x}_{k}^{c},\,r_{k}\right) \tag{32}\]  
that is a nonlinear function of the vector \(\mathbf{x}_{k}^{c}\) and the reference \(r_{k}\). However, in general, with reference to Equation (32), it is difficult to determine the analytical expression of the inverse function \(f^{-1}(\cdot)\). Therefore, the methodology proposed in this work suggested to exploit the identified fuzzy TS prototype of Equation (31) to provide the particular state \(\mathbf{x}_{k}^{(m)}\) at each time step \(k\). In this way, from this mapping, the inverse mapping \(u_{k+1}=f^{-1}\left(\mathbf{x}_{k}^{(c)},\,r_{k}\right)\) is directly identified the form of Equation (14), if the controlled system is stable, and in particular in the form of Equation (33):  
\[u_{k+1}=\frac{\sum_{i=1}^{M}u_{i}^{(c)}\left(\mathbf{x}_{k}^{(c)}\right)\, \left(\mathbf{a}_{i}^{(c)}\,\mathbf{x}_{k}^{(c)}+b_{i}^{(c)}\right)}{\sum_{i= 1}^{M}u_{i}^{(c)}\left(\mathbf{x}_{k}^{(c)}\right)} \tag{33}\]  
where the state \(\mathbf{x}_{k}^{(c)}=[\mathbf{x}_{k}^{(m)},\,r_{k-1},\,\dots,\,r_{k-n+1}]^{T}\) and the reference \(r_{k}\) signal represent the inputs of the identified controller model. The model of Equation (33) contains the estimated membership functions \(\mu_{i}^{(c)}\) and the parameters \(\mathbf{a}_{i}^{(c)},\,b_{i}^{(c)}\) of the identified controller model, that are denoted by the superscript \((c)\). The complete scheme is outlined in Figure 5.  
Note that Figure 5 sketches the general principle of the design of the controller for a system with input \(u_{k}\) and output \(y_{k}\). On the other hand, the signal \(r_{k}\) represents the generic set-point to be tracked (i.e., \(P_{r}\) or \(\omega_{nom}\)) depending on the working region (1 or 2) and the controlled output (i.e., the sampled signal \(P_{g}(t)\) or \(\omega_{g}(t)\)). Under this assumption, the identification of the fuzzy controller parameters leads to minimize the difference between \(r_{k}\) and \(y_{k}\). It is not required that \(r_{k}\) equals \(y_{k}\), thus making the problem feasible within the selected degree of accuracy. Figure 5 highlights the series connection between the controller models (i.e., the inverse process model identified using the fuzzy systems) and the process model itself (described by means of fuzzy models), which should lead to an identity mapping as in Equation (34):  
\[y_{k+1}=f\left(\mathbf{x}_{k}^{(m)},\,u_{k}\right)=f\left(\mathbf{x}_{k}^{(m) },\,f^{-1}\left(\mathbf{x}_{k}^{(c)},\,r_{k}\right)\right)=r_{k+1} \tag{34}\]  
where \(r_{k+1}=f\left(\mathbf{x}_{k}^{(m)},\,u_{k}\right)\) for a proper value of \(u_{k}\). However, the expression of Equation (34) holds in ideal conditions. However, the model-reality mismatch and measurement errors are properly managed by means of the fuzzy modeling scheme recalled in Section 3. In this way, the difference \(|r_{k+1}-f\left(\mathbf{x}_{k}^{(m)},\,u_{k}\right)|\) can be made arbitrarily small by a suitable selection of the model parameters, i.e., the fuzzy membership functions \(\mu_{i}^{(c)}\), the number of clusters \(M\), and the regressand \(\mathbf{a}_{i}^{(c)}\), \(b_{i}^{(c)}\). Moreover, as highlighted in Figure 5, the fuzzy model of the process is used for providing the state  
Figure 5: The fuzzy controller based on the inverse process model principle. vector \(\mathbf{x}_{k}^{(m)}\).