# A Study on \(H_{\infty}\)-Fuzzy Controller for a Non-Linear Wind Turbine with Uncertainty
## 3 Control Algorithms
### Fuzzy Inference Algorithm  
The T-S fuzzy model can approximate the entire fuzzy model using a combination of linear sub-fuzzy models [29]. Therefore, the control sub-command \(u^{\prime}(t)\) of the augmented linear plant (6) by wind speed region, i.e., the region mentioned in Figure 1, is interpolated using fuzzy logic, which can lead to an approximation of the control command \(u(t)\) of the non-linear entire plant. The control sub-command \(u^{\prime}(t)\) is calculated from the membership function \(\mu\) of the wind speed nonlinearity, i.e., the sub-optimized \(H_{\infty}\) control command \(u_{subopt}(t)\). The subscript below \(i\) indicates the number of sub-systems. The control sub-command \(u^{\prime}(t)\) can be expressed as:  
\[u^{\prime}(t)=\frac{\sum\Big{\{}\mu_{i}(z(t))\cdot u_{subopt,i}(t)\Big{\}}}{ \sum\mu_{i}(z(t))} \tag{9}\]  
The final control command \(u(t)\) is calculated from the membership function \(\kappa\) of the uncertainty of the estimated wind speed, the control sub-command \(u^{\prime}(t)\), and the control command of the previous step \(u_{0}(t)\). The calculation formula is given in Equation (10). Equation (11) describes the system decision variable \(z(t)\) given by the estimated wind speed information.  
\[u(t)=\kappa(z(t))u_{0}(t)+(1-\kappa(z(t)))u^{\prime}(t) \tag{10}\]  
\[z(t)=\left\{\begin{array}{ccc}\hat{\sigma}(t)&for&\mu,&(0\leq\mu<1)\\ \hat{\sigma}(t)&for&\kappa,&(0\leq\kappa<1)\end{array}\right. \tag{11}\]  
Figure 3 shows the block diagram of the T-S fuzzy controller. As shown in the figure, the wind speed and wind speed rate are used as inputs for the T-S fuzzy model. As shown in the figure, wind speed and wind speed rate are used as inputs to induce fuzzification in the T-S fuzzy model. In the T-S fuzzy model, interpolation calculations are performed using membership functions and fuzzy rules. In addition, defuzzification is performed to calculate the weight values \(\mu_{1}\), \(\mu_{2}\), and \(\mu_{3}\) of the control sub-commands for each wind speed region and the weight value \(\kappa\) of the control command for the uncertainty of the estimated wind speed.  
The fuzzy rules and membership functions for the fuzzy inference algorithm were designed with MATLAB's Fuzzy Logic Toolbox [30]. The non-linear membership functions (trapmf, trimf, and trapmf) were applied to the wind speed and wind speed rate corresponding to the input of the T-S fuzzy model. The constant-type membership functions (0, 0.3333, 0.5, 0.6667, and 1) were applied to each wind speed region (below-rated region (4.0 m/s), rated region (5.5 m/s), and above-rated region (8.0 m/s)) of the T-S fuzzy model. The constant-type membership functions (0, 0.5, 1) were applied to determine the uncertainty of the estimated wind speed.  
Since the uncertainty of estimated wind speed can be easily categorized using the degree to which the wind speed changes rapidly, the membership function for the wind speed were simply divided into three points between 0 and 1. In the case of the membership function for the wind speed, three points were divided between 0 and 1, and two points were added near the middle value to prevent transience caused by pitch and torque control conversion in the rated wind speed region during the interpolation process of the control command values of each sub-system. The membership functions and fuzzy rules are shown in detail in Figure 11 and Table 12 of Appendix B.