# A Study on \(H_{\infty}\)-Fuzzy Controller for a Non-Linear Wind Turbine with Uncertainty
## 3 Control Algorithms
### Mixed-Sensitivity \(H_{\infty}\) Control Algorithm  
\(H_{\infty}\)-based loop-shaping techniques were used to improve the uncertainty effect in the control system of wind turbines. In the control system, the plant is given as:  
\[\begin{array}{l}\dot{x}=Ax+B\;u\\ y=C\;x+D\;u\end{array} \tag{5}\]  
That is, plant (5) is a generalized representation of Equations (1) and (2). However, when the uncertainty effect of the plant is expressed through perturbation in \(H_{\infty}\) control theory, plant (5) can be expressed as an augmented plant (6) using linear fractional transformation (LFT) [27]. Due to the intervention of the disturbance input \(w\), the input matrix \(B\) of the plant (5) was augmented to \(B_{1}\) and \(B_{2}\). In addition, due to the intervention of the performance signal z, the output matrix \(C\) of the plant (5) was augmented to \(C_{1}\) and \(C_{2}\), and the direct transmission matrix \(D\) was augmented to \(D_{11}\), \(D_{12}\), \(D_{21}\), and \(D_{22}\). In this study, the augmented plant (6) was implemented using the mixed-sensitivity method. The mixed-sensitivity method represents the performance signal z by representing input sensitivity \(S\), control sensitivity \(KS\), output sensitivity \(T\), and each weight function, i.e., \(W_{1}\), \(W_{2}\), and \(W_{3}\).  
\[\begin{array}{l}\dot{x}=Ax+B_{1}w+B_{2}u\\ z=C_{1}x+D_{11}w+D_{12}u\\ y=C_{2}x+D_{21}w+D_{22}u\end{array} \tag{6}\]  
Figure 2 shows a closed-loop transfer function representing the \(H_{\infty}\) control system. \(K\) is the searched controller that minimizes the \(H_{\infty}\) norm of the augmented plant \(P\), i.e., plant (6), and \(G\) is the nominal form of the augmented plant \(P\). The \(H_{\infty}\) control uses a state feedback gain that minimizes the \(H_{\infty}\) norm of the mixed-sensitivity functions \(W_{1}S\), \(W_{2}KS\), and \(W_{3}T\).  
In addition, the value when the \(H_{\infty}\) norm of the mixed-sensitivity functions \(W_{1}S\), \(W_{2}KS\), and \(W_{3}T\) is at a minimum can be expressed as the optimal performance level \(\gamma_{0}\):  
\[\gamma_{0}=\min_{K\;\text{stabilizing}}\left\|\begin{array}{l}W_{1}S\\ W_{2}KS\\ W_{3}T\end{array}\right\|_{\infty} \tag{7}\]  
However, since complex systems such as uncertain non-linear wind turbines do not guarantee a unique solution to the optimal control problem (7), non-negative scalar performance level \(\gamma\), i.e., tolerance, is selected through iteration. Accordingly, the optimization problem (7) can be represented as a suboptimal control problem (8) using arbitrary perfor  
Figure 2: Block diagram of the closed-loop transfer function with an augmented plant using the mixed-sensitivity method.  
mance level \(\gamma\). In addition, through Figure 2, the sensitivity functions \(S\), \(KS\), and \(T\) can be represented by \(\left(I+GK\right)^{-1}\), \(K(I+GK)^{-1}\), and \(GK(I+GK)^{-1}\) respectively.  
\[\left\|\begin{array}{c}W_{1}\Big{(}I+GK\Big{)}^{-1}\\ W_{2}K\Big{(}I+GK\Big{)}^{-1}\\ W_{3}GK\Big{(}I+GK\Big{)}^{-1}\end{array}\right\|_{\infty}<\gamma,(\gamma>\gamma_ {0}) \tag{8}\]  
It is known that a given suboptimal control problem (8) can be solved using two algebraic Riccati equations (ARE) [27]. With the selection of performance level \(\gamma\), this suboptimal control problem was solved using MATLAB's hinfsyn function. The weight functions were applied using low-pass and high-pass filters [28].