## 3 LMI Performance Requirements for Fuzzy Controllers  
The design procedure of the controller is based on the Parallel Distributed Compensation technique (PDC), which is used to design state feedback controllers based on T-S fuzzy models [13]. This methodology consists in associating each control rule with the corresponding rule of the fuzzy model, as follows:  
\[R_{i}:\textbf{If}\ \ \delta_{1}\ \ \textit{is}\ \ M_{i1}\ \ \textit{and}\ \ldots\textit{and}\ \ \delta_{j}\ \ \textit{is}\ \ M_{ji}\ \ \textbf{then}\ \ \ u(t)=-\textbf{F}_{i}x(t)\quad i=1,\ldots,r \tag{15}\]  
where \(\textbf{F}_{i}\) are lineal feedback gain vectors associated with each rule. Then, the output controller is deduced as:  
\[u(t)=-\sum_{i=1}^{r}h_{i}\textbf{F}_{i}x(t) \tag{16}\]  
Substituting the control law (16) in the fuzzy model (4), the closed loop system dynamics is given by:  
\[\dot{x}(t)=\sum_{i=1}^{r}\sum_{j=1}^{r}h_{i}(\delta)h_{j}(\delta)\big{[}A_{i}+ B_{i}\textbf{F}_{j}\big{]}x(t) \tag{17}\]  
In order to find the feedback gain vectors properly \(\textbf{F}_{i}\) in an operating range, Lyapunov stability and performance constraints in form of LMIs are imposed. In this way, these performance constraints, taken from [13], are expressed by the following theorems. **Theorem 1**. : _The system defined by (4) is quadratically stable for some feedback gains \(\mathbf{F}_{i}\) and \(\frac{\|z\|_{2}}{\|w\|_{2}}<\gamma\) if there is a common positive definite matrix \(\mathbf{W}\) and \(\mathbf{Y}_{i}\) such that [13]:_  
\[\left[ \begin{array}{ccc}\left(\begin{array}{c}\frac{1}{2}\Big{\{}A_{i} \mathbf{W}+\mathbf{W}A_{i}^{T}+A_{j}\mathbf{W}+\mathbf{W}A_{j}^{T}+\\ -B_{i}\mathbf{Y}_{j}-\mathbf{Y}_{j}^{T}B_{i}^{T}-B_{j}\mathbf{Y}_{i}-\mathbf{ Y}_{i}^{T}B_{j}^{T}\Big{\}}\end{array}\right)&\frac{B_{\mathbf{Y}_{i}}+B_{ \mathbf{Y}_{j}}}{2}&-\mathbf{W}\frac{\left(\mathbb{C}_{i}+\mathbb{C}_{z_{j}} \right)^{T}}{2}\\ \frac{\left(B_{\mathbf{Y}_{i}}+B_{\mathbf{Y}_{j}}\right)^{T}}{2}&-\gamma I& \mathbf{0}\\ -\frac{\left(C_{i}+C_{z_{j}}\right)}{2}\mathbf{W}&\mathbf{0}&-I\end{array}  \right]<\mathbf{0} \tag{18}\]  
_where \(I\) is the identity matrix, \(\mathbf{W}=\mathbf{P}^{-1}\) and \(\mathbf{F}_{i}=\mathbf{Y}_{i}\mathbf{W}^{-1}\)._  
**Theorem 2**. : _The eigenvalues of \(\left(A_{i}-B_{i}\mathbf{F}_{j}\right)\) in each linear fuzzy system are inside the region \(\mathbf{S}(\alpha)\) (see Figure 4) if there is a common positive definite matrix \(\mathbf{W}\) such that [13]:_  
\[ \begin{array}{ccc}A_{i}\mathbf{W}+\mathbf{W}A_{i}^{T}-B_{i}\mathbf{Y}_{i}- \mathbf{Y}_{i}^{T}B_{i}^{T}+2\alpha\mathbf{W}<0,&i=1,\ldots r\\ A_{i}\mathbf{W}+\mathbf{W}A_{i}^{T}+A_{j}\mathbf{W}+\mathbf{W}A_{j}^{T}-B_{i} \mathbf{Y}_{j}-\mathbf{Y}_{j}^{T}B_{i}^{T}-B_{j}\mathbf{Y}_{i}-\mathbf{Y}_{i} ^{T}B_{j}^{T}+4\alpha\mathbf{W}\leq 0,&i<j\leq r\end{array}  \tag{19}\]  
_being \(\mathbf{Y}_{i}=\mathbf{F}_{i}\mathbf{W}\) so that for \(\mathbf{W}>0\), it is had \(\mathbf{F}_{i}=\mathbf{Y}_{i}\mathbf{W}^{-1}\)._  
**Theorem 3**. : _Assume that initial condition \(x(0)\) is known. The constraint \(\|d(t)\|_{2}\leq\mu\) is enforced at all times \(t\geq 0\) if the LMIs (20) hold [13]:_  
\[\left[ \begin{array}{cc}1&x(0)^{T}\\ x(0)&\mathbf{W}\end{array} \right]\geq 0, \tag{20}\] \[\left[ \begin{array}{cc}\mathbf{W}&\mathbf{Y}_{j}^{T}\\ \mathbf{Y}_{i}&\mu^{2}I\end{array} \right]\geq 0\]  
_where \(\mathbf{W}=\mathbf{P}^{-1}\) and \(\mathbf{Y}_{i}=\mathbf{F}_{i}\mathbf{W}\)._  
A detailed proof of the Theorems 1, 2 and 3 are shown in [13]. Thus, from the previous LMIs, the procedure proposed is to find the minimum norm \(H_{\infty}\left(\gamma\right)\) between the disturbance input and the regulated output, ensuring at the same time stability in a wide  
Figure 4: Pole location region \(S(\alpha)\). algorithm (21) through MATLAB's LMI toolbox [24], the fuzzy state feedback gains are obtained \(\mathbf{F}_{LMI-Fuzzy}=\{\mathbf{F}_{1},\ldots,\mathbf{F}_{4}\}\):  
\[\mathbf{F}_{1}=\left[ \begin{array}{cc}-0.1094&0.1537&-165.2341\end{array}  \right] \mathbf{F}_{2}=\left[