## 3 LMI Performance Requirements for Fuzzy Controllers
### LMI-Fuzzy Control of a Boost Converter  
As in the previous subsection, the regulation of the output voltage for a step-up converter is presented, taking into account its corresponding fuzzy model (12). This design consists of solving the optimization algorithm (21) for the set of parameters shown in Table 3, where the nominal load of the converter is equal to 10 \(\Omega\) and the complementary duty cycle in steady state is equal to 0.5. As it has been established, the control synthesis procedure consists of finding a vector of feedback gains \(F_{LMI-Fuzzy}\) such that the parameter \(\gamma\) is minimized in the LMI (20), while the constraints on the decay rate (\(\alpha\)) and the control effort are satisfied for the four linear submodels that build the T-S fuzzy model (12). The values of the control parameters, i.e., \((\alpha,\mu)\), are the same used for the case of the buck-boost converter. In order to demonstrate the advantage of the fuzzy control, as in the previous example, a comparison is made with the results of the LMI robust linear control law proposed in [9]. Therefore, for the set of parameters in Tables 2 and 3, the feedback gain vectors \(\mathbf{F}_{LMI-Fuzzy}\) and \(\mathbf{F}_{LMI}\) are obtained. In this way, for the case of LMI-Fuzzy control, the fuzzy state feedback gains correspond to:  
\[\mathbf{F}_{1}=\left[ \begin{array}{ccc}-0.6&-0.982&1229.7\end{array} \right] \mathbf{F}_{2}=\left[ \begin{array}{ccc}-0.7&-1.272&1498.7\end{array} \right]\] \[\mathbf{F}_{3}=\left[ \begin{array}{ccc}-0.97&-1.67&2053.7\end{array} \right] \mathbf{F}_{4}=\left[ \begin{array}{ccc}-1.01&-1.824&2143.6\end{array} \right] \tag{24}\]  
This controller ensures a \(H_{\infty}\) gain of the output voltage with respect to the output current disturbance of \(\gamma=1.0145\) (0.1250 dB). Furthermore, it is shown that the matrix \(\mathbf{P}\) is positive definite, ensuring that the asymptotic stability of the converter is fulfilled, as shown in (25). \[\mathbf{P}=\mathbf{W}^{-1}=\left[ \begin{array}{ccc}4.292\times 10^{-4}&7.699 \times 10^{-4}&-0.919\\ 7.699\times 10^{-4}&1.4094\times 10^{-4}&-1.668\\ -0.9193&-1.668&2223.954\end{array} \right]\mathbb{R}(\lambda(\mathbf{P}))= \left[ \begin{array}{c}6.663\times 10^{-6}\\ 1.9979\times 10^{-4}\\ 2223.9\end{array} \right]>0 \tag{25}\]  
The LMI robust feedback gain vector \(\mathbf{F}_{LMI}\) obtained corresponds to:  
\[\mathbf{F}_{LMI} = [-0.5555\quad-0.6090\quad 743.8420] \tag{26}\]  
This controller ensures a minimum level of disturbance rejection equivalent to the inverse of \(\gamma=2.31\) (7.2 dB). In order to verify the dynamic behavior of the boost converter under the laws LMI robust (26) and LMI-Fuzzy (24), some numerical simulations were performed via PSIM. Figure 8a depicts waveforms of the output current \(i_{o}\) and the regulated voltage \(v_{o}\) of the converter, controlled by the feedback gains (24). The bottom waveform shows a change from 2.4 A to 3.6 of the output current \(i_{o}\) at \(t=4\) ms, and the opposite transition at \(t=24\) ms. As expected, the settling time is 8.9 ms below, which corresponds to the minimum decay rate set in the Table 2. Now, the same current disturbance \(i_{o}\) is applied to the boost converter controlled by the feedback gain (26), where it can be seen from the waveform that the settling time is once again within the chosen design limit. As in the case of the buck-boost converter, it is worth noting that the results of the LMI-Fuzzy control show better dynamic behaviour than the LMI robust control, both in terms of settling time and disturbance rejection. Figure 9a,b illustrate the responses of the output voltage \(v_{o}\) to an input voltage variation, for both the law of LMI-Fuzzy control and the law of LMI robust control, respectively. The input voltage \(V_{g}\) changes from 12 V to 10 V at \(t=4\) ms and returns to 12 V at \(t=24\) ms. Again, the waveforms have a settling time below the