# Robust LQR Control for PWM Converters with Parameter-Dependent Lyapunov Functions
## 5 Simulation Results  
This section illustrates the properties of the different controllers \(K,K_{eq}\), \(K_{perf}\) and \(K_{rob}\). We have performed a set of PSIM [26] simulations of the switched DC-DC boost converter, according to Figure 1. The first set of simulations is useful to establish the performance of the controllers, by analyzing the response of the converter with respect to changes in the output current. The second set aims to establish the robustness of the different controllers when there is a change in the operating point, by modifying the supply voltage. First, the waveforms of the simulations with changes in the load are grouped in Figure 5. The top waveforms in each subfigure correspond to the output voltage \(v_{0}(t)\), whereas the bottom waveform represents the output current \(i_{load}(t)\). In all simulations, the converter load is initially the nominal value \(R=25\Omega\). At time \(t=1\) ms, the load changes to \(R=10\Omega\), which is the maximum load allowed by design in all polytopes. The load returns to \(R=25\Omega\) at \(t=6\) ms. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(D^{{}^{\prime}}\) & \(\frac{1}{D^{2}R}\) & \(\frac{1}{R}\) \\ \hline \(\theta_{1}(t)\) & 0.1 & 1/0.1 & 1/0.1 & 1/10 \\ \hline \(\theta_{2}(t)\) & 0.1 & 1/0.1 & 1/0.5 & 1/50 \\ \hline \(\theta_{3}(t)\) & 0.165 & 1.8 & 3/10 & 1/10 \\ \hline \(\theta_{4}(t)\) & 0.165 & 1.8 & 3/50 & 1/50 \\ \hline \(\theta_{5}(t)\) & 0.165 & 5.2 & 1 & 1/10 \\ \hline \(\theta_{6}(t)\) & 0.165 & 5.2 & 1/5 & 1/50 \\ \hline \(\theta_{7}(t)\) & 1 & 1 & 1/10 & 1/10 \\ \hline \(\theta_{8}(t)\) & 1 & 1 & 1/50 & 1/50 \\ \hline \end{tabular}
\end{table} 
Table 2: Vertices of the polytopic covering of \(P_{2020}\)As a baseline for the comparison, Figure 5a shows the performance of controller \(K\), as in [7], and the performance of controller \(K_{\text{eq}}\) obtained with the proposed method and the same polytope used in [7], \(P_{2009}\). It can be seen that the disturbance rejection properties and the settling time are nearly identical. In contrast, Figure 5b shows a comparison with controller \(K_{\text{perf}}\), which exhibits a tight regulation of the output voltage, such that the maximum error of \(v_{0}(t)\) and its settling time are reduced to approximately one half of what is achieved with \(K\). As expected, the robust controller \(K_{rob}\) presents loose regulation and a slower response, as shown in Figure 5c, when compared to \(K\). Note that Figure 5a-c shows the response at the nominal operating point, when \(v_{g}(t)=12V\) and \(\text{D}^{\prime}=0.5\). In order to evaluate the performance at a different operating point, Figure 5d shows the response of \(K\), \(K_{perf}\) and \(K_{rob}\) under an input voltage variation of \(-40\%\), such that the operating point is now \(\text{D}^{\prime}=0.3\). Again, \(K_{perf}\) is the controller that achieves excellent regulation properties, maintaining its robustness in the expected region of operation. If \(K_{perf}\) is the controller that demonstrates that the proposed method can be used to improved regulation while maintaining the same robustness properties, \(K_{rob}\) is the controller that demonstrates that the method can also be employed to enlarge the stability region. Figure 6 shows the waveforms of the simulations in which the input voltage is stepped, such that the operating point of the converter is modified in time. Figure 6a shows a voltage step of \(-40\%\), which corresponds to a step in the duty cycle from \(\text{D}^{\prime}=0.5\) to \(\text{D}^{\prime}=0.3\) (D = 0.7). Since all polytopes considered such a region, the three controllers maintain the stability, with \(K_{perf}\) exhibiting the best regulation performance.