# Optimal Robust LQI Controller Design for Z-Source Inverters
## 5 Discussion  
To assess the performance of the presented controller, the closed-loop response of the system is derived in the presence of load disturbance under nominal and perturbed conditions. Figure 7 presents the voltage across the capacitor and output current reaction to a load disturbance 4 A, when the inverter operates at the nominal condition (\(D=0.4374\) and \(R_{o}=27\)). The corresponding performance indices are summarized in Table 3 for both servo and regulatory problems. It can be recognized that all of the controllers exhibit stable servo and regulatory responses without a steady-state error, due to utilizing an extended integral action. However, the proposed robust LQI controller provides a fast yet no overshoot servo response besides a less undershoot regulatory response. It can be seen that the steady-state voltage across the capacitor is 89.8 V, which has a subtle difference from the designed value of 89.8146, and the output current reached 4.236, which is very close to the designed value of 4.2362. Moreover, the control signals of all comparative methods are smooth enough for the successful operation of the developed ZSI. According to Figure 7, the presented method provides more initial control effort to properly decrease the undershoot value. To investigate the robustness to variations in plant parameters, simulations are repeated by keeping the previous parameter setting for the perturbed conditions. Figure 8 presents the voltage across the capacitor and output current reaction to a load disturbance 4 A at the perturbed condition (\(D=0.4\) and \(R_{o}=60\)). Table 4 indicates the corresponding performance indices for both servo and regulatory problems. While the SF and LQI-based controllers yield stable performance, the PI-based control system produces a near unstable response. Furthermore, the SF-based control scheme results in unacceptable oscillations in regulatory response, whereas the proposed LQI-based robust controller shows better robustness against the parameter variations by preserving its closed-loop response. To analyze more the robust performance, simulations are repeated at the perturbed condition (\(D=0.45\) and \(R_{o}=60\)). The corresponding performance indices are provided in Table 5 for both servo and regulatory problems. As can be understood from Table 5 and Figure 9, PI and SF controllers degraded their transient responses and showed high-frequency small or low-frequency large oscillations in disturbance rejection. The results reveal that despite the perturbed condition, the synthesized controller preserves no overshoot and less undershoot transient responses. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**Servo Response**} & \multicolumn{2}{c}{**Regulatory Response**} \\ \hline
**Controller** & **TV** & **IAE** & **MP(\%)** & **IAE** & **Peak** \\ \hline PI & 0.1285 & 1.5059 & 2.21 & 0.7733 & 29.1953 \\ SF & 0.0835 & 1.5620 & 0 & 1.1866 & 62.0276 \\ LQI & 0.0955 & 1.3253 & 0 & 0.6370 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance indices of the nominal closed-loop responses (\(D=0.4374\) and \(R_{o}=27\)). Process noise needs to be included in a model, due to modeling approximations and model integration errors, which shall be investigated when considering closed loop control. For simulations, the process noise is taken into account as Gaussian distribution with zero means and different covariance matrices. The closed-loop response under PI and SF controllers result in unstable responses for a process noise with zero means and covariance matrices of 1. Figure 10 depicts the corresponding voltage across the capacitor and output current signal produced by the proposed controller. It is clear that the proposed controller keeps the performance and robustness of the closed loop system under a process noise with zero means and covariance matrices of 100. Figure 8: Simulation results for a load disturbance of 4 A under perturbed condition (\(D=0.45\) and \(R_{o}=60\)).