# Optimal Robust LQI Controller Design for Z-Source Inverters
## 4 Simulation Results  
A three-phase 50-Hz 55-V (line) 5-A Y-connected load with a lagging power factor of 0.8 is fed with an input voltage of 20 V through a ZSI operating at 5 kHz using simple-boost PWM. On the basis of guidelines presented in [38] to design impedance networks, the unknown parameters were selected as listed in Table 1. It worth noting that the range of the uncertain parameters are \(R_{o}\in[10,60]\) and \(D\in[0.3,1]\).  
The proposed controller aim and objective is to minimize (18), in which weighting matrices are adjusted by the BA. Table 2 provides the required parameters for the successful operation of the BA algorithm. Using (24), the optimization processes are executed and the evolution of the fitness function besides the convergence curves of weighting matrices are illustrated in Figure 6. It can be seen that the tuning parameters quickly converged toward the best solutions after 150 iterations.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **Symbol** & **Value** \\ \hline Input voltage (V) & \(V_{in}\) & 20 \\ Capacitor voltage (V) & \(V_{C}\) & 89.8146 \\ Inductor current (A) & \(I_{L}\) & 19.05 \\ Output current (A) & \(I_{o}\) & 4.2362 \\ Capacitor (\(\mu\)F) & \(C\) & 92.25 \\ Inductor (\(m\)H) & \(L\) & 2.1 \\ Load (\(\Omega\)) & \(R_{o}\) & 27 \\ Shoot-through duty cycle & \(D\) & 0.4374 \\ Coupling inductor (\(m\)H) & \(L_{o}\) & 6.6 \\ Parasitic resistance (\(\Omega\)) & \(r\) & 0.05 \\ Switching frequency (KHz) & \(f\) & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Nominal inverter parameters.  
Accordingly, the global best of the bats, the optimal weighting matrices, are calculated as \(\mathcal{R}=1\) and \(\mathcal{Q}=diag[0.01,0.01,0.01,500]\). Then, using Matlab to solve (19), the proposed robust LQI controller is computed as (25).  
\[K=[0.6241\ \ 0.0153\ \ -0.1468\ \ \ -22.3607] \tag{25}\]  
Thus, the corresponding control signal generating the shoot-through duty cycle is calculated as (26).  
\[d(t)=0.6241i_{L}+0.0153v_{C}-0.1468i_{o}-22.3607x_{ext} \tag{26}\]  
To compare the performance and robustness of the proposed controller, the ZSI is also simulated with PI and SF controllers within dual loop schemes presented in Figure 3a,b, respectively. The outer PI controller in Figure 3a is replaced with a simple integral action while the inner PI controller is calculated as (27).  
\[\text{PI}=\frac{k_{i}}{s},\ \ k_{i}=0.0564 \tag{27}\]  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Loudness & 0.5 \\ Pulse rate & 1 \\ Population size & 50 \\ Frequency range & \([0,2]\) \\ Maximum iteration & 400 \\ Dimension of search variables & 3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Bat algorithm (BA) parameters.  
Figure 6: (**a**) Evolution of the fitness function and (**b**) convergence curves of weighting matrices.  
To design the SF controller, the well-known [PERSON]'s formula given by (28) is adopted.  
\[K=[0\ \ 0\ \ \cdots\ \ 1]\varphi_{\mathbb{C}}^{-1}\theta(A_{tot}) \tag{28}\]  
where, \(\varphi_{\mathbb{C}}\) is the controllability matrix, and \(\theta(s)=(s+w_{0})^{n+1}\) is the desired characteristic equation with order of \(n\). Thus, by setting the desired bandwidth as \(\omega_{0}=300\), the SF gain with corresponding control law are derived as (29) and (30), respectively.  
\[K=[-0.0007\ \ 0.0031\ \ -0.071\ \ -0.0211] \tag{29}\]  
\[d(t)=-0.0007i_{L}+0.0031v_{\mathbb{C}}-0.071i_{o}-0.0211x_{ext} \tag{30}\]