## 2 Fundamentals of the Performance Improvement  
The boost inverter depicted in Figure 1 is a fourth order DC-AC converter which is composed by two conventional boost converter cells. The sub-indices \(x=\{1,2\}\) were defined to differentiate left and right sides of the converter, respectively. Each cell is integrated by one inductor (\(L\)), one capacitor (\(C\)) and two controlled switches (\(S_{x1}\): high-side and \(S_{x2}\): low-side). The two switches into a cell assemble a bridge leg and then the complete circuit topology is quite similar to the conventional full-bridge inverter. The input voltage \(v_{in}\) is applied to the inputs of the two cells and the output voltage (\(v_{o}\)) is obtained as the difference between the voltages of the output capacitors of the cells (\(v_{1}\) and \(v_{2}\)). The study is developed considering a resistive load \(r_{o}\). Stand-alone applications demand that inverter produces a pure sinusoidal signal from a DC input voltage which in the case of the boost inverter can be lower that the amplitude of the desired output. Then consider the following output voltage:  
\[v_{\alpha}=V_{m}\sin\omega t=2V_{s1}\sin\omega t \tag{1}\]  
being \(V_{m}\) the desired amplitude and \(\omega=2\pi f\), for \(f\) being the desired output frequency. To obtain this voltage, majority of control proposals define as objective the tracking of DC-biased sinusoidal references in the capacitor voltages. These references have the form:  
\[v_{1e}=V_{dc}-V_{s1}\sin\omega t \tag{2}\]  
\[v_{2e}=V_{dc}+V_{s1}\sin\omega t \tag{3}\]  
Considering that Equation (1) is obtained from the difference between Equations (3) and (2), it is easy to note that the same result can be produced using references \(v_{1e}\) and \(v_{2e}\) with enriched harmonic content as follows:  
\[v_{1e}=V_{dc}-V_{s1}\sin\omega t+V_{c1}\cos\omega t+\sum_{n=2}^{\infty}V_{sn} \sin n\omega t+V_{cn}\cos n\omega t \tag{4}\]  
\[v_{2e}=V_{dc}+V_{s1}\sin\omega t+V_{s1}\cos\omega t+\sum_{n=2}^{\infty}V_{sn} \sin n\omega t+V_{cn}\cos n\omega t \tag{5}\]  
As it was developed in [16], the voltage stress of the semiconductors in the boost inverter is directly related with the shape of the capacitor voltages. Then, that work shows how enforcing a DC-biased half-wave rectified sinusoidal is optimal to alleviate the switching losses improving the  
Figure 1: Schematic diagram of the boost inverter circuit. efficiency of the converter. However, in that work, the desired behavior is obtained by using two additional switches increasing the number of semiconductors and the required auxiliary circuitry. Then, the concept developed in this study consists of enforcing the desired shape in the capacitor voltages through independent controllers for each boost cell of the inverter simultaneously ensuring an output voltage of high quality. Then, two objectives are defined: (a) generating the adequate voltage references and (b) guaranteeing a robust tracking of these references. Consider that references are defined to enforce the DC-biased half-wave rectified sinusoidal shape in the capacitor voltages as follows:  
\[v_{1e}=V_{dc}-V_{s1}+2V_{s1}\sin\omega t[1+\text{sign}(\sin\omega t)] \tag{6}\]  
\[v_{2e}=V_{dc}+V_{s1}+2V_{s1}\sin\omega t[1+\text{sign}(\sin\omega t)] \tag{7}\]  
Although references Equations (6) and (7) can be easily produced, they can introduce negative effects into the control loop because of the discontinuity of their derivatives. Then, it is proposed to build approximate, but smooth references by adding one or two harmonic components to Equations (2) and (3). To sake of simplicity, analysis is shown only for reference signal \(v_{2e}\) as follows:  
\[v_{2e}=V_{dc}-V_{s1}+V_{c2}+V_{s1}\sin\omega t-V_{c2}\cos 2\omega t \tag{8}\]  
\[v_{2e}=V_{dc}-V_{s1}+V_{c2}+V_{c4}+V_{s1}\sin\omega t-V_{c2}\cos 2\omega t-V_{c4} \cos 4\omega t \tag{9}\]  
Figure 2 depicts a comparison between signals obtained from numeric evaluation of Equations (7)-(9).