## 1 Introduction
Moreover, differential inverter topologies seem to prevail in price and size due to the utilization of small passive elements of DC-DC converters hence improving the efficiency [11]. Some inverter topologies with voltage boosting capability such as the Z-source inverter [12] and the quasi-Z-source inverters [13; 14] have been also developed. Another topology with voltage boosting capability is the DC-AC differential boost inverter that has been presented in [15]. This topology was obtained by connecting the load differentially across two identical DC-DC boost converters sharing the same input. The same concept can be applied to any DC-DC converter to form a differential inverter. A study of boost, buck, buck-boost and Cuk differential inverters has been presented in [16]. The differential boost inverter has been used for power processing stage fuel-cell energy system [17; 18]. It has been also used for high quality sine wave generation with a high oscillation frequency [19] where a dynamic modulator has been adopted to control the capacitor voltage of the boost inverter. Since its introduction in [15], many control strategies have been applied to the differential boost inverter. The nonlinear control-to-output behavior in large-signal dynamic sense and the non-minimum phase nature in the small-signal sense of the boost inverter make the direct tracking of the sinusoidal voltage reference in this kind of topologies a difficult task [20]. Sliding mode control was first applied in which a switching surface composed by weighted errors in the output capacitor voltages and the inductor currents is used [15]. The inverter model with a constant voltage source and a resistive load system and its controller design has been addressed in [21] using a sliding mode control of the inner current loop and a PI controller for the AC output voltage. Adaptive control for the boost inverter has been designed in [22] where by using energy shaping methodology with a suitable Hamiltonian function an autonomous oscillator is generated without an external reference signal hence being the approach suitable for resistive loads. Other Pulse Width Modulation (PWM) techniques, mostly based on two-loop linear control scheme, have been used with a slow outer loop for the capacitor voltages and a fast inner loop for inductor currents. Two separate control loops are usually used for each converter. Examples of this control strategy can be found in [17; 20]. Most of the previous studies reported on the differential boost inverters have been tested by a resistive load. Grid-connected boost inverter was also considered in [23; 24] for integrating energy storage devices such as a battery, fuel cell or supercapacitor to a single-phase AC grid. A load-adaptation mechanism for the boost inverter has been carried out to design a state observer for some of the converter variables in order to estimate the load [22] leading to a fast system response and successful adaptation of the load parameter as confirmed by numerical simulations in the same paper. A current control strategy with zero steady-state tracking error has been applied to a differential grid-connected boost inverter supplied from an ideal input voltage source [25]. In that work, based on the internal model principle, Proportional-Resonant (PR) controllers instead of Proportional-Integral (PI) controllers were used to achieve zero phase and amplitude tracking error demonstrating experimentally that the proposed control meet grid-connection harmonics standards. The grid-connection of the differential boost inverter and the use of PR controller for the grid current loop was also addressed in [26] where a sliding mode control for the inner inductor current loop and a PI controller was used. In most of the published works, a decoupling between the two branches of the differential inverter is performed with the aim to simplify the control design.