## 1 Introduction
Using such an approach, the inverter is not taken as a whole system but two identical decoupled converters and the output of each branch is controlled individually and separately from the other branch. The control of the inverter is performed by controlling each DC-DC boost converter separately. In this approach, each boost converter is controlled to generate a DC biased sinusoidal voltage with a phase shift of 180\({}^{\circ}\) such that the difference between both output voltages is an AC sine wave signal that is used to supply the AC load. The DC-DC boost converter cannot generate a voltage lower than its DC input voltage, the DC component of the voltage generated by each DC-DC boost converter must be larger than the DC input voltage. For that, two controllers for each loop and two modulators are used for the entire system. Hence increasing the number of components in the control system. Moreover, the two branches of the inverter are coupled both at the input and the output sides. In particular, in PV application, the input side is no more a constant voltage but a state variable that is coupled to both branches of the inverter. Model predictive control was applied to the differential boost inverter in [27] used for PV grid-connected system applications where it was demonstrated by numerical simulations that a simple control comparable to classical control techniques can be used. Most of the previous works have converted the problem of controlling the boost inverter to the problem of controlling two identical DC-DC boost converters. An exception where the inverter was controlled as an individual system is [28] where a strategy for generating a sinusoidal voltage at the output of the inverter with resistive load directly from the difference between the capacitor voltages of both converters. The approach was finally implemented by a fixed frequency PWM approach based on a sliding mode control theory leading to a robust system under load and input voltage variations with simple implementation. One advantage of controlling the system as whole is a decrease in the number of required sensors and simplicity of the control scheme to be implemented. Sliding mode control has many advantages for nonlinear systems. Nevertheless, the main drawback of the sliding mode control in power electronics field is that it is ultimately implemented by using hysteretic comparators leading to variable switching frequency. PR controllers have been used to overcome the issues related to the use of PI control in single-phase inverters [11; 29]. Nevertheless, since the AC signals are much slower than the switching frequency, conventional controllers can still be used as will be shown in this paper. However, PI controllers can lead to steady-state errors and phase shifts when used to control sinusoidal signals. The bandwidth must be enough large but high bandwidth could lead to either instability due to the phase lag introduced by the PI controller. Therefore, a phase boosting is needed near the desired crossover frequency and this can be accomplished by using type III controllers widely adopted in the control of DC-DC converters [30]. In most of the published works, resistive passive loads and constant input voltage were considered. Except [21], real nonlinear real PV sources and grid connection have not been fully addressed for the differential boost inverter due particularly to its complexity when it is used in PV systems such as in grid-connected microinverter applications where the cost and the simplicity are to be taken into account. When the system is supplied by a constant input voltage and loaded by a resistive load, its design can be roughly divided in the design of two identical boost converters resulting eventually to order reduction of the overall dynamics.