## 1 Introduction  
The demand for small-scale renewable energy systems is increasingly growing [1, 2, 3, 4]. DC-AC inverters are essential elements for these systems [4]. They are used for efficient energy conversion either to supply local AC loads or to inject part of the electrical power into the AC grid [3]. In conventional PV systems, the PV arrays are formed using multiple series-connected PV modules to attain high enough DC voltage at the PV side. The resulting arrays are then arranged in parallel according to the needed power to be generated [4]. With this approach the partial shadowing of some of the PV modules in the arrays causes the operating point of shaded modules to be different from the rest of modules and therefore the total generated power is significantly decreased [2, 4]. Furthermore, a high irradiance mismatch between PV sources may lead to damaging over-voltages in the modules [2, 4]. To mitigate this problem, the AC module approach (known in the literature as micro inverter approach) has been proposed [3, 4]. With this approach, a small DC-AC inverter is used for each individual PV module. PV systems based on AC module approach has many advantages over central and string inverter systems and it is foreseen to be the future trend of PV system architectures. These advantages include small size, good quality of supplied electrical power, high conversion efficiency, good reliability, robust stability, simple and flexible installation options and enhanced safety. There exist mainly two kinds of architectures for integrating AC modules into a power generation system. The first one uses a single-stage inverter with a step-up transformer [3]. The unique stage is in charge for regulating the DC PV voltage, performing Maximum Power Point Tracking (MPPT) and carrying out DC-AC conversion. The second one is based on a double-stage approach where the first stage is a front-end DC-DC converter used to sufficiently stepping-up the DC voltage at the PV side while performing MPPT and the second stage is a DC-AC inverter used in to generate AC voltage [3; 4]. Both topologies suffer from decreased efficiency and increased size. Single-stage transformer-less DC-AC boost inverters have recently received a considerable attention from many researchers, see [3] and references therein. These structures apart from performing DC-AC conversion they also step-up the voltage being therefore excellent candidates for microinverter applications. The price of single-stage inverter topologies is also low, and their size is reduced because of using small reactive components. Furthermore, their efficiency is better [5]. The differential boost inverter has been introduced in [6] and since then many studies have dealt with its control design [6; 7; 8; 9; 10]. In most of the studies, inverters with a constant input voltage and an ohmic resistance as a load have been considered and their control design has been accordingly tackled. In [7; 10; 11], a grid-connected boost inverter has been considered and its control design has been addressed. Like many other practical engineering systems whose dynamics are governed by switching among different circuit topologies, power conversion systems can be classified into to the category of nonlinear piecewise smooth systems [12]. The main research works on the nonlinear dynamics of this kind of systems have hitherto dealt with the behavior of simple DC-DC converters with linear and mostly resistive loads [13; 14; 15; 16; 17; 18] due to the relative simplicity of their models. The nonlinear modeling of pulse width modulated DC-DC converters is relatively simple in the sense that their behavior is governed by a single constant switching frequency. In contrast, the dynamics of AC-DC rectifiers and DC-AC inverters are governed by two vastly distant frequencies: the switching high frequency \(f_{s}\) and the grid low frequency \(f_{g}\)[19; 20; 21; 22; 23; 24; 25]. Their nonlinear modeling is therefore mathematically more involved.