## 1 Introduction  
Distributed energy resource (DER) systems are extensively used in micro-grids for the connection of renewable energy sources and storage [1]. However, this system must be able to maintain in its output voltage the same characteristics of the voltage provided by the electrical network to which they are connected, i.e., constant frequency and voltage amplitude and sine waveform, even when the primary network presents any failure (distortion, or even cut in the blackout supply). In practice, it is sought that, in the event of a power outage, for example, the equipment connected to the DER output will continue to operate normally to feed the locally connected loads. A basic method to obtain the proposed results is to compare the DER output voltage with a sinusoidal reference signal of the same waveform than the expected signal of the primary electrical network in normal operation [2]. However, the tracking of the reference signal becomes complex due to the non-linearities induced by the association of the different types of loads and the LCL filter connected at the output of the inverter. For example, the wide majority of electronic devices require direct current sources, equipped with a rectifier stage with full diodes bridge, which represents a great source of harmonic distortions due to the current waveform, and consequently the voltage in the grid [3]. Due to the critical nature of the loads, DER performance is regulated by international (ANSI/IEEE, 1986; IEC, 2011) standards [4], imposing conditions for the system's transient and steady state performance. During transient, standards require a small variation in the output voltage amplitude and a fast recovery time when a load is added or removed from the system. In steady state, the output voltage of a DER must be a sinusoidal signal with constant amplitude and frequency. To be considered as a sinusoidal signal when subjected to periodic disturbances caused by non-linear loads, the output voltage THD (Total Harmonic Distortion) and IHD (Individual Harmonic) must be within limits defined in these standards. A fundamental performance requirement of a distributed energy resources (DER) system with sinusoidal output is to provide a voltage with low harmonic distortion, which operates even under uncertainties, parametric variations and non-linearities induced by phenomena such as delay and saturation and disturbances, which are very common in practice. This objective can be fulfilled by choosing an adequate control law. Recently, considerable literature has been produced around the theme of DER control in microgrids to solve the above-cited problems. A major solution consists in the use of resonant compensators, adjusted to act on the fundamental frequency of the output and most significant harmonics. This method is based on the addition of harmonic modes (sine or cosine functions) in the direct control loop. The ideal resonant controller presents infinite gain at the resonance frequency, which allows us to reject the other components. The use of a set of resonant controllers in parallel allows the rejection of disturbances and reference tracking at selected frequencies. However, a high number of resonant compensators makes control tuning more complicated. Furthermore, it is necessary to take into account the ranges of the load and its variation in the problem formulation. Although there is a large number of published results dealing with DER system control, topics that still deserve further investigation are the proposition of controller design methodologies according to specific standards and the use of conditions based on Lyapunov functions for a more rigorous analysis of the stability and robust performance of the closed-loop system. In the literature, different controllers have been applied to control the distributed energy system based on the inverter with an LC filter. In Reference [5], PID (Proportional Integral Derivative) controllers were proposed. However, these ones were based on the voltage RMS value to obtain the feedback signal.