## 1 Introduction
Thus, their use solves the problem of voltage amplitude, but, in general, it does not solve the problem of distortion of the output voltage waveform [6]. A cascaded voltage/current control based on a virtual impedance concept has been developed in [7]. This topology simultaneously improves the injected grid current and local load voltage in island and grid connected microgrids. An optimized proportional resonant (PR) controller taking account of the computational delay from the digital control system was proposed in [8]. A controller based on a polytopic model was proposed in [9] for the current controller development by taking into account the grid inductance variations and soft saturation. To minimize output current ripples and reduce the steady-state current deviation, a quasi proportional resonant controller was proposed in [10]. Generally, the proportional resonant controller parameters are tuned manually. However, in [8], the authors proposed an optimization tuning of the RCs controller. An adaptive PR controller was proposed in [11] to ensure a current tracking under grid parameter variations. The authors used a fourth order filter with the adaptive RC to estimate online the resonance frequency, and then the grid impedance variations. In order to reduce the steady state current error, a modified proportional resonant controller used for the inner current control loop and indirect vector controller at the rotating reference frame was proposed in [12]. A controller to compensate the resonance phenomena without knowing the system parameters and without affecting the controller bandwidth in island microgrids was proposed in [13]. A three degrees of freedom cascaded voltage/current controller based on \(\mathcal{H}_{\infty}\) theory was proposed in [14; 15]. The main advantage of this controller is to ensure the stability of the closed-loop system and the robustness against parameter uncertainties. To solve the above problems, the proposed method in this paper is based on the solution of optimization problems addressed by linear matrix inequalities (LMIs) formulation, which allowsto concatenate the parameters of different resonant compensators acting in different frequencies. The proposed design method includes an easy solution to synthetizing the gains of the resonant controller, for a given capacity of the applied load range to distributed generation system units in the microgrid. The determination of gains is formalized through a convex optimization problem under a set of linear matrix inequalities constraints. In this paper, state feedback is used in conjunction with multiple resonant compensators that ensure the elimination of the steady state error and guarantee the rejection of multiple harmonics of the fundamental frequency produced by non-linear loads. In order to ensure the desired performances, the LMIs constraints construction is performed using a performance criterion throughout a robust regional poles placement [16; 17]. Following this paper, a mathematical modeling of a DER system is proposed in Section 2. In Section 3 the proposed controller is presented, as well as the augmented model of the DER system and proportional resonant controllers including harmonic compensator and disturbance. Followed by the fundamental concepts for stability by [PERSON] and D-stability based on LMIs, which are discussed in Section 4. The simulation results are presented in Section 5, where the obtained values of the controller gains are given as well as considered DER system parameters, the controller discretization method, and the structure of the digital implementation of the controller. In Section 5, the results obtained with the use of the controller designed for the simulated system are discussed, with the description of the tests used and discussion for each of the results. Finally, in Section 6, the conclusions reached from the developments and the simulation of this contribution are presented, as well as the definition of possible lines of development for the future.