Microgrids (MGs) interest is growing very fast due to the environment urgency and their capability to integrate renewable energy in a flexible way. In particular, islanded MGs in which distributed energy resources (DERs) are connected to the infrastructure with power electronic converters have attracted the interest of many researchers of both academia and industry because management, control and protection of such systems is quite different from the case of traditional networks. According to their operation mode, MGs that power electronic converters can be divided into grid-forming, grid-feeding and grid-supporting inverters. In particular, grid forming inverters are asked to impose voltage and frequency in the MG. This paper aims to propose a model predictive control (MPC) based approach for grid-forming inverters in an islanded MG. The MPC strategy is implemented because of its capability to define the optimal control actions that contemporarily track the desired reference signals and accounts for equality and inequality constraints. The overall problem formulation (objective function and relevant constraints) is described step by step and considers the specificity of the considered DC source. The proposed approach allows for the obtaining of very good results in terms of readiness against disturbances, even if it requires being fed only by local measurements. In order to validate the developed method, this paper proposes an experimental validation of the designed MPC controller in order to show its correct operation on a real system in a power hardware in the loop set-up using a rapid control prototyping approach.