This article proposes an adaptive virtual inertia control system for stable operation of microgrids: it theoretically improves recent related results in the literature. The overall control system is conceived to provide ancillary services support (namely, enhanced frequency regulation with inertial response and voltage support) to an ac/dc microgrid that consists of a dc bus providing power to the ac one, being, in turn, composed of a diesel generator and loads. The ac voltages and currents are controlled via a nonlinear algorithm based on control-induced time-scale separation and singular perturbation analysis, whose use greatly simplifies the control design and the choice of the tuning gains, when compared with the classical linear controls. With the aim of reducing oscillations from the grid by assuring suitable short circuit ratio (SCR) and proper $X/R$ characteristics, a virtual impedance is also implemented, along with frequency and voltage droops: they constitute high-level controllers that give references to the virtual inertia scheme and compose a comprehensive control. Detailed simulations illustrate, in different relevant settings, the performance of the proposed scheme.