Shipboard AC/DC microgrids are used for power supply and electric propulsion in vessels. An indicative form of such a microgrid comprises diesel engines or gas turbines that provide power for the rotation of synchronous or asynchronous generators. Next, the AC output voltage of the generators is turned into DC voltage with the use of AC to DC converters and is distributed through DC voltage buses to the ship's compartments. Besides, with the use of DC to AC inverters voltage excitation is provided to synchronous or asynchronous motors which can be used in turn for the vessel's propulsion. The dynamic model of the considered shipboard AC/DC microgrid, being initially expressed in a nonlinear and multivariable state-space form, undergoes approximate linearization around a temporary operating point that is recomputed at each time-step of the control method. The linearization relies on first-order Taylor series expansion and on the computation of the associated Jacobian matrices. For the linearized state-space model of the shipboard AC/DC microgrid a stabilizing optimal (H-infinity) feedback controller is designed. This controller stands for the solution to the nonlinear optimal control problem of the AC/DC microgrid under model uncertainty and external perturbations. To compute the controller's feedback gains an algebraic Riccati equation is repetitively solved at each iteration of the control algorithm. The global stability properties of the control method are proven through Lyapunov analysis. Finally, to implement state estimation-based control of the shipboard AC/DC microgrid, without the need to measure its entire state vector, the H-infinity Kalman Filter is used as a robust state estimator. The article's method provides one of the few algorithmically simple and computationally efficient solutions for the nonlinear optimal control problem of shipboard microgrids.