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Solid Oxide Fuel Cells (SOFC) are energy conversion devices that produce electrical energy via the reaction of a fuel with an oxidant. Although SOFCs have become credible alternatives to non-renewable energy sources, efforts are still needed to extend their applicability to a broader scope of applications, such as domestic appliances. SOFCs are typically operated continuously and are characterized by the presence of stringent operating constraints. Particularly, violating the constraint on the cell potential can severely damage a cell, while violating the upper bound on the fuel utilization can also induce negative effects due to fuel starvation. Hence, control and optimization are required to improve cost effectiveness, while respecting operational constraints. Among the numerous control strategies available in the literature, Model-Predictive Control (MPC) is an excellent candidate because it can handle constraints explicitly. Furthermore, the control inputs are obtained via the solution of a model-based optimization problem. Only the first moves of the resulting input profiles are applied to the process, and the procedure is repeated at the next sampling time. Constraints are often handled by penalizing the cost function for any constraint violation rather than by including constraints in the optimization problem. This approach, referred to as soft-constraint MPC, is advantageous since (i) the computational load is reduced, and (ii) it avoids the instabilities that MPC with hard output constraints typically induces. However, it also presents several drawbacks: (i) the constraints can be violated, (ii) an oscillatory behavior is often observed, and (iii) the performance is weight dependent. Consequently, hard-constraint MPC will be considered in this study. Because of the aforementioned stability issues, it is proposed to linearize the nonlinear output constraints with respect to the inputs, thus resulting in linear hard constraints on the inputs. In addition, a bias term is introduced in these linearized constraints to handle inaccuracies by artificially reducing the size of the feasible region. The bias term is then adapted using measurements, which leads to improved performance via a progressive, yet safe, expansion of the feasible region. This hard-constraint MPC approach is validated experimentally.