Steam methane reforming in solid oxide proton conducting membranes is a state-of-the-art process capable of initiating methane reforming reactions, electrochemical hydrogen separation, and the compression of purified hydrogen product within a single electrochemical processing unit. Given the many process variables involved, a model predictive controller is needed to safely operate a protonic membrane reformer (PMR) under dynamic operational conditions by employing physically relevant constraints that protect the reactor materials of construction and maximize the stability of the process. This work derives, and experimentally validates, physics-based models for a PMR process and integrates an overall process model into centralized and decentralized model predictive control schemes. The performance of control actions from classical proportional-integral controllers and model predictive controllers are surveyed, and the decentralized model predictive control algorithm, developed here, obeys practical constraints, reaches the target variables' setpoints quickly, and lowers computational costs relative to the centralized predictive controller. Finally, the addition of a disturbance observer (DOB) ensures robust controller performance when subject to incomplete and infrequent process measurements or common system disturbances.