The present work focuses on applying model predictive methods to control nonlinear vibrations. Various approaches to derive the model used within the controller, including linearization (modal and Carleman/Koopman) and nonlinear model reduction techniques, are reviewed and compared. The structure used for the numerical and experimental investigations is a straight cantilever beam with geometric nonlinearity and a 1:3 internal resonance between its first two modes, resulting in the usual saddle-node and Neimark–Sacker bifurcations and the presence of an isola. Approximate analytical and finite element models of the beam are first derived and then used to construct the linearized and nonlinear reduced-order models. The models are subsequently exploited by the model predictive controller to reach a range of stable and unstable periodic responses of the structure under harmonic excitation.