  A nonlinear frequency response based adaptive vibration controller is
proposed for a class of nonlinear mechanical systems. In order to obtain the
nonlinear Frequency Response Function (FRF), the convergence properties of the
system are studied by using the convergence (contraction) theory. If the system
under consideration is: 1) convergent, it directly enables to derive a
nonlinear FRF for a band of excitation inputs, 2) non-convergent, first a
controller is used to obtain the convergence and then the corresponding FRF for
a band of excitation inputs is derived. Now the gains of the proposed adaptive
controller are tuned such that a desired closed-loop frequency response, in the
presence of excitation inputs is achieved. Finally, a building structure with
nonlinear cubic stiffness and a satellite system are considered to illustrate
the theoretical results.
