Adaptive backstepping control of some uncertain nonlinear oscillators

A backstepping-based adaptive controller is designed for a class of uncertain second order nonlinear systems under the strict-feedback form. It is shown that the closed loop is globally uniformly ultimately bounded and we give explicit bounds on both the asymptotic and transient performance. The control strategy is applied to a system typically found in base isolation schemes for seismic active protection of building structures. This system exhibits a hysteretic nonlinear behavior which is described analytically by the so-called Bouc-Wen model. Unlike other control schemes, the developed backstepping control does not require an exact knowledge of the model parameters. They are only defined within known intervals. The practical effectiveness of the controller is illustrated by numerical simulations.