Adaptive nonlinear vibration damping inspired by the concept of fractional derivatives

In this paper a simple nonlinear, adaptive approach inspired by the CRONE method is presented for vibration control. It replaces the fractional derivatives with time-invariant Green functions. Being completed by a nonlinear feedback term it makes the positive definite weighted moving average of the square of the error converge to zero in the kinematic design of the desired damping the realization of which is guaranteed by the controller's adaptive nature. The burden of designing a sophisticated linear controller is evaded. The applicability of the approach is illustrated via simulations for a damped linear oscillator under external excitation at its resonance frequency. The adaptive loop simply successively maps the observed system behavior to the desired one without exerting any effort to identify the reasons of the differences. It is expected to be useful for solving even more complicated vibration damping problems with unmodeled and uncontrolled internal degrees of freedom.N/