Fractional order adaptive active vibration damping designed on the basis of simple finematic considerations

In this paper a simple nonlinear, adaptive control using fractional order derivatives is applied for damping the vibration of a car forced during passing along a bumpy road. Its key idea is the partial replacement of the integer order derivatives in a traditional PD controller with a time-shift invariant, causal approximation of Caputo\´s fractional derivative that behaves like a Green function. Since its physical essence is rather frequency filtering than providing integer order derivatives in limit cases, the approximation applied numerically is quite convenient. In this way simple kinematic design of the desired damping becomes possible. The adaptive part of the controller guarantees the realization of this kinematic design without making it necessary for the designer to have accurate and complete dynamic model of the system to be controlled or to design a sophisticated linear "CRONE" controller that has to take the responsibility for the unknown dynamics of the system. The applicability of the approach is illustrated via simulations for a paradigm that is a rough model of a car. It was found that both adaptivity and the use of fractional order derivatives in the control are essential parts of the success of the method