The optimal form of the fractional-order difference feedbacks in enhancing the stability of a sdof vibration system

Many practical systems including the slightly damped mechanical systems, even they are already stable, are required to be controlled, in order to get better performance or better stability. In this paper, the concept of fractional-order difference feedback that generalizes the displacement difference feedback, velocity difference feedback and acceleration difference feedback, is proposed for improving the stability of a sdof vibration system. It is found that among the various state difference feedbacks, some fractional-order difference feedbacks including fractional-order integrators and fractional-order differentiators improve the stability of vibration systems best. Fractional-order integrator/differentiator is a controller with memory for the whole time history, its implementation is usually more complicated than the classical PID control and acceleration control. Thus, proper classical controller is suggested for improving the stability of the vibration system with small damping and small delay. If a displacement sensor is used, then the optimal form of state difference feedbacks for enhancing stability is the displacement difference feedback with k>0k>0. If an acceleration sensor is used, then the optimal form of state difference feedbacks for enhancing stability is the acceleration difference feedback with k<0k<0. Moreover, on the basis of the principal of stability switch, the admissible feedback gains and delay governing the asymptotical stability and γγ-stability are studied in detail, and illustrated with numerical experiments.