Increasing the damping of oscillatory systems with an arbitrary number of time varying frequencies using fractional-order collocated feedback

This paper studies the active damping of the oscillations of lightly damped linear systems whose parameters are indeterminate or may change through time. Systems with an arbitrary number of vibration modes are considered. Systems described by partial differential equations, that yield an infinite number of vibration modes, can also be included. In the case of collocated feedback, i.e. the sensor is placed at the same location of the actuator, a simple fractional order differentiation or integration of the measured signal is proposed that provides an effective control: (1) it guarantees a minimum phase margin or damping of the closed-loop system at all vibration modes, (2) this feature is robustly achieved, i.e., it is attained for very large variations or uncertainties of the oscillation frequencies of the system and (3) it is robust to spillover effects, i.e., to the unstabilizing effects of the vibration modes neglected in the controller design (especially important in infinite dimensional systems). Moreover, the sensitivity of the gain crossover frequency to such variations is assessed. Finally, these results are applied to the position control of a single link flexible robot. Simulated results are provided.