Time-delay pre-filter design for vibration free tracking of periodic reference signals

The aim of this work is the development of time-delay filters that enable tracking of periodic signals with zero phase error. Time-Delay control or input-shaping is a proven technique to reduce motion induced vibrations for systems with lightly damped modes subject to point-to-point movements. An extension is proposed for systems subject to periodic reference signals, for which a closed-form solution is provided for second-order systems, followed by a generalization for higher order systems and multi-harmonic reference signals. It is shown that the theory enables a significant reduction of tracking errors for non-differentiable reference signals. The paper concludes with the presentation of an approach to desensitize the pre-filter to uncertainties in the forcing frequencies.