Can a time invariant input shaping technique eliminate residual vibrations of LTV systems?

This paper proves that the conventional time invariant input shaping technique (IST) can eliminate the residual vibrations in linear time varying (LTV) systems completely when IST parameters are chosen properly. Through quasi-static analysis, the response of a LTV system to an arbitrary input is derived. Then it is shown that the residual vibration in the LTV system cancels the response to the reference input with time delay if the responses have opposite phase and same amplitude. This work also presents a method to obtain the proper IST parameters, with which an arbitrary reference trajectory can be shaped to produce vibration free motion. This proof is validated by simulations for a second order system.