Residual vibration reduction of time-varying flexible structures by input shaping

Abstract

Control strategies to accomplish precise point-to-point motion for time-varying flexible systems are discussed. Defining the residual vibration vector due to the time-bounded control input, the necessary and sufficient conditions for zero residual vibration for linear time-varying systems are derived. It is said that the residual vibration vanishes if and only if the residual vibration vector due to the input is eliminated at all system poles. The response can be made robust to possible system error by using the residual vibration vector. The differences in input shaping for time-varying and time-invariant systems are discussed. The Fourier or Laplace transform of shaped input for the time-varying system does not indicate the condition of zero residual vibration. Instead the elimination of the residual vibration vector does guarantee zero residual vibration. An input to move a nonlinear system from one position to another without residual vibrations is found assuming that the motions consist of a dominant motion associated with the ideal rigid body modes and a relatively small motion associated with elastic modes. The input related to the former (rigid body dominant motion) is optimally designed, and a method to obtain the additional input related to the latter (small flexible motion) for suppressing the residual vibration is then suggested.