Distributed delays have proved beneficial in eliminating vibrations in higher frequency ranges compared to the modeled frequency, but not in improving the robustness against the modeled frequency to modeling errors. In this paper, the extra-insensitive shaper with distributed delays is proposed to improve robustness and efficiency for systems with a large variation of modal frequencies. The extra-insensitive shaper with distributed delays is derived by distributing two zeros in the radial direction of the designed zero of the distributed zero-vibration shaper. Two methods of distributing zeros are provided to parameterize the extra-insensitive shaper with distributed delays. Asymmetric extra-insensitive shaper with distributed delays and symmetric extra-insensitive shaper with distributed delays are two types of the extra-insensitive shaper with distributed delays parameterized by the asymmetric distribution method and the symmetric distribution method. Properties of the Asymmetric extra-insensitive shaper with distributed delays and the symmetric extra-insensitive shaper with distributed delays in spectral, sensitivity and step response are analyzed and compared with those of the classical extra-insensitive shaper and distributed zero-vibration-derivative shaper. The results present larger robustness in variations of modeled and higher frequencies, suppressing the vibration of the modeled mode and unmodeled higher multiple modes to a tolerable level. The implement in the time-varying double-pendulum crane verifies the effectivity for vibration suppress of multi-mode systems.
Estimates for solutions to one class of nonlinear nonautonomous systems with time-varying concentrated and distributed delays
