Adaptive input shaping for nonlinear systems: A case study

Input shaping is a technique that seeks to reduce residual vibrations of lightly damped systems through modification of the command input to the system. Although several input shaping techniques have been derived primarily from linear system theory, theoretical results are hard to be traced for their application to nonlinear systems. In most of the reported cases, a fixed shaper is designed based on the linearized version around an operating point of the nonlinear system. In this paper, an adaptive form of the input shaper is proposed for a class of nonlinear lightly damped systems. The adaptive shaper adjusts the magnitude and relative time difference between its impulses according to the instant frequency and damping of the linearized systems. The efficacy of the proposed scheme and its comparison to a fixed shaper is investigated through its application to a pendulum system. The adaptive shaper's parameters vary according to the pendulum's angle. The illustrative examples indicate the deficiencies of the fixed case and demonstrate the efficacy of the designed controller.