Frequency weighted H2 optimization of multi-mode input shaper

An optimization-based design of a robust multi-mode input shaper is presented with an embedded weighting of the frequency response. By prefixing the delay length in the shaper, the optimization design task is formulated as a standard quadratic programming problem. The shaper robustness in terms of zeroed or upper-bounded residual vibration function for the modes to be precompensated is imposed via the equality and inequality constraints. As the main result, an additional degree of freedom in the shaper design is introduced via parametrization of the quadratic form cost matrix and utilized to modulate the amplitude frequency response over a given frequency range. In particular, this allows to embed the high-frequency roll-off to the shaper and improve its insensitivity to modelling errors by adjusting its stop-band response. The proposed shaper design is demonstrated and experimentally validated on a problem of positioning a flexible manipulator arm with multiple oscillatory modes