Phase-plane based feedback scheme for negative amplitude shapers

In this note we extend the phase-plane based closed-loop scheme of implementing commands shaped with vibration-reduction filters. A generalized shaping filter is considered in this work which can have negative impulse intensities and different acceleration and deceleration limits. Switching conditions are derived in terms of the filter parameters for both convolution-based and closed-form based shaping techniques. Analytical expressions are provided for the switching curves and various schemes are discussed for selecting appropriate phase-planes and implementing shaped-commands on real-time servomechanisms.