Characterizing approximated differentiators in digital control

Characterizations of the approximated differentiator used in digital control are derived. The gap between the pure differentiator and its approximation is modeled as an additive perturbation. Two types of quadratic characterizations are obtained. One is the L2 gain and the other is related to the passivity property. The results guarantee that the perturbation is small when the sampling period is small enough, and can be used for robust control design when the perturbation is not negligible.