Proper discretization of homogeneous differentiators

Homogeneous sliding-mode-based differentiators (HD) are known to provide for the high-accuracy robust estimation of derivatives in the presence of sampling noises and discrete measurements, provided that the differentiator dynamics evolve in continuous time. The popular one-step Euler discrete-time implementation is proved to cause differentiation accuracy deterioration, if the differentiation order exceeds 1. A novel discrete-time realization of the HD is proposed, which preserves the ultimate accuracy of the continuous-time HD also with discrete measurements.