An exact robust hyperexponential differentiator

A simple differentiator is proposed, which is modeled by a second order time-varying linear differential equation. It is shown that for any signal of interest, whose second derivative is an essentially bounded function of time, the differentiation error converges to zero with a hyperexponential rate (faster than any exponential). An implicit discretization scheme of the differentiator is given, which preserves all main properties of the continuous-time counterpart. In addition, the differentiation error is robustly stable with respect to the measurement noise with a linear gain. The efficiency of the suggested differentiator is illustrated through comparison in numeric experiments with popular alternatives.