Time-varying high-gain observers for numerical differentiation

In this note, we propose high-gain numerical differentiators for estimating the higher derivatives of a given signal. We consider time-varying high-gain vectors converging exponentially to the high-gain vectors introduced by Esfandiari and Khalil in an earlier paper. The dynamics of these time-varying high-gain vectors can be chosen in order to achieve specific objectives, such as peaking attenuation and low sensitivity with respect to noise disturbance. In particular, we show that the numerical differentiator introduced in an earlier paper avoids the peaking phenomenon in the sense of Sussmann and Kokotovic, i.e., there is no unbounded overshoot of the error estimate during the initial times. We also propose another numerical differentiator which filters the reference signal with respect to a very simple quadratic cost.