Prescribed-Time Robust Differentiator Design Using Finite Varying Gains

A novel hybrid differentiator is proposed for any time-varying signal, whose second derivative is uniformly bounded. The exact real-time differentiation is obtained in prescribed time, and it is based on the robust observer design for the perturbed double integrator. The proposed observer strategy is in successive applications of rescaled and standard supertwisting observers with finite (time-varying and respectively constant) gains. The former observer aims to nullify the observation error dynamics in prescribed time whereas the latter observer is to extend desired robustness features to the infinite horizon. The resulting real-time differentiator uses the current signal measurement only and inherits the observer features of robust convergence to the estimated signal derivative in prescribed time regardless of the initial differentiator state. Tuning conditions to achieve the exact signal differentiation in prescribed time are explicitly derived. Theoretical results are supported by an experimental study of the exact prescribed-time velocity estimation of an oscillating pendulum, operating under uniformly bounded disturbances. The developed approach is additionally discussed to admit an extension to the sequential arbitrary order differentiation.