A predefined-time first-order exact differentiator based on time-varying gains

TL;DR

This paper introduces a predefined-time first-order exact differentiator using time-varying gains, ensuring convergence within a user-defined time, suitable for time-critical applications, and improves upon existing finite-time differentiators.

Contribution

The paper redesigns a known finite-time differentiator to achieve fixed-time convergence with a user-defined upper bound on settling time using time base generators.

Findings

Achieves guaranteed convergence before a desired time.

Demonstrates effectiveness through numerical examples.

Improves state-of-the-art differentiators with fixed-time convergence.

Abstract

Recently, a first-order differentiator based on time-varying gains was introduced in the literature, in its non recursive form, for a class of differentiable signals $y(t)$, satisfying $|\ddot{y}(t)|\leq L(t-t_0)$, for a known function $L(t-t_0)$, such that $\frac{1}{L(t-t_0)}\left|\frac{d {L}(t-t_0)}{dt}\right|\leq M$ with a known constant $M$. It has been shown that such differentiator is globally finite-time convergent. In this paper, we redesign such an algorithm, using time base generators (a class of time-varying gains), to obtain a differentiator algorithm for the same class of signals, with guaranteed convergence before a desired time, i.e., with fixed-time convergence with an a priori user-defined upper bound for the settling time. Thus, our approach can be applied for scenarios under time-constraints. We present numerical examples exposing the contribution with respect to state-of-the-art algorithms.