An arbitrary-order predefined-time exact differentiator for signals with exponential growth bound

TL;DR

This paper introduces a novel arbitrary-order differentiator with predefined convergence time for signals with exponential growth bounds, overcoming limitations of existing methods by using bounded time-varying gains.

Contribution

A new methodology using bounded time-varying gains for arbitrary-order differentiators with predefined upper bound on settling time, applicable to a broader class of signals.

Findings

Achieves exact differentiation with bounded gains at convergence.

Handles signals with exponential growth bounds.

Improves robustness to measurement noise.

Abstract

There is a growing interest in differentiation algorithms that converge in fixed time with a predefined Upper Bound on the Settling Time (UBST). However, existing differentiation algorithms are limited to signals having an $n$-th order Lipschitz derivative. Here, we introduce a general methodology based on time-varying gains to circumvent this limitation, allowing us to design $n$-th order differentiators with a predefined UBST for the broader class of signals whose $(n+1)$-th derivative is bounded by a function with bounded logarithmic derivative. Unlike existing methods whose time-varying gain tends to infinity, our approach yields a time-varying gain that remains bounded at convergence time. We show how this last property maintains exact convergence using bounded gains when considering a compact set of initial conditions and improves the algorithm's performance to measurement noise.