Arbitrary Order Fixed-Time Differentiators

TL;DR

This paper introduces a novel class of fixed-time differentiators that guarantee convergence within a user-defined time, regardless of initial errors, enhancing the robustness and predictability of derivative estimation.

Contribution

It extends Levant's differentiator to achieve fixed-time convergence and develops a unified Lyapunov framework for analysis and design.

Findings

Achieves arbitrary fixed-time convergence independent of initial conditions

Provides a family of continuous differentiators with guaranteed convergence time

Develops a unified Lyapunov analysis framework for differentiator design

Abstract

Differentiation is an important task in control, observation and fault detection. Levant's differentiator is unique, since it is able to estimate exactly and robustly the derivatives of a signal with a bounded high-order derivative. However, the convergence time, although finite, grows unboundedly with the norm of the initial differentiation error, making it uncertain when the estimated derivative is exact. In this paper we propose an extension of Levant's differentiator so that the worst case convergence time can be arbitrarily assigned independently of the initial condition, i.e. the estimation converges in \emph{Fixed-Time}. We propose also a family of continuous differentiators and provide a unified Lyapunov framework for analysis and design.