Arbitrary-Order Fixed-Time Differentiators

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 article, 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fixed time</i> . We propose also a family of continuous differentiators and provide a unified Lyapunov framework for analysis and design.