Proper Implicit Discretization of Arbitrary-Order Robust Exact Differentiators

TL;DR

This paper introduces a new implicit discretization method for arbitrary-order robust exact differentiators that avoids bias errors and chattering, ensuring stable and accurate numerical differentiation in noisy measurement scenarios.

Contribution

A novel proper implicit discretization approach for Levant's differentiator that eliminates bias and chattering, with stability analysis and noise robustness considerations.

Findings

The new discretization avoids bias errors.

It prevents discretization chattering.

Numerical simulations confirm stability and robustness.

Abstract

This paper considers the implicit Euler discretization of Levant's arbitrary order robust exact differentiator in presence of sampled measurements. Existing implicit discretizations of that differentiator are shown to exhibit either unbounded bias errors or, surprisingly, discretization chattering despite the use of the implicit discretization. A new, proper implicit discretization that exhibits neither of these two detrimental effects is proposed by computing the differentiator's outputs as appropriately designed linear combinations of its state variables. A numerical differentiator implementation is discussed and closed-form stability conditions for arbitrary differentiation orders are given. The influence of bounded measurement noise and numerical approximation errors is formally analyzed. Numerical simulations confirm the obtained results.