Proper Implicit Discretization of Arbitrary-Order Robust Exact Differentiators—Without and With Noise Filtering

Abstract

This paper considers the implicit Euler discretization of Levant's arbitrary order robust exact differentiator (RED) and Levant and Livne's arbitrary order robust exact filtering differentiator in the presence of sampled measurements. Existing implicit discretizations of the RED 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 for both the nonfiltering and the filtering differentiator by computing the differentiator's outputs as appropriately designed linear combinations of the state variables. Numerical differentiator implementations are discussed, and closed-form stability conditions for arbitrary differentiation orders are given. The influence of numerical approximation errors and bounded measurement noise in the nonfiltering case, or more general noise of appropriate filtering order in the filtering case, is formally analyzed. Numerical simulations confirm the obtained results.