Implicit and explicit discrete‐time realizations of homogeneous differentiators

This article presents explicit and implicit discrete‐time realizations for a class of homogeneous sliding mode‐based differentiators. The proposed approach relies on the method of exact discretization for linear systems with a zero‐order holder. The two discrete‐time schemes present the homogeneity property and, after a finite time, the accuracy of its continuous‐time counterpart, even in the presence of bounded noise. In comparison to the explicit realization, which makes it possible to determine the state of the system later from the state at present, for this case, the implicit method requires finding a solution by solving generalized equations that involve the current state of the system and two support variables. Therefore, this document proves that the polynomial's unique positive root defines the required solution as part of the main results. Hence, it is possible to introduce a non‐anticipative method to implement the implicit discrete‐time realization, including an appropriate root‐finding method for the polynomial. Finally, the simulation results include comparisons between the proposed implicit and explicit discrete methods with other existing schemes. Numerical studies clearly show that the implicit method supersedes the explicit one, consistent with the implicit and explicit time discretization of other continuous‐time algorithms.