Exact Robust Filtering and Differentiation Based on Sliding Modes and Homogeneity

2026-07-06

Jaime A. Moreno *

Arie Levant

DOI：10.1002/rnc.70635

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This article addresses the problem of online estimation of the derivatives of a signal corrupted by measurement noise. The measured signal is modeled as the sum of a smooth nominal component and a uniformly bounded noise term. A key objective is to attenuate the effect of high-frequency noise. A natural approach is to filter the measured signal before feeding it to the differentiator. However, this generally prevents exact estimation, even in the noise-free case. Recently, filtering differentiators have been proposed to attenuate measurement noise while preserving exact derivative estimation in the absence of noise. This article extends existing filtering differentiator designs in several directions. First, we propose and analyze filtering differentiators based on homogeneity in the bi-limit, which combine the robustness properties of bi-limit homogeneous estimation with the noise-attenuation capabilities provided by filtering. Second, we consider a broader class of filters, including general strongly observable LTI systems and nonlinear alternatives. We show that, in the absence of noise, the proposed differentiators recover the exact derivatives of the base signal. In addition, we provide a rigorous analysis of the effect of measurement noise and derive estimation-error bounds for the corresponding bi-homogeneous differentiators. The results are established through Lyapunov-based arguments and illustrated by numerical examples.

Keywords:

filtering differentiator

finite-time convergence

fixed-time convergence

high order sliding mode differentiation

homogeneous differentiator

Exact Robust Filtering and Differentiation Based on Sliding Modes and Homogeneity

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