Two-step differentiator for delayed signal

This paper presents a high-order differentiator for delayed measurement signal. The proposed differentiator not only can correct the delay in signal, but aslo can estimate the undelayed derivatives. The differentiator consists of two-step algorithms with the delayed time instant. Conditions are given ensuring convergence of the estimation error for the given delay in the signals. The merits of method include its simple implementation and interesting application. Numerical simulations illustrate the effectiveness of the proposed differentiator.

💡 Research Summary

The paper addresses a common problem in real‑time control and signal‑processing applications: the measured signal is often delayed by a known constant amount τ, yet many algorithms require the undelayed signal and its higher‑order derivatives. Existing solutions either compensate the delay through prediction or use high‑order differentiators that assume an instantaneous measurement, but they do not combine both capabilities in a single, easily implementable structure.

To fill this gap, the authors propose a two‑step high‑order differentiator that (1) processes the delayed measurement x(t‑τ) with a conventional high‑order differentiator and (2) applies an analytical correction based on the known delay τ to recover the derivatives of the undelayed signal x(t). The first step uses a set of constant coefficients a₀,…,aₙ, identical to those of a standard robust differentiator, and can be implemented as a linear differential (or difference) equation of order n. The output of this stage, denoted (\hat{x}^{(i)}_1(t)), approximates the i‑th derivative of x(t) but contains a systematic phase error proportional to τ.

The second step eliminates this error by adding a correction term derived from a Taylor expansion of the delayed signal around the current time. For each derivative order i, the corrected estimate is

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