Observer-based Differentiators for Noisy Signals

TL;DR

This paper addresses the challenge of robust real-time differentiation of noisy signals for unknown-input observers. It analyzes two observer-based differentiators—sliding mode differentiators with a super-twisting dynamic and high-gain observer differentiators—detailing their design, stability, and performance under bounded measurement noise. The sliding mode approach offers finite-time convergence with error bounds that scale with the Lipschitz constant of the base signal and the noise bound, while the high-gain method provides a tunable speed-versus-noise trade-off and explicit epsilon-dependent error bounds. Through simulations, the work provides practical tuning rules, demonstrates performance against noise, and discusses phenomena such as peaking in high-gain schemes, underscoring the methods' applicability to robust derivative estimation in noisy environments.

Abstract

We present a collection of different types of observation systems that work as differentiators. These observer-based differentiators can produce estimates for derivatives of a given signal, even though the given signal is prone to noise.

Figures (6)

• Figure 1: Steps of executing the sliding mode differentiators.
• Figure 2: Asymptotic convergence of the estimate by the sliding mode differentiator when there is no noise.
• Figure 3: The estimate by the sliding mode differentiator when there is noise.
• Figure 4: Steps of executing the high-gain differentiators influenced by noise.
• Figure 5: Convergence of the estimate by the high gain differentiator when there is no noise. The convergence is not asymptotic.