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Optimal robust exact first-order differentiators with Lipschitz continuous output

Rodrigo Aldana-Lopez, Richard Seeber, Hernan Haimovich, David Gomez-Gutierrez

TL;DR

The paper tackles online differentiation of a signal from noisily observed data when the second-derivative bound $L$ is known but the noise bound $N$ is only roughly estimated. It integrates a regularized, optimal exact differentiator with a first-order sliding-mode filter to ensure a Lipschitz-continuous output while preserving the optimal worst-case accuracy $2\sqrt{2NL}$, robustness from the start, and fixed-time exactness. A continuous-time design is complemented by a sample-based implementation via implicit discretization, yielding a quasi-exact, discretely robust differentiator whose Lipschitz constant is tunable through $\gamma$. The approach is validated theoretically and via simulations, showing smooth, fast convergence with guarantees on worst-case error and convergence time, making it suitable for real-time control and fault-diagnosis tasks.

Abstract

The signal differentiation problem involves the development of algorithms that allow to recover a signal's derivatives from noisy measurements. This paper develops a first-order differentiator with the following combination of properties: robustness to measurement noise, exactness in the absence of noise, optimal worst-case differentiation error, and Lipschitz continuous output where the output's Lipschitz constant is a tunable parameter. This combination of advantageous properties is not shared by any existing differentiator. Both continuous-time and sample-based versions of the differentiator are developed and theoretical guarantees are established for both. The continuous-time version of the differentiator consists in a regularized and sliding-mode-filtered linear adaptive differentiator. The sample-based, implementable version is then obtained through appropriate discretization. An illustrative example is provided to highlight the features of the developed differentiator.

Optimal robust exact first-order differentiators with Lipschitz continuous output

TL;DR

The paper tackles online differentiation of a signal from noisily observed data when the second-derivative bound is known but the noise bound is only roughly estimated. It integrates a regularized, optimal exact differentiator with a first-order sliding-mode filter to ensure a Lipschitz-continuous output while preserving the optimal worst-case accuracy , robustness from the start, and fixed-time exactness. A continuous-time design is complemented by a sample-based implementation via implicit discretization, yielding a quasi-exact, discretely robust differentiator whose Lipschitz constant is tunable through . The approach is validated theoretically and via simulations, showing smooth, fast convergence with guarantees on worst-case error and convergence time, making it suitable for real-time control and fault-diagnosis tasks.

Abstract

The signal differentiation problem involves the development of algorithms that allow to recover a signal's derivatives from noisy measurements. This paper develops a first-order differentiator with the following combination of properties: robustness to measurement noise, exactness in the absence of noise, optimal worst-case differentiation error, and Lipschitz continuous output where the output's Lipschitz constant is a tunable parameter. This combination of advantageous properties is not shared by any existing differentiator. Both continuous-time and sample-based versions of the differentiator are developed and theoretical guarantees are established for both. The continuous-time version of the differentiator consists in a regularized and sliding-mode-filtered linear adaptive differentiator. The sample-based, implementable version is then obtained through appropriate discretization. An illustrative example is provided to highlight the features of the developed differentiator.
Paper Structure (13 sections, 8 theorems, 49 equations, 1 figure)

This paper contains 13 sections, 8 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Performance Measures for Differentiators
  4. Optimal exact differentiation
  5. Problem statement
  6. Continuous-time design
  7. Differentiator output regularization
  8. Exact differentiator with Lipschitz-continuous output
  9. Sample-based implementation
  10. Simulation results
  11. Conclusions
  12. Auxiliary lemmata
  13. Proofs of all results

Key Result

Lemma 1

Let $L \in {\mathbb R}_{> 0}$ and consider the differentiator $\mathcal{D}\sb{\mathrm{w}}$ defined in eq:proposed:diff with parameter $\bar{T} \in {\mathbb R}_{>0} \cup \{ \infty \}$. Then, for all $u \in \mathcal{U}$ with noise bounds less than $\overline{N}=\frac{L\overline{T}^2}{2}$ and all $t >

Figures (1)

  • Figure 1: Simulation results with sampling time $\Delta = 0.01$, comparing the RED, the non-filtered differentiator \ref{['eq:meas:sampled']} proposed in seeber2023, and the proposed differentiator \ref{['eq:explicit:sol']} with $L = 1, N = 0.08$, using $k_0=0$ and $k_0=25$for two different scenarios. The plots show the differentiation error $|\dot{f}-y|$ in each case. Left: $f(t)=t^2/2+t + 0.5(t-10)\mathds{1}_{t\geq 10}$ and uniformly distributed random noise $\eta(t)$ over $[-N, N]$. Right: $f(t)=(t^2/2+t+\cos(t))/2$ and piece-wise constant noise $\eta(t)$ with different levels of $N$.

Theorems & Definitions (21)

  • Definition 1: Worst-case error seeber2023
  • Definition 2: Exactness seeber2023
  • Definition 3: Robustness seeber2023
  • Definition 4: Accuracy
  • Remark 1
  • Lemma 1
  • Proposition 1
  • Theorem 1
  • Remark 2
  • Definition 5
  • ...and 11 more