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Proper Implicit Discretization of Arbitrary-Order Robust Exact Differentiators

Richard Seeber

TL;DR

This work tackles the problem of differentiating signals from sampled, noisy measurements by refining Levant's robust exact differentiator to a proper implicit discretization. It introduces the implicit robust exact differentiator (IRED), where outputs are carefully designed linear combinations of state variables, avoiding prior discretization chattering and bias. The authors derive finite-time stability conditions, rigorous error bounds under measurement noise, and provide a numerical implementation with approximate schemes and root-finding guidance. Simulations demonstrate superior accuracy and robustness relative to existing implicit methods, confirming the approach's practical impact for high-order, noise-robust differentiation in digital control systems.

Abstract

This paper considers the implicit Euler discretization of Levant's arbitrary order robust exact differentiator in presence of sampled measurements. Existing implicit discretizations of that differentiator are shown to exhibit either unbounded bias errors or, surprisingly, discretization chattering despite the use of the implicit discretization. A new, proper implicit discretization that exhibits neither of these two detrimental effects is proposed by computing the differentiator's outputs as appropriately designed linear combinations of its state variables. A numerical differentiator implementation is discussed and closed-form stability conditions for arbitrary differentiation orders are given. The influence of bounded measurement noise and numerical approximation errors is formally analyzed. Numerical simulations confirm the obtained results.

Proper Implicit Discretization of Arbitrary-Order Robust Exact Differentiators

TL;DR

This work tackles the problem of differentiating signals from sampled, noisy measurements by refining Levant's robust exact differentiator to a proper implicit discretization. It introduces the implicit robust exact differentiator (IRED), where outputs are carefully designed linear combinations of state variables, avoiding prior discretization chattering and bias. The authors derive finite-time stability conditions, rigorous error bounds under measurement noise, and provide a numerical implementation with approximate schemes and root-finding guidance. Simulations demonstrate superior accuracy and robustness relative to existing implicit methods, confirming the approach's practical impact for high-order, noise-robust differentiation in digital control systems.

Abstract

This paper considers the implicit Euler discretization of Levant's arbitrary order robust exact differentiator in presence of sampled measurements. Existing implicit discretizations of that differentiator are shown to exhibit either unbounded bias errors or, surprisingly, discretization chattering despite the use of the implicit discretization. A new, proper implicit discretization that exhibits neither of these two detrimental effects is proposed by computing the differentiator's outputs as appropriately designed linear combinations of its state variables. A numerical differentiator implementation is discussed and closed-form stability conditions for arbitrary differentiation orders are given. The influence of bounded measurement noise and numerical approximation errors is formally analyzed. Numerical simulations confirm the obtained results.
Paper Structure (22 sections, 16 theorems, 97 equations, 2 figures, 1 table)

This paper contains 22 sections, 16 theorems, 97 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Continuous-Time Robust Exact Differentiation
  4. Signal Differentiation from Sampled and Noisy Measurements
  5. Motivating Examples -- Chattering and Bias of Existing Implicit Differentiators
  6. Proper Implicit Discretization of the Robust Exact Differentiator
  7. Proposed Differentiator and Main Results
  8. Implicit Robust Exact Differentiator (IRED)
  9. Numerical Implementation
  10. Exactness Properties and Absence of Discretization Chattering
  11. Stability Conditions and Robustness to Noise
  12. Stability Analysis
  13. Differentiation Error System
  14. Transformed Error System
  15. Lyapunov Function
  16. ...and 7 more sections

Key Result

Proposition 3.3

Let $m \in \mathbb{N}$ and $L, R, T, \lambda_1, \ldots, \lambda_{m+1} \in \mathbb{R}_{> 0}$. Consider the sample-based sliding-mode differentiator $\mathcal{D}^{(m)}_T$ defined in eq:diff and its numerical implementation $\hat{\mathcal{D}}^{(m)}_T$ with identical initial condition and output equatio

Figures (2)

  • Figure 1: Comparison of I-HDD (green, dotted), HIDD (blue, dashed), and proposed ISHD (red) without measurement noise along with the bound obtained from Theorem \ref{['th:exactness']} for $M = \frac{17}{16}$; for comparison with further approaches, see also the simulation comparison by Carvajal-Rubio et al.carvajal2021implicit with the same parameter setting
  • Figure 2: Comparison of I-HDD (green, dotted), HIDD (blue, dashed), and proposed ISHD (red) with simulation setup as in Figure \ref{['fig:noise-free']} but with additional, independently uniformly distributed measurement noise $\eta_k \in [-N, N]$ with bound $N = 0.1$

Theorems & Definitions (44)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Example 3.4
  • Theorem 3.5
  • ...and 34 more