Główne pojęcia
The paper develops a first-order differentiator that combines the following advantageous properties: robustness to measurement noise, exactness in the absence of noise, optimal worst-case differentiation error, and Lipschitz continuous output with a tunable Lipschitz constant.
Streszczenie
The paper presents a novel differentiator that addresses the limitations of existing approaches. The key highlights are:
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The differentiator is robust and exact, meaning it can recover the signal's derivative from noisy measurements, and its output converges to the true derivative in the absence of noise.
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It achieves the optimal worst-case differentiation error, which is not shared by other existing differentiators.
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The differentiator's output is Lipschitz continuous, allowing for a smooth derivative estimate. The Lipschitz constant can be tuned as a trade-off between convergence speed and output smoothness.
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Both continuous-time and sample-based (discrete-time) versions of the differentiator are developed, with theoretical guarantees established for both.
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The continuous-time version consists of a regularized and sliding-mode-filtered linear adaptive differentiator, while the sample-based version is obtained through appropriate discretization.
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An illustrative example is provided to highlight the features of the developed differentiator, including its superior performance compared to existing approaches.
Statystyki
The paper does not provide explicit numerical data, but rather focuses on theoretical analysis and properties of the proposed differentiator. The key figures used to support the analysis are:
The worst-case differentiation error bound:
|yw(t) - ̇f(t)| ≤ (Lt + 2N)/t if t ∈ (0, √(2N/L)), and 2√(2NL) if t ≥ √(2N/L)
The convergence time function in the presence of noise:
T̂(R, N) = 2√(2N/L) + R/(γ - L) for the case of t0 = 0, and
T̂(R, N) = t0/Δ if N ≤ L(Δt0)^2/2, otherwise
T̂(R, N) = 2√(2N/L) + 2N/(Δt0 - γΔt0)/(γ - L) + 3Δ(γ - L)/(2(γ - L))
for the case of t0 > 0
Cytaty
"The only exact differentiator known to achieve the optimal worst-case accuracy 2√(2NL) was proposed recently in [8]; this differentiator is exact from the beginning and robust almost from the beginning."
"The proposed differentiator is obtained by combining the differentiator in [8] with a first-order sliding-mode filter to obtain robustness and a Lipschitz continuous output while retaining the optimal accuracy of [8]."