发表机构
Graz University of Technology; Centro Internacional Franco-Argentino de Ciencias de la Información y Sistemas (CIFASIS), UNR-CONICET（格拉茨工业大学; 法阿信息与系统科学国际中心 (CIFASIS)，罗萨里奥国立大学-国家科学研究委员会）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对任意阶导数估计，首次确立因果微分器的理论误差下限，并构造出兼具精确性与强鲁棒性的确定性最优精确微分器。
AI 中文摘要
本文在确定性框架下，考虑了当函数的高阶导数有界且测量噪声有界时，对该函数导数的估计问题。首次针对任意微分阶数，建立了因果微分器在可实现的最差情况微分误差下限方面的理论基本限制，以及期望性质，如精确性（无噪声时导数的零误差估计）和鲁棒性（小扰动下估计的小敏感性）。正式定义了达到这些理论上最低微分误差界的微分器，全面刻画了其特性，并研究了它们的性质。特别地，通过一种新颖的构造，证明了确定性最优微分器（即在精确微分器类别中具有最优微分误差界的微分器）的存在，该构造除了从一开始就具有精确性外，还展现出非常强的鲁棒性形式。
英文摘要
Estimation of the derivatives of a function with bounded high-order derivative in the presence of bounded measurement noise is considered, in a deterministic setting. Theoretical fundamental limitations of causal differentiators in terms of the lowest achievable worst-case differentiation error and desired properties such as exactness---the zero-error estimation of derivatives in the absence of noise---and robustness---the small sensitivity of the estimate under small perturbations---are established for the first time for arbitrary differentiation order. Differentiators that achieve these theoretically lowest bounds on their differentiation error are formally defined, fully characterized, and their properties are studied. In particular, deterministically optimal differentiators---those featuring optimal differentiation error bounds among the class of exact differentiators---are shown to exist by means of a novel construction exhibiting, in addition to exactness from the beginning, a very strong form of robustness.
CommentsSubmitted to SIAM
