The problem of differentiating a function with bounded second derivative in the presence of bounded measurement noise is considered in both continuous-time and sampled-data settings. Fundamental performance limitations of causal differentiators, in terms of the smallest achievable worst-case differentiation error, are shown. A robust exact differentiator is then constructed via the adaptation of a single parameter of a linear differentiator. It is demonstrated that the resulting differentiator exhibits a combination of properties that outperforms existing continuous-time differentiators: it is robust with respect to noise, it instantaneously converges to the exact derivative in the absence of noise, and it attains the smallest possible—hence optimal—upper bound on its differentiation error under noisy measurements. For sample-based differentiators, the concept of quasi-exactness is introduced to classify differentiators that achieve the lowest possible worst-case error based on sampled measurements in the absence of noise. A straightforward sample-based implementation of the proposed linear adaptive continuous-time differentiator is shown to achieve quasi-exactness after a single sampling step as well as a theoretically optimal differentiation error bound that, in addition, converges to the continuous-time optimal one as the sampling period becomes arbitrarily small. A numerical simulation illustrates the presented formal results.

在连续时间和采样数据设置中都考虑了在存在有界测量噪声的情况下对具有有界二阶导数的函数进行微分的问题。显示了因果微分器的基本性能限制，根据可实现的最小最坏情况微分误差。然后通过调整线性微分器的单个参数来构建稳健的精确微分器。结果证明，由此产生的微分器表现出优于现有连续时间微分器的特性组合：它对噪声具有鲁棒性，它瞬时在没有噪声的情况下收敛到精确导数，并且它在噪声测量下获得微分误差的最小可能（因此是最优）上限。对于基于样本的微分器，引入了准精确性的概念来对微分器进行分类，这些微分器在没有噪声的情况下基于采样测量值实现尽可能低的最坏情况误差。所提出的线性自适应连续时间微分器的直接基于样本的实现被证明可以在单个采样步骤之后实现准精确性以及理论上最优的微分误差界限，此外，收敛到连续时间最优的采样周期变得任意小。数值模拟说明了所呈现的正式结果。