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.

翻译：本文研究了在采样测量条件下，Levant任意阶鲁棒精确微分器的隐式欧拉离散化问题。研究表明，现有该微分器的隐式离散化方法要么存在无界偏差误差，要么（令人意外地）尽管采用了隐式离散化，仍会出现离散化颤振现象。本文提出了一种新的、恰当的隐式离散化方法，该方法通过将微分器输出计算为其状态变量经适当设计的线性组合，从而避免了上述两种不利影响。文中讨论了数值微分器的实现方式，并给出了任意微分阶数的闭式稳定性条件。此外，还正式分析了有界测量噪声和数值近似误差的影响。数值仿真结果验证了所得结论。