Optimal Gain Selection for the Arbitrary‐Order Homogeneous Differentiator

Differentiation of noisy signals is a relevant and challenging task. Widespread approaches are the linear high‐gain observer acting as a differentiator and Levant's robust exact differentiator with a discontinuous right‐hand side. We consider the family of arbitrary‐order homogeneous differentiators, which includes these special cases. The effect of noise and disturbance on the differentiation estimation error of interest is estimated in terms of the homogeneous ‐gain. For this purpose, we construct a solution to the homogeneous differential dissipation inequality based on a Lyapunov function. Analogous to the classical ‐gain, the homogeneous ‐gain is not defined for the discontinuous case when a disturbance acts on the last channel. Hence, only continuous differentiators are considered. In a high‐gain setup, we show not only the existence of an optimal gain‐scaling that globally minimizes the homogeneous ‐gain estimate for any admissible , including the worst‐case considerations , but also how to obtain it. The theoretical results are underlined with extensive numerical evaluation and simulation examples.