Exact differentiator with Lipschitz continuous output and optimal worst-case accuracy under bounded noise

The online differentiation of a signal contaminated with bounded noise is addressed. A differentiator is developed that generates a Lipschitz continuous output, is exact in the absence of noise, and provides the optimal worst-case accuracy among all possible exact differentiators when noise is present. This combination of features is not shared by any previously existing differentiator. Tuning of the developed differentiator is very simple, requiring only the knowledge of a bound for the second-order derivative of the signal. The approach consists in regularizing the possibly highly noisy output of a
recently introduced linear adaptive robust exact differentiator and feeding it to a first-order sliding-mode filter designed to maintain optimal accuracy. The proposed regularization and filtering of this output allows trading the speed with which exactness is obtained for the feature of a Lipschitz continuous, hence less noisy, output. An illustrative example is provided to highlight the features of the developed differentiator.