Optimal Robust Exact First-Order Differentiators With Lipschitz-Continuous Output

The signal differentiation problem involves the development of algorithms that allow to recover a signal's derivatives from noisy measurements. This article develops a first-order differentiator with robustness to measurement noise, exactness in the absence of noise, optimal worst-case differentiation error, and Lipschitz-continuous output, where the output's Lipschitz constant is a tunable parameter. This combination of advantageous properties is not shared by any existing differentiator. A sample-based implementation by implicit discretization is obtained, which is quasi-exact, inherits the optimal worst-case error bound, and the Lipschitz constant translates to a discrete-time increment bound. Illustrative examples are provided to highlight the features of the developed differentiator.