On numerical differentiation algorithms for nonlinear estimation

Practical implementations of the operation which consists of differentiating a signal, a real valued function of the time, known only through its on line samples is much wanted, given the numerous areas in control theory where it is encountered. But this problem is known as ill-posed. In this communication we present further studies on its regularizations which are candidates for use in control practices. The so-called Savitzky-Golay differentiation scheme is closely re-examined. We have been able to relate its regularization to more practical issues such as the time lag/window size of the sampled data. In control literature some authors invoke continuous time linear filters such as s/(λs+ 1) as numerical differentiation methods. They usually suggest to choose λ as small as possible. Here we provide a thorough analysis of the regularization conditions of these filters, and show that making λ too small may degrade the estimation in some conditions where the signal being differentiated is noisy. The parameter λ needs to be assigned a specific value.