An effective algorithm for filtering and observer design

We propose a new numerical algorithm for calculating the continuous filtered signal and its reliable higher derivatives from noisy discrete data. This algorithm is conceived to be implemented as a filter or as an observer for dynamical systems where the measurements are supposed to be available at equally-spaced instants and whose states could be expressed in terms of finite numbers of their higher derivatives. The presented algorithm uses the Generalized cross-validation criterion to compute the optimal value of the smoothing parameter which translates the compromize between the smoothing and the closeness to measurable data. The algorithm has the advantage of being used in a recursive manner, like an adaptive filter, so as to improve the quality of filtering. Numerical simulations are presented to show its efficiency.