On regularized numerical observers

Diop et al. previously (1994) proposed a general observer design methodology based upon numerical differentiation and the interpretation of observability of a system as the solvability of the system´s dynamical equations for the state vector in terms of a finite number of derivatives of the output and input. Numerical differentiation-base observers are alternatives to asymptotic observers for nonlinear systems. Various techniques are known to be efficient for the estimation of the few first derivatives from data with low frequency content, such as polynomial- and spline-based least squares, and averaged central differences. The main advantages of such observers are intuitiveness, flexibility and speed. However, as is the case of many inverse problems, differentiation is an ill-posed operator. This communication proposes the use of regularization to partially overcome the noise sensitivity that is inherent in the standard numerical differentiation. Regularization of numerical derivatives from experimental data consists of two operations: filtering and differentiation. Mollification is a method of filter design that is fairly well amenable to a mathematical analysis, including computation of estimation error bounds. In this method experimental data is projected onto Sobolev spaces of signals with less high frequency content, which may then be differentiated stably. The filters in question are infinite dimensional. They can be implemented approximately by means of digital Fourier transformation on finite moving windows of the data