Robust estimate of excitations in mechanical systems using M-estimators—Theoretical background and numerical applications

Model-based methods are often used along with least squares to estimate (or to identify in equivalent but in more engineering terms) dynamic forces, parameters and malfunctions in mechanical systems, starting from experimental vibrations. The effectiveness of these methods, broadly proven and documented by several cases of study, can be reduced if the model of the system is not accurate or if the experimental data are corrupted by noise, especially if the mean value of the noise is not null or if biases are present.A possible solution is the use of robust estimation techniques instead of traditional least squares in the ambit of model-based identification. The author proposes the application of the M-estimators and discusses the problems related to their application to excitation identification in mechanical systems.In this paper, the necessary theory is presented in detail, introducing several concepts of statistics, in order to properly introduce the concept of robust estimation and the required algorithms (based on iterative re-weighted least squares) are described. Then the different types of M-estimators proposed in literature are introduced. Their performances with regard to mechanical applications are evaluated by means of a theoretical analysis and a couple of simple numerical examples: a single input–single output and a multiple inputs–multiple outputs systems. Moreover, the problem of the scale parameter, which is not discussed in literature for complex numbers, as the vibrations are conveniently represented, is analyzed and a solution is proposed using a concept related to the data depth.