A two-stage sparse algorithm for localization and characterization of local nonlinear structures

Abstract

This paper considers the localization and characterization of nonlinear components in local nonlinear structures with noisy measurements, and a novel two-stage sparse identification method is proposed. In this algorithm, one set of candidate nonlinear basis functions are leveraged to characterize system nonlinearities. To effectively localize and characterize nonlinear components, the algorithm introduces two sparse stages. At the first sparse stage, the adaptive group Lasso is adopted to localize nonlinear components in structures. At the second sparse stage, the adaptive Lasso is used to further characterize the contributing nonlinear basis functions in the identified nonlinear positions. Since the first sparse stage can greatly decrease the number of candidate nonlinear basis functions, the contributing functions can be effectively selected and characterized at the second sparse stage. Compared with the one-stage sparse algorithms, the proposed algorithm is more robust to noise, i.e., the new method can correctly identify the local nonlinear structures with a lower Signal to Noise Ratio (SNR). In addition, many existed nonlinear identification methods, simultaneously require the synchronous data of vibration displacements, velocities and accelerations. If only vibration displacements are measured, the first and second order derivatives of displacement are conducted to estimate velocities and accelerations, respectively. Nevertheless, the noise mixed into the displacement measurement will be severely amplified in the process of derivative calculation, which will make the estimation of velocities and accelerations deviating from the actual value and reduce the identification accuracy. To avoid performing the second order derivative, the Duhamel's integral is adopted to rewrite the dynamic relationship between the system input and output. In addition, a non-parametric de-noising method based on reproducing kernel Hilbert space (RKHS) is further developed to reduce the effect of noise in vibration displacements and obtain vibration velocities, which can effectively reduce the effect of noise and improve the identification accuracy. The numerical and experimental studies verify the potential and effectiveness of the proposed algorithm in localizing and characterizing nonlinear components in local nonlinear structures.