Time–frequency analysis for parametric and non-parametric identification of nonlinear dynamical systems

Abstract

This paper points out the differences between linear and nonlinear system identification tasks, shows that time–frequency analysis is most appropriate for nonlinearity identification, and presents advanced signal processing techniques that combine time–frequency decomposition and perturbation methods for parametric and non-parametric identification of nonlinear dynamical systems. Hilbert–Huang transform (HHT) is a recent data-driven adaptive time–frequency analysis technique that combines the use of empirical mode decomposition (EMD) and Hilbert transform (HT). Because EMD does not use predetermined basis functions and function orthogonality for component extraction, HHT provides more concise component decomposition and more accurate time–frequency analysis than the short-time Fourier transform and wavelet transform for extraction of system characteristics and nonlinearities. However, HHT's accuracy seriously suffers from the end effect caused by the discontinuity-induced Gibbs' phenomenon. Moreover, because HHT requires a long set of data obtained by high-frequency sampling, it is not appropriate for online frequency tracking. This paper presents a conjugate-pair decomposition (CPD) method that requires only a few recent data points sampled at a low-frequency for sliding-window point-by-point adaptive time–frequency analysis and can be used for online frequency tracking. To improve adaptive time–frequency analysis, a methodology is developed by combining EMD and CPD for noise filtering in the time domain, reducing the end effect, and dissolving other mathematical and numerical problems in time–frequency analysis. For parametric identification of a nonlinear system, the methodology processes one steady-state response and/or one free damped transient response and uses amplitude-dependent dynamic characteristics derived from perturbation analysis to determine the type and order of nonlinearity and system parameters. For non-parametric identification, the methodology uses the maximum displacement states to determine the displacement–stiffness curve and the maximum velocity states to determine the velocity–damping curve. Numerical simulations and experimental verifications of several nonlinear discrete and continuous systems show that the proposed methodology can provide accurate parametric and non-parametric identifications of different nonlinear dynamical systems.