System Identification of Dynamic Systems With Cubic Nonlinearities Using Linear Time-Periodic Approximations

Abstract

This work develops methods to identify parametric models of nonlinear dynamic systems from response measurements using tools for Linear Time Periodic (LTP) systems. The basic approach is to drive the system periodically in a stable limit cycle and then measure deviations of the response from that limit cycle. Under certain conditions, the resulting response can be well approximated as that of a linear-time periodic system. In the analytical realm it is common to linearize a system about a periodic trajectory and then use Floquet analysis to assess the stability of the limit cycle. This work is concerned with the inverse problem, using a measured time-periodic response to derive a nonlinear dynamic model for the system. Recently, a few new methods were developed that facilitate the experimental identification of linear time periodic systems, and those methods are exploited in this work. The proposed system identification methodology is evaluated by applying it to a Duffing oscillator, demonstrating that the nonlinear force-displacement relationship can be identified without a priori knowledge of its functional form. The proposed methods are also applied to simulated measurements from a cantilever beam with a cubic nonlinear spring on its tip, revealing that the model order of the system and the displacement dependent stiffness can be readily identified.

Method for identifying models of nonlinear systems using linear time periodic approximations

“…To be exact, the Fourier expansion may require an infinite number of terms, but in practice a finite number of Fourier terms, m¼ ÀN B , y, N B , will provide sufficient accuracy [9,20]. The equation above reveals that the free response of the time periodic system consists of n(2N B þ1) complex exponential components l r þimo T .…”

Subspace Identification of Linear Time-Periodic Systems With Periodic Inputs

“…[1], [2], [3]. More importantly, LTP systems serve as an intermediate step to capture more general representations than linear time-invariant (LTI) systems, for example, linear parameter-varying (LPV) systems [4], [5], [6] and nonlinear systems along limit cycles [7].…”

“…This paper focuses on identifying state-space LTP models from input-output data. Based on early work from [8], methods were developed to estimate nonparametric models of the harmonic transfer function [7], [9]. This input-output model can then be realized as state-space form [5].…”

Output-only modal analysis of linear time-periodic systems with application to wind turbine simulation data

“…one can obtain the analytical STM by using initial conditions of (18) where ψ STM (t 0 ) is the eigenvector of the STM at t=T A . Then, because the Floquet mode vectors are periodic, they can be expanded with a Fourier Series as given in eq.…”

“…[14][15][16][17]). Allen & Sracic have even proposed that a general nonlinear system might be more easily identified if excited such that an LTP model is appropriate [18]. Other researchers have sought to exploit LTP effects to detect defects, such as a cracked shaft, in a rotating system [8][9][10].…”