Robust identification of harmonic oscillator parameters using the adjoint Fokker-Planck equation

摘要

We present a model-based output-only method for identifying from time seriesthe parameters governing the dynamics of stochastically forced oscillators. Inthis context, suitable models of the oscillator's damping and stiffnessproperties are postulated, guided by physical understanding of the oscillatoryphenomena. The temporal dynamics and the probability density function of theoscillation amplitude are described by a Langevin equation and its associatedFokker-Planck equation, respectively. One method consists in fitting thepostulated analytical drift and diffusion coefficients with their estimatedvalues, obtained from data processing by taking the short-time limit of thefirst two transition moments. However, this limit estimation loses robustnessin some situations - for instance when the data is band-pass filtered toisolate the spectral contents of the oscillatory phenomena of interest. In thispaper, we use a robust alternative where the adjoint Fokker-Planck equation issolved to compute Kramers-Moyal coefficients exactly, and an iterativeoptimisation yields the parameters that best fit the observed statisticssimultaneously in a wide range of amplitudes and time scales. The method isillustrated with a stochastic Van der Pol oscillator serving as a prototypicalmodel of thermoacoustic instabilities in practical combustors, where systemidentification is highly relevant to control.