Recovery of Differential Equations from Impulse Response Time Series Data for Model Identification and Feature Extraction

Abstract

Time recordings of impulse-type oscillation responses are short and highly transient. These characteristics may complicate the usage of classical spectral signal processing techniques for (a) describing the dynamics and (b) deriving discriminative features from the data. However, common model identification and validation techniques mostly rely on steady-state recordings, characteristic spectral properties and non-transient behavior. In this work, a recent method, which allows reconstructing differential equations from time series data, is extended for higher degrees of automation. With special focus on short and strongly damped oscillations, an optimization procedure is proposed that fine-tunes the reconstructed dynamical models with respect to model simplicity and error reduction. This framework is analyzed with particular focus on the amount of information available to the reconstruction, noise contamination and nonlinearities contained in the time series input. Using the example of a mechanical oscillator, we illustrate how the optimized reconstruction method can be used to identify a suitable model and how to extract features from uni-variate and multivariate time series recordings in an engineering-compliant environment. Moreover, the determined minimal models allow for identifying the qualitative nature of the underlying dynamical systems as well as testing for the degree and strength of nonlinearity. The reconstructed differential equations would then be potentially available for classical numerical studies, such as bifurcation analysis. These results represent a physically interpretable enhancement of data-driven modeling approaches in structural dynamics.

Reconstruction of Governing Equations from Vibration Measurements for Geometrically Nonlinear Systems

“…SINDy also requires state time derivatives that can be measured or generated numerically. As proposed in [24,25]. Total Variation Regularized Numerical Differentiation (TVRegDiff) [30] is used to compute derivatives numerically without noise amplification.…”

“…SINDy tries to reconstruct the time series accurately, sometimes generating models of high complexity that reproduce the given data perfectly but rely on a larger number of active functions than the one of the actual underlying governing equation. Tools proposed by Stender et al [25] are used to improve and automate the identification of a sparse system with SINDy. The first algorithm introduced finds a correct value of the sparsification parameter λ, on which the population of the coefficient matrix Ξ depends; λ is varied between the full range of non-zero entries (NZE) of Ξ from NZE = 100% to NZE = 0% and it selects the optimal value of the sparsification parameter λ that minimizes the error between the input signal and the one obtained through time integration of the identified set of ODEs.…”

“…The resulting over-determined system of equations is solved by sparsity-promoting regression techniques to yield a sparse reconstruction of the underlying dynamical system in terms of ordinary differential equations. From ( [25], Figure 1).…”

“…For the last test with time series computed numerically from the Duffing equation, only the time series of displacement x are assumed as known, such as in any real experiment where only a limited number of motions can be measured. Time-delay embedding [25,26] is used to enrich the univariate data such that the phase space is reconstructed using the embedding dimension m = 3, delay d = The test has been iterated 10 times for each value of η. MATLAB function 'randn' is used to compute the matrix Z used to add noise to data. The value of SNR plotted for each η is the logarithm of the mean value of SNR of the 10 repetitions.…”

“…Taking vibration recordings as an input, SINDy outputs the ODE reconstruction, i.e., a set of differential equations that describe the observed dynamics. Stender et al [25] proposed an optimization technique which aims at automating the system reconstruction, enforcing further sparsification of the reconstructed ODEs and at the same time reducing the error. They have tested this approach extensively for a linear oscillator and for the free vibration of a Duffing oscillator.…”