# Linear Time-Varying System Identification in the Presence of Nonlinear Distortions

In this research we consider time-varying systems that are nonlinear to some extent. The output spectrum of such a system, when excited by a multisine, consists of skirts around the excited frequencies, but also could have nonlinear contributions at multiples of the multisine’s fundamental frequency, as shown in the figure below.

The purpose of this research is to find a linear time-varying model, with uncertainty bounds due to noise and nonlinear distortions, for this nonlinear time-varying system.

The advantage of using a sparse multisine as an excitation signal, is that one can obtain the set of frequencies where potential nonlinear distortions could be present. For estimating the noise on the measurement we don’t consider the data at these frequencies and are able to estimate the variance of the noise with a local polynomial and hyperbola method. Then, we estimate a linear time-varying model for the system by not considering this set of frequencies where nonlinear distortions could be present. Then, from the noise and nonlinear distortion estimates on the output spectrum, we compute uncertainties on the time-varying transfer function due to on the one hand the noise and on the other hand nonlinear distortions.

*Reference:*

- N. Hallemans, R. Pintelon, X. Zhu, T. Collet, R. Claessens, B. Wouters, A. Hubin, J. Lataire. Detection, Classification, and Quantification of Nonlinear Distortions in Time-Varying Frequency Response Function Measurements. IEEE Transactions on Instrumentation and Measurement. Vol. 70pp 1-14. August 2020.