Full nonparametric identification of the instantaneous dynamics of linear time-periodic systems

As (quasi) time-periodic (TP) systems are encountered in many engineering applications, ranging from reciprocating devices in the field of mechanics, through harmonic distortions in power distribution networks, to cardio-vascular monitoring in the bio-medical science, the extraction of experimental linear time-periodic (LTP) models meant for physical interpretation, analysis, prediction or control can be a useful step for the practicing engineer. Most of the identification methods available in the LTP literature are (non-)parametric-in-the-dynamics and parametric-in-the-time-variations. Because a full nonparametric model avoids a model order selection for the dynamics as well as for the time-variation part, it is more than welcome to have full nonparametric identification tools at hand. Estimation schemes, which are both nonparametric-in-the-dynamics as well as in-the-time-variations, for slowly varying dynamics are based on the short-time Fourier transform (STFT) principle. However, this can be a very restrictive assumption for applications with fast time-variations. To circumvent this problem, we show that when the excitation is a stationary random process the identification problem boils down to the estimation of the time-periodic cross-power spectral density (PSD) in the extended Wiener-Hopf relation: time-periodic cross-PSD = instantaneous dynamics × input auto-PSD. This input-output relationship is always fulfilled irrespective of the speed and strength of the cyclic variations. As there is quite a lot of research done on how to estimate cyclo-/non-stationary auto-PSDs from noisy data, ideas can be used to estimate the time-periodic cross-PSD. For instance, by measuring a sufficient amount of system cycles an unbiased nonparametric estimate can be constructed for the time-periodic cross-PSD through synchronous averaging.