Frequency-Domain Identification of Linear Time-Periodic Systems Using LTI Techniques

A variety of systems can be faithfully modeled as linear with coefficients that vary periodically with time or linear time-periodic (LTP). Examples include anisotropic rotor-bearing systems, wind turbines, and nonlinear systems linearized about a periodic trajectory. Many of these have been treated analytically in the literature, yet few methods exist for experimentally characterizing LTP systems. This paper presents a set of tools that can be used to identify a parametric model of a LTP system, using a frequency-domain approach and employing existing algorithms to perform parameter identification. One of the approaches is based on lifting the response to obtain an equivalent linear time-invariant (LTI) form and the other based is on Fourier series expansion. The development focuses on the preprocessing steps needed to apply LTI identification to the measurements, the postprocessing needed to reconstruct the LTP model from the identification results, and the interpretation of the measurements. This elucidates the similarities between LTP and LTI identification, allowing the experimentalist to transfer insight between the two. The approach determines the model order of the system and the postprocessing reveals the shapes of the time-periodic functions comprising the LTP model. Further postprocessing is also presented, which allows one to generate the state transition and time-varying state matrices of the system from the output of the LTI identification routine, so long as the measurement set is adequate. The experimental techniques are demonstrated on simulated measurements from a Jeffcott rotor mounted on an anisotropic flexible shaft supported by anisotropic bearings.