Identification of Volterra time-varying parameters in dynamical systems under random excitation

This paper deals with the estimation of the parameters of non-linear dynamical systems when they are time-varying and the excitation is not measured. This problem is relevant for the structural health monitoring (SHM) field, where input signals are unknown, and system properties can evolve over time due to damage or degradation. In this research, a new identification procedure is proposed, based on the parametrisation of the power spectral density (PSD) of the system response given the PSD of the input. The approach relies on the Volterra series formulation, here extended to time-varying systems, and employs time-frequency transforms that support a running time-domain window interpretation, producing a representation that is causal (i.e., the analysis at any point only uses past/present data, not future data), except for the window’s lag. When used for the estimation of time-varying parameters, such transforms lead to an estimation bias that can be related to the analysis window. A benchmark numerical study is conducted on classical Duffing oscillators. Results indicate that the proposed procedure is robust to measurement noise and parameter estimates show negligible bias provided that the Volterra series convergence is satisfied.