Probabilistic System Identification for Nonlinear Systems with Uncertain Input

The problem of system identification of structural or mechanical systems using dynamic data has received much attention over the years because of its importance in response prediction, control and health monitoring. However, the results of system identification studies are usually restricted to the "optimal" estimates of the modal or model parameters, whereas there is additional information related to the uncertainty associated with these estimates which is very important. For example, how precisely are the values of the individual parameters pinned down by the measurements made on the system? Probability distributions may be used to describe this uncertainty quantitatively and so avoid misleading results. Also, if the identification results are used for damage detection, this probability distribution for the identified model parameters may be used to compute the probability of damage. An important special case of system identification is where the input is unknown so only response measurements are available. In particular, this is the case in ambient vibrations surveys (AVS) where the naturally occurring vibrations of a structure (due to traffic, micro-tremors and structural operations) are measured. AVS have attracted much interest because they offer a means of obtaining dynamic data in an efficient manner, without requiring the set-up of special dynamic experiments which are usually costly and time consuming. The uncertain input excitation is usually modeled as a broad-band stationary stochastic process such as white noise. Usually a linear structural model is then employed to estimate the parameters of the contributing modes of vibration. System identification using linear models is appropriate for the small-amplitude ambient vibrations of a structure that are continuously occurring. There is, however, a number of cases in recent years where the strong-motion response of a structure has been recorded but not the corresponding seismic excitation. In some cases this is because of inadequate instrumentation of the structure and in other cases it is because the free-field or base sensors malfunctioned during the earthquake. A literature search reveals relatively few papers that deal with system identification using nonlinear models and measurements of only the system response and these few papers tend to favor a stochastic spectral approach. In this paper, this subject is tackled using a stochastic model for the uncertain input and a Bayesian probabilistic approach to quantify the uncertainties in the model parameters. This Bayesian probabilistic system identification framework was first presented for the case of measured input and it has been recently extended to the case of unknown input and linear structural models. The proposed spectral-based approach utilizes important statistical properties of the Fast Fourier Transform (FFf) and the robustness of the FFf in terms of probability distribution of the response signal, i.e., regardless of the stochastic model for this signal, its FFf is approximately Gaussian distributed. It allows for the direct calculation of the probability density function for the parameters of a nonlinear model conditional on the measured response. It turns out that this probabilistic approach is well suited for the identification of nonlinear systems and does not require huge amounts of dynamic data, which is in contrast to most other methods for nonlinear models and unknown input. For simplicity, the formulation is presented for single-degree-of-freedom systems. Examples using simulated data are presented to illustrate the proposed approach