Likelihood updating of random process load and resistance parameters by monitoring

Spectral parameters for a stationary Gaussian process are most often estimated by Fourier transformation of a realization followed by some smoothing procedure. This smoothing is often a weighted least square fitting of some prespecified parametric form of the spectrum. In this paper it is shown that maximum likelihood estimation is a rational alternative to an arbitrary weighting for least square fitting. The derived likelihood function gets singularities if the spectrum is prescribed with zero values at some frequencies. This is often the case for models of technically relevant processes. The numerical problem caused by these singularities is easily overcome by adding simulated low intensity white noise to the realization. Without changing its parameters the spectrum is hereby lifted above zero by an amount equal to the white noise intensity. The knowledge of an explicit likelihood function, even though it is of complicated mathematical form, allows an approximate Bayesian updating and control of the time development of the parameters. Some of these parameters can be structural parameters that by too much change reveal progressing damage or other malfunctioning. Thus current process monitoring and updating, for example administered in a Bayesian network system, can be a useful aid for the operation of a complicated technical system (large important structure, ship, wind power engine, etc.). Keywords: Parametric spectral estimation, likelihood of spectral parameters, response monitoring