The variation of non-parametric estimates in closed-loop. (Q1421410)

The paper deals with non-parametric estimates of linear transfer functions in the frequency domain where the experiments are carried out in a closed loop. The probability density function of such estimates is characterized under the assumption that the corresponding closed-loop estimate has complex normal distribution in the frequency domain. It is shown that the probability density function may be described as a horseshoe encircling the inverse of the controller, with a global maximum on the line between the true value and the inverse of the controller. The expected value of the absolute value of such estimates is finite, and it is proposed as a measure of variation. A new expression for the variance, when an exclusion zone is introduced around the singularity, is also derived and discussed.