The variation of non-parametric estimates in closed-loop

Authors:
W. P. Heath
Affiliations:
Centre for Integrated Dynamics and Control, School of Electrical Engineering and Computer Science, University of Newcastle, Callaghan, NSW 2308, Australia
Venue:
Automatica (Journal of IFAC)
Year:
2003

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Abstract

It is known that non-parametric transfer function estimates in closed-loop often have infinite variance. We characterise the probability density function of such estimates under the assumption that the corresponding closed-loop system estimate has complex normal distribution in the frequency domain. The probability density function can 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 we propose it as a measure of variation. We also derive and discuss new expressions for the variance when an exclusion zone is introduced around the singularity.