Covariance averaging in the analysis of uncertain systems

The effects of parametric uncertainty in stable state-space systems are analyzed by averaging the state covariance over the statistics of the uncertain parameters. For natural frequency uncertainty, this computation is related to the Fourier transform of the probability density function of the uncertain parameter. Equipartition and incoherence are illustrated for a single-mode oscillator. Averaging over a discrete uncertainty model yields the Bourret design equations while averaging over a Cauchy uncertainty distribution yields the maximum-entropy covariance equation of D.C. Hyland (1982)