Covariance averaging in the analysis of uncertain systems

To analyze the effects of parametric uncertainty in state space systems, the state covariance is averaged over the statistics of the uncertain parameters. For natural frequency uncertainty, this computation is shown to be related to the Fourier transform of the probability density function of the uncertain parameter. In the case of a single mode oscillator, the average covariance exhibits both equipartition and incoherence phenomena. The analysis is carried out for several probability distributions. It is shown that 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 Hyland. For arbitrary rational distributions, the average covariance can be evaluated as the solution of a set of coupled Lyapunov equations. In addition, incoherence is demonstrated for the case of multiple modes, and the problem of robust controller synthesis with the maximum entropy design technique is considered