On distributional robustness of systems with complex uncertainty

The main result of this paper addresses the minimum and maximum expected values of a gain measure for the transfer function of a system which depends on a vector Δ of independent random complex gains. In the distributional robustness framework of this paper, the probability density function for Δ is not completely specified. It is assumed only that the distribution of each component Δi is non-increasing with respect to |Δi|, radially symmetric and supported on the disc of radius ri centered at zero in the complex plane. Under these conditions, the expected value of the magnitude-squared of the gain function at a fixed frequency ω⩾0 is shown to be maximized when each Δi is uniformly distributed over the disc of radius ri and minimized when each Δi has the impulse distribution. This result is extended to show that an H2 measure of the gain is also maximized and minimized in the same way. These results apply to quotients of multilinear functions of Δ, which includes system transfer functions obtained using Mason's gain formula.