Distributionally robust gain analysis for systems containing complex uncertainty

The main result of the paper addresses the minimum and maximum expected values of various gain measures for the transfer function of a system which depends on a vector Δ of independent complex random gains. In the distributional robustness framework of the paper, the probability density function for Δ is not completely specified. It is assumed only that the distribution of each component Δ_i is non-increasing with respect to |Δ_i|, radially symmetric and supported on the disc of radius r_i 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 seen to be maximized when each Δ_i is uniformly distributed over the disc of radius r_i and minimized when each Δ_i has the impulse distribution. The 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 formula