On distributional robustness of systems with complex uncertainty

Abstract

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 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 shown 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. This result is extended to show that an H 2 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.