A worst-case estimate of stability probability for polynomials with multilinear uncertainty structure

Considers the probability of stability for an uncertain polynomial which has as coefficients multilinear functions of real, random, independent parameters q_i. The result requires little a priori information about the probability distributions of these uncertain parameters. We only require that the distributions are symmetric about zero, non-increasing as |q_i| increases, and supported on a given interval [-r_i,r_i]. The probability estimate is sharp in the sense that the estimated probability of stability is pˆ*=1 when the uncertainty bounds r_i are below the deterministic robustness radius r_map obtained with the Mapping Theorem. To obtain the probabilistic estimate, we recast the problem so that the following characterization of stability is applicable: if the Nyquist curve for a proper plant lies to the right of a frequency-dependent separating line through -1+j0 at each frequency, then stability is guaranteed. The result is applied in a numerical example, illustrating a common amplification phenomenon: even when the magnitude of uncertainty is significantly greater than the deterministic robustness bound, the risk of instability is small