The uniform distribution: rigorous justification for its use in robustness analysis

Consider a control system with admissible uncertain parameter values which exceed the bounds specified by classical robustness theory. The article concerns trade-offs between performance degradation risk and uncertainty tolerance. If a large increase in the uncertainty bound can be established, an acceptably small risk may be justified. Since robustness formulations do not include statistical descriptions of the uncertainty, the question arises whether it is possible to provide such assurances for any distribution. The paper concentrates on problems associated with Kharitonov´s theorem and the edge theorem. If p(s, q) denote the uncertain polynomial under consideration and 𝒫(ω) a frequency dependent convex target set for the uncertain values p(jω, q), 𝒫(ω) symmetric with respect to the nominal p(jω, 0), the uncertain parameters q_i zero-mean independent random variables with known support interval, and for each uncertainty ℱ consists of symmetric nonincreasing density functions, then, for fixed frequency ω, the first theorem indicates that the probability that p(jω, q) is in 𝒫(ω) is minimized by the uniform distribution for q. The second theorem, a generalization of the first, indicates that the same result holds uniformly with respect to frequency. Then, probabilistic guarantees for robust stability are given in the third theorem. In many cases, one can far exceed classical robustness margins while keeping the risk of instability small