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

Consider a control system which is operated with admissible values of uncertain parameters which exceed the bounds specified by classical robustness theory. In this case it is important to quantify the tradeoffs between risk of performance degradation and increased tolerance of uncertainty. If a large increase in the uncertainty bound can be established, an acceptably small risk may often be justified. Since robustness problem formulations do not include statistical descriptions of the uncertainty, the question arises whether it is possible to provide such assurances in a "distribution-free" manner. In other words, if ℱ denotes a class of possible probability distributions for the uncertainty q, we seek some worst-case f* ∈ ℱ having the following property: The probability of performance satisfaction under f* is smaller than the probability under any other f ∈ ℱ. Said another way, f* provides the best possible guarantee. This new framework is illustrated on robust stability problems associated with Kharitonov's theorem and the Edge Theorem. The main results are straightforward to describe: Let p(s, q) denote the uncertain polynomial under consideration and take ℘(ω) to be a frequency-dependent convex target set (in the complex plane) for the uncertain values p(jω, q). Consistent with value set analysis, ℘(ω) is assumed to be symmetric with respect to the nominal p(jω, 0). The uncertain parameters q_i are taken to be zero-mean independent random variables with known support interval. For each uncertainty, the class ℱ is assumed to consist of density functions which are symmetric and nonincreasing on each side of zero. Then, for fixed frequency ω, the first theorem indicates that the probability that p(jω, q) is in ℘(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. It turns out that in many cases, classical robustness margins can be far exceeded while keeping the risk of instability surprisingly small. Finally, for a much more general class of uncertainty structures, this paper also establishes the fact that f* can be estimated by a truncated uniform distribution.