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 ro bustness theory. In this case it is important to quantify the tradeoff s between risk of performance degradation and increased tolerance of u ncertainty. If a large increase in the uncertainty bound can be establ ished, an acceptably small risk may often be justified. Since robustne ss problem formulations do not include statistical descriptions of the uncertainty, the question arises whether it is possible to provide su ch assurances in a ''distribution-free'' manner. In other words, if F denotes a class of possible probability distributions for the uncertai nty q, we seek some worst-case f is an element of F having the follow ing property: The probability of performance satisfaction under f is smaller than the probability under any other f is an element of F. Sai d another way, f provides the best possible guarantee. This new frame work is illustrated on robust stability problems associated with Khari tonov's theorem and the Edge Theorem. The main results are straightfor ward to describe: Let p(s, q) denote the uncertain polynomial under co nsideration and take P(omega) to be a frequency-dependent convex targe t set (in the complex plane) for the uncertain values p(j omega, q). C onsistent with value set analysis, P(omega) is assumed to be symmetric with respect to the nominal p(j omega, 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 cons ist of density functions which are symmetric and nonincreasing on each side of zero. Then, for fixed frequency omega, the first theorem indi cates that the probability that p(j omega, q) is in P(omega) is minimi zed by the uniform distribution for q. The second theorem, a generaliz ation of the first, indicates that the same result holds uniformly wit h respect to frequency. Then probabilistic guarantees for robust stabi lity are given in the third theorem. It turns out that in many cases, classical robustness margins can be far exceeded white keeping the ris k 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.