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Stability of polynomial families with multiaffine coefficient perturbations

In many control applications one is confronted with the following problem: 'Given a region Q2 in the complex plane, and a family of polynomials P determine whether or not all the roots of the polynomials are located in f. Important examples of the regions Q are the left half plane, and the unit disc. The famous Kharitonov theorem [K] provides an elegant solution for the problem when the region 12 is the left half plane, and P contains polynomials whose coefficients are allowed to vary independently in specified intervals. Numerous results concerning efficient tests for Q2 stability of polynomials whose coefficients depend affinely on physical parameters have been derived in recent years. An extraction of a testing subset P' C P, so that the Q? stability of P' would imply the SI stability of the whole family P has been an important objective of the research. The famous Edge Theorem [BHL] is the best known general result concerning this problem. A number of results concerning technical conditions that lead to the "easily testable" criteria have been reported recently by Polyak [P], Anderson, Kraus, Mansour, Dasgupta [AKMD), and Tsing, Tits [TT]. The goal of this note is to provide robust stability conditions under "minimal assumptions"-our criterion is less conservative, and the results presented in this note are implied by those given in [(], [AKMD], and [TT). The outline of the paper is as follows. In Section 2 we introduce the basic notations, and state the problem. In Section 3 we consider the image of a box under a multiaffine mapping, and introduce principal vertices of the box. A simple robust stability criterion completes the section. In Section 4 we recall and compare results reported in [P], [AKMD], and [TT]. 2. Notations and the problem