The robust root locus of polynomial families with multilinear parameter dependence

The mapping theorem by Zadeh and Desoer (1963) is a sufficient condition for the zero exclusion of the image or value set of an m-dimensional box B under a multilinear mapping f : R^m→C, where R and C denote the real line and the complex plane, respectively. In this paper, we present a sufficient condition for the zero inclusion of the value set f( B). On the basis of these two conditions and the iterative subdivision of the box B, we propose a numerical procedure for testing whether or not the value set f(B) includes the origin. The procedure is easy to implement and is more efficient than that based on constructing the value set f(B) explicitly. As an application, the proposed zero inclusion test procedure is used along with a homotopy continuation algorithm to trace out the boundary curves of the robust root loci of polynomial families with multilinear parametric uncertainties