On mappable nonlinearities in robustness analysis

When carrying out robustness analysis in the frequency domain, the following fundamental problem arises: Given a description of the uncertain quantities entering the system, at each frequency ω, we need to carry out a mapping into the complex plane. For the special case of multilinear uncertainty structures, the mapping theorem greatly facilitates this process and leads to the convex hull of the value set of interest. In this paper, we generalize the class of nonlinear uncertainty structures for which the convex hull can be generated, the so-called generalized mapping theorem which goes considerably beyond the multilinear setting. For example, this new framework leads to mappability for large classes of polynomial and nonlinear uncertainty structures. The formulas associated with convex hull generation can be easily implemented in two-dimensional graphics