Robustness Analysis for Multilinear Perturbations

Abstract

An image f(K) of a box $K\subset {{R}^{m}}$ under multilinear transformation $f:{{R}^{m}}\to G$ is studied. For a class of functions called D — functions it is proved to be a convex polygon with 2m edges. Under less restrictive assumptions f(K) is a noncovex polygon. In the most general case f(K) has curvilinear parts of the boundary.

This result has numerous applications in robustness analysis. Edge and Kharitonovlike theorems as well as frequency criteria for polytopes of polynomials can be extended to multilinear families of polynomials. Systems with a cascade of uncertain blocks or with uncertain plant and controller are considered as examples.