Value sets of polynomial families with coefficients depending nonlinearly on perturbed parameters

The problem of constructing the value set of an m-dimensional box Q under a nonlinear differentiable mapping f:R^m→C is considered, where R and C denote the sets of real and complex numbers respectively. More precisely, the notions of principal points and generalised principal points are exploited to characterise the boundary of the value set f(Q) and to present an effective and noniterative procedure for obtaining the set of principal points P whose image covers the boundary of the value set f(Q). The value-set construction procedure involves using a simplicial algorithm to trace out the set of generalised principal points G, which is the superset of P. A theorem for explicitly identifying the set of principal points P from the set of generalised principal points G is presented. As an application, the proposed value-set construction procedure is applied to robust D-stability analysis of a family of polynomials whose coefficients depend nonlinearly on a set of perturbed parameters