A unified approach for the stability robustness of polynomials in a convex set

A polynomial p(s, k) that is affine in the parameter perturbation k is considered. It is assumed that the vector k is uncertain but belongs to a convex set which contains the origin, and a polynomial is called stable if all of its roots are contained in a prespecified stability region in the complex plane. Then the stability robustness of p(s, k) can be measured by the maximal nonnegative number ρ with the property that if the gauge (or the Minkowski functional) of k with respect to the convex set is less than ρ, the polynomial pk is always stable. A unified approach is presented for computing the robustness measure ρ. The approach imbeds the problem considered into the framework of convex analysis so that some powerful tools in convex analysis can be used. The procedure for computing ρ that results from this approach is easy to implement. Various examples are included to illustrate the type of results which may be obtained