Parameter Partitioning via Shaping Conditions for the Stability of Families of Polynomials

Let ¿(s,a) = ¿0(s) + q(s,a) be a family of polynomials in s, with coefficients which are continuous functions in the real k-vector parameter a, where a lies in the hyperectangle ¿a in R^k with O¿¿a. Assume further that the leading coefficient in s of ¿(s,a) is independent of a, ¿0 is stable and that q(s,O) = 0. It is shown that if a parameter partition exists, defined by certain shaping conditions, the locus of q(jw,a) in the complex plane for each frequency ¿, - ¿ ≪ ¿ ≪ ¿ as a varies in ¿a is a polytope. This facilitates the study of robust stability for a large class of systems and leads to simple conditions for stability of families of polynomials.