Parameter partitioning via shaping conditions for the stability of families of polynomials

It is shown that if parameter uncertainty enters a polynomial in a manner which allows an appropriate partitioning generated by certain shaping conditions, then the q locus becomes, at each frequency, a convex parpolygon in the complex plane. The results generalize some earlier work and expand the class of problems for which simple robust stability tests exist. These results imply that when uncertainty affects polynomial coefficients in a linear manner, the q locus is guaranteed to be a convex parpolygon at each frequency