Applications of optimization methods to robust stability of linear systems

We study the robust stability problem for a family of polynomials. We allow for all the coefficients of the polynomials to be affinely perturbed, where the size of the perturbation is measured by an arbitrary convex function. We apply optimization techniques, and in particular convex duality methods, to derive simple formulas for the stability radius, to find a minimal perturbation which destroys stability, and to obtain necessary and sufficient conditions for robust stability. Our framework is general enough to cover many applications. As special cases, we obtain many results recently reported in the literature.The work of the first author was partially supported by AFOSR Grant 91-008 and NSF Grant DMS-92-01297.