ROOT DISTRIBUTION INVARIANCE OF POLYNOMIAL FAMILIES

This paper treats the problem of root distribution invariance of polynomial families. We first establish the generalized zero exclusion principle for root distribution of polynomial families, and prove the complex boundary theorem and complex edge theorem for robust stability in parameter space. Based on the generalized zero-exclusion principle, the corresponding results on root distribution in parameter and coefficient space are obtained. Moreover, for real edge theorem on robust stability in coefficient space, We show that the assumptions made on the stability region can further be weakened. For some more geometrically characterized polytopes, and some specific stability regions, the edge theorem can be improved. i.e. the number of edges to be checked is independent of the number of edges of the polytope. Finally, a Nyquist-type criterion is proposed for verification of the root distribution of an edge.