A general theory for matrix root-clustering in subregions of the complex plane

We consider the general problem of root-clustering of a matrix in the complex plane: LetA ∈ Cn × nandS ⊂ C. Find the largest class ofSand an algebraic criterion which is necessary and sufficient forλᵢ[A] ∈ S, i=1,2,..., n. We introduce two types of regions which constitute the largest class ofSknown to date. The criterion is presented both for open regions and closed ones. The results are used to define a design methodology for control systems. Moreover, all classical results are shown to be special cases of the present theory.