A necessary and sufficient condition for Schur invariance and generalized stability of polytopes of polynomials

Polytopes of (characteristic) polynomials arise when systems experience real parameter variations. A necessary and sufficient condition (requiring only a finite number of computations) is presented for determining whether all the roots of a polytope of polynomials are contained in a desired region. This can be any region that is the image of the open left-half plane under a real linear fractional transformation. Since the open unit cycle is such a region, this result establishes a finite algorithm for strict Schur invariance. Finite algorithms for generalized stability regions such as maximum bandwidth and maximum setting time are also established