Necessary and sufficient conditions for root clustering of a polytope of polynomials in a simply connected domain

Linear time-invariant systems with uncertain parameters are considered. A test is given which is both necessary and sufficient for root clustering of a family of polytopic polynomials in a simply connected domain. The test does not require checking a continuum of polynomials. Special cases include stability testing of continuous-time systems and of discrete-time systems with uncertain parameters, as well as `relative stability´ tests of such systems. Interval polynomials can also be handled as a special case