The edge theorem and graphical tests for robust stability of neutral time-delay systems

Alrstract--We consider the robust stability problem for a class of uncertain neutral time-delay systems where the characteristic equations involve a polytope ~ of quasipolynomials of neutral type. Given a stability region D in the complex plane our goal is to find a constructive technique to verify the D-stability of ~ (i.e. to verify whether the roots of every quasipolynomial in ~ all belong to D). We first show that, under a certain assumption on the stability region D, is D-stable if and only if the edges of ~ are D-stable. Hence, the D-stability problem of a higher dimensional polytope is reduced to the D-stability problem of a finite number of pairwise convex combinations of vertices. Based on this result, we then give an effective graphical test for checking the D-stability of a polytope of quasipolynomials of neutral type