Robust stability for time-delay systems: the edge theorem and graphical tests

The authors consider the D-stability problem for a class of uncertain delay systems where the characteristic equations involve a polytope of quasipolynomials. Their first result shows that under a mild assumption on the set D, a polytope of quasipolynomials is D-stable if and only if the edges of the polytope are D-stable. This extends the edge theorem developed by A.C. Bartlett, et. al. (1987) and M. Fu and B.R. Barmish (1988) for the D -stability of a polytope of polynomials. The second result provides a polar-plot-based graphical test for checking the D-stability of a polytope of quasipolynomials. In a special case in which the vertical quasipolynomials are in a factored form, the graphical test is further simplified by a special mapping. As shown in an example, the graphical tests provided here are quite useful in applications, making it possible to handle examples with many uncertain parameters easily