Robust stability of distributed parameter systems

The authors consider the robust stability problem for a class of uncertain distributed parameter systems where the characteristic equations involve a polytope P of quasipolynomials. Given a stability region D in the complex plane, the objective is to find a constructive technique for verifying whether all roots of every quasipolynomial in P belong to D (that is, to verify the D-stability of P). The first result is that, under a certain assumption on the stability region D, P is D-stable if and only if the edges of P are D-stable. Hence the D-stability problem of a high dimensional polytope is reduced to the D-stability problem of a finite number of pairwise convex combinations of vertices. The second result gives effective graphical tests for checking the D-stability of a polytope of quasipolynomials when the set D is any open left half plane. The graphical test is based on the frequency-response plots of some transfer functions associated with the vertices