Destabilizing effects of small time delays on feedback-controlled descriptor systems Original Research Article

In the last 15 years the problem of stabilizability and stabilization of descriptor systems have received considerable attention. In this paper it is shown that if a descriptor system image exhibits impulsive behavior, then the stability of the closed-loop system is extremely sensitive to small delays. More precisely, if F is the feedback which leads to a stable and impulsive-free closed-loop system, then there exist numbers var epsilonj > 0 and sj ε image with limj → ∞ var epsilonj = 0 and limj → ∝ Re sj = + ∝ and such that the delayed closed-loop system obtained by applying the feedback u(t) = Fx(t − var epsilonj) has a pole at sj. Moreover, if the open-loop system does not have impulsive behavior, the same phenomenon occurs, provided that the spectral radius of the matrix lims → ∝ F(sE − A)−1B is greater than 1. If this spectral radius is smaller than 1, it is shown that the closed-loop stability is robust with respect to small delays.