DESTABILIZING EFFECTS OF SMALL TIME DELAYS ON FEEDBACK-CONTROLLED DESCRIPTOR SYSTEMS

In the last 15 years the problem of stabilizability and stabilization of descriptor systems have received considerable attention. In this pa per it is shown that if a descriptor system E (x)over dot = As + Bu ex hibits impulsive behavior, then the stability of the closed-loop syste m 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 epsilon(j) > 0 and s(j) epsilon C with lim( j-->infinity) epsilon(j) = 0 and lim(j-->infinity)Re s(j) = +infinity and such that the delayed closed-loop system obtained by applying the feedback u(t) = Fx(t - epsilon(j)) has a pole at s(j). Moreover, if th e open-loop system does not have impulsive behavior, the same phenomen on occurs, provided that the spectral radius of the matrix lim(\s\-->i nfinity) F(sE - A)B-1 is greater than 1. If this spectral radius is sm aller than 1, it is shown that the closed-loop stability is robust wit h respect to small delays. (C) 1998 Elsevier Science Inc.