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Discrete Dynamics in Nature and Society
Volume 2006 (2006), Article ID 76361, 8 pages

Stability and assignment of spectrum in systems with discrete time lags

Instituto de Investigacion y Desarrollo de Procesos (IIDP), Departamento de Electricidad y Electrónica, Facultad de Ciencias, Universidad del País Vasco, Leioa (Bizkaia), Aptdo, 644 de Bilbao, Bilbao 48080, Spain

Received 8 November 2005; Accepted 13 March 2006

Copyright © 2006 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The asymptotic stability with a prescribed degree of time delayed systems subject to multiple bounded discrete delays has received important attention in the last years. It is basically proved that the α-stability locally in the delays (i.e., all the eigenvalues have prefixed strictly negative real parts located in Resα<0) may be tested for a set of admissible delays including possible zero delays either through a set of Lyapunov's matrix inequalities or, equivalently, by checking that an identical number of matrices related to the delayed dynamics are all stability matrices. The result may be easily extended to check the ε-asymptotic stability independent of the delays, that is, for all the delays having any values, the eigenvalues are stable and located in Resε0. The above referred number of stable matrices to be tested is 2r for a set of distinct r point delays and includes all possible cases of alternate signs for summations for all the matrices of delayed dynamics. The manuscript is completed with a study for prescribed closed-loop spectrum assignment (or “pole placement”) under output feedback.