Mathematical Problems in Engineering
Volume 2005 (2005), Issue 1, Pages 123-140

On pole-placement controllers for linear time-delay systems with commensurate point delays

M. de la Sen

Departamento de Ingeniería de Sistemas y Automática, Instituto de Investigación y Desarrollo de Procesos (IIDP), Facultad de Ciencias, Universidad del País Vasco, Campus de Bizkais, Aptdo. 644, Bilbao 48080, Spain

Received 12 December 2003; Revised 18 May 2004

Copyright © 2005 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.


We investigate the exact and approximate spectrum assignment properties associated with realizable output-feedback pole-placement-type controllers for single-input single-output linear time-invariant time-delay systems with commensurate point delays. The controller synthesis problem is discussed through the solvability of a set of coupled Diophantine equations of polynomials. An extra complexity is incorporated in the above design to cancel extra unsuitable dynamics being generated when solving the above Diophantine equations. Thus, the complete controller tracks any arbitrary prefixed (either finite or delay-dependent) closed-loop spectrum. However, if the controller is simplified by deleting the above-mentioned extra complexity, then robust stability and approximated spectrum assignment are still achievable for a certain sufficiently small amount of delayed dynamics. Finally, the approximate spectrum assignment and robust stability problems are revisited under plant disturbances if the nominal controller is maintained. In the current approach, the finite spectrum assignment is only considered as a particular case of the designer's choice of a (delay-dependent) arbitrary spectrum assignment objective.