Mathematical Problems in Engineering
Volume 2008 (2008), Article ID 790530, 19 pages
Research Article

On Minimal Realizations and Minimal Partial Realizations of Linear Time-Invariant Systems Subject to Point Incommensurate Delays

M. De la Sen

Department of Electricity and Electronics, Institute for Research and Development of Processes, Campus of Leioa (Bizkaia), Apartado 644, 48080 Bilbao, Spain

Received 1 June 2007; Revised 22 November 2007; Accepted 28 November 2007

Academic Editor: Giuseppe Rega

Copyright © 2008 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.


This paper investigates key aspects of realization and partial realization theories for linear time-invariant systems being subject to a set of incommensurate internal and external point delays. The results are obtained based on the use of formal Laurent expansions whose coefficients are polynomial matrices of appropriate orders and which are also appropriately related to truncated and infinite block Hankel matrices. The above-mentioned polynomial matrices arise in a natural way from the transcendent equations associated with the delayed dynamics. The results are linked to the properties of controllability and observability of dynamic systems. Some related overview is given related to robustness concerned with keeping the realization properties under mismatching between a current transfer matrix and a nominal one.