Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 212053, 17 pages
Research Article

Computation of All Stabilizing PID Gain for Second-Order Delay System

Research Unit on Systems Analysis and Control, National School of Engineering of Tunis, BP, 37, Le Belvédère, 1002 Tunis, Tunisia

Received 2 April 2009; Revised 19 June 2009; Accepted 15 July 2009

Academic Editor: Ji Huan He

Copyright © 2009 Rihem Farkh et al. 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 problem of stabilizing a second-order delay system using classical proportional-integral-derivative (PID) controller is considered. An extension of the Hermite-Biehler theorem, which is applicable to quasipolynomials, is used to seek the set of complete stabilizing PID parameters. The range of admissible proportional gains is determined in closed form. For each proportional gain, the stabilizing set in the space of the integral and derivative gains is shown to be either a trapezoid or a triangle.