Journal of Applied Mathematics
Volume 2013 (2013), Article ID 926430, 9 pages
Research Article

Characterization of the Stabilizing PID Controller Region for the Model-Free Time-Delay System

1College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China
2Zhuji Sitong Mechanical and Electrical Equipment Manufacturing Co., LTD, Zhuji 311824, China

Received 25 January 2013; Accepted 21 March 2013

Academic Editor: Mamdouh M. El Kady

Copyright © 2013 Linlin Ou 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.


For model-free time-delay systems, an analytical method is proposed to characterize the stabilizing PID region based on the frequency response data. Such characterization uses linear programming which is computationally efficient. The characteristic parameters of the controller are first extracted from the frequency response data. Subsequently, by employing an extended Hermite-Biehler theorem on quasipolynomials, the stabilizing polygon region with respect to the integral and derivative gains and is described for a given proportional gain in term of the frequency response data. Simultaneously, the allowable stabilizing range of is derived such that the complete stabilizing set of the PID controller can be obtained easily. The proposed method avoids the complexity and inaccuracy of the model identification and thus provides a convenient approach for the design and tuning of the PID controller in practice. The advantage of the proposed algorithm lies in that the boundaries of the stabilizing region consist of several simple straight lines, the complete stabilizing set can be obtained efficiently, and it can be implemented automatically in computers.