Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2013, Special Issue (2013), Article ID 926430, 9 pages.

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

Linlin Ou, Yuan Su, and Xuanguang Chen

Full-text: Open access

Abstract

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 ( k i and k d ) is described for a given proportional gain ( k p ) in term of the frequency response data. Simultaneously, the allowable stabilizing range of k p 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.

Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 926430, 9 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807781

Digital Object Identifier
doi:10.1155/2013/926430

Mathematical Reviews number (MathSciNet)
MR3064961

Zentralblatt MATH identifier
1271.93103

Citation

Ou, Linlin; Su, Yuan; Chen, Xuanguang. Characterization of the Stabilizing PID Controller Region for the Model-Free Time-Delay System. J. Appl. Math. 2013, Special Issue (2013), Article ID 926430, 9 pages. doi:10.1155/2013/926430. https://projecteuclid.org/euclid.jam/1394807781


Export citation

References

  • K. Gu, V. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Birkhauser, Boston, Mass, USA, 2003.
  • Y. X. Qing, Y. Q. Liu, and L. Wang, Stabilization of Dynamic System with Time-Delay, Academic Science, Beijing, China, 1989.
  • L. Xie, E. Fridman, and U. Shaked, “Robust H$_{\infty }$ control of distributed delay systems with application to combustion control,” IEEE Transactions on Automatic Control, vol. 46, no. 12, pp. 1930–1935, 2001.
  • W. L. Zang, Y. G. Wang, Z. Guo, and Y. X. Wang, “Satisfactory PID design for servo systems based on iterative LMI technique,” Control Theory and Applications, vol. 23, no. 6, pp. 967–975, 2006.
  • S. Bernt, “Model-free tracking of cars and people based on color regions,” Image and Vision Computing, vol. 24, no. 11, pp. 1172–1178, 2006.
  • Q. D. Qing and R. J. Li, “Particle swarm optimization algorithm mimicking biological ideal free distribution model,” Control and Decision, vol. 26, no. 12, pp. 46–51, 2011.
  • J. Z. Ziegler, N. B. Nichols, and N. Y. Rochester, “Optimum settings for automatic controllers,” Transactions of the ASME, vol. 64, pp. 759–768, 1942.
  • A. Datta, M. T. Ho, and S. P. Bhattacharyya, Structure and Synthesis of PID Controllers, Springer, London, UK, 2000.
  • M. T. Söylemez, N. Munro, and H. Baki, “Fast calculation of stabilizing PID controllers,” Automatica, vol. 39, no. 1, pp. 121–126, 2003.
  • D. Ackermann and D. Kaesbauer, “Stable polyhedra in parameter space,” Automatica, vol. 39, no. 5, pp. 937–943, 2003.
  • L. L. Ou, W. D. Zhang, and L. Yu, “Low-order stabilization of LTI systems with time delay,” IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 774–787, 2009.
  • L. H. Keel and S. P. Bhattacharyya, “PID controller synthesis free of analytical models,” in Proceedings of the 16th Triennial World Congress of International Federation of Automatic Control (IFAC '05), pp. 367–372, Prague, Czech Republic, July 2005.
  • L. H. Keel and S. P. Bhattacharyya, “Direct synthesis of first order controllers from frequency response measurements,” in Proceedings of the American Control Conference (ACC '05), pp. 1192–1196, Portland, Ore, USA, June 2005.
  • Y. Li, A. Sheng, and Y. Wang, “Synthesis of PID-type controllers without parametric models: a graphical approach,” Energy Conversion and Management, vol. 49, no. 8, pp. 2392–2402, 2008.
  • Q. G. Wang, C. C. Hang, and Q. Bi, “A technique for frequency response identification from relay feedback,” IEEE Transactions on Control Systems Technology, vol. 7, no. 1, pp. 122–128, 1999.