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Mathematical Problems in Engineering
Volume 2017, Article ID 1241545, 25 pages
https://doi.org/10.1155/2017/1241545
Research Article

MIMO PI Controllers for LTI Systems with Multiple Time Delays Based on ILMIs and Sensitivity Functions

National Institute of Applied Sciences and Technology (INSAT), Centre Urbain Nord, BP 676, 1080 Tunis Cedex, Tunisia

Correspondence should be addressed to Olfa Boubaker; nt.unr.tasni@rekabuob.aflo

Received 12 August 2016; Revised 9 November 2016; Accepted 5 December 2016; Published 20 February 2017

Academic Editor: Tamas Kalmar-Nagy

Copyright © 2017 Wajdi Belhaj and Olfa Boubaker. 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.

Linked References

  1. K. J. Astrom and T. Hagglund, “Advanced PID control,” in ISA-The Instrumentation, Systems, and Automation Society, 2005. View at Google Scholar
  2. R. Vilanova and A. Visioli, PID Control in the Third Millennium: Lessons Learned and New Approaches, Springer, New York, NY, USA, 2012.
  3. F. D. Bianchi, R. J. Mantz, and C. F. Christiansen, “Multivariable PID control with set-point weighting via BMI optimisation,” Automatica, vol. 44, no. 2, pp. 472–478, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. X. Luan, Q. Chen, and F. Liu, “Equivalent transfer function based multi-loop PI control for high dimensional multivariable systems,” International Journal of Control, Automation and Systems, vol. 13, no. 2, pp. 346–352, 2015. View at Publisher · View at Google Scholar · View at Scopus
  5. S. Boyd, M. Hast, and K. J. Åström, “MIMO PID tuning via iterated LMI restriction,” International Journal of Robust and Nonlinear Control, vol. 26, no. 8, pp. 1718–1731, 2016. View at Publisher · View at Google Scholar · View at Scopus
  6. A. N. Gündeş, H. Özbay, and A. B. Özgüler, “PID controller synthesis for a class of unstable MIMO plants with I/O delays,” Automatica, vol. 43, no. 1, pp. 135–142, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. T. N. L. Vu and M. Lee, “Analytical design of robust multi-loop PI controller for multi-time delay processes,” in Advances in Machine Learning and Data Analysis, vol. 48 of Lecture Notes in Electrical Engineering, pp. 95–108, Springer, Berlin, Germany, 2010. View at Publisher · View at Google Scholar
  8. A. N. Gündeş and H. Özbay, “Reliable decentralised control of delayed MIMO plants,” International Journal of Control, vol. 83, no. 3, pp. 516–526, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. Q. Jin, F. Hao, and Q. Wang, “IMC-PID controller based on diagonal equivalent model for multivariable systems with multiple time delays,” Journal of Chemical Engineering of Japan, vol. 46, no. 3, pp. 209–218, 2013. View at Publisher · View at Google Scholar · View at Scopus
  10. C. G. Economou and M. Morari, “Internal Model Control: multi-loop design,” Industrial and Engineering Chemistry Process Design and Development, vol. 25, no. 2, pp. 411–419, 1986. View at Google Scholar
  11. S. Cha, D. Chun, and J. Lee, “Two-step IMC-PID method for multiloop control system design,” Industrial and Engineering Chemistry Research, vol. 41, no. 12, pp. 3037–3041, 2002. View at Publisher · View at Google Scholar · View at Scopus
  12. M. Lee, K. Lee, C. Kim, and J. Lee, “Analytical design of multiloop PID controllers for desired closed-loop responses,” AIChE Journal, vol. 50, no. 7, pp. 1631–1635, 2004. View at Publisher · View at Google Scholar · View at Scopus
  13. T. N. L. Vu and M. Lee, “Multi-loop PI controller design based on the direct synthesis for interacting multi-time delay processes,” ISA Transactions, vol. 49, no. 1, pp. 79–86, 2010. View at Publisher · View at Google Scholar · View at Scopus
  14. F. Zheng, Q.-G. Wang, and T. H. Lee, “On the design of multivariable PID controllers via LMI approach,” Automatica, vol. 38, no. 3, pp. 517–526, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. C. Lin, Q.-G. Wang, and T. H. Lee, “An improvement on multivariable PID controller design via iterative LMI approach,” Automatica, vol. 40, no. 3, pp. 519–525, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. Y. He and Q. G. Wang, “An improved ILMI method for static output feedback control with application to multivariable PID control,” IEEE Transactions on Automatic Control, vol. 51, no. 10, pp. 1678–1683, 2006. View at Publisher · View at Google Scholar
  17. W. Belhaj and O. Boubaker, “On MIMO PID control of the quadruple-tank process via ILMIs approaches: minimum and non-minimum case studies,” in Proceedings of the 10th IFAC Symposium on Dynamics and Control of Process Systems (DYCOPS '13), pp. 481–486, Mumbai, India, December 2013. View at Publisher · View at Google Scholar · View at Scopus
  18. W. Belhaj and O. Boubaker, “Multivariable PID controller via LMIs: performances assessement,” International Journal of Smart Sensing and Intelligent Systems, vol. 8, no. 4, pp. 1896–1916, 2015. View at Google Scholar
  19. D. McFarlane and K. Glover, Robust Controller Design Using Normalized Coprime Factor Plant Descriptions, vol. 138 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 1990. View at Publisher · View at Google Scholar
  20. K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control, Prentice Hall, Upper Saddle River, NJ, USA, 1996.
  21. H. Panagopoulos and K. J. Astrom, “PID control design and loop shaping,” in Proceedings of the IEEE International Conference on Control Applications, pp. 103–108, Honolulu, Hawaii, USA, 1999.
  22. W. Tan, T. Chen, and H. J. Marquez, “Robust controller design and PID tuning for multivariable processes,” Asian Journal of Control, vol. 4, no. 4, pp. 439–451, 2002. View at Google Scholar · View at Scopus
  23. B. Vanavil, K. K. Chaitanya, and A. S. Rao, “Improved PID controller design for unstable time delay processes based on direct synthesis method and maximum sensitivity,” International Journal of Systems Science, vol. 46, no. 8, pp. 1349–1366, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. J.-P. Richard, “Time-delay systems: an overview of some recent advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667–1694, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. P. J. Davis, Interpolation and Approximation, Blaisdell, Boston, Mass, USA, 1975. View at MathSciNet
  26. O. Boubaker, J. P. Babary, and M. Ksouri, “MIMO sliding mode control of a distributed parameter denitrifying biofilter,” Applied Mathematical Modelling, vol. 25, no. 8, pp. 671–682, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  27. L. Lefèvre, D. Dochain, S. Feyo De Azevedo, and A. Magnus, “Optimal selection of orthogonal polynomials applied to the integration of chemical reactor equations by collocation methods,” Computers and Chemical Engineering, vol. 24, no. 12, pp. 2571–2588, 2000. View at Publisher · View at Google Scholar · View at Scopus
  28. M. T. Nihtila, T. Damak, and J. P. Babary, “Recursive input delay estimation in linear systems,” in Proceedings of the IFAC Symposium on Identification and System Parameter Estimation, July 1994.
  29. Y.-Y. Cao, L. James, and Y.-X. Sun, “Static output feedback stabilization: an ILMI approach,” Automatica, vol. 34, no. 12, pp. 1641–1645, 1998. View at Publisher · View at Google Scholar · View at Scopus
  30. J. Löfberg, “YALMIP: a toolbox for modeling and optimization in MATLAB,” in Proceedings of the IEEE International Symposium on Computer Aided Control Systems Design, pp. 284–289, Taipei, Taiwan, September 2004. View at Scopus
  31. T. Kailath, Linear Systems, Prentice-Hall, New Jersey, NJ, USA, 1980. View at MathSciNet
  32. A. N. Mete, A. N. Gundes, and H. Özbay, “Decoupled PID controller synthesis for MIMO plants with I/O delays,” in Proceedings of the 45th IEEE Conference on Decision and Control (CDC '06), pp. 852–857, San Diego, Calif, USA, December 2006. View at Scopus
  33. B. Friedland, Control System Design: An Introduction to State Space Methods, Dover Publications, New York, NY, USA, 1986.
  34. I.-L. Chien, H.-P. Huang, and J.-C. Yang, “A simple multiloop tuning method for PID controllers with no proportional kick,” Industrial and Engineering Chemistry Research, vol. 38, no. 4, pp. 1456–1468, 1999. View at Publisher · View at Google Scholar · View at Scopus
  35. K. H. Johansson, “The quadruple-tank process: a multivariable laboratory process with an adjustable zero,” IEEE Transactions on Control Systems Technology, vol. 8, no. 3, pp. 456–465, 2000. View at Publisher · View at Google Scholar · View at Scopus
  36. F. El Haoussi, E. H. Tissir, F. Tadeo, and A. Hmamed, “Delay-dependent stabilisation of systems with time-delayed state and control: application to a quadruple-tank process,” International Journal of Systems Science, vol. 42, no. 1, pp. 41–49, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus