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Fractional Order IMC Controller Design for Two-input-two-output Fractional Order System

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  • Control Theory and Applications
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Abstract

Research on the fractional order system is becoming more and more popular. Most of the fractional order controller design methods focus on single-input-single-output processes. In this paper, a fractional order internal model controller with inverted decoupling is proposed to handle non-integer order two-input-two-output systems with time delay. The fractional order two-input-two-output (FO-TITO) process is decoupled by inverted decoupling method. The fractional order internal model control (IMC) is then used to simplify the tuning process. Because of the complexity of multiple time delay, the condition of FO-TITO process with time delay is discussed. In order to ensure the robustness of the system, a Maximum sensitivity function is used to tune the parameters. Then Lyapunov stability theory is applied to verify the stability of the system. The proposed controller provides ideal performance for both set point-tracking and disturbance rejection and is robust to process gain variations. Numerical results show the performance of the proposed method.

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References

  1. Y. Luo and Y. Q. Chen, Fractional Order Motion Controls, John Wiley & Sons, 2012.

    Book  Google Scholar 

  2. Y. Q. Chen, I. Petras, and D. Xue, “Fractional order control-A tutorial,” Proc. of the American Control Conference (ACC), pp. 1397–1411, 2009.

    Google Scholar 

  3. D. Li, P. Ding, and Z. Gao, “Fractional active disturbance rejection control,” ISA Transactions, vol. 62, pp. 109–119, 2016.

    Article  Google Scholar 

  4. A. Oustaloup, La Commande CRONE: Commande Robuste d’Ordre Non Entier, Hermes, 1991.

    MATH  Google Scholar 

  5. I. Polubny, “Fractional-order systems and PIlDm controller,” IEEE Transactions on Automatic Control, vol. 44, pp. 208–214, 1999.

    Article  MathSciNet  Google Scholar 

  6. D. Valério, and J. S. da Costa, “Tuning of fractional PID controllers with ZieglerNichols-type rules,” Signal Processing, vol. 86, no. 10, pp. 2771–2784, 2006.

    Article  MATH  Google Scholar 

  7. C. A. Monje, B. M. Vinagre, and V. Feliu, “Tuning and auto-tuning of fractional order controllers for industry applications,” Control Engineering Practice, vol. 16, no. 7, pp. 798–812, 2008.

    Article  Google Scholar 

  8. M. Zamani, M. Karimi-Ghartemani, and N. Sadati, “Design of a fractional order PID controller for an AVR using particle swarm optimization,” Control Engineering Practice, vol. 17, no. 12, pp. 1380–1387, 2009.

    Article  Google Scholar 

  9. Y. Luo and Y. Q. Chen, “Fractional order [proportional derivative]_controller for a class of fractional order systems,” Automatica, vol. 45, no. 10, pp. 2446–2450, 2009.

    Article  MathSciNet  MATH  Google Scholar 

  10. H. S. Li, Y. Luo, and Y. Q. Chen, “A fractional order proportional and derivative (FOPD) motion controller: tuning rule and experiments,” IEEE Transactions on Control Systems Technology, vol. 18, no. 2, pp. 516–520, 2010.

    Article  Google Scholar 

  11. F. Padula and A. Visioli, “Tuning rules for optimal PID and fractional-order PID controllers,” Journal of Process Control, vol. 21, no. 1, pp. 69–81, 2011.

    Article  MATH  Google Scholar 

  12. T. N. L. Vu and M. Lee, “Analytical design of fractionalorder proportional-integral controllers for time-delay processes,” ISA transactions, vol. 52, no. 5, pp. 583–591, 2013.

    Article  Google Scholar 

  13. S. Victor, P. Melchior, and A. Oustaloup, “Robust path tracking using flatness for fractional linear MIMO systems: a thermal application,” Computers&Mathematics with Applications, vol. 59, no. 5, pp. 1667–1678, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. N. Gruel and P. Lanusse, “Oustaloup a. robust control design for multivariable plants with time-delays,” Chemical Engineering Journal, vol. 146, no. 3, pp. 414–427, 2009.

    Article  Google Scholar 

  15. I. Pan and S. Das, “Frequency domain design of fractional order PID controller for AVR system using chaotic multiobjective optimization,” International Journal of Electrical Power & Energy Systems, vol. 51, pp. 106–118, 2013.

    Article  Google Scholar 

  16. D. Li, L. Liu, and Q. Jin, “Maximum sensitivity based fractional IMCPID controller design for non-integer order system with time delay,” Journal of Process Control, vol. 31, pp. 17–29, 2015.

    Article  Google Scholar 

  17. D. Z. Li and X. Y. He, “The inverted decoupling based fractional order two-input-two-output IMC controller,” Proc. of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Cleaveland, 2017.

    Google Scholar 

  18. S. G. Samko, A. A. Kilbas, and O. I. Marichev, “Fractional integrals and derivatives,” Theory and Applications, Gordon and Breach, Yverdon, 1993.

    Google Scholar 

  19. M. Morari and E. Zafiriou, Robust Process Control, Prentice Hall, Englewood Cliffs, NJ, 1989.

    MATH  Google Scholar 

  20. D. E. Rivera, M. Morari, and S. Skogestad, “Internal model control: PID controller design,” Industrial & Engineering Chemistry Process Design and Development, vol. 25, no. 1, pp. 252–265, 1986.

    Article  Google Scholar 

  21. B. J. West, “Fractional calculus in bioengineering,” Journal of Statistical Physic, vol. 126, no. 6, pp. 1285–1286, 2007.

    Article  Google Scholar 

  22. L. Sun, J. Dong, and D. Li, “A practical multivariable control approach based on inverted decoupling and decentralized active disturbance rejection control,” Industrial & Engineering Chemistry Research, vol. 55, no. 7, pp. 2008–2019, 2016.

    Article  Google Scholar 

  23. Z. Li and Y. Q. Chen, “Ideal, simplified and inverted decoupling of fractional order TITO processes,” IFAC Proceedings Volumes 2014., vol. 47, no. 3, pp. 2897–2902, 2014.

    Article  Google Scholar 

  24. H. Sheng, Y. Q Chen, and T. S. Qiu, Fractional Processes and Fractional-order Signal Processing: Techniques and Applications, Springer Science & Business Media, 2011.

    MATH  Google Scholar 

  25. Y. Li, Y. Q. Chen, and Y. Cao, “Fractional order universal adaptive stabilization,” Proc. of the 3rd IFAC Workshop on Fractional Differentiation and Its Applications, Ankara, Turkey. 2008.

    Google Scholar 

  26. L. Sun, J. Dong, and D. Li, “A practical multivariable control approach based on inverted decoupling and decentralized active disturbance rejection control,” Industrial & Engineering Chemistry Research, vol. 55, no. 7, pp. 2008–2019, 2016.

    Article  Google Scholar 

  27. Z. Li and Y. Q. Chen, “Ideal, simplified and inverted decoupling of fractional order TITO processes,” IFAC Proceedings Volumes 2014., vol. 47, no. 3, pp. 2897–2902, 2014.

    Article  Google Scholar 

  28. H. Sheng, Y. Q Chen, and T. S. Qiu, Fractional Processes and Fractional-order Signal Processing: Techniques and Applications, Springer Science & Business Media, 2011.

    MATH  Google Scholar 

  29. K. Vladimír, “Optimal decoupling controllers for singular systems,” Proc. of the European Control Conference (ECC), Zürich, pp. 306–311, 2013.

    Google Scholar 

  30. C. A. Lin, “Necessary and sufficient conditions for existence of decoupling controllers,” IEEE Transactions on Automatic Control, vol. 42, no. 8, pp. 1157–1161, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  31. Q. Jin, X. Du, and Q. Wang, “Analytical design 2 DOF IMC control based on inverted decoupling for nonsquare systems with time delay,” The Canadian Journal of Chemical Engineering, vol. 94, no. 7, pp. 1354–1367, 2016.

    Article  Google Scholar 

  32. M. A. Pakzad, S. Pakzad, and M. A. Nekoui, “Stability analysis of time-delayed linear fractional-order systems,” International Journal of Control, Automation and Systems, vol. 11, no. 3, pp. 519–525, 2013.

    Article  Google Scholar 

  33. M. A. Pakzad and M. A. Nekoui, “Stability map of multiple time delayed fractional order systems,” International Journal of Control, Automation and Systems, vol. 12, no. 1, pp. 37–43, 2014.

    Article  Google Scholar 

  34. D. Valério, and J. S. da Costa, “A review of tuning methods for fractional PIDs,” Proc. of the 4th IFAC Workshop on Fractional Differentiation and Its Applications, vol. 10, 2011.

    Google Scholar 

  35. B. Sundara Vadivoo, Raja Ramachandran, Jinde Cao, Hai Zhang, and Xiaodi Li, “Controllability analysis of nonlinear neutral-type fractional-order differential systems with state delay and impulsive effects,” International Journal of Control, Automation and Systems, vol. 16, no. 2, pp. 659–669, 2018.

    Article  Google Scholar 

  36. H. Zhang, M. Ye, J. Cao, and A. Alsaedi, “Synchronization control of Riemann-Liouville fractional competitive network systems with time-varying delay and different time scales,” International Journal of Control, Automation and Systems, vol. 16, no. 3, pp. 1404–1414, 2018.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Automation, College of Information Science and Technology, Beijing University of Chemical Technology, Beijing, 100029, China

    Dazi Li, Xingyu He, Tianheng Song & Qibing Jin

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  1. Dazi Li
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  2. Xingyu He
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  3. Tianheng Song
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  4. Qibing Jin
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Corresponding author

Correspondence to Dazi Li.

Additional information

Recommended by Associate Editor Niket Kaisare under the direction of Editor Young IL Lee. This research has been supported in part by the National Natural Science Foundation of China (61873022, 61573052), in part by the Beijing Natural Science Foundation (4182045) and in part by the Fundamental Research Funds for the Central Universities under Grant XK1802-4.

Dazi Li received the Ph.D. degree in engineering from the Department of Electrical and Electronic Systems, Kyushu University, Fukuoka, Japan, in 2004. She is currently a Full Professor and Chair of Department of Automation, College of Information Science and Technology, Beijing University of Chemical Technology, Beijing, China. Her current research interests include machine learning and artificial intelligence, advanced process control, complex system modeling and optimization. Dr. Li is currently an Associate Editor of ISA Transactions.

Xingyu He received the B.S. degree in automation from the Department of Automation, Beijing University of Chemical Technology, Beijing, China, in 2011, where he is currently pursuing the M.S. degree with the College of Information Science and Technology. His current research interests include fractional order control.

Tianheng Song received the M.S. and Ph.D. degrees in control science and engineering from the Department of Automation, Beijing University of Chemical Technology, Beijing, China, in 2011 and 2016, respectively, where he is currently a postdoctoral fellow with the College of Information Science and Technology. His current research interests include reinforcement learning.

Qibing jin received the Ph.D. degree in control theory and engineering from the Northeastern University, Shenyang, Liaoning, China, in 1999. He is currently a Full Professor with the college of Information Science and Technology, the Director of the Institute of Automation with the Beijing University of Chemical Technology, Beijing, China. His main research interests include advanced control, intelligent instrument, Intelligent optimization algorithm, multivariable system identification and control theory. He has rich experience in control engineering, and his many research results have been applied in petroleum and chemical industry.

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Li, D., He, X., Song, T. et al. Fractional Order IMC Controller Design for Two-input-two-output Fractional Order System. Int. J. Control Autom. Syst. 17, 936–947 (2019). https://doi.org/10.1007/s12555-018-0129-3

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  • DOI: https://doi.org/10.1007/s12555-018-0129-3

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