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Robust Fractional-order PID Tuning Method for a Plant with an Uncertain Parameter

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  • Control Theory and Applications
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Abstract

The robust design of fractional-order proportional-integral-differential (FOPID) controllers for controlled plants with uncertainty is a popular research topic. The well-studied “flat phase” condition is effective for the gain variation but not for variations in other parameters. This paper addresses the problem of tuning a robust FOPID controller for a plant with a known structure and an uncertain parameter (a coefficient or order in the plant transfer function). The method is based on preserving the phase margin of the open-loop system when the plant parameter varies around the nominal value. First, the partial derivatives of the gain crossover frequency with respect to the plant parameters are calculated. Then, the partial derivatives of the phase margin with respect to the plant parameters are obtained as the robust performance indexes. In addition, the equations needed to compute FOPID parameters that meet the specifications in the frequency domain are obtained and used as nonlinear constraints. Finally, the FOPID parameters can be obtained by optimizing the robust performance indexes under these constraints. Simulation experiments are carried out on examples with different types of uncertain parameters to verify the effectiveness of the tuning method. The results show that the requirements are fulfilled and that the system with the proposed FOPID controller is stable and robust to variations in the uncertain parameters. Comparisons clearly show that the controllers designed by the proposed method provide relatively robust performance.

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References

  1. J. T. Sawicki and J. Padovan, “Frequency driven phasic shifting and elastic-hysteretic partitioning properties of fractional mechanical system representation schemes,” Journal of the Franklin Institute, vol. 336, no. 3, pp. 423–433, April 1999.

    Article  Google Scholar 

  2. R. Hotzel and M. Fliess, “On linear systems with a fractional derivation: Introductory theory and examples,” Mathematics & Computers in Simulation, vol. 45, no. 3–4, pp. 385–395, 1998.

    Article  MathSciNet  Google Scholar 

  3. T. Chekari, R. Mansouri, and M. Bettayeb, “IMC-PID fractional order filter multi-loop controller design for multivariable systems based on two degrees of freedom control scheme,” International Journal of Control, Automation and Systems, vol. 16, no. 2, pp. 689–701, April 2018.

    Article  Google Scholar 

  4. V. Mohan, H. Chhabra, A. Rani, and V. Singh, “An expert 2DOF fractional order fuzzy PID controller for nonlinear systems,” Neural Computing and Applications, vol. 31, no. 8, pp. 4253–4270, January 2018.

    Article  Google Scholar 

  5. F. Merrikh-Bayat, N. Mirebrahimi, and M. R. Khalili, “Discrete-time fractional-order PID controller: Definition, tuning, digital realization and some applications,” International Journal of Control, Automation and Systems, vol. 13, no. 1, pp. 81–90, February 2015.

    Article  Google Scholar 

  6. H. Ma and Y. Li, “Fractional order exponential type discrete-time sliding mode control,” International Journal of Control, Automation and Systems, vol. 18, no. 2, pp. 374–383, February 2020.

    Article  Google Scholar 

  7. J. Cheng, J. H. Park, X. Zhao, H. Karimi, and J. Cao, “Quantized nonstationary filtering of networked Markov switching RSNSs: A multiple hierarchical structure strategy,” IEEE Transactions on Automatic Control, vol. 65, no. 11, pp. 4816–4823, 2020.

    Article  MathSciNet  Google Scholar 

  8. J. Cheng, J. H. Park, X. Zhao, J. Cao, and W. Qi, “Static output feedback control of switched systems with quantization: A nonhomogeneous sojourn probability approach,” International Journal of Robust and Nonlinear Control, vol. 29, no. 17, pp. 5992–6005, November 2019.

    Article  MathSciNet  Google Scholar 

  9. J. Cheng and Y. Zhan, “Nonstationary l2 — l∞ filtering for Markov switching repeated scalar nonlinear systems with randomly occurring nonlinearities,” Applied Mathematics and Computation, vol. 365, pp. 124714, January 2020.

    Article  MathSciNet  Google Scholar 

  10. J. Cheng, J. H. Park, J. Cao, and W. Qi, “A hidden mode observation approach to finite-time SOFC of Markovian switching systems with quantization,” Nonlinear Dynamics, vol. 100, pp. 509–521, 2020.

    Article  Google Scholar 

  11. X. Zhang and Z. Wang, “Stability and robust stabilization of uncertain switched fractional order systems,” ISA Transactions, vol. 103, pp. 1–9, August 2020.

    Article  Google Scholar 

  12. J. Zhang and Z. Wei, “A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising,” Applied Mathematical Modelling, vol. 35, no. 5, pp. 2516–2528, May 2011.

    Article  MathSciNet  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, March 2010.

    Article  MathSciNet  Google Scholar 

  14. D.-Y. Liu, Y. Tian, D. Boutat, and T. M. Laleg-Kirati, “An algebraic fractional order differentiator for a class of signals satisfying a linear differential equation,” Signal Processing, vol. 116, no. 7, pp. 78–90, November 2015.

    Article  Google Scholar 

  15. J. Tian, R. Xiong, and Q. Yu, “Fractional-order modelbased incremental capacity analysis for degradation state recognition of lithiumion batteries,” IEEE Transactions on Industrial Electronics, vol. 66, no. 2, pp. 1576–1584, February 2019.

    Article  Google Scholar 

  16. X. Shi, Y. Chen, and J. Huang, “Application of fractional-order active disturbance rejection controller on linear motion system,” Control Engineering Practice, vol. 81, no. pp. 207–214, December 2018.

    Article  Google Scholar 

  17. K. J. Åström and T. Hägglund, “The future of PID control,” Control Engineering Practice, vol. 9, no. 11, pp. 1163–1175, November 2001.

    Article  Google Scholar 

  18. A. K. Heong, G. Chong, and L. Yun, “PID control system analysis, design, and technology,” IEEE Transactions on Control Systems Technology, vol. 13, no. 4, pp. 559–576, July 2005.

    Article  Google Scholar 

  19. Q. H. Seer and J. Nandong, “Stabilization and PID tuning algorithms for second-order unstable processes with time-delays,” ISA Transactions, vol. 67, no. pp. 233–245, March 2017.

    Article  Google Scholar 

  20. I. Podlubny, “Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208–214, January 1999.

    Article  MathSciNet  Google Scholar 

  21. K. Saadaoui, S. Testouri, and M. Benrejeb, “Robust stabilizing first-order controllers for a class of time delay systems,” ISA Transactions, vol. 49, no. 3, pp. 277–282, July 2010.

    Article  Google Scholar 

  22. M. Ben Hariz, F. Bouani, and M. Ksouri, “Robust controller for uncertain parameters systems,” ISA Transactions, vol. 51, no. 5, pp. 632–640, September 2012.

    Article  Google Scholar 

  23. Z. Wang, Q. Tian, and H. Hu, “Dynamics of flexible multi-body systems with hybrid uncertain parameters,” Mechanism and Machine Theory, vol. 121, pp. 128–147, March 2018.

    Article  Google Scholar 

  24. B. Saidi, M. Amairi, S. Najar, and M. Aoun, “Bode shaping-based design methods of a fractional order PID controller for uncertain systems,” Nonlinear Dynamics, vol. 80, no. 4, pp. 1817–1838, June 2015.

    Article  MathSciNet  Google Scholar 

  25. Y. Luo, Y. Q. Chen, C. Y. Wang, and Y. G. Pi, “Tuning fractional order proportional integral controllers for fractional order systems,” Journal of Process Control, vol. 20, no. 7, pp. 823–831, August 2010.

    Article  Google Scholar 

  26. A. Chevalier, C. Francis, C. Copot, C. M. Ionescu, and R. de Keyser, “Fractional-order PID design: Towards transition from state-of-art to state-of-use,” ISA Transactions, vol. 84, no. pp. 178–186, January 2019.

  27. A. Tepljakov, B. B. Alagoz, C. Yeroglu, E. Gonzalez, S. H. HosseinNia, and E. Petlenkov, “FOPID controllers and their industrial applications: A survey of recent results,” IFAC-PapersOnLine, vol. 51, no. 4, pp. 25–30, 2018.

    Article  Google Scholar 

  28. F. Meng, S. Liu, and K. Liu, “Design of an optimal fractional order PID for constant tension control system,” IEEE Access, vol. 8, pp. 58933–58939, March 2020.

    Article  Google Scholar 

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

    Article  Google Scholar 

  30. R. Azarmi, M. Tavakoli-Kakhki, A. K. Sedigh, and A. Fatehi, “Robust fractional order PI controller tuning based on Bode’s ideal transfer function,” IFAC-PapersOnLine, vol. 49, no. 9, pp. 158–163, 2016.

    Article  Google Scholar 

  31. Y. Jin, Y. Chen, and D. Xue, “Time-constant robust analysis of a fractional order [proportional derivative] controller,” IET Control Theory & Applications, vol. 5, no. 1, pp. 164–172, February 2011.

    Article  Google Scholar 

  32. V. Feliu-Batlle and F. J. Castillo-García, “On the robust control of stable minimum phase plants with large uncertainty in a time constant. A fractional-order control approach,” Automatica, vol. 50, no. 1, pp. 218–224, January 2014.

    Article  MathSciNet  Google Scholar 

  33. S. P. Jadhav, R. H. Chile, and S. T. Hamde, “A simple method to design robust fractional-order lead compensator,” International Journal of Control, Automation and Systems, vol. 15, no. 3, pp. 1236–1248, May 2017.

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  35. H. Li, Y. Luo, and Y. 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, March 2010.

    Article  Google Scholar 

  36. C. A. Monje, A. J. Calderon, B. M. Vinagre, Y. Chen, and V. Feliu, “On fractional PIλ controllers: Some tuning rules for robustness to plant uncertainties,” Nonlinear Dynamics, vol. 38, no. 1, pp. 369–381, December 2004.

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Intelligent Machines, Chinese Academy of Sciences, No. 350 Shushan Road, Hefei, 230031, China

    Xu Li & Lifu Gao

  2. Department of Automation, University of Science and Technology of China, No. 96 JinZhai Road Baohe District, Hefei, 230026, China

    Xu Li

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  1. Xu Li
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Xu Li received his M.S. degree from the Hefei University of Technology, Hefei, China, in 2014. He is currently pursuing a Ph.D. degree in the Department of Automation, University of Science and Technology of China, Hefei, China. His research interests include fractional order system, nonlinear control, and industrial process control.

Lifu Gao received his M.S. degree in pattern recognition and intelligent system from Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei, China, in 1999 and a Ph.D. degree in pattern recognition and intelligent system from University of Science and Technology of China, Hefei, China, in 2002. From 2005 to 2011, He was a Postdoctoral Fellow and a Senior Engineer with European Molecular Biology Laboratory, Hamburg, Germany. Since 2011, he has been a Research Fellow with Institute of Intelligent Machines, Chinese Academy of Sciences, Hefei, China. His research interests include robotics and sensor.

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Li, X., Gao, L. Robust Fractional-order PID Tuning Method for a Plant with an Uncertain Parameter. Int. J. Control Autom. Syst. 19, 1302–1310 (2021). https://doi.org/10.1007/s12555-019-0866-y

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  • DOI: https://doi.org/10.1007/s12555-019-0866-y

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