IMC-PID Fractional Order Filter Multi-loop Controller Design for Multivariable Systems Based on Two Degrees of Freedom Control Scheme

  • Tassadit Chekari
  • Rachid Mansouri
  • Maamar Bettayeb
Regular Paper Control Theory and Applications


An IMC-PID fractional order filter multi-loop controller design method based on two degrees of freedom paradigm is proposed for Multiple Input-Multiple Output systems with time delays. The interactions among the control loops are considered as disturbances. Thus, a two degrees of freedom control scheme, used for monovariable system to ensure the disturbance rejection, is extended to multivariable systems in order to reduce the effect of the coupling among the control loops. The proposed controller design method requires the control pairing selection with the least interactions and a set-point controller is calculated. An interactions reduction effect controller is calculated for each loop by defining a suitable complementary sensitivity function. The proposed controller design method is simple and systematic in relation with the desired closed loop specifications of each loop. The controllers obtained ensure robustness to process variations. Two illustrative examples are presented to show the merits of the proposed method.


Bode’s ideal transfer function IMC control MIMO systems multiloop control 2DOF structure 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. D. S. Coelho and V. C. Mariani, “Firefly algorithm approach based on chaotic Tinkerbell map applied to multivariable PID controllers tuning,” computers & mathematics with applications, vol. 64, no. 8, pp. 2371–2382, 2012. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    X. Luan, Q. Chen, and F. Liu, “Centralized PIcontrol for high dimensional multivariable systems based on equivalent transfer function,” ISA Transactions, vol. 53, no. 5, pp. 1554–1561, 2014. [click]CrossRefGoogle Scholar
  3. [3]
    T. Liu, W. Zhang, and D. Gu, “Analytical design of decoupling internal model control (IMC) scheme for two-input two-output (TITO) processes with time delays,” Industrial & engineering chemistry research, vol. 45, no. 9, pp. 3149–3160, 2006.CrossRefGoogle Scholar
  4. [4]
    J. Garrido, F. Vázqueza, and F. Morrila, “Inverted decoupling internal model control for square stable multivariable time delay systems,” Journal of Process Control, vol. 24, no. 11, pp. 1710–1719, 2014. [click]CrossRefGoogle Scholar
  5. [5]
    W. Zhang, Y. Yang, T. liu, and W. Zhang, “Multivariable disturbance observer-based H2 analytical decoupling control design for multivariable systems,” International Journal of Systems Science, vol. 47, no. 1, pp. 179–193, 2016. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. K. Maghade and B. M. Patre, “Decentralized PI/PID controllers based on gain and phase margin specifications for TITO processes,” ISA Transactions, vol. 51, no. 4, pp. 550–558, 2012. [click]CrossRefGoogle Scholar
  7. [7]
    E. Bristol, “On a new measure of interaction for multivariable process control,” IEEE Transactions on automatic control, vol. 11, no. 1, pp. 133–134, 1966. [click]CrossRefGoogle Scholar
  8. [8]
    A. Niederlinski, “Two-variables distillation control: decouple or not decouple,” AICHE journal, vol. 17, no. 5, pp. 1261–1263, 1971. [click]CrossRefGoogle Scholar
  9. [9]
    T. O. Ajayi and I. S. Ogboh, “Determination of control pairing for higher order multivariable systems by the use of multi-ratios,” International Journal of Scientific and Engineering Research, vol. 3, no. 3, 2012.Google Scholar
  10. [10]
    R. H. Naik, D. V. A. Kumar, and K. S. R. Anjaneyulu, “Control configuration selection and controller design for multivariable processes using normalized gain,” International Journal of Electrical, Computer, Electronics and Communication Engineering, vol. 8, no. 10, pp. 1636–1640, 2014.Google Scholar
  11. [11]
    W. Luyben, “Simple method for tuning SISO controllers in multivariable systems,” Industrial & Engineering Chemistry Process Design and Development, vol. 25, no. 3, pp. 654–660, 1986. [click]CrossRefGoogle Scholar
  12. [12]
    B. C. Ji, E. M. Lee, Y. Han, and J. Lee, “Computation of multiloop controllers having desired closed loop responses,” Korean Journal of Chemical Engineering, vol. 24, no. 4, pp. 562–566, 2007. [click]CrossRefGoogle Scholar
  13. [13]
    C. G. Economou and M. Morari, “Internal model control: multiloop design,” Industrial & Engineering Chemistry Process Design and Development, vol. 25, no. 2, pp. 411–419, 1986.CrossRefGoogle Scholar
  14. [14]
    Z. Zhao, B. Hu, and J. Liang, “Multiloop adaptive internal model control based on a dynamic partial least squares model,” Journal of Zhejiang University-SCIENCE (Applied Physics and Engineering), vol. 12, no. 3, pp. 190–200, 2011.CrossRefGoogle Scholar
  15. [15]
    T. N. L. Vu, J. Lee, and M. Lee, “Design of Multiloop PID Controllers Based on the Generalized IMC-PID Method with Mp Criterion,” International Journal of Control, Automation, and Systems, vol. 5, no. 2, pp. 212–217, 2007.Google Scholar
  16. [16]
    M. Morari and E. Zafiriou, Robust Process Control, Prentice hall, Englewood Cliffs, NJ, 1989.zbMATHGoogle Scholar
  17. [17]
    T. N. L. Vu and M. Lee, “Independent design of multi-loop PI/PID controllers for interacting multivariable processes,” Journal of Process Control, vol. 40, no. 8, pp. 922–933, 2010. [click]CrossRefGoogle Scholar
  18. [18]
    H. P. Huang, J. C. Jeng, C. H. Chiang, and W. Pan, “A direct method for multi-loop PI/PID controller design,” Journal of Process Control, vol. 13, no. 8, pp. 769–786, 2003. [click]CrossRefGoogle Scholar
  19. [19]
    X. Luan, Q. Chen, and F. Liu, “Equivalent transfer function based multiloop PIcontrol for high dimensional multivariable systems,” International Journal of Control, Automation, and Systems, vol. 13, no. 2, pp. 1–7, 2015.CrossRefGoogle Scholar
  20. [20]
    Y. Wei, J. Qiu, H. R. Karimi, and M. Wang, “New results on H dynamic output feedback control for Markovian jump systems with time varying delay and defective mode information,” Optimal Control Applications and Methods, vol. 35, no. 6, pp. 656–675, 2014. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Y. Wei, J. Qiu, and S. Fu, “Mode-dependent nonrational output feedback control for continuous time semi Markovian jump systems with time varying delay,” Non linear Analysis: Hybrid Systems, vol. 16, pp. 51–71, 2015.MathSciNetzbMATHGoogle Scholar
  22. [22]
    Y. Wei, X. Peng, and J. Qiu, “Robust and non-fragile static output feedback control for continuous-time semi- Markovian jump systems,” Transactions of the Institute of Measurement and Control, vol. 38, no. 9, pp. 1136–1150, 2016.CrossRefGoogle Scholar
  23. [23]
    Y. Wei, J. Qiu, P. Shi, and H. K. Lam, “A new design of H-infinity piecewise filtering for Discrete-time nonlinear time-varying delay systems via TSfuzzy affine models,” IEEE Transactions on Systems, Man, and Cybernetics: systems, 2016.Google Scholar
  24. [24]
    Y. Wei, J. Qiu, H. K. Lam, and L. Wu, “Approaches to TSfuzzy-affine-model-based reliable output feedback control for nonlinear Itô stochastic systems,” IEEE Transactions on Fuzzy Systems, 2016.Google Scholar
  25. [25]
    I. Podlubny, “Fractional-order-systems and PIaDm controller,” IEEE Transactions on automatic control, vol. 44, no. 1, pp. 208–214, 1999. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    R. S. Barbosa, J. A. T. Machado, and I. M. Ferreira, “Tuning of PID controllers based on Bode’s ideal transfer function,” Nonlinear Dynamics, vol. 38, no. 1-4, pp. 305–321, 2004. [click]CrossRefzbMATHGoogle Scholar
  27. [27]
    J. Yi and B. G. Cao, “Design of fractional order controller based on particle swarm optimisation,” International Journal of Control, Automation, and Systems, vol. 4, no. 6, pp. 775–781, 2006.Google Scholar
  28. [28]
    M. Bettayeb and R. Mansouri, “IMC-PID-fractional-orderfilter controller design for integer order systems,” ISA Transactions, vol. 53, no. 5, pp. 1620–1628, 2014. [click]CrossRefGoogle Scholar
  29. [29]
    M. Bettayeb and R. Mansouri, “Fractional IMC-PID-filter controllers design for non integer order systems,” Journal of Process Control, vol. 24, no. 4, pp. 261–271, 2014.CrossRefGoogle Scholar
  30. [30]
    K. Titouche, R. Mansouri, M. Bettayeb, and U. M. Al-Saggaf, “Internal model control proportional integral derivative fractional-order filter controllers design for unstable delay systems,” Journal of Dynamic Systems, Measurement and Control, vol. 138, no. 2, pp. 451–461, 2016.Google Scholar
  31. [31]
    D. N. Gruel, P. Lanusse, and A. Oustaloup, “Robust control design for multivariable plants with time-delay,” Chemical Engeneering Journal, vol. 146, no. 3, pp. 414–427, 2009. [click]CrossRefGoogle Scholar
  32. [32]
    D. N. Gruel, P. Lanusse, and A. Oustaloup, “Commande CRONE des systemes multi-entrés multi-sorties non carrés retardés,” e-STA Sciences et Technologies de l’Automatique, vol. 6, no. 1, pp. 21–28, 2009.Google Scholar
  33. [33]
    P. Lanusse, D. N. Gruel, A. Lamara, A. Lesobre, X. Wang, Y. Chamaillard, and A. Oustaloup, “Development of a fractional order based MIMO controller for high dynamic engine testbeds,” Control Engineering Practice, vol. 56, pp. 174–189, 2016.CrossRefGoogle Scholar
  34. [34]
    C. I. Muresan, A. Dutta, E. H. Dulf, Z. Pinar, A. Maxim, and C. M. Ionescu, “Tuning algorithms for fractional order internal model controllers for time delay processes,” International Journal of Control, vol. 89, no. 3, pp. 579–593, 2016. [click]MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    C. I. Muresan, R. De Keyser, and C. M. Ionescu, “Autotuning method for a fractional order controllers for a multivariable 13C isotope separation column,” Proc. of European Control Conference, Denmark, pp. 358–363, 2016.Google Scholar
  36. [36]
    C. I. Muresan, E. H. Dulf, and C. M. Ionescu, “Robustness evaluation of a multivariable fractional order PIcontroller for time delay processes,” Control and Intelligent Systems, vol. 42, no. 2, pp. 112–118, 2014.CrossRefGoogle Scholar
  37. [37]
    Z. Li and Y. Q. Chen, “Ideal simplified and inverted decoupling of fractional order TITO processes,” IFAC Proceedings, vol. 47, no. 3, pp. 2897–2902, 2014. [click]CrossRefGoogle Scholar
  38. [38]
    U. S. Banu and S. K. Lakshmanaprabu, “Adaptive multiloop fractional order PID controller tuning using bat colony optimization for quadruple tank process,” Proc. of International Conference on Robotics, Automation, Control and Embedded Systems (RACE), 18-20 February, Chennai, India, pp. 1–8, 2015.Google Scholar
  39. [39]
    F. G. Prakash and V. Alamelumangai, “Design of predictive fractional order PIcontroller for the quadruple tank process,” WSEAS, Transactions on Systems and Control, vol. 10, pp. 85–94, 2015Google Scholar
  40. [40]
    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. [click]CrossRefGoogle Scholar
  41. [41]
    J. E. Normey-Rico and E. F. Camacho, Control of Deadtime Processes, Springer Science & Business Media, 2007.Google Scholar
  42. [42]
    T. Hélie, “Simulation of fractional-order low-pass filters,” IEEE/ACM Transactions on Audio, Speech and Language Processing (TASLP), vol. 22, no. 11, pp. 1636–1647, 2014. [click]CrossRefGoogle Scholar
  43. [43]
    I. Petras, “The fractional-order controllers: methods for their synthesis and application,” arXiv preprint math/0004064, 2000.Google Scholar
  44. [44]
    B. A. Orgunaike, J. P. Lemaire, M. Morari, and W. H. Ray, “Advanced multivariable control of a pilot plant distillation column,” AICHE Journal, vol. 29, no. 4, pp. 632–640, 1983. [click]CrossRefGoogle Scholar

Copyright information

© Institute of Control, Robotics and Systems and The Korean Institute of Electrical Engineers and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Tassadit Chekari
    • 1
  • Rachid Mansouri
    • 1
  • Maamar Bettayeb
    • 2
  1. 1.L2CSP LaboratoryMouloud Mammeri UniversityTizi OuzouAlgeria
  2. 2.Electrical and Computer Engineering Department, University of Sharjah, United Arab Emirates and Distinguished Adjunct ProfessorCenter of Excellence in Intelligent Engineering Systems (CEIES) King Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations