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Licensed Unlicensed Requires Authentication Published by De Gruyter August 23, 2021

Multivariable fractional-order PID tuning by iterative non-smooth static-dynamic H synthesis

  • Atefeh Saeedian , Farshad Merrikh-Bayat EMAIL logo and Abolfazl Jalilvand


This paper proposes a new method for tuning the parameters of multi-input multi-output (MIMO) fractional-order PID (FOPID) controller. The aim of the proposed method is to calculate the parameters of this controller such that the rise time and steady-state errors of the feedback system are minimized without violating the predetermined stability margins. Mathematically, this problem is formulated as maximizing the spectral norm of the open-loop transfer matrix at zero frequency subject to a constraint on the H-norm of the sensitivity function. This problem is nonlinear in parameters of the MIMO FOPID, which can be solved using the iterative algorithm developed in this paper based on non-smooth H synthesis.


[1] P. Apkarian, D. Noll, H.D. Tuan, Fixed-order H control design via a partially augmented Lagrangian method. IFAC Proc. Volumes 36, No 8 (2003), 69–74.10.1016/S1474-6670(17)35762-2Search in Google Scholar

[2] P. Apkarian, D. Noll, J.B. Thevenet, H.D. Tuan, A spectral quadratic-SDP method with applications to fixed-order H2 and H synthesis. European J. of Control 10, No 6 (2004), 527–538.10.3166/ejc.10.527-538Search in Google Scholar

[3] P. Apkarian, D. Noll, Nonsmooth H synthesis. IEEE Trans. on Automatic Control 51, No 1 (2006), 71–86.10.1109/TAC.2005.860290Search in Google Scholar

[4] P. Apkarian, V. Bompart, D. Noll, Non-smooth structured control design with application to PID loop-shaping of a process. Internat. J. of ℝobust and Nonlinear Control: IFAC - Affiliated J. 17, No 14 (2007), 1320–1342.10.1002/rnc.1175Search in Google Scholar

[5] M. Beschi, F. Padula, A. Visioli, Fractional robust PID control of a solar furnace. Control Engineering Practice 56, (2016), 190–199.10.1016/j.conengprac.2016.04.005Search in Google Scholar

[6] S. Boyd, M. Hast, K.J. Åström, MIMO PID tuning via iterated LMI restriction. Internat. J. of ℝobust and Nonlinear Control 26, No 8 (2016), 1718–1731.Search in Google Scholar

[7] H. Chao, Y. Luo, L. Di, Y. Chen, ℝoll-channel fractional order controller design for a small fixed-wing unmanned aerial vehicle. Control Engineering Practice 18, No 7 (2010), 761–772.10.1016/j.conengprac.2010.02.003Search in Google Scholar

[8] Z. Chen, X. Yuan, B. Ji, P. Wang, H. Tian, Design of a fractional order PID controller for hydraulic turbine regulating system using chaotic non-dominated sorting genetic algorithm II. Energy Conversion and Management 84 (2014), 390–404.10.1016/j.enconman.2014.04.052Search in Google Scholar

[9] J.C. Doyle, K. Glover, P.P. Khargonekar, B.A. Francis, State-space solutions to standard H2 and H control problems. 1988 American Control Conference (1988), 1691–1696.10.23919/ACC.1988.4789992Search in Google Scholar

[10] ℝ. El-Khazali, Fractional-order PIλDμ controller design. Computers & Mathematics with Appl. 66, No 5 (2013), 639–646.10.1016/j.camwa.2013.02.015Search in Google Scholar

[11] F. Leibfritz, E.M.E. Mostafa, An interior point constrained trust region method for a special class of nonlinear semidefinite programming. SIAM J. on Optimization 12, No 4 (2002), 1048–1074.10.1137/S1052623400375865Search in Google Scholar

[12] H. Li, Y. Luo, Y. Chen, A fractional order proportional and derivative (FOPD) motion controller: tuning rule and experiments. IEEE Trans. on Control Systems Technology 18, No 2 (2009), 516–520.10.1109/TCST.2009.2019120Search in Google Scholar

[13] Y. Luo, Y. Chen, C.Y. Wang, Y.G. Pi, Tuning fractional order proportional integral controllers for fractional order systems. J. of Process Control 20, No 7 (2010), 823–831.10.1016/j.jprocont.2010.04.011Search in Google Scholar

[14] F. Merrikh-Bayat, M. Karimi-Ghartemani, Method for designing PIλDμ stabilisers for minimum-phase fractional-order systems. IET Control Theory & Appl. 4, No 1 (2010), 61–70.10.1049/iet-cta.2008.0062Search in Google Scholar

[15] F. Merrikh-Bayat, N. Mirebrahimi, M.R. Khalili, Discrete-time fractional-order PID controller: Definition, tuning, digital realization and some applications. Internat. J. of Control, Automation and Systems 13, No 1 (2015), 81–90.10.1007/s12555-013-0335-ySearch in Google Scholar

[16] F. Merrikh-Bayat, A. Salimi, Performance enhancement of non-minimum phase feedback systems by fractional-order cancellation of non-minimum phase zero on the ℝiemann surface: New theoretical and experimental results. arXiv Preprint arXiv:1611.09121 (2016).Search in Google Scholar

[17] F. Merrikh-Bayat, A uniform LMI formulation for tuning PID, multi-term fractional-order PID, and Tilt-Integral-Derivative (TID) for integer and fractional-order processes. ISA Trans. 68 (2017), 99–108.10.1016/j.isatra.2017.03.002Search in Google Scholar

[18] F. Merrikh-Bayat, An iterative LMI approach for H synthesis of multivariable PI/PD controllers for stable and unstable processes. Chemical Engin. ℝes. and Design 132 (2018), 606–615.10.1016/j.cherd.2018.02.012Search in Google Scholar

[19] ℝ. Mohsenipour, M. Jegarkandi, Fractional order MIMO controllers for robust performance of airplane longitudinal motion. Aerospace Science and Technology 91 (2019), 617–626.Search in Google Scholar

[20] D. Mustafa, K. Glover, D.J.N. Limebeer, Solutions to the H general distance problem which minimize an entropy integral. Automatica 27, No 1 (1991), 193–199.10.1016/0005-1098(91)90021-SSearch in Google Scholar

[21] P. ℝoy, B.K. ℝoy, Fractional order PI control applied to level control in coupled two tank MIMO system with experimental validation. Control Engin. Practice 48 (2016), 119–135.Search in Google Scholar

[22] J.B. Thevenet, D. Noll, P. Apkarian, Nonlinear spectral SDP method for BMI-constrained problems: Applications to control design. Informatics in Control, Automation and ℝobotics I (2006), 61–72.10.1007/1-4020-4543-3_7Search in Google Scholar

[23] H.H. Vahab, C.A. Monje, Fractional-order PID control of a MIMO distillation column process using improved bat algorithm. Soft Computing 23, No 18 (2018), 8887–8906.Search in Google Scholar

[24] W.Y. Chiu, Method of reduction of variables for bilinear matrix inequality problems in system and control designs. IEEE Trans. on Systems, Man, and Cybernetics: Systems 47, No 7 (2016), 1241–1256.10.1109/TSMC.2016.2571323Search in Google Scholar

[25] M. Zhang, X. Lin, W. Yin, An improved tuning method of fractional order proportional differentiation (FOPD) controller for the path tracking control of tractors. Biosystems Engin. 116, No 4 (2013), 478–486.10.1016/j.biosystemseng.2013.10.001Search in Google Scholar

Received: 2020-01-21
Revised: 2021-05-29
Published Online: 2021-08-23
Published in Print: 2021-08-26

© 2021 Diogenes Co., Sofia

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