This paper focuses on computing the stabilizing region of fractional-order proportional integral derivative (FOPID) controllers for interval fractional-order plants. In this study, first, a theorem is presented to investigate the robust stability of the given closed-loop system. Then, an algorithm is proposed to solve the problem of robustly stabilizing the interval fractional-order plant using FOPID controllers. Also, two auxiliary functions are proposed to achieve the control requirements on the disturbance rejection and the noise reduction. Finally, four numerical examples are presented to confirm the obtained results.
BonnetCPartingtonJR (2002) Analysis of fractional delay systems of retarded and neutral type. Automatica38(7): 1133–1138.
2.
BrownJWChurchillRV (2009) Complex Variables and Applications Eighth Edition. New York, NY: McGraw-Hill Book Company.
3.
BusłowiczM (2008) Stability of linear continuous-time fractional order systems with delays of the retarded type. Bulletin of the Polish Academy of Sciences: Technical Sciences56(4): 158–163.
4.
ChenPLuoYPengY, et al. (2021) Optimal robust fractional order controller synthesis for first order plus time delay systems. ISA Transactions114: 136–149.
5.
DulfEH (2019) Simplified fractional order controller design algorithm. Mathematics7(12): 1166.
6.
DwivediPPandeyS (2021) Tuning rules: Graphical analysis and experimental validation of a simplified fractional order controller for a class of open-loop unstable systems. Asian Journal of Control23(5): 2293–2310.
7.
EbrahimiMAsgariM (2021) Robust fractional-order fixed-structure controller design for uncertain non-commensurate fractional plants using fractional Kharitonov theorem. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering235(8): 1375–1387.
8.
Feliu-BatlleVRivas-PerezRCastillo-GarcíaF (2021) Design of a PI controller for the robust control of the steam pressure in the steam drum of a Bagasse-fired boiler. IEEE Access9: 95123–95134.
9.
GaoSMeteredHBassuinyA, et al. (2017) Multi-objective genetic algorithm fractional-order PID controller for semi-active magnetorheologically damped seat suspension. Journal of Vibration and Control23(8): 1248–1266.
10.
GaoZ (2018) An analytical method on the stabilization of fractional-order plants with one fractional-order term and interval uncertainties using fractional-order PI D controllers. Transactions of the Institute of Measurement and Control40(15): 4133–4142.
11.
GaoZ (2015) Robust stabilization criterion of fractional-order controllers for interval fractional-order plants. Automatica61: 9–17.
12.
GaoZ (2019) Analytical method on stabilisation of fractional-order plants with interval uncertainties using fractional-order PI D controllers. International Journal of Systems Science50(5): 935–953.
13.
GaoZZhaiL-RYan-DongL (2017) Robust stabilizing regions of fractional-order PI controllers for fractional-order systems with time-delays. International Journal of Automation and Computing14(3): 340–349.
14.
GeorgeMAKamathDV (2020) Design and tuning of fractional order PiD (FOPID) controller for speed control of electric vehicle on concrete roads. In: 2020 IEEE international conference on power electronics, smart grid and renewable energy (PESGRE), Cochin, India, 2–4January2020, pp. 1–6. New York: IEEE.
15.
GhorbaniMTavakoli-KakhkiM (2021a) Robust stability analysis of uncertain incommensurate fractional order quasi-polynomials in the presence of interval fractional orders and interval coefficients. Transactions of the Institute of Measurement and Control43: 1117–1125.
16.
GhorbaniMTavakoli-KakhkiM (2021b) Robust stability analysis of a general class of interval delayed fractional order plants by a general form of fractional order controllers. Mathematical Methods in the Applied Sciences44: 10172–10189.
17.
GhorbaniMTavakoli-KakhkiMEstaramiAA (2019) Robust FOPID stabilization of retarded type fractional order plants with interval uncertainties and interval time delay. Journal of the Franklin Institute356(16): 9302–9329.
18.
GhorbaniMTavakoli-KakhkiMTepljakovA, et al. (2021) Robust stability analysis of interval fractional-order plants with interval time delay and general form of fractional-order controllers. IEEE Control Systems Letters6: 1268–1273.
19.
HamamciSE (2007) An algorithm for stabilization of fractional-order time delay systems using fractional-order PID controllers. IEEE Transactions on Automatic Control52(10): 1964–1969.
20.
HamamciSE (2008) Stabilization using fractional-order PI and PID controllers. Nonlinear Dynamics51(1): 329–343.
21.
IdirAKidoucheMBensafiaY, et al. (2018) Speed control of DC motor using PID and FOPID controllers based on differential evolution and PSO. Evolutionary Computation20: 21.
22.
LeeYKWatkinsJM (2012) Determination of all stabilizing fractional-order PID controllers that satisfy a weighted sensitivity constraint. In: 2012 IEEE 51st IEEE conference on decision and control (CDC), Maui, HI, 10–13December, pp. 254–259. New York: IEEE.
23.
LevineWS (2018) The Control Systems Handbook: Control System Advanced Methods. Boca Raton, FL: CRC Press.
24.
LiangTChenJZhaoH (2013) Robust stability region of fractional order controller for fractional order interval plant. International Journal of Systems Science44(9): 1762–1773.
25.
LuoYChenYQ (2011) Stabilizing and robust FOPI controller synthesis for first order plus time delay systems. In: 2011 50th IEEE conference on decision and control and European control conference, Orlando, FL, 12–15December, pp. 2040–2045. New York: IEEE.
26.
LuoYChenYQ (2012) Stabilizing and robust fractional order PI controller synthesis for first order plus time delay systems. Automatica48(9): 2159–2167.
27.
MatusuRŞenolBPekařL (2020) Robust PI control of interval plants with gain and phase margin specifications: Application to a continuous stirred tank reactor. IEEE Access8: 145372–145380.
28.
MonjeCAVinagreBMFeliuV, et al. (2008) Tuning and auto-tuning of fractional order controllers for industry applications. Control Engineering Practice16(7): 798–812.
29.
MoornaniKAHaeriM (2009) On robust stability of linear time invariant fractional-order systems with real parametric uncertainties. ISA Transactions48(4): 484–490.
30.
MoornaniKAHaeriM (2010a) On robust stability of LTI fractional-order delay systems of retarded and neutral type. Automatica46(2): 362–368.
31.
MoornaniKAHaeriM (2010b) Robust stability testing function and Kharitonov-like theorem for fractional order interval systems. IET Control Theory & Applications4(10): 2097–2108.
32.
OustaloupALevronFMathieuB, et al. (2000) Frequency-band complex noninteger differentiator: Characterization and synthesis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications47(1): 25–39.
33.
Rezaei EstakhrouiyehMGharaveisiAValiM (2018) Fractional order proportional-integral-derivative controller parameter selection based on iterative feedback tuning. Case study: Ball Levitation system. Transactions of the Institute of Measurement and Control40(6): 1776–1787.
34.
SimfukweDDPalBC (2012) Multivariable feedback control: Analysis and design. IEEE Transactions on Power Systems28(2): 1599–1608.
TanNOzguvenOFOzyetkinMM (2009) Robust stability analysis of fractional order interval polynomials. ISA Transactions48(2): 166–172.
37.
Tavakoli-KakhkiMHaeriM (2011) Fractional order model reduction approach based on retention of the dominant dynamics: Application in IMC based tuning of FOPI and FOPID controllers. ISA Transactions50(3): 432–442.
38.
TepljakovAAlagozBBYerogluC, et al. (2021) Towards Industrialization of FOPID controllers: A survey on milestones of fractional-order control and pathways for future developments. IEEE Access9: 21016–21042.
39.
TepljakovAPetlenkovEBelikovJ (2015) Robust FOPI and FOPID controller design for FFOPDT plants in embedded control applications using frequency-domain analysis. In: 2015 American control conference (ACC), Chicago, IL, 1–3July, pp. 3868–3873. New York: IEEE.
40.
TuYWHoMT (2011) Robust second-order controller synthesis for model matching of interval plants and its application to servo motor control. IEEE Transactions on Control Systems Technology20(2): 530–537.
41.
YerogluCTanN (2011) Note on fractional-order proportional–integral–differential controller design. IET Control Theory & Applications5(17): 1978–1989.
42.
ZávackáJBakošováM (2011) Robust controller design for a laboratory process with uncertainties. In: Proceedings of the 18th international conference on process control, Tatranska Lomnica, Slovakia, 14–17 June 2011, pp. 72–76.
43.
ZhengSTangXSongB (2014) A graphical tuning method of fractional order proportional integral derivative controllers for interval fractional order plant. Journal of Process Control24(11): 1691–1709.
44.
ZhengWLuoYChenYQ, et al. (2021a) A simplified fractional order PID controller’s optimal tuning: A case study on a PMSM speed servo. Entropy23(2): 130.
45.
ZhengWLuoYChenYQ, et al. (2021b) Synthesis of fractional order robust controller based on Bode’s ideas. ISA Transactions111: 290–301.
