A new class of fractional proportional–integral (PI) and PI–derivative (PID) controllers is defined, based on a closed-loop reference model expressing the dynamical and robust performances. Moreover, the filtering of the derivative action is explicitly formulated using a generalized second-order reference model. Numerical simulations with simple systems illustrate some general principles that can be used as guidelines for the tuning of fractional PID controllers. For more complex or delayed systems, a tuning method based on the time moments approach is also proposed. Numerical examples illustrate the capabilities of this tuning technique.
MonjeC.ChenY.VinagreB.XueD.FeliuV.Fractional-order systems and controls: Fundamentals and applications, 2010 (Springer, Berlin).
2.
BodeH. W.Network analysis and feedback amplifier design, 1945 (Van Nostrand, Princeton, New Jersey).
3.
ManabeS.The non integer integral and its application to control systems. JIEE (Japanese Institute of Electrical Engineers) J., 1960, 80, 589–597.
4.
HorowitzI.Application of quantitative feedback theory (QFT) to highly uncertain non linear time varying plants. In Proceedings of the European Control Conference (ECC’91), Grenoble, France, 1991.
OustaloupA.MathieuB.La commande CRONE, du scalaire au multivariable, 1995 (Hermes, Paris).
7.
SabatierJ.OustaloupA.Garcia IturrichaA.LanusseP.CRONE control: Principles and extension to time-variant plants with asymptotically constant coefficients. Nonlin. Dynam., 2002, 29(1-4), 363–385.
8.
ValerioD.Da CostaJ.A review of tuning methods for fractional PIDs. In Proceedings of the 4th IFAC Workshop on Fractional differentiation and its applications, FDA’10, Badajoz, Spain, 2010.
9.
PodlubnyI.DorcakL.KostialI.On fractional derivatives, fractional order systems and PIαDβ. control. In Proceedings of the Conference on Decision and control, San Diego, 1997.
10.
BarbosaR.MachadoJ. R.FerreiraI.Tuning of PID controllers based on Bode’s ideal transfer function. Nonlin. Dynam., 2004, 38, 305–321.
11.
MonjeC.VinagreB. M.ChenY.FeliuV.LanusseP.SabatierJ.Optimal tunings for fractional PIλDμ controllers. In Fractional differentiation and its applications (Eds Le MehauteA.Tenreiro MachadoJ. A.TrigeasouJ.-C.SabatierJ.), 2005, Part 3, Chapter 13, pp. 675–686.
12.
ValerioD.Da CostaJ.Tuning rules for fractional PIDs. In Theoretical developments and applications in physics and engineering, 2007 (Springer, Berlin).
13.
ZhaoC.XueD.ChenY. Q.A fractional order PID tuning algorithm for a class of fractional order plants. In Proceedings of the IEEE International Conference on Mechatronics and automation, Niagara, Canada, 2005, pp. 216–221.
14.
MonjeC. A.CalderonA. J.VinagreB. M.ChehnY.FeliuV.On fractional PIα controllers: Some tuning rules for robustness to plant uncertainties. Nonlin. Dynam., 2004, 38, 369–381.
15.
LuoaY.ChencY. Q.WangC. Y.PiY. G.Tuning fractional order proportional integral controllers for fractional order systems. J. Process Contr., 2010, 20, 823–831.
16.
VinagreP.PodlubnyI.DorcakL.FeliuV.On fractional PI controllers: a frequency domain approach. In Proceedings of the IFAC Workshop, Terrasa, Spain, 2000, pp. 53–58.
17.
TrigeassouJ.-C.Une méthode de réglage des régulateurs PID. Revue Automatisme, 1978, mars/avril.
18.
MaamriN.Identification et commande des systèmes à représentation continue par la méthode des moments. PhD Thesis, University of Poitiers, France, 1991.
19.
MillerK.RossB.An introduction to the fractional calculus and fractional differential equations, 1993 (Wiley, New York).
20.
OustaloupA.La dérivation non entière: théorie, synthèse et applications,1995 (Hermes, Paris).
21.
PodlubnyI.Fractional differential equations, 1999 (Academic, San Diego).
22.
TrigeassouJ.-C.PoinotT.LinJ.OustaloupA.LevronF.Modelling and identification of a non integer order system. In Proceedings of the ECC’99, Karlsruhe, Germany, 1999.
23.
TrigeassouJ.-C.MaamriN.State space modelling of fractional differential equations and the initial condition problem. In Proceedings of the IEEE SSD’09, Djerba, Tunisia, 2009.
24.
PodlubnyI.Fractional-order systems and PID-controllers. IEEE Trans. Autom. Contr., 1999, 44, 208–213.
25.
MaamriN.TrigeassouJ.-C.TenoutitM.On fractional PI and PID controllers. In Proceedings of the 4th IFAC Workshop on Fractional differentiation and its applications, FDA’10, Badajoz, Spain, 2010.
26.
TrigeassouJ.-C.La méthode des moments en automatique. In Proceedings of CIFA, Lille, France, 2000.
27.
TenoutitM.MaamriN.TrigeassouJ.-C.A Time Moments approach to the design of robust fractional PID controllers. In Proceedings of the 8th International Multi-Conference on Systems, signals and devices, IEEE SSD’11, Sousse, Tunisia, 2011.
28.
MonrouxF.Méthodologie générale de synthèse de correcteurs par la méthode des moments, approche mixte: Fréquentielle et temporelle. PhD Thesis, University of Poitiers, France, 1999.
29.
TenoutitM.MaamriN.TrigeassouJ.-C.An output feedback approach to the design of robust fractional PI and PID controllers. In Proceedings of the 18th World Congress of the International Federation of Automatic Control (W-C IFAC), Milan, Italy, 2011.
