This paper proposes a scheme for computing the stable regions from which the parameters of fractional-order proportional–integral (PI) and proportional–derivative (PD) controllers can be selected. In particular, a previously investigated method for plotting the stability regions of PIλ and PDμ controllers is extended to their integer-order approximations. Next, the two stability regions are compared and their overlapping sections, referred to as the implementable stability region, are derived. In order for the final closed-loop system to be stable, the designer must choose the controller parameters from the overlapping section of the two stability regions. Simulation results for two example systems are provided to illustrate the effectiveness of the proposed method. The authors believe that the devised scheme can facilitate the design procedure for fractional-order PI and PD controllers.
PodlubnyI.Fractional differential equation1999Academic Press, San Diego
2.
BarbosaR.MachadoJ. T.FerreiraI. M.Tuning of PID controllers based on Bodes ideal transfer function. Nonlinear Dynamics200438, (1), 305–321
3.
ChenY. Q.MooreK. L.VinagreB. M.PodlubnyI.Robust PID controller autotuning with a phase shaper. In Proceedings of the 1st IFAC Symposium on Fractional differentiation and its applications (FDA04), Bordeaux, France 2004.
4.
ChenY.HuC.MooreK.Relay feedback tuning of robust PID controllers with iso-damping property. In Proceedings of the 42nd IEEE Conference on Decision and control, Maui, HI, USA December 2003, vol. 3, pp. 2180–2185
5.
ÅströmK. J.HägglundT.Advanced PID control2005The Instrumentation, Systems, and Automation Society, Research Triangle Park, North Carolina, USA
6.
ÅströmK. J.HägglundT.PID controllers: Theory, design and tuning, second edition 1995The Instrumentation, Systems, and Automation Society, Research Triangle Park, North Carolina, USA
7.
PodlubnyI.Fractional-order systems and PIλ Dμ-controllers. IEEE Trans. Autom. Control199944, (1), 208–214
8.
ValerioD.da CostaJ. S.Tuning of fractional PID controllers with Ziegler-Nichols-type rules. Signal Processing200686, (10), 2771–2784, Special section: Fractional calculus applications in signals and systems.
9.
MonjeC. A.VinagreB. M.FeliuV.ChenY.Tuning and auto-tuning of fractional order controllers for industry applications. Control Engng Practice200816, (7), 798–812
10.
MonjeC. A.CalderonA. J.VinagreB. M.ChenY.FeliuV.On fractional PIλ controllers: Some tuning rules for robustness to plant uncertainties. Nonlinear Dynamics200438, (1), 369–381
11.
ChenY.BhaskaranT.XueD.Practical tuning rule development for fractional order proportional and integral controllers. J. Comput. Nonlinear Dynamics20083, (2), 021 403.1–021 403.8
12.
CastilloF. J.FeliuV.RivasR.SnchezL.Design of a class of fractional controllers from frequency specifications with guaranteed time domain behavior. Computers Math. Applic.201059, (5), 1656–1666
13.
BiswasA.DasS.AbrahamA.DasguptaS.Design of fractional-order PIλ Dμ controllers with an improved differential evolution. Engng Applic. Artif. Intell.200922, (2), 343–350
14.
ZhaoC.XueD.ChenY.A fractional order PID tuning algorithm for a class of fractional order plants. In IEEE International Conference Proceedings of Mechatronics and automation July–August 2005, vol. 1, pp. 216–221
15.
ÅströmK. J.HägglundT.The future of PID control. Control Engng Practice20019, (11), 1163–1175
16.
BouafouraM. K.BraiekN. B.PIλ Dμ controller design for integer and fractional plants using piecewise orthogonal functions. Communs. Nonlinear Sci. Numer. Simulation201015, (5), 1267–1278
17.
MaC.HoriY.Fractional-order control: Theory and applications in motion control [past and present]. Ind. Electronics Mag., IEEE20071, (4), 6–16
18.
FangJ.ZhengD.RenZ.Computation of stabilizing PI and PID controllers by using Kronecker summation method. Energy Conversion Mgmnt200950, (7), 1821–1827
19.
HamamciS.An algorithm for stabilization of fractional-order time delay systems using fractional-order PID controllers. IEEE Trans. Autom. Control200752, (10), 1964–1969
20.
TanN.KayaI.YerogluC.AthertonD. P.Computation of stabilizing PI and PID controllers using the stability boundary locus. Energy Conversion Mgmnt200647, (18–19), 3045–3058
21.
CaponettoR.DongolaG.FortunaL.GalloA.New results on the synthesis of FO-PID controllers. Communs Nonlinear Sci. Numer. Simulation201015, (4), 997–1007
22.
HohenbichlerN.All stabilizing PID controllers for time delay systems. Automatica200945, (11), 2678–2684
23.
SoylemezN.MunroM. T.BakiH.Fast calculation of stabilizing PID controllers. Automatica200339, 121–126
24.
HohenbichlerN.AckermannJ.Computing stable regions in parameter spaces for a class of quasipolynomials. In Proceedings of the 4th IFAC Workshop on Time delay systems, Rocquencourt, France 2003.
25.
HamamciS. E.Stabilization using fractional-order PI and PID controllers. Nonlinear Dynamics200851, (1–2), 329–343
26.
HamamciS. E.KoksalM.Calculation of all stabilizing fractional-order PD controllers for integrating time delay systems. Computers and Math. Applic.201059, (5), 1621–1629, DOI: 10.1016/j.camwa.2009.08.049.
27.
TavazoeiM. S.HaeriM.Rational approximations in the simulation and implementation of fractional-order dynamics: A descriptor system approach. Automatica201046, 94–100
28.
OustaloupA.LevronF.MathieuB.NanotF. M.Frequency-band complex noninteger differentiator: Characterization and synthesis. IEEE Trans. Circuits and Systems I: Fundamental Theory and Applic.200047, (1), 25–39
29.
MatsudaK.FujiiH.H∞-optimized wave-absorbing control: Analytical and experimental results. J. Guidance, Control, Dynamics199316, (6), 1146–1153
30.
CharefA.Analogue realisation of fractional-order integrator, differentiator and fractional PIλ Dμ controller. IEEE Proc. Control Theory Applic.2006153, (6), 714–720
31.
CarlsonG.HalijakC.Approximation of fractional capacitors (1/s)(1/n) by a regular Newton process. IEEE Trans. Circuit Theory196411, (2), 210–213
32.
RoyS. D.On the realization of a constant-argument immittance or fractional operator. IEEE Trans. Circuit Theory196714, (3), 264–274
33.
ChenY.VinagreB. M.PodlubnyI.Continued fraction expansion approaches to discretizing fractional order derivatives an expository review. Nonlinear Dynamics200438, (1–2), 155–170
34.
PetrasI.PodlubnyI.O'LearyP.DorcakL.VinagreB. M.Analogue realization of fractional order controllers2002FBERG, Technical University of Kosice, Kosice
35.
VinagreB.PodlubnyI.HernándezA.FeliuV.Some approximations of fractional order operators used in control theory and applications. Fractional Calculus and Appl. Analysis20003, (3), 231–248
36.
PodlubnyI.PetrasI.VinagreB. M.O'LearyP.DorcakL.Analogue realizations of fractional-order controllers. Nonlinear Dynamics200229, (1), 281–296
37.
TavazoeiM. S.HaeriM.BoloukiS.SiamiM.Stability preservation analysis for frequency-based methods in numerical simulation of fractional order systems. SIAM J. Numer. Analysis200847, 321–338
38.
BonnetC.PartingtonJ. R.Coprime factorizations and stability of fractional differential systems. Systems & Control Lett.200041, (3), 167–174
39.
HohenbichlerN.AckermannJ.Synthesis of robust PID controllers for time delay systems. In Proceedings of the European Control Conference, Cambridge 2003.
40.
OgataK.Modern control engineering, 5th edition 2009Prentice Hall PTR, Upper Saddle River, NJ
41.
PetrasI.ChenY.VinagreB.PodlubnyI.Stability of linear time invariant systems with interval fractional orders and interval coefficients. In Proceedings of the 2nd IEEE International Conference on Computational cybernetics (ICCC 2004), Vienna, Austria, August 2004, pp. 341–346
42.
CharefA.SunH.TsaoY.OnaralB.Fractal system as represented by singularity function. Trans. Autom. Control, IEEE199237, (9), 1465–1470
43.
ChengY.-C.HwangC.Stabilization of unstable first-order time-delay systems using fractional-order pd controllers. J. Chin. Inst. Engrs200629, (2), 241–249
