ijetae_0312_66

7/28/2019 IJETAE_0312_66

1/7

International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 3, March 2012)

383

Trends in Fractional Order ControllersKaranjkar D. S.1,, Chatterji. S.2, Venkateswaran P.R.3

1Head, Instrumentation Engineering Department,Institute of Petrochemical Engineering,

Lonere, Raigad, Maharastra, India.2Professor & Head, Electrical Engineering Department,

National Institute of Technical Teachers Training and Research,

Chandigarh, India.3 Senior Development Engineer,

Bharat Heavy Electricals Limited, Tiruchirappalli, India.

[email protected]

Abstract Use of fractional order integral and derivativeoperators became very popular among many research areas

during the last decade. This paper reviews the trends in

fractional order controller and ongoing research on tuning of

its parameters. Using n integer toolbox with

MATLAB/SIMULINK, series and parallel connected

fractional order PID controllers have designed for first-order

system and simulation results have been compared. The

application of fractional order calculus in controller design

introduces superior performance than a conventionalcontroller.

Keywords Fractional order controllers, PID, nintegertoolbox.

I. INTRODUCTIONFractional order integral and derivative operators have

found several applications in large areas of research duringthe last decade. Application of fractional order calculus toconventional controller design extends the opportunity ofimproved performance. Outline of the paper is as follows.Section two discusses the introduction and development of

fractional calculus. Third section deals with applications ofthe fractional calculus in control systems and introductionof four types of fractional controller and tuning offractional PID controller have been discussed. In fifthsection the design of series and parallel FOPID controllersare presented using ninteger toolbox. Conclusions andfuture scope of work are discussed in the sixth section.

II. INTRODUCTION TO FRACTIONAL CALCULUSThe concept of fractional calculus (calculus of integrals

and derivatives of any arbitrary real or complex order) wasraised in year 1695 by Marquis de LHopital to GottfriedWilhelm Leibniz regarding solution of non-integer orderderivative. On September 30 th1695, Leibniz replied to LHopital This is an apparent paradox from which one day,useful consequences will be drawn. Between 1695 and

1819 several mathematicians (Euler in 1730, Lagrange in1772, Laplace in 1812, and so on..) mentioned it. Thequestion raised in 1695 was only partly answered 124 yearslater! [1] in 1819, by S. F. Lacroix. The real journey ofdevelopment of fractional calculus started in 1974 when thefirst monograph on fractional calculus was published byacademic press [2]. Since then many books were publisheduntil now [3-12]. Fractional calculus is presently beingapplied in the field of mathematics, physics, engineering,chemistry, computer science, mechanics, pharmacology,material science, neuroscience and neurology. Percentageutilization of fractional calculus in mathematics is around25 percentage followed by that of in physics around 20percentage while in engineering field it is around 14percentage [13]. Fractional calculus will perhaps be thecalculus of twenty-first century. The fractional-orderdifferentiator can be denoted by a general fundamental

operator qa tD as a generalization of the differential and

integral operators, which is defined as follows [14]:

7/28/2019 IJETAE_0312_66

2/7



384

, ( ) 0

1 , ( ) 0

( ) , ( ) 0

q

q

q

a t

t

q

a

d R qdt

D R q

d R q

(1)where, q is the fractional order which can be a complex

number, the constant a is related to the initial conditions.

There are two commonly used definitions for the general

fractional differentiation and integration, i.e., the

GrnwaldLetnikov (GL) and the Riemann Liouville (RL).

The GL definition is as mentioned below:

( )

00

1( ) lim ( 1) ( )

t q h

q ja t qh

j

qD f t f t jh

jh

(2)

where, . is a flooring-operator. On the other hand, the RL

definition is given by: (3)

11 ( )( )

( ) ( )

tn

qa t n q n

a

d fD f t dn q dt t

, for ( 1 )n q n

where, ( )x is the well known Eulers Gamma function.

Their is another definition of fractional differintegral

introduced by Caputo in [15] which can be written as:

( )

1

1 ( )( ) , 1

( ) ( )

tn

qa t q n

a

fD f t d n q n

q n t

(4)

The stability of fractional order differential equations can be

supposed to be as equal as that of their integer orders

counterparts, this is because; systems with memory

are typically more stable as compared to their memory-less

alternatives [16].

III. APPLICATIONS OF FRACTIONAL CALCULUS INCONTROL

Classification of dynamic systems according to the orderof the plant and the controller can be done as: i) integerorder system - integer order controller ii) integer ordersystem - fractional order controller iii) fractional ordersystem - integer order controller and iv) fractional ordersystem - fractional order controller. In fractional ordercontroller given by, Gc(s) = Kp + Ki s

- + Kd s (where Kp,

Ki and Kd are proportional, integral and derivative gainsrespectively) more parameters need to be tuned. Its unfairbut, theoretically, always better than integer order

controller.

Fig.1 PID controller: from points to plane

Many control objects are fractional-order ones, so thatthe fractional approach for control of the fractional-ordersystems becomes a meaningful work. This approach (asshown in fig.1) has changed the point based control schemeto plane based scheme. Achieving something better isalways the major concern from control engineering point ofview. Existing evidences have confirmed that the bestfractional order controller outperforms the best integerorder controller. It has also been answered in the literaturewhy to consider fractional order control even when integerorder control works comparatively quite well [17]. Sinceinteger-order PID control dominates the industry, it can bebelieved that fractional order-PID control will gain

increasing impact and wide acceptance.The use of fractional-order calculus in dynamic system

control was initiated in year 1960 [18]. Since then

application of fractional calculus was extended to

distributed control system[19], to linear feedback control

[20], for linear approximation of transfer function [21] and

to process control strategy with fractional derivatives

through recursivity [22]. In year 1990 patent was registered

for robust fractional controller (CRONE-Controle RobustedOrdre Non Entier) approach [23].

=1

PI

PD

PID

P

0

=1

0

=1

=1

PID

PDP

PI

7/28/2019 IJETAE_0312_66

3/7



385

Robust regulators with fractional structure and fractional

order state equations for the control of visco-elasticdamped structures was presented in year 1991 [24, 25]. The

concept of tilt-integral-derivative (TID) (refer fig. 2) was

presented and patent was registered in 1994 [26]. The first

practical application of CRONE (refer to CRONE Groupsintroduction and the demo of MATLAB CRONE Toolbox)

control was presented in year 1995 [27]. Fractional order

lead-lag compensator was presented in year 2000 [28].

Application of fractional calculus in control theory was

accelerated after year 2002. In the year 2002 a special issue

on fractional order calculus and its applications was

published in an international journal of Nonlinear

Dynamics and Chaos in Engineering Systems [29]. A

tutorial Workshop in IEEE International Conference on

Fractional Order Calculus in Control and Robotics was

organized in Las Vegas in the year 2002 [30]. Tuning of

PID controllers is a new research subject during last few

years. Some important articles on design and parameter

optimization of fractional controllers are presented here.

Fig. 2. Tilt-integral-derivative controller

Fig.3. FO-PID (PI

D

) controller (where, 0 1 & 0 1)

Articles [31-36] present various methodologies todesign and optimize the fractional order controller viz.PSO, GA, minimization of ISE etc. Many tuningtechniques for obtaining the parameters of controllers wereintroduced since inception of PID controller. The most wellknown tuning rules for classical controllers are given byZiegler-Nichols [37] and strm-Hgglund [38] whichhave been the milestones for developments of many othermethods.Several tuning rules similar to Ziegler and Nicholsfor integer PID, were presented [39,40,41] in the year 2006.

Quantitative Feedback Theory (QFT) and evolutionary

algorithms have been used for tuning of a fractional PIDcontroller.[42-46].

A paper [47] presents a different method for parameteradjustment scheme to improve the robustness of fractionalfuzzy adaptive sliding-mode control by the use of anANFIS architecture for two degree of freedom robot. Thepaper [48] describes an application of PSO to the problemof designing a fractional-order PID controller based onITAE. A fractional high-gain adaptive controller for a classof linear systems was presented in the paper [49]. Thepaper [50] deals with the design of FO-PI for a coupledtank system. A global search optimization method withbacterial foraging technique oriented by particle swarmoptimization was applied for optimizing five parameters offractional controllers in the year 2009 [51]. The self-tuningregulators form an important sub-class of conventionaladaptive controllers. Design of a fractional order self-tuning regulator has been presented in [52]. The particleswarm optimization algorithm has been utilized for theonline identification of parameters of the dynamicfractional order process while the tuning of the controllerparameters has been performed by differential evolution.The paper [53] presents adaptive genetic algorithm (AGA)for the multi-objective optimization design of a fractionalPID controller. The article [54] describes an application of

differential evolution (DE) to the design of fractional-ordercontroller. The paper [55] presents a robust adaptivecontrol using fractional order systems as parallel feed-forward in the adaptation loop based on the AlmostStrictly Positive Realness (ASPR) property of the plant. Inpaper [56] the tuning of FOPID controller ofelectromagnetic actuator (EMA) system for aerofin control(AFC) using particle swarm optimization (PSO) has beenpresented. Another method for tuning of fractionalcontroller based on the relay feedback technique has beenpresented in [57]. Paper [58] reports about the design ofFOPID using multi-objective optimization based geneticalgorithm.

Another work [59] published in the year 2010, onfractional PID controller tuning, explains the process oftuning by internal model control (IMC) based method withprocess described by fractional transfer function. Agraphical tuning method of PID controllers for fractional-order processes with time-delay has been presented in [60].In this work a random search optimization method has beenintroduced for fractional order model reduction and theparameters of fractional order controllers have been tunedby internal model control (IMC) based method. Fractionalorder sliding mode control has been presented in [61].

Ref

I

T

D

1/S

I/S(1/n)

s

+ Plant

Ref

P

I

D

I/S

s

+ Plant

7/28/2019 IJETAE_0312_66

4/7



386

The control objectives have been achieved by adopting

the reaching law approach of sliding mode control. In asimilar work published in the year 2010 [62], an optimalfractional order controller has been designed and theoptimal values of FOPID controller parameters forminimizing the cost function have been tuned using anevolutionary algorithm. Design of FOPD for motioncontrol and minimum integral squared error (ISE) criterionapproach is presented in [63,64]. In another paper [65] aservo control strategy for tuning of fractional order PIcontrollers is presented for fractional order system modelswith and without time delays. Recent article of the year2011 [66], presents the design of internal model controller(IMC) based fractional order two-degrees of freedomcontroller with robust control. The features of the IMCbased PID controller have been combined with fractionalorder controller. Tuning of FOPID controller using Taylorseries expansion of desired closed-loop and actual closed-loop transfer functions has been presented in [67]. Thisliterature survey reveals that the fractional controller givesbetter results as compared to the conventional PIDcontroller. Tuning of FOPID controller is difficult as fivetuning parameters need to be tuned and theoreticallyinfinite memory is essential for its digital implementation.In next section simulation example of FOPID is presented.

IV.

SIMULATION RESULTSThree types of PID configurations have been mostly used

in industrial applications- parallel, series and ideal. Parallel

configuration is commonly used in process control

applications. The response of the conventional PID with

parallel configuration can be represented as:

KP .e(s) + KC .(1/TI.s).e(s)+KD.TD.s.e(s) (5)

Where, KP, KC, and KD are proportional, integral and

derivative gains respectively, TI and TD are integral and

derivative time, and e(s) is error between measured variable

and set-point. Transfer function of fractional order PID

(series configuration) can be represented by:

KPKC [1+(1/Ti.salpha)]KD.s

beta (6)

Where, alpha and beta are the fractional powers of

integral and derivative terms respectively. Response of

Fractional order PID (parallel configuration) can be

represented by:

KP .e(s) + KC .(1/Ti.salpha).e(s)+KD.TD.s

beta.e(s) (7)

In this section, series and parallel configuration of

fractional order PID is designed in MATLAB/ SIMULINKusing nid block of ninteger toolbox, which uses croneformula with mcltime expansion for fractional operatorimplementation. Simulink model (Fig.4) is designed for

implementation of conventional (parallel) PID, parallel-

FOPID and series- FOPID controllers for first order system

with transfer function s/(s+1). Parameters of conventional

PID have been optimized using SIMULINK control design

as shown in Table.1. Design of parallel and series-FOPID

is done according to the parameters shown in table 2.

Fig.4. Simulink implementation of series, parallel and

conventional PID for first order system.

TABLEIOPTIMIZED PARAMETERS FORPARALLEL PIDWITH MINIMUM ERROR

CRITERIA FORFIRST ORDERSYSTEM

KP TI TD

1.3185 2.2306 -0.2735

7/28/2019 IJETAE_0312_66

5/7



387

TABLEIII

PARAMETERS FORPARALLEL AND SERIES FO-PIDFORFIRST ORDERSYSTEM

KP KI KD alpha beta

Parallel

FOPID

11 11 03 0.5 0.4

Series

FOPID (case

I)

11 11 03 0.5 0.4

Series

FOPID (case

II)

11 1 01 0.75 0.5

Simulation results (Fig.5) shows that fractional order PID

with parallel configuration gives better response to step

input in terms of peak overshoot and settling time, even

when the parameters of FOPID were not optimized.

Parallel FOPID do not offer any overshoot and settling time

is also much less as compared to the conventional PID

controller. Response of series FOPID - case I, with

(similar parameters as that of parallel FOPID) higher

integral gain and lower values of alpha exhibit high peakovershoot and damping. The integral gain can be lowered

while fractional order of integration can be increased in

order to minimize the peak overshoot and damping (as inseries FOPID-case II). Fractional order controller provides

additional flexibility in terms of two additional control

parameters- alpha and beta. For second order system it is

seen that series configuration of FOPID also gives better

response like parallel configuration. Based on process

under control one can select configuration of controller.

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

Responce of PID-optimized

Series FOPID responce with Kp=11,Ki=11,alfa=0.5,beta=0.4 & Kd=3

Series FOPID responce with Kp=11,Ki=1,alfa=0.75,beta=0.5 & Kd=1

Parallel FOPID responce with Kp=11,Ki=11,alfa=0.5,beta=0.4 & Kd=3

Fig.5. Responses of Fractional (Series & Parallel) and

conventional PID to step input

V. CONCLUSION AND FUTURE SCOPE OF WORKThe use of fractional calculus has gained popularity

among many research areas during the last decade. Its

theoretical and practical interests are well established

nowadays, and its applicability to science and engineering

can be considered as an emerging new analytical approach.

The introduction of fractional order calculus to

conventional controller design extends the scope of added

performance improvement. Many existing control schemes

can be modified with the notion of fractional order

calculus. Conventional PID performance has many limits

due to dead time, disturbances, noise, etc. Fractional-orderPID control is the development of the integer-order PID

control. Design of fractional order controller is an on-going

research topic now a days. Significant work has been done

on CRONE controller and its industrial applications.

Compared to integer PID controller, fractional PID

controller has more advantages, but the difficulty of tuning

methods of the fractional PID is still a challenge to be

resolved. Fractional order PID controllers could benefit the

industry significantly with a wide spread impact when

FOPID parameter tuning techniques have been well

developed. In order to achieve better results and to make

industry acceptable FOPID, a need is felt for designing newmethods for auto-tuning and self-tuning the parameters of

PID controllers and development digital algorithm for

implementation of FOPID using microprocessor or

microcontroller.

References

[1] Chen Y.Q., Fractional Order Dynamic System Control AHistorical Note and A Comparative Introduction of Four FractionalOrder Controllers, Tutorial Workshop on Fractional Order Calculusin Control and Robotics IEEE International Conference on Decisionand Control, Las Vegas, NE, USA, Dec. 2002.

[2] Oldham K. and Spanier J., The fractional calculus: Theory andapplications of differentiation and integration to arbitrary order,Academic Press, 1974.

[3] Samko S.G., Kilbas A.A., Fractional integrals and derivatives:theory and applications Gordon and Bench Science Publishers,1993. (This book was published in Russian in 1987).

[4] MillerK.S. and Ross B., An introduction to the fractional calculusand fractional differential equations, John Wiley and Sons, 1993.

[5] Carpinteri A. and Mainardi F., Fractals and fractional calculus incontinuum mechanics, Springer-Verlag, 1998.

[6] Podlubny I., Fractional differential equations, Academic press,1999.

[7] Hilfer R., Applications of fractional calculus in physics, WorldScintific, 2000.

7/28/2019 IJETAE_0312_66

6/7



388

[8] Kilbas A. A., Srivastava H.M. and Trujillo J.J., Theory andapplications of fractional differential equations, Elsevier, 2006.[9] Magin R. L., Fractional calculus in bioengineering, Begell HousePublishers, 2006.

[10] Biagini F., Zhang T.. Stochastic calculus for fractional Brownianmotion and applications, Springer, 2008.

[11] Monje C.A., Chen Y.Q., Vinagre B.M., Xue D. and Feliu V.,Fractional order systems and controls, Springer, 2010.

[12] Chen W., Sun H., and Li X., Fractional derivative in mechanics andengineering modeling, Science press, 2010.

[13] Chen Yang Quan, Applied Fractional Calculus in Controls andSignal Processing, in 49th IEEE Conf. on- Decision and Control,Atlanta, USA, Dec, 2010.

[14] Caldern, A.J., Vinagre B.M., and Feliu V., Fractional order controlstrategies for power electronic buck converters, Signal Processing,2006, pp. 28032819.

[15] Caputo, M., Linear models of dissipation whose Q is almostfrequency independent II, Geophys. J. Royal Astron. Soc. 1967, pp.

529539.[16] Ahmed. E., El-Sayed A.M.A., El-Saka H.A.A., Equilibrium points,

stability and numerical solutions of fractional order predatorpreyand rabies models, J. Math. Anal., pp. 542553, 2007.

[17] Chen Yang Quan, Applied Fractional Calculus in Control,American Control Conference-ACC2009, St. Louis, Missouri, USA,June 10-12, 2009,

[18] Manabe S., The non-integer integral and its application to controlsystems, JIEE (Japanese Institute of Electrical Engineers) Journal,vol. 6, no. 3/4, pp. 8387, 1961. Translation from JIEE J., vol. 80,no. 860, 1960, pp. 589-597.

[19] Chen C. F., Tsay Y. T. and Wu T. T., Walsh operational matricesfor fractional calculus and their application to distributed system, J.of Franklin Institute, vol. 303, pp. 267-284, 1977.

[20] Oustaloup A., Linear feedback control systems of fractional orderbetween 1 and 2, in Proc. of the IEEE Symp. on Circuit andSystems, Chicago, USA, 1981.

[21] Sun H., Abdelwahab A. and Onaral B., Linear approximation oftransfer function with a pole of fractional power, IEEE Trans.Automation and Control, vol. 29, May 1984, pp. 441-444.

[22] Outstaloup, A., From fractality to non integer derivation throughrecursivity, a property common to these two concepts: Afundamental idea from a new process control strategy, Proc. of 12thIMACS World Congress, Paris, France, vol. 3, July 1988, pp. 203-208.

[23] Oustaloup A., Nouveau systeme de suspension: La suspensionCRONE, INPI Patent 90046 13, 1990.

[24] Kmetek P. and Prokop J., Robust regulators with fractionalstructure, in Proc. of 2nd Int. Symp. - DAAAM, FlexibleAutomation, Dec. 1991, pp.24-25.

[25] Bagley R. L. and Calico R. A., Fractional-order state equations forthe control of viscoelastic damped structures, J. Guidance, Controland Dynamics, vol. 14, no. 2, 1991, pp. 304 -311.

[26] Lurie B. J., Three-parameter tunable tilt-integral-derivative (TID)controller, US Patent US5371670, 1994.[27] Oustaloup A., Mathieu B., and Lanusse P., The CRONE control of

resonant plants: application to a flexible transmission, EuropeanJournal of Control, vol. 1, no. 2, 1995.

[28] Raynaud H. F. and Zerganoh A., State-space representation forfractional order controllers, Automatica, vol. 36, 2000, pp. 10171021.

[29] Tenreiro Machado J.A., Nonlinear dynamics- Special issue onfractional order calculus and its applications, An InternationalJournal of Nonlinear Dynamics and Chaos in Engineering Systems,Kluwer Academic Publishers. vol. 29, July 2002, pp.1-4.

[30] Chen Yang Quan, Fractional order dynamic system control Ahistorical note and a comparative introduction of four fractionalorder controllers, Tutorial Workshop on Fractional Order Calculusin Control and Robotics, IEEE International Conference on Decisionand Control (IEEE CDC2002), Las Vegas, NE, USA, Dec. 2002.

[31] Cao J. Y., Liang J., and Cao B. G., Optimization of fractiona l orderPID controllers based on genetic algorithms, in Proceedings of the4th International Conference on Machine Learning and Cybernetics,2005, pp. 56865689.

[32] Xue D., Zhao C., and Chen Y.Q., Fractional order PID control of ADC-motor with elastic shaft: a case study, in Proceedings of theAmerican Control Conference, 2006, pp. 31823187.

[33] Chen Y.Q., Ubiquitous fractional order controls in Proceedings ofthe 2nd IFAC Workshop on Fractional Differentiation and ItsApplications, vol. 2, July 2006.

[34] Agrawal O. P., A formulation and a numerical scheme forfractional optimal control problems, in Proceedings of the 2ndIFAC Workshop on Fractional Differentiation and Its Applications,

vol. 2, July 2006.[35] Sadati N., Zamani M., and Mohajerin P., Optimum design of

fractional order PID for MIMO and SISO systems using particleswarm optimization techniques, in Proceedings of the 4th IEEEInternational Conference on Mechatronics (ICM 07), 2007, pp. 15.

[36] Caponetto R. and Porto D., Analog implementation of non in tegerorder integrator via field programmable analog array, inProceedings of the 2nd IFAC Workshop on FractionalDifferentiation and Its Applications, vol. 2, July 2006.

[37] Ogata K., Modern Control Engineering, Prentice Hall, NewJersey, 2002.

[38] strm K. and Hgglund T., PID controllers: Theory, Design andTuning, Instrument society of America, North Carolina, 1995.

[39] Monje C.A., Calderon A.J., Vinagre B.M., Chen Y.Q. and Feliu V.,On fractional PID controllers: some tuning rules for robustness toplant uncertainties, Nonlinear Dynamics, vol. 38, 2004, pp. 369 -381.

[40] Valerio Duarte, Jose Sa da Costa, Tuning rules for fractional PIDcontrollers, in Proceedings of the 2nd IFAC Workshop onFractional Differentiation and its applications, Porto, Portugal, July19-21, 2006.

[41] Maione G. and Lino P., New tuning rules for fractional PIDcontrollers, Nonlinear Dynamics, 2006.

[42] Joaquin Cervera, Alfonso Baos , Concha A. Monje, Vinagre BlasM. Tuning of fractional PID controllers by using QFT, in IEEEDigital Library, 2006.

[43] Valrio Duarte, Jos S da Costa , Tuning of fractional controllersminimising H2 and H norms, Acta Polytechnica Hungarica, vol. 3,no. 4, 2006.

[44] Arman Kiani B., Naser Pariz, Fractional PID controller designbased on evolutionary algorithms for robust two-inertia speedcontrol, First Joint Congress on Fuzzy and Intelligent Systems,Ferdowsi University of Mashhad, Iran, Aug. 2007, pp. 29-31.

[45] Monje C. A., Vinagre B. M., Feliu V., Chen Y.Q., Tuning and auto -tuning of fractional order controllers for industry applications,Control Engineering Practice, vol. 16, 2008, pp. 798812.

[46] Bettou K., Charef A., Mesquine F., A new design method forfractional PID controller, IJ-STA, vol. 2, 2008, pp. 414-429.

[47] Mehmet nder Efe, Fractional fuzzy adaptive sliding-mode controlof a 2-DOF direct-drive robot arm, IEEE Trans.on Systems, Man,and Cybernetics-Part B: Cybernetics, vol. 38, no. 6, Dec. 2008.

[48] Deepyaman Maiti, Ayan Acharya, Mithun Chakraborty, AmitKonar, Tuning PID and PID controllers using the in tegral timeabsolute error criterion, IEEE Digital Library, 2008.

[49] Samir Ladaci, Jean Jacques Loiseau, Abdelfatah Charef, Fractionalorder adaptive high gain controllers for a class of linear systems,Science Direct - Communications in Nonlinear Science andNumerical Simulation, 2008, pp.707714.

7/28/2019 IJETAE_0312_66

7/7



389

[50] Bhambhani Varsha and Chen Yang Quan, Experimental study offractional order proportional integral (FOPI) controller for waterlevel control, in Proceedings of the 47th IEEE Conference onDecision and Control, Cancun, Mexico, Dec. 9-11, 2008.

[51] Youxin Luo, Jianying Li, The controlling parameters tuning and itsapplication of fractional order PID bacterial foraging based oriented

by particle swarm optimization, IEEE Digital Library, 2009.[52] Deepyaman Maiti, Mithun Chakraborty, Ayan Acharya, and Amit

Konar, Design of a fractional-order self-tuning regulator usingoptimization algorithms, IEEE Digital Library, 2009.

[53] Long Yi Chang Hung, Cheng Chen , Tuning of fractional PIDcontrollers using adaptive genetic algorithm for active magnetic

bearing system, WSEAS Trans. on Systems, issue 1, vol. 8, Jan.2009.

[54] Arijit Biswas, Swagatam Das, Ajith Abraham, Sambarta Dasgupta,Design of fractional-order PID controllers with an improveddifferential evolution, Science Direct -Engineering Applications ofArtificial Intelligence, 2009, pp. 343350.

[55] Samir Ladaci , Abdelfatah Charef, Jean Jacques Loiseau, Robustfractional adaptive control based on the strictly positive realnesscondition, Int. J. Appl. Math. Comput. Sci., vol. 19, no. 1, 2009,pp. 6976.

[56] Venu Kishore Kadiyala, Ravi Kumar Jatoth, Sake Pothalaiah,Design and implementation of fractional order PID controller foraerofin control system, World Congress on Nature & BiologicallyInspired Computing, Sept. 2009.

[57] Guillermo E. Santamaria, Tejado I., Vinagre Blas M., Fullyautomated tuning and implementation of fractional PID controllers,in Proceedings of the ASME- International Design EngineeringTechnical Conferences & Computers and Information inEngineering Conference, San Diego, California, USA, Aug. 30th Sept. 2nd, 2009.

[58] Li Meng, Dingy Xue, Design of an optimal fractional-order PIDcontroller using multi-objective GA optimization, IEEE DigitalLibrary, 2009. Available at: http://ieeexplore.ieee.org/ xpls/abs_all.jsp? arnumber=5191796&tag=1

[59] LI Dazi, Fan Weiguang, Jin Qibing, Tan Tianwei, An IMC PIDcontroller design for fractional calculus system, in Proceedings ofthe 29th Chinese Control Conference, Beijing, China, July 29-31,2010.

[60] Jianghui Zhang, Dejin Wang, A graphical tuning of PIDcontrollers for fractional-order systems with time-delay, ChineseControl and Decision Conference, 2010, pp. 729-734.

[61] Mehmet Onder Efe, Fractional order sliding mode control withreaching law approach, Turk J Elec Eng & Comp Sci, vol.18, No.5,2010, pp. 731-747.

[62] Ammar A. Aldair and Weiji J. Wang, Design of fractional ordercontroller based on evolutionary algorithm for a full vehiclenonlinear active suspension systems, International Journal ofControl and Automation, vol. 3, no. 4, Dec. 2010.

[63] HongSheng Li, Ying Luo, and Chen Yang Quan, Fractional orderproportional and derivative (FOPD) motion controller: Tuning ruleand experiments, IEEE Transactions on Control SystemsTechnology, vol. 18, no. 2, March 2010.

[64] Bettou Khalfa, Charef Abdelfatah, Parameter tuning of fractionalPID controllers with integral performance criterion, Confrence

Nationale surles Systmes dOrdre Fractionnaire et leursApplications, Skikda, Algrie, May 2010, pp.18-19. Available at:http://www.univ-chlef.dz/seminaires/seminaires_2010/bettou_khalfa.pdf

[65] Anuj Narang, Sirish L. Shah and Tongwen Chen, Tuning offractional PI controllers for fractional order system models with andwithout time delays, American Control Conference, Baltimore,MD, USA, June 30-July 02, 2010.

[66] Vinopraba T., Sivakumaran N., Narayanan S., IMC based fractionalorder PID controller, IEEE Digital Library, 2011,http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=5754348.

[67] Ali Akbar Jalali, Shabnam Khosravi, Tuning of FOPID controllerusing Taylorseries expansion, International Journal of Scientific &Engineering Research, vol. 2, issue 5, May 2011.
http://ieeexplore.ieee.org/http://ieeexplore.ieee.org/

ijetae_0312_66

Documents