International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 3, March 2012)

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

Head, Instrumentation Engineering Department, Institute of Petrochemical Engineering, Lonere, Raigad, Maharastra, India. 2 Professor & Head, Electrical Engineering Department, National Institute of Technical Teachers’ Training and Research, Chandigarh, India. 3 Senior Development Engineer, Bharat Heavy Electricals Limited, Tiruchirappalli, India. 1

[email protected]

II. INTRODUCTION TO FRACTIONAL CALCULUS

Abstract— Use of fractional order integral and derivative operators 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 conventional controller.

The concept of fractional calculus (calculus of integrals and derivatives of any arbitrary real or complex order) was raised in year 1695 by Marquis de L‘Hopital to Gottfried Wilhelm Leibniz regarding solution of non-integer order derivative. On September 30th 1695, Leibniz replied to L‘ Hopital ―This is an apparent paradox from which one day, useful consequences will be drawn‖. Between 1695 and 1819 several mathematicians (Euler in 1730, Lagrange in 1772, Laplace in 1812, and so on..) mentioned it. The question raised in 1695 was only partly answered 124 years later! [1] in 1819, by S. F. Lacroix. The real journey of development of fractional calculus started in 1974 when the first monograph on fractional calculus was published by academic press [2]. Since then many books were published until now [3-12]. Fractional calculus is presently being applied in the field of mathematics, physics, engineering, chemistry, computer science, mechanics, pharmacology, material science, neuroscience and neurology. Percentage utilization of fractional calculus in mathematics is around 25 percentage followed by that of in physics around 20 percentage while in engineering field it is around 14 percentage [13]. Fractional calculus will perhaps be the calculus of twenty-first century. The fractional-order differentiator can be denoted by a general fundamental

Keywords— Fractional order controllers, PID, ninteger toolbox.

I. INTRODUCTION Fractional order integral and derivative operators have found several applications in large areas of research during the last decade. Application of fractional order calculus to conventional controller design extends the opportunity of improved performance. Outline of the paper is as follows. Section two discusses the introduction and development of fractional calculus. Third section deals with applications of the fractional calculus in control systems and introduction of four types of fractional controller and tuning of fractional PID controller have been discussed. In fifth section the design of series and parallel FOPID controllers are presented using ‗ninteger‘ toolbox. Conclusions and future scope of work are discussed in the sixth section.

operator a Dtq as a generalization of the differential and integral operators, which is defined as follows [14]:

383

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 3, March 2012)  q d  dt q  q a Dt  1 t  ( d )  q  a

, R(q)  0

III. APPLICATIONS OF FRACTIONAL CALCULUS IN CONTROL

, R(q)  0



Classification of dynamic systems according to the order of the plant and the controller can be done as: i) integer order system - integer order controller ii) integer order system - fractional order controller iii) fractional order system - integer order controller and iv) fractional order system - fractional order controller. In fractional order controller given by, Gc(s) = Kp + Ki s-λ + Kd sµ (where Kp, Ki and Kd are proportional, integral and derivative gains respectively) more parameters need to be tuned. It‘s unfair but, theoretically, always better than integer order controller.

, R(q)  0

(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 Grünwald–Letnikov (GL) and the Riemann Liouville (RL). The GL definition is as mentioned below:

q a Dt

f (t )  lim

h 0

1 h

 (t  q ) h 

q

 (1)  j  f (t  jh) j

q

λ

PI

PID

PI

PID

λ=1

λ =1

j 0

λ

(2) P

where, . is a flooring-operator. On the other hand, the RL (3)

definition is given by:

q a Dt f (t ) 

1 dn (n  q) dt n

0

t

d , for (n  1  q  n) q  n 1

a

where, ( x) is the well known Euler‘s Gamma function. Their is another definition of fractional differintegral introduced by Caputo in [15] which can be written as: t

q a Dt

1 f (t )   ( q  n)

f ( n) ( )

 (t  )

q  n 1

d ,

P

µ 0 µ=1 λ λ µ Fig.1 PI D controller: from points to plane µ=1

PD

µ λ

Many control objects are fractional-order ones, so that the fractional approach for control of the fractional-order systems becomes a meaningful work. This approach (as shown in fig.1) has changed the point based control scheme to plane based scheme. Achieving something better is always the major concern from control engineering point of view. Existing evidences have confirmed that the best fractional order controller outperforms the best integer order controller. It has also been answered in the literature why to consider fractional order control even when integer order control works comparatively quite well [17]. Since integer-order PID control dominates the industry, it can be believed 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-Contrˆole Robuste d‘Ordre Non Entier) approach [23].

f ( )

 (t  )

PD

n  1  q  n (4)

a

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]. 384

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 3, March 2012) Quantitative Feedback Theory (QFT) and evolutionary algorithms have been used for tuning of a fractional PID controller.[42-46]. A paper [47] presents a different method for parameter adjustment scheme to improve the robustness of fractional fuzzy adaptive sliding-mode control by the use of an ANFIS architecture for two degree of freedom robot. The paper [48] describes an application of PSO to the problem of designing a fractional-order PID controller based on ITAE. A fractional high-gain adaptive controller for a class of linear systems was presented in the paper [49]. The paper [50] deals with the design of FO-PI for a coupled tank system. A global search optimization method with bacterial foraging technique oriented by particle swarm optimization was applied for optimizing five parameters of fractional controllers in the year 2009 [51]. The self-tuning regulators form an important sub-class of conventional adaptive controllers. Design of a fractional order selftuning regulator has been presented in [52]. The particle swarm optimization algorithm has been utilized for the online identification of parameters of the dynamic fractional order process while the tuning of the controller parameters has been performed by differential evolution. The paper [53] presents adaptive genetic algorithm (AGA) for the multi-objective optimization design of a fractional PID controller. The article [54] describes an application of differential evolution (DE) to the design of fractional-order controller. The paper [55] presents a robust adaptive control using fractional order systems as parallel feedforward in the adaptation loop based on the ―Almost Strictly Positive Realness (ASPR) property‖ of the plant. In paper [56] the tuning of FOPID controller of electromagnetic actuator (EMA) system for aerofin control (AFC) using particle swarm optimization (PSO) has been presented. Another method for tuning of fractional controller based on the relay feedback technique has been presented in [57]. Paper [58] reports about the design of FOPID using multi-objective optimization based genetic algorithm. Another work [59] published in the year 2010, on fractional PIλDμ controller tuning, explains the process of tuning by internal model control (IMC) based method with process described by fractional transfer function. A graphical tuning method of PIλDµ controllers for fractionalorder processes with time-delay has been presented in [60]. In this work a random search optimization method has been introduced for fractional order model reduction and the parameters of fractional order controllers have been tuned by internal model control (IMC) based method. Fractional order sliding mode control has been presented in [61].

Robust regulators with fractional structure and fractional order state equations for the control of visco-elastic damped 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 Group‘s introduction 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 PIλDμ controllers is a new research subject during last few years. Some important articles on design and parameter optimization of fractional controllers are presented here. I

Ref

±

T

1/S

I/S(1/n)

D

+

Plant

s

Fig. 2. Tilt-integral-derivative controller P

Ref

±

I

D

1 I/S

+

Plant

s

Fig.3. FO-PID (PIλDμ) controller (where, 0 ≤ λ ≤ 1 & 0 ≤ µ ≤ 1)

Articles [31-36] present various methodologies to design and optimize the fractional order controller viz. PSO, GA, minimization of ISE etc. Many tuning techniques for obtaining the parameters of controllers were introduced since inception of PID controller. The most well known tuning rules for classical controllers are given by Ziegler-Nichols [37] and Åström-Hägglund [38] which have been the milestones for developments of many other methods.Several tuning rules similar to Ziegler and Nichols for integer PID, were presented [39,40,41] in the year 2006. 385

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 3, March 2012) The control objectives have been achieved by adopting the reaching law approach of sliding mode control. In a similar work published in the year 2010 [62], an optimal fractional order controller has been designed and the optimal values of FOPID controller parameters for minimizing the cost function have been tuned using an evolutionary algorithm. Design of FOPD for motion control and minimum integral squared error (ISE) criterion approach is presented in [63,64]. In another paper [65] a servo control strategy for tuning of fractional order PI controllers is presented for fractional order system models with and without time delays. Recent article of the year 2011 [66], presents the design of internal model controller (IMC) based fractional order two-degrees of freedom controller with robust control. The features of the IMC based PID controller have been combined with fractional order controller. Tuning of FOPID controller using Taylor series expansion of desired closed-loop and actual closedloop transfer functions has been presented in [67]. This literature survey reveals that the fractional controller gives better results as compared to the conventional PID controller. Tuning of FOPID controller is difficult as five tuning parameters need to be tuned and theoretically infinite memory is essential for its digital implementation. In next section simulation example of FOPID is presented.

In this section, series and parallel configuration of fractional order PID is designed in MATLAB/ SIMULINK using ‗nid‘ block of ‗ninteger‘ toolbox, which uses ‗crone‘ formula with ‗mcltime‘ expansion for fractional operator implementation. Simulink model (Fig.4) is designed for implementation of conventional (parallel) PID, parallelFOPID 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.

IV. SIMULATION RESULTS Three 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, T I 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.sbeta

Fig.4. Simulink implementation of series, parallel and conventional PID for first order system. TABLE I OPTIMIZED PARAMETERS FOR PARALLEL PID WITH MINIMUM ERROR CRITERIA FOR FIRST ORDER SYSTEM

(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.sbeta.e(s)

(7) 386

KP

TI

TD

1.3185

2.2306

-0.2735

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 3, March 2012) TABLE III PARAMETERS FOR PARALLEL AND SERIES FO-PID FOR FIRST ORDER SYSTEM 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

V. CONCLUSION AND FUTURE SCOPE OF WORK The 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-order PID 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 new methods for auto-tuning and self-tuning the parameters of PIλDμ controllers and development digital algorithm for implementation of FOPID using microprocessor or microcontroller.

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 peak overshoot 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 in series 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.

References [1]

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

1.2 1

[2]

0.8 Responce of PID-optimized 0.6

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

0.4

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

0.2 0

[3]

0

50

100

150

200

250

[4] [5]

300

[6]

Fig.5. Responses of Fractional (Series & Parallel) and conventional PID to step input

[7]

387

Chen Y.Q., ―Fractional Order Dynamic System Control – A Historical Note and A Comparative Introduction of Four Fractional Order Controllers‖, Tutorial Workshop on Fractional Order Calculus in Control and Robotics IEEE International Conference on Decision and Control, Las Vegas, NE, USA, Dec. 2002. Oldham K. and Spanier J., ―The fractional calculus: Theory and applications of differentiation and integration to arbitrary order‖, Academic Press, 1974. 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). Miller K.S. and Ross B., ―An introduction to the fractional calculus and fractional differential equations‖, John Wiley and Sons, 1993. Carpinteri A. and Mainardi F., ―Fractals and fractional calculus in continuum mechanics‖, Springer-Verlag, 1998. Podlubny I., ―Fractional differential equations‖, Academic press, 1999. Hilfer R., ―Applications of fractional calculus in physics‖, World Scintific, 2000.

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 3, March 2012) [8] [9] [10] [11] [12] [13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23] [24]

[25]

[26] [27]

[28]

[29]

Kilbas A. A., Srivastava H.M. and Trujillo J.J., ―Theory and applications of fractional differential equations‖, Elsevier, 2006. Magin R. L., ―Fractional calculus in bioengineering‖, Begell House Publishers, 2006. Biagini F., Zhang T.. ―Stochastic calculus for fractional Brownian motion and applications‘, Springer, 2008. Monje C.A., Chen Y.Q., Vinagre B.M., Xue D. and Feliu V., ―Fractional order systems and controls‖, Springer, 2010. Chen W., Sun H., and Li X., ―Fractional derivative in mechanics and engineering modeling‖, Science press, 2010. Chen Yang Quan, ―Applied Fractional Calculus in Controls and Signal Processing‖, in 49th IEEE Conf. on- Decision and Control, Atlanta, USA, Dec, 2010. Calderón, A.J., Vinagre B.M., and Feliu V., ―Fractional order control strategies for power electronic buck converters‖, Signal Processing, 2006, pp. 2803–2819. Caputo, M., ―Linear models of dissipation whose Q is almost frequency independent II‖, Geophys. J. Royal Astron. Soc. 1967, pp. 529–539. Ahmed. E., El-Sayed A.M.A., El-Saka H.A.A., ―Equilibrium points, stability and numerical solutions of fractional order predator–prey and rabies models‖, J. Math. Anal., pp. 542–553, 2007. Chen Yang Quan, ―Applied Fractional Calculus in Control‖, American Control Conference-ACC2009, St. Louis, Missouri, USA, June 10-12, 2009, Manabe S., ―The non-integer integral and its application to control systems‖, JIEE (Japanese Institute of Electrical Engineers) Journal, vol. 6, no. 3/4, pp. 83–87, 1961. Translation from JIEE J., vol. 80, no. 860, 1960, pp. 589-597. Chen C. F., Tsay Y. T. and Wu T. T., ―Walsh operational matrices for fractional calculus and their application to distributed system‖, J. of Franklin Institute, vol. 303, pp. 267-284, 1977. Oustaloup A., ―Linear feedback control systems of fractional order between 1 and 2,‖ in Proc. of the IEEE Symp. on Circuit and Systems, Chicago, USA, 1981. Sun H., Abdelwahab A. and Onaral B., ―Linear approximation of transfer function with a pole of fractional power,‖ IEEE Trans. Automation and Control, vol. 29, May 1984, pp. 441-444. Outstaloup, A., ―From fractality to non integer derivation through recursivity, a property common to these two concepts: A fundamental idea from a new process control strategy‖, Proc. of 12th IMACS World Congress, Paris, France, vol. 3, July 1988, pp. 203208. Oustaloup A., ―Nouveau syst´eme de suspension: La suspension CRONE,‖ INPI Patent 90046 13, 1990. Kmetek P. and Prokop J., ―Robust regulators with fractional structure‖, in Proc. of 2nd Int. Symp. - DAAAM, Flexible Automation, Dec. 1991, pp.24-25. Bagley R. L. and Calico R. A., ―Fractional-order state equations for the control of viscoelastic damped structures‖, J. Guidance, Control and Dynamics, vol. 14, no. 2, 1991, pp. 304 -311. Lurie B. J., ―Three-parameter tunable tilt-integral-derivative (TID) controller,‖ US Patent US5371670, 1994. Oustaloup A., Mathieu B., and Lanusse P., ―The CRONE control of resonant plants: application to a flexible transmission,‖ European Journal of Control, vol. 1, no. 2, 1995. Raynaud H. F. and Zergaınoh A., ―State-space representation for fractional order controllers,‖ Automatica, vol. 36, 2000, pp. 1017– 1021. Tenreiro Machado J.A., ―Nonlinear dynamics- Special issue on fractional order calculus and its applications‖, An International Journal 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– A historical note and a comparative introduction of four fractional order controllers‖, Tutorial Workshop on Fractional Order Calculus in Control and Robotics, IEEE International Conference on Decision and Control (IEEE CDC‘2002), Las Vegas, NE, USA, Dec. 2002. [31] Cao J. Y., Liang J., and Cao B. G., ―Optimization of fractional order PID controllers based on genetic algorithms,‖ in Proceedings of the 4th International Conference on Machine Learning and Cybernetics, 2005, pp. 5686–5689. [32] Xue D., Zhao C., and Chen Y.Q., ―Fractional order PID control of A DC-motor with elastic shaft: a case study,‖ in Proceedings of the American Control Conference, 2006, pp. 3182–3187. [33] Chen Y.Q., ―Ubiquitous fractional order controls‖ in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006. [34] Agrawal O. P., ―A formulation and a numerical scheme for fractional optimal control problems,‖ in Proceedings of the 2nd IFAC 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 particle swarm optimization techniques,‖ in Proceedings of the 4th IEEE International Conference on Mechatronics (ICM ‘07), 2007, pp. 1–5. [36] Caponetto R. and Porto D., ―Analog implementation of non integer order integrator via field programmable analog array,‖ in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, vol. 2, July 2006. [37] Ogata K., ―Modern Control Engineering‖, Prentice Hall, New Jersey, 2002. [38] Åström K. and Hägglund T., ―PID controllers: Theory, Design and Tuning‖, 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 to plant uncertainties‖, Nonlinear Dynamics, vol. 38, 2004, pp. 369381. [40] Valerio Duarte, Jose Sa da Costa, ―Tuning rules for fractional PID controllers‖, in Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and its applications, Porto, Portugal, July 19-21, 2006. [41] Maione G. and Lino P., ―New tuning rules for fractional PID controllers‖, Nonlinear Dynamics, 2006. [42] Joaquin Cervera, Alfonso Baños , Concha A. Monje, Vinagre Blas M. ―Tuning of fractional PID controllers by using QFT‖, in IEEE Digital Library, 2006. [43] Valério Duarte, José Sá da Costa , ―Tuning of fractional controllers minimising H2 and H∞ norms, Acta Polytechnica Hungarica, vol. 3, no. 4, 2006. [44] Arman Kiani B., Naser Pariz, ―Fractional PID controller design based on evolutionary algorithms for robust two-inertia speed control‖, 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 autotuning of fractional order controllers for industry applications‖, Control Engineering Practice, vol. 16, 2008, pp. 798–812. [46] Bettou K., Charef A., Mesquine F., ―A new design method for fractional PID controller‖, IJ-STA, vol. 2, 2008, pp. 414-429. [47] Mehmet Önder Efe, ―Fractional fuzzy adaptive sliding-mode control of 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, Amit Konar, ―Tuning PID and PIλDδ controllers using the integral time absolute error criterion‖, IEEE Digital Library, 2008. [49] Samir Ladaci, Jean Jacques Loiseau, Abdelfatah Charef, ―Fractional order adaptive high gain controllers for a class of linear systems‖, Science Direct - Communications in Nonlinear Science and Numerical Simulation, 2008, pp.707–714.

388

International Journal of Emerging Technology and Advanced Engineering Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 3, March 2012) [50] Bhambhani Varsha and Chen Yang Quan, ―Experimental study of fractional order proportional integral (FOPI) controller for water level control‖, in Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico, Dec. 9-11, 2008. [51] Youxin Luo, Jianying Li, ―The controlling parameters tuning and its application 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 using optimization algorithms, IEEE Digital Library, 2009. [53] Long Yi Chang Hung, Cheng Chen , ―Tuning of fractional PID controllers 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 improved differential evolution‖, Science Direct -Engineering Applications of Artificial Intelligence, 2009, pp. 343–350. [55] Samir Ladaci , Abdelfatah Charef, Jean Jacques Loiseau, ―Robust fractional adaptive control based on the strictly positive realness condition‖, Int. J. Appl. Math. Comput. Sci., vol. 19, no. 1, 2009, pp. 69–76. [56] Venu Kishore Kadiyala, Ravi Kumar Jatoth, Sake Pothalaiah, ―Design and implementation of fractional order PID controller for aerofin control system‖, World Congress on Nature & Biologically Inspired Computing, Sept. 2009. [57] Guillermo E. Santamaria, Tejado I., Vinagre Blas M., ―Fully automated tuning and implementation of fractional PID controllers‖, in Proceedings of the ASME- International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, San Diego, California, USA, Aug. 30th – Sept. 2nd, 2009. [58] Li Meng, Dingy Xue, ―Design of an optimal fractional-order PID controller using multi-objective GA optimization‖, IEEE Digital Library, 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 PIλDμ controller design for fractional calculus system‖, in Proceedings of the 29th Chinese Control Conference, Beijing, China, July 29-31, 2010. [60] Jianghui Zhang, Dejin Wang, ―A graphical tuning of PIλDμ controllers for fractional-order systems with time-delay‖, Chinese Control and Decision Conference, 2010, pp. 729-734. [61] Mehmet Onder Efe, ―Fractional order sliding mode control with reaching 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 order controller based on evolutionary algorithm for a full vehicle nonlinear active suspension systems‖, International Journal of Control and Automation, vol. 3, no. 4, Dec. 2010. [63] HongSheng Li, Ying Luo, and Chen Yang Quan, ―Fractional order proportional and derivative (FOPD) motion controller: Tuning rule and experiments‖, IEEE Transactions on Control Systems Technology, vol. 18, no. 2, March 2010. [64] Bettou Khalfa, Charef Abdelfatah, ―Parameter tuning of fractional PIλDμ controllers with integral performance criterion‖, Conférence Nationale surles Systèmes d‘Ordre Fractionnaire et leurs Applications, Skikda, Algérie, May 2010, pp.18-19. Available at: http://www.univchlef.dz/seminaires/seminaires_2010/bettou_khalfa.pdf [65] Anuj Narang, Sirish L. Shah and Tongwen Chen, ―Tuning of fractional PI controllers for fractional order system models with and without time delays‖, American Control Conference, Baltimore, MD, USA, June 30-July 02, 2010.

[66] Vinopraba T., Sivakumaran N., Narayanan S., ―IMC based fractional order 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 controller using Taylor series expansion‖, International Journal of Scientific & Engineering Research, vol. 2, issue 5, May 2011.

389