by Donato Cafagna

Past and Present

The history of fractional calculusbegins at the end of the 17thcentury, and it should not be sur-

prising if its birth was due to a letterexchange. At that time scientific jour-nals did not exist, and the usual way ofexchanging information among scien-tists was mainly by letter. Every scien-tist exchanged letters with otherscientists who possibly had the samelevel of knowledge and were trustwor-thy persons. This method of knowl-edge exchange had two maindrawbacks: the problem of plagiarismand the fact that letters took a longtime to reach the final destination.

During those years, even scientificmatters were taught by letter. Thiswas the case of Johann Bernoulli. In1691 Bernoulli moved to Paris andthere he met mathematicians in Male-branche's circle, where the cream ofFrench mathematics gathered at thattime. There Bernoulli met Guillaumede l'Hôpital, and they engaged in deepmathematical conversations (con-trary to what is often believed, del’Hôpital was a fine mathematician,perhaps the best in Paris at that time,although not of the same class asBernoulli). During their conversa-tions, de l’Hôpital discovered thatBernoulli had deeply understood thenew methods related to both differen-tial and integral calculus that Got-tfried Leibniz (Figure 1) had publishedin the Acta Eruditorum journal in 1684and 1686, respectively. De l’Hôpitalasked Bernoulli to teach him thesemethods. Bernoulli agreed to do thisduring his stay in Paris, receiving gen-erous payment from de l’Hôpital.

When Bernoulli moved to Basel hestill continued his calculus lessons bycorrespondence. Thanks to this learn-ing, in 1696 de l’Hôpital published thefirst textbook written on differentialcalculus (with no mention of the lec-tures of Bernoulli). Around 1695,when de l’Hôpital was already skilledin differential and integral calculus, aletter exchange between Leibniz andde l’Hôpital began. In particular, inone of these letters, Leibniz put to del’Hôpital the following question [1]:

Can the meaning of derivatives withinteger order be generalized to deriva-tives with non-integer orders?

That strange question aroused del’Hôpital’s curiosity, and he replied toLeibniz with another question:

What if the order will be 1/2?Leibniz in a reply letter dated

30 September 1695 wrote the famouswords:

Thus it follows that will be equal tox 2

√dx : x, an apparent paradox, from

which one day useful consequences willbe drawn.

Nowadays, on the basis of such far-sighted words, many scientists consid-er 30 September 1695 as the exactbirthday of fractional calculus and Got-tfried Leibniz as its father [2]. It isworth noting that the current name of“fractional calculus” is actually a mis-nomer and the designation of “integra-tion and differentiation of arbitraryorder” would be more appropriate.

Defining Fractional CalculusAfter that first intuition, it wasn’t until1730 that Euler mentioned fractionalcalculus when he studied the interpo-lation between integer orders of a

derivative and, successively in 1772,Lagrange referred to it in one of hisstudies. However, the earliest more orless systematic studies on fractionalcalculus were made only at the begin-ning of the 19th century. In particular,in 1812, Laplace defined a fractionalderivative by means of an integral,and in 1819 the first discussion on aderivative of fractional order in a cal-culus text was written by S.F. Lacroix.Nevertheless, Lacroix’s method ofgeneralizing from a case of integerorder did not offer hints for possibleapplications, and Lacroix himself con-sidered this question as a mere math-ematical exercise. In 1822, Fourier wasthe next to mention a derivative ofarbitrary order, but like his famouspredecessors, he gave no application.A fractional operation was applied forthe first time by Niels Abel in 1823 forsolving an integral equation arisen

Fractional Calculus: A Mathematical Tool from the Past for Present Engineers

FIGURE 1—Gottfried Wilhelm von Leibniz, in apainting by Bernhard Christoph Francke (about1700).

Digital Object Identifier 10.1109/MIE.2007.901479

SUMMER 2007 ■ IEEE INDUSTRIAL ELECTRONICS MAGAZINE 351932-4529/07/$25.00©2007IEEE

during the demonstration of thetautochrone problem [3].

The first systematic study on frac-tional calculus was made in 1832 byJoseph Liouville, who was probablyattracted to this topic by the briefcomments of Laplace and Fourier orby the Abel’s demonstration. Initially,Liouville defined a derivative of arbi-trary order as an infinite series. Thisdefinition had the disadvantage thatthe order must be restricted to thosevalues for which the series converges.Aware of the restrictive nature of hisfirst definition, he formulated a sec-ond definition in which he was able togive a fractional derivative of x−a ,whenever both x, a, and n are posi-tive. Successively, Liouville appliedthe fractional derivative to problemsin potential theory and was also thefirst scientist who attempted to solvedifferential equations by means offractional operations [2].

After Liouville, Bernhard Riemannwas to develop a somewhat differenttheory of fractional calculus duringhis student days (but this theory waspublished only posthumously in1876). Riemann, probably the great-est mathematician of all time, used ageneralization of a Taylor series toderive a formula for integration ofarbitrary order.

At this point of our history we havetwo different approaches for the inte-gration of arbitrary order, but it can bedemonstrated that the approaches pro-posed by Liouville and Riemann can beabridged in a single formula (nowadaysknown as the Riemann-Liouville Frac-tional Integral Formula) [4]:

J αc f(t ) = 1

�(α)

∫ t

c

f(τ)

(t − τ)1−αdτ (1)

where J α represents the fractional inte-gral operator of order α ∈ R+, f(t) is acausal function of time (i.e., identicallyvanishing for t < 0), c is a proper lowerlimit of integration, and � is the Gammafunction. In particular, if c = 0 then (1)gives the Riemann Formula, whereas ifc = −∞ then (1) becomes the LiouvilleFormula (it should be noted that inmany articles and books on fractionalcalculus, the Riemann-Liouville Frac-

tional Integral Formula is given by con-sidering c = 0 and therefore the Rie-mann Formula is commonly given).

After the notion of a fractional inte-gral, one may ask whether the defini-tion of fractional derivative of orderα(> 0) can be deduced from (1) bymerely substituting α with −α . Theanswer is that the application of thistrick is allowed, but it needs some caresince the convergence of the integralsand the properties of the ordinaryderivative of integer order have to beguaranteed and preserved. To this pur-pose, denoting by D n(n ∈ N) the oper-ator of the derivative of integer ordern, note that D n J n = I but J nD n �= I;i.e., D n is left-inverse (and not right-inverse) to the corresponding integraloperator J n . As a consequence, oneexpects that D α is defined as left-inverse to J α . Therefore, by introduc-ing the positive integer m such thatm − 1 < α ≤ m, the Fractional Deriva-tive Formula of order α > 0 is obtainedstarting from (1) for c = 0 [4]:

D α f(t ) := D m J m−α f(t ) = d m

dt m

×[

1�(m − α)

∫ t

0

f(τ)

(t − τ)α+1−mdτ

].

(2)

Note that if α = m then (1) reduces tothe formula for integer orderderivative D α f(t ) := D m J m−α f(t ) =d m

dt m f(t ).In 1867, Anton Karl Grunwald pro-

posed perhaps the most difficult, yetin some way the most natural,approach for fractional differentiation.His method was based on the general-ization of the finite difference quo-tients for fractional derivatives,obtaining the formula (nowadaysknown as Grunwald-Letnikov Fraction-al Derivative Formula) [5]:

D α f(t ) = limh →01

hα

t−ah∑

m=0

(−1)m

× �(α + 1)

m!�(α − m + 1)f(t − mh) (3)

where t and a are the upper andlower limits of differentiation, respec-tively. Unfortunately, the derivation of

Grunwald’s approach, even if correct,was not mathematically rigorous. Thefirst rigorous demonstration of the for-mula (3) was given by the A.V. Letnikovin 1868 [6], only one year after theappearance of Grunwald’s paper. Letnikov was perhaps the most prolificwriter on the subject of fractional dif-ferentiation. His papers were the firstto exhibit a strong attention to detailsand mathematical rigor. It should benoted that, unlike the Riemann-Liouville approach, which derives defi-nition (1) from the repeated integral ofa function, the Grunwald-Letnikov for-mulation (3) faces the problem fromthe derivative side.

It has been shown that the Rie-mann-Liouville definition (2) of thefractional integral can be adapted todefine the fractional derivative. Also,in the case of the Grunwald-Letnikovderivative formula, it could be shownthat (3) can be modified for use in analternate definition of the fractionalintegral. The most natural change isto consider the Grunwald-Letnikovderivative for negative α . Namely,with proper mathematical adjust-ments the Grunwald-Letnikov Fraction-al Integral Formula is obtained [5]:

D−α f(t) = limh →0hα

t−ah∑

m=0

× �(α + m)

m!�(α)f(t − mh). (4)

At this point, two formulations of frac-tional calculus have been presented,the one proposed by Riemann and Liou-ville, and the other by Grunwald andLetnikov, respectively. The existence ofdifferent formulations of the same con-cept poses the question of whetherthese formulations are also equivalent.The answer to this question is affirma-tive but its mathematical proof isbeyond the scope of this article. How-ever, simple guidelines about the use ofthe two different approaches can begiven. The Riemann-Liouville definitionfor the fractional integral and deriva-tive, by virtue of its form, is well suitedfor finding the analytical solution of rel-atively simple functions (x a, ex, sin(x),etc.). Conversely, the Grunwald-

36 IEEE INDUSTRIAL ELECTRONICS MAGAZINE ■ SUMMER 2007

Letnikov definition is successfully uti-lized for numerical evaluations.

In the last two centuries, besidesprevious definitions, several differentfractional differation-integral formulaehave been proposed. But the scope ofthis article is to present those defini-tions that could be useful from an engi-neering (read applicative) point ofview. To this purpose, the last defini-tion of fractional derivative presentedin this article is that originally intro-duced by Caputo in 1967; i.e., the so-called Caputo Fractional Derivative oforder α > 0 (m − 1 < α ≤ m, m is apositive integer) [7]:

D α∗ f(t ) := J m−α D m f(t ) = 1�(m − α)

×∫ t

0

f (m)(τ )

(t − τ) α+1−md τ . (5)

The five formulas presented so farare summarized in Table 1.

Also, in the case of (5), it can bedemonstrated that the Riemann-Liouville derivative and the Caputoderivative are both equivalent underthe same set of homogenous initialconditions. In many engineering appli-cations the Caputo derivative (5) ispreferred to the commonly used(among mathematicians) Riemann-Liouville derivative (2), since the firstone guarantees a straightforward con-nection between the type of the initialcondition and the type of the fractionalderivative. In particular, in the Rie-mann-Liouville derivative (2), it is nec-essary to specify the values of certainfractional derivatives of the function fat the initial point t = 0. When a real

physical application is considered, thephysical meaning of such fractionalderivative of f could be unknown, andhence not measurable. On the con-trary, when the Caputo derivative isconsidered, the initial valuesf( 0 ), f ′( 0 ), . . . , f (m−1)( 0 ) , i.e., thefunction value itself and integer-orderderivatives, have to be specified.These data typically have a well-under-stood physical meaning and can bemeasured [4].

It is worth introducing one impor-tant property that all the definitionsof fractional derivative share; namely,the fractional derivative operators arenot local. This means that the value ofD α

t0f(t ) depends on all the values of f

in the interval [t0, t ]; i.e., on theentire history of the function f [3]. Onthe contrary, it is well known that dif-ferential operators of integer orderare always local.

The Applicative Point of ViewAs shown so far, fractional calculus hasa long history, but, from an applicativepoint of view, it fell into oblivion formany years since it was considered notuseful for solving problems in physicsand engineering. Actually this oblivionwas due to its high complexity and thelack of an acceptable physical andgeometric interpretation. Just in 2002,Podlubny proposed a convincing ex-planation of the fractional phenomena[8]. He suggested both a geometricinterpretation of the Riemann-Liouvillefractional integral (based on the projec-tion of a very fascinating “shadow onthe wall”) and a physical interpretationfor the Riemann-Liouville (and Caputo)fractional differentiation (based on the

fact that, as in the theory of relativity,two time scales should be consideredsimultaneously: the ideal, equably flow-ing homogeneous time and the cosmicinhomogeneous time).

The subject of fractional calculusand its applications has re-attractedconsiderable interest in the pastthree/four decades. Several applica-tions based on fractional modeling innumerous seemingly various and wide-spread fields of science and engineer-ing have been proposed. A list of thefields of application may include [9]:acoustic wave propagation in inhomo-geneous porous material, diffusivetransport, fluid flow, dynamical pro-cesses in self-similar structures, dyna-mics of earthquakes, optics, geology,viscoelastic materials, bio-sciences,medicine, economics, probability andstatistics, astrophysics, chemical engi-neering, physics, electromagneticwaves, nonlinear control, signal pro-cessing, control of power electronicconverters, chaotic dynamics.

Note that the above list coversalmost all the fields of research inscience and engineering. Scientistshave indeed discovered that many phe-nomena, not completely understoodbefore, have complex microscopicbehaviors, and that their macroscopicdynamics cannot be modeled anymorevia integer-order derivatives. In particu-lar, it has been found that most of theprocesses associated with complexsystems have nonlocal dynamics involv-ing long-term memory effects; i.e., pre-cisely the property that characterizesfractional derivative operators.

In order to better appreciate thepotentialities of fractional calculus

TABLE 1—FRACTIONAL ORDER “DIFFERINTEGRAL” FORMULAS.

FRACTIONAL CALCULUS (α ∈ R+) RIEMANN-LIOUVILLE GRUNWALD-LETNIKOV CAPUTO

FRACTIONAL J αc f (t ) = 1

�(α)

∫ t

c

f (τ)

(t − τ)1−αdτ D−αf (t) = limh →0hα

t−ah∑

m=0

�(α + m)

m!�(α)f (t − mh) —

INTEGRAL

FRACTIONAL

D αf (t ) := D mJ m−αf (t )

= d m

dt m

[1

�(m − α)

∫ t

0

f (τ)

(t − τ)α+1−m dτ

] D αf (t ) =limh →01

h α

t−ah∑

m=0

(−1)m

× �(α + 1)

m!�(α − m + 1)f (t − mh)

D α∗ f (t ) :=J m−αD mf (t )

= 1�(m − α)

∫ t

0

f (m)(τ )

(t − τ)α+1−m d τDERIVATIVE

SUMMER 2007 ■ IEEE INDUSTRIAL ELECTRONICS MAGAZINE 37

from an applicative point of view, in thefollowing some examples of complexsystems modeled by means of fraction-al differential equations are given.

The first example comes from thefield of bioengineering. Inthis field of research thereis an ongoing need todevelop efficient, high-fidelity material models tosimulate the stress re-sponse of biological mate-rials. To this purpose,fractional calculus is beingused to develop fractional-order viscoelastic equa-tions, which could beuseful for modeling softbiological tissues. In [10],it is asserted that adescription based on frac-tional-order differentialequations has potentiallysuperior accuracy and gives the possibil-ity of correlating the hierarchical struc-ture of biological tissues to the fractionalorder of derivative. In particular, theauthors formulate a one-dimensionalversion of fractional-order viscoelasticequations, called quasilinear fractional-order viscoelasticity, and apply it formodeling the stress response of porcineaortic valve tissues [10].

The second example presentedherein comes from the study of poly-meric materials. In organic dielectricmaterials, such as semi-crystallinepolymers, the intimate mixturebetween crystals and amorphousphase gives rise to a complex structurewhose fractal features have been exper-imentally detected. This propertymakes the polymers very difficult tohandle analytically, and fractional cal-culus seems to represent the onlymathematical tool able to describefractal functions. In particular, itenables one to obtain constitutiveequations that can be linked to molecu-lar theories describing the macroscop-ic behavior of polymeric materials. In[11] a dielectric fractional model isgiven, which is based on the idea ofobtaining an intermediate electricalbehavior between a resistance (Ohm’slaw) and a capacitor. The authors havecalled this new electrical element cap-

resistor: when the order of derivativeequals one, an electrical resistor isobtained, whereas when the order ofderivative equals zero, a capacitor isobtained. Using this new electric ele-

ment, the authors proposeelectric circuits able tomodel relaxation processesin organic dielectric materi-als (semi-crystalline poly-mers) [11].

Recently, in order toattain an effective controlof the physical systemsdescribed by fractional-order models, the study offractional-order controllershas attracted a lot of inter-est. Some interesting exam-ples related to such field ofresearch are now reported.

In [12] a fractional-order P ID k controller

designed for a torsional system’sbacklash vibration suppression con-trol is proposed. Previous methods,applied to the speed control of inertiasystems, have always failed in sup-pressing the vibrations caused bygear backlash nonlinearity. On theother hand, the proposed robustP ID k controller gives the possibilityof directly tuning the fractional-orderk and, therefore, this enables one toadjust the control system’s frequencyresponse [12]. Additionally, anapproximation method based on theshort memory principle (a principleoften utilized for reducing the compu-tational complexity of fractional calcu-lus) is introduced for realizing thediscrete D k controller.

In [13] both linear and sliding-modecontrol have been applied to powerdc/dc buck converters by means of frac-tional-order controllers. It is well knownthat such converters are fast and, due tothe pulse-width modulation process,nonlinear. Therefore, [13] shows howfractional-order controllers can be uti-lized to reach an effective control ofsuch devices and that the practicalimplementation of these controllers isfeasible using a digital processor.

Interesting research on fractional-order controllers has been conductedalso by the CRONE (French acronym

for Controle Robuste d’Ordre Non-Entier) team that has effectivelyapplied fractional-order controllersinto car suspension control [14], flexi-ble transmission [15], and hydraulicactuator [16].

In [17] and [18], the theoreticalaspects of fractional-order controllers,such as stability, controllability, andobservability, have been analyzed andsome useful results have been derived.

Igor Podlubny and other re-searchers are giving an important con-tribution from the point of view ofmodeling, realizing and implementingfractional-order controllers [19], [20].

The last application of fractionalcalculus presented in this articlecomes from the study of chaotic phe-nomena. In the last decade, manyresearchers have found that severalnonlinear fractional systems can gen-erate complex bifurcations and chaot-ic phenomena very similar to thosedisplayed by their integer-order coun-terparts (e.g., the fractional-orderChua circuit [21], the fractional-orderChen system, the fractional-order jerksystem, the fractional-order Lorenzsystem, and many others). In particu-lar, it has been demonstrated thatchaotic attractors appear in nonlin-ear fractional systems with ordermuch lower than three, when anappropriate adjustment of the valuesof system parameters is taken intoaccount. It is worth noting that,according to Poicarè-Bendixon theo-rem, chaos cannot occur in auto-nomous continuous-time systems ofinteger order less than three. Theresults of the current research on thecomplex phenomena in fractional sys-tems enables one to make the follow-ing considerations: 1) fractional-ordersystems can be viewed as a general-ization of the integer-order case,where the latter is retrieved when thederivative of fractional-order approach-es the integer-order; 2) nonlinear frac-tional systems can generate at leastthe same variety of dynamic phe-nomena found in nonlinear integer-order systems; 3) the occurrence ofchaos in fractional-order systemsdoes not depend on the order of sys-tem equations [21].

NOWADAYS, ONTHE BASIS OF SUCHFAR-SIGHTEDWORDS, MANYSCIENTISTSCONSIDER 30SEPTEMBER 1695AS THE EXACTBIRTHDAY OFFRACTIONALCALCULUS ANDGOTTFRIED LEIBNIZAS ITS FATHER.

38 IEEE INDUSTRIAL ELECTRONICS MAGAZINE ■ SUMMER 2007

Solving FractionalDifferential EquationsThe last argument related to fractionalcalculus that it is worth introducing isthe following: after having modeled anonlocal phenomenon, a not trivial taskconsists in finding a solution of theobtained fractional differential equa-tions. From an engineering point ofview, two main classes of methods forsolving fractional differential equationscan be identified: the frequency-domainmethods and the time-domain methods.

One well-known frequency-domainmethod is based on the approximationof the transition function (1/s) α

(obtained by applying the Laplacetransform) using a specified error indecibels and a bandwidth to generatea continuous sequence of pole-zeropairs for the system with a single frac-tional power pole [21]. This frequency-domain method presents two maindrawbacks: 1) the approximations ofthe transition function are availableonly for few values of α and 2) the dif-ference between the original and theapproximate systems is still unknown.

Recent studies have shown thattime-domain methods may represent abetter choice. In particular, in this arti-cle three time-domain methods will bebriefly introduced.

The first one is the Adams-Bashforth-Moulton method [22]. Thismethod is based on the Volterra inte-gral equation and seems to be an effi-cient technique for studying fractional-order dynamical systems. However,this numerical method encounters twomain difficulties: 1) the size of the com-putational work can be burdensome,and 2) the rounding-off error can causeloss of accuracy.

The second time-domain approachis based on the direct application of theGrunwald-Letnikov derivative formula(3) [8]. This formula is written as asummation of virtually infinite termsand, therefore, is well suited for usewith a computer-based numericalsolver. Contrary to what one may think,the limiting factor to the accuracy ofthis approach is not the speed of thecomputer simulation but rather thecapacity of the computer to accuratelystore and move the large numbers that

SUMMER 2007 ■ IEEE INDUSTRIAL ELECTRONICS MAGAZINE 39

can result from the computation of theGamma function.

The last, but not least, is the numer-ical method based on the Adomiandecomposition [21]. This method pro-vides the solution of the fractional-order system in the form of a powerseries with easily computed terms.The decomposition method has someadvantages over the previous tech-niques; i.e., 1) it preserves the systemnonlinearities (this property makes itthe first choice when the nonlinearsolution for systems of fractional dif-ferential equation have to be found); 2)it avoids discretization in time (if thenumber of the terms of the series isgreat enough); 3) it provides an effi-cient numerical solution with highaccuracy. Finally, note that the Adomi-an decomposition method has beeneffectively applied to the analysis ofthe chaotic phenomena (e.g., for gener-ating the Double-Scroll attractorreported in Figure 2) in the nonlinearfractional-order Chua’s circuit [21].

ConclusionsIn 1831, somebody asked Michael Faraday to talk about the usefulness ofhis discovery: the electromagneticinduction phenomenon. Faraday simplyanswered: What is the usefulness of achild? He grows into a man! In 2007, I liketo conclude this article thinking of frac-tional calculus as an adolescent whoshows promise to become a great man.

BiograhyDonato Cafagna received the Dr. Eng.in electronic engineering (with hon-ors) and the Ph.D. in electrical engi-neering from the Politecnico di Bari(Italy) in 1995 and 1999, respectively.In 2001, after two postdoctoral yearsat the Politecnico di Bari, he joinedthe Dipartimento di Ingegneria dell’In-novazione, Università del Salento(Italy), where he is currently assistantprofessor of electrical engineering. Hisresearch interests include analysisand design of chaotic circuits withapplication to synchronization andcontrol, study of chaotic phenomenain power converters, and analysis offractional order circuits and systems.He is a Member of the IEEE.

References[1] K.B. Oldham and J. Spanier, The Fractional Calcu-

lus. New York: Academic Press, 1974.

[2] B. Ross, “Fractional calculus,” Mathemat. Mag.,vol. 50, no. 3, pp. 115–122, 1977.

[3] K.S. Miller and B. Ross, An Introduction to the Frac-tional Calculus and Fractional Differential Equa-tions. New York: Wiley, 1993.

[4] R. Gorenflo and F. Mainardi, “Fractional calcu-lus: Integral and differential equations of frac-tional order,” in Fractal and Fractional Calculusin Continuum Mechanics. Berlin: Springer-Verlag,pp. 223–276, 1997.

[5] K.S. Miller, “Derivatives of noninteger order,”Mathemat. Mag., vol. 68, no. 3, pp. 183–192, 1995.

[6] A.V. Letnikov, “Theory of differentiation of frac-tional order,” Mat. Sb., vol. 3, pp. 1–68, 1868.

[7] M. Caputo, “Linear models of dissipation whoseQ is almost frequency independent. Part II,” J.

Roy. Astr. Soc., vol. 13, pp. 529–539, 1967.

[8] I. Podlubny, Fractional Differential Equations. NewYork: Academic Press, 1999.

[9] R. Hilfer (Ed.), Applications of Fractional Calculusin Physics. Singapore: World Scientific, 2001.

[10] E.O. Carew, T.C. Doehring, J.E. Barber, A.D.Freed, and I. Vesely, “Fractional-order viscoelas-ticity applied to heart valve tissues,” in Proc.2003 Summer Bioengineering Conf., Key Biscayne,FL, June 2003, pp. 721–722.

[11] M.E. Reyes-Melo, J.J. Martinez-Vega, C.A.Guerrero-Salazar, and U. Ortiz-Mendez, “Applica-tion of fractional calculus to modelling of relax-ation phenomena of organic dielectric materials,”in Proc. IEEE Int. Conf. Solid Dielectrics, Toulouse,France, July 2004, pp. 530–533.

[12] C. Ma and Y. Hori, “Backlash vibration suppres-sion control of torsional system by novel fraction-al order P ID k controller,” IEEJ Trans. Ind.Applicat., vol. 124-D, no. 3, pp. 312–317, 2004.

[13] A.J. Calderon, B.M. Vinagre, and V. Feliu, “Frac-tional order control strategies for power electron-ic buck converters,” Signal Processing, vol. 86, no. 10, pp. 2803–2819, 2006.

[14] A. Oustaloup, X. Moreau, and M. Nouillant, “TheCRONE suspension,” Control Eng. Practice, vol. 4,no. 8, pp. 1101–1108, 1996.

[15] A. Oustaloup, B. Mathieu, and P. Lanusse, “TheCRONE control of resonant plants: Application toa flexible transmission,” Eur. J. Contr., vol. 1, no. 2,pp. 113–121, 1995.

[16] P. Lanusse, V. Pommier, and A. Oustaloup, “Frac-tional control-system design for a hydraulic actu-ator,” in Proc. 1st IFAC Conf. Mechatronics Systems,Darmstadt, Germany, 2000.

[17] D. Matignon, “Stability properties for general-ized fractional differential systems,” in Proc.ESAIM, vol. 5, pp. 145–158, 1998.

[18] D. Matignon and B. d’Andrea-Novel, “Someresults on controllability and observability offinite-dimensional fractional differential systems,”in Proc. Computational Engineering in Systems andApplication Multiconference, Lille, France, vol. 2,pp. 952–956, Lille, France, July 1996.

[19] I. Podlubny. “Fractional-order systems andPIλDμ controller,” IEEE Trans. Automat. Contr., vol. 44, no. 1, pp. 208–214, 1999.

[20] L. Dorcak, I. Petras, and I. Kostial, “The model-ling and analysis of fractional-order regulatedsystems in the state space,” in Proc. 1st Int. Carpa-tian Control Conf., Podbanske, Slovak Republic,pp. 185–188, 2000.

[21] D. Cafagna and G. Grassi, “Fractional-orderChua’s circuit: Time-domain analysis, bifurcation,chaotic behaviour and test for chaos,” Int. J. Bifur-cations Chaos, to be published.

[22] H. Sun, A. Abdelwahed, and B. Onaral, “Linearapproximation for transfer function with a poleof fractional order,” IEEE Trans. Automat. Contr.,vol. 29, no. 5, pp. 441–444, 1984.

[23] K. Diethelm, N.J. Ford, and A.D. Freed, “A pre-dictor-corrector approach for the numericalsolution of fractional differential equations,”Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22,2002.

[24] G. Adomian, Solving Frontier Problems ofPhysics: The Decomposition Method. Boston:Kluwer 1994.

40 IEEE INDUSTRIAL ELECTRONICS MAGAZINE ■ SUMMER 2007

FIGURE 2—Double-scroll chaotic attractor in the fractional-order Chua’s circuit (order equal to 1.05).

1.5

1

0.5

0

−0.5

−1

−1.5−1

−0.5 0

x

y

z

0.51 0.2

0.10

−0.1−0.2