This paper demonstrates the feasibility of modeling any dynamical system using a set of fractional order di®erential equations, including distributed and lumped systems. Fractional order differentiators and integrators are the basic elements of these equations representing the real model of the dynamical system, which in turn implies the necessity of using fractional order controllers instead of controllers with integer order. This paper proves that fractional order differential equations can be used to model any dynamical system whether it is continuous or lumped

Kahalil, Islam

Naskali, Ahmet Teoman

Sabanovic, Asif

Şabanoviç, Asif

English

Sabanci University Research Database

On Fraction Order Modeling and Controlof Dynamical Systems ?Islam S. M. Khalil ∗ A. Teoman Naskali ∗ Asif Sabanovic ∗∗ Faculty of Engineering and Natural Science, Sabanci University,34956 Turkey (e-mail: kahalil,teoman@su.sabanciuniv.edu,asif@sabanciuniv.edu).Abstract: This paper demonstrates the feasibility of modeling any dynamical system using a setof fractional order differential equations, including distributed and lumped systems. Fractionalorder differentiators and integrators are the basic elements of these equations representing thereal model of the dynamical system, which in turn implies the necessity of using fractional ordercontrollers instead of controllers with integer order. This paper proves that fractional orderdifferential equations can be used to model any dynamical system whether it is continuous orlumped.Keywords: Fraction calculus, Continuous and lumped systems, Fraction order control, Laplacetransform.1. INTRODUCTIONDynamics of any lumped system is descried by a set ofdifferential equations that can be obtained by the new-tonian approach based on the entire system forces com-putations or the variational approach that is based onsystem energies computations. On the other hand, contin-uous systems such as beams and flexible manipulators aremodeled using a set of partial differential equations. Andas its well known that these ordinary differential equationscan be represented as a set of linear equations in thelaplace domain. Surprisingly enough, acting by the laplaceoperator on a set partial differential equations turns theminto ordinary differential equations, and if it was appliedtwice on the same set of partial differential equations,we end up with another set of fraction order differentialequations. It turns out that, in the laplace domain anydynamical system whether it is lumped or continuous iscomposed of a set of fraction order differentiators andintegrators, that describe the exact systems dynamics. Inother words, if the dynamical system is described by aset of partial differential equations, fraction order transferfunction has to be obtained, on the other hand, if thesystem dynamics are described by ordinary differentialequations, we may end up with a fractional order transferfunction when we represent these equations in the laplacedomain. However, dynamical systems can be classified intofractional or integer order systems. Simply, fractional ordersystems are systems described by fractional order transferfunctions.Ma and Hori (2007) pointed out that integer ordermodel is equivalent to the fraction order model in the lowfrequency range, while at the high frequency range integerorder model doesn’t describe the system dynamics at all.Moreover, the control system or the compensator have to? This work was supported by the financial assistance provided bySanteZ project number 00183.STZ.2007-2.be with a fraction order as it was presented in Podlunbnyet al. (1997).Fractional order controllers show better transient responseresults compared with integer order controllers when theyare applied on fractional order systems Podlubny (1999).Minimum over shoot, less sensitivity to system parametersvariations and controller parameters are shown in Pod-lunbny (1999) when fraction-order control(FOC) was ap-plied to a fraction order system.The idea of using fractional-order controllers belongs toOustaloup, who developed the so called Command Robusted’Ordre Non Entire (CRONE) Outaloup (1995). Fractionorder PID controllers were investigated in Podlunbny et al.(1997) and instead of jumping in the P-I-D plane by usinginteger order differentiator and integrator. The fraction or-der PID controller makes it possible to move continuouslyin the P-I-D plane. The FOC concept was restricted dueto the unfamiliar idea of taking fractional order and thecomputational efficiency of computing the fraction orderdifferentiators and integrators, that is basically based onapproximations and memory length, that indicates howthis approximation is close to the fraction ordered opera-tor Chen and Moore (2002).This paper proves that dynamical systems whether theyare lumped or continuous can be modeled using fractionalorder differential equations. In other words, using inte-ger order differentiators and integrators is nothing buta special case of the general fraction order operators.The paper is organized as follows, in section 2, fractionorder transfer function is derived for both lumped andcontinuous dynamical systems. In section 3, the fractionorder differentiator or integrator is approximated by adiscretization process. A comparison between integer andfractional order controller for a plant with a fractionalorder transfer function is included in section 4.©2009 IFAC2. FRACTION ORDER TRANSFER FUNCTION2.1 Continuous SystemsFlexible beam A flexible beam is modeled by the follow-ing partial differential equationEI∂4w(x, t)∂x4+ ρA∂2w(x, t)∂t4= f(x, t) (1)where E, I, ρ and A are respectively the elasticity modu-lus, moment of inertia, material density and beams crosssection. f(x, t) is the external forcing function. Taking thelaplace transform of (1) we getEId4w(x, s)dx4+ ρAs2w(x, s) = f(x, s) (2)where (2) is an ordinary differential equation, solvingfor the homogenous solution of w(x, s) and making thefollowing definitionc , ρAEI.The characteristic equation of (2) isλ4 + cs2 = 0 (3)and the roots areλ1 = i1/2 c1/4s1/2 (4)λ2 =−i1/2 c1/4s1/2λ3 = i3/2 c1/4s1/2λ3 =−i3/2 c1/4s1/2 .The homogenous solution describing the beams transientresponse isw(x, s) = c1eλ1s + c2eλ2s + c3eλ3s + c4eλ4s (5)where c1,2,3,4 are constants that depend on beam’s bound-ary conditions. The previous equation indicates that thetransient response of the beam is governed by a fractionorder differentiator s3/2, which in turn implies that flexiblebeam can be described precisely using a fraction ordertransfer function.Rotor attached to a flexible shaft the partial differentialequation describing this system in Manabe (2002)I1l∂2θ(x, t)∂t2− kl∂2θ(x, t)∂x2= 0 (6)where I1,l are the rotor inertia and the shaft length, kis the torsional stiffness of the shaft. Taking the laplacetransform of (6) we getI1ls2θ(x, s)− kl∂2θ(x, s)∂x2= 0 (7)and the fraction order transfer function between the inputtorque τ and the rotor angular position θ isτ(x, s)θ(x, s)= I1s2 + klµs tanh(µls) (8)whereµ2 , I1klthat is nothing but a fraction order transfer function, andthe results obtained from (5) and (8) were expected asthey depend on fraction order differentiator or integrators,because of the continuous nature of both systems.Fig. 1. Lumped flexible system2.2 Lumped SystemsFor the lumped mass spring system shown in Fig.1, themotion of each mass can be modeled by an ordinary differ-ential equation, and the entire system can be modeled bya set of coupled linear or nonlinear differential equations.The motion of each mass is related to its neighbor massby the following equation W.J.O’Connar (2007b).Xi+1(s) = G(s)Xi(s) (9)where G(s) is the transfer function relating any particularmass with its neighbor mass. And the equation of motionfor the ith massmx¨i = k(xi−1 − 2xi + xi+1) . (10)Taking laplace transform we get a quadratic equation inG(s)G2(s)− (ms2 + 2k)G(s) + k = 0 (11)solving the quadratic equation we getG1,2(s) = 1 +12s22ω2n±√s22ω2n(1 +s22ω2n) (12)whereω2n =kmand the general solution for the motion of each mass is a re-sult of the superposition of two components W.J.O’Connar(2007a)Xi(s) = αi(s)G1(s) + βi(s)G2(s) . (13)Where αi(s) and βi(s) are arbitrary, and the transferfunction (12) has a non-integer order and its also nonra-tional. On one hand this transfer function is hard to workwith, but on the other hand it can be approximated bya second order transfer function that will not represent aproblem in case of feedback control as Occonar pointed outin W.J.O’Connar (2007b). In any event, the point here isto prove that the exact model that describes the dynamicsof these systems is with fractional order, that is not easyto work with in the sense of mathematical computation.3. FRACTION ORDER OPERATORDISCRETIZATION3.1 Generating Function MethodAs the fraction order transfer function is based on a set offraction order differentiators sr and integrators s−r, wherer is any arbitrary real number not necessarily integer,they can be approximated using the following generatingfunctions = ω(z−1) (14)(ω(z−1))±r = (2T)rAn(z−1, r)An(z−1,−r) (15)= (2T)r lim∆→0An(z−1, r)An(z−1,−r)where n represent the memory length of the generatingfunction, the larger n we select the closer approximationto the exact operator we get with more computations.Ao(z−1, r) = 1 (16)An(z−1,r) =An−1(z−1, r)− cnznAn−1(z, r)c ={r/n, n is odd;0, n is even. (17)Assuming that the fraction order differentiator s0.5 isrequired to be approximated using the previous generatingfunction we get the following. Maione (2006).For n=1 we get the following discrete approximationA1(z−1, 0.5) = 1− 0.5z−1 (18)A1(z−1,−0.5) = 1 + 0.5z−1G1(z) =44.72z − 22.36z + 0.5. (19)For n=3 we get the following discrete approximationG3(z) =44.72z3 − 22.36z2 + 3.727z − 7.454z3 + 0.5z2 + 0.0833z + 0.1667. (20)For n=7 we get the following discrete approximationG7(z) =44.72z7 − 22.36z6+4.792z5 − 7.986z4 + 2.795z3−4.792z2 + 1.597z − 3.194z7 + 0.5z6+0.107z5 + 0.1786z4 + 0.0625z3+0.107z2 + 0.0357z + 0.07143(21)For n=9 we get the following discrete approximationG9(z) =44.72z9 − 22.36z8 + 4.969z7 − 8.075z6+3.061z5 − 4.947z4 + 2.041z3 − 3.461z2+1.242z − 2.485z9 + 0.5z8 + 0.111z7 + 0.1806z6+0.0684z5 + 0.1106z4 + 0.0456z3+0.0773z2 + 0.02778z + 0.05556(22)Fig.2 shows the frequency responses of that discrete ap-proximations of the fractional order operator, it turnsout that increasing the memory length n makes the ap-proximation closer the exact fraction differentiator, Fig.2doesn’t show the actual frequency response of the frac-tion order operator s0.5, but as the generating functiondescribing this operator is an infinite series, the length ofthe memory can be adjusted when the high order termsstarts to have no pioneer impact on the frequency response.Therefore a generating function with memory length n = 9is suitable approximation for the fraction order differentia-tor s0.5.1020304050Magnitude (dB)  100 101 102 103 104−3003060Phase (deg)Bode DiagramFrequency  (rad/sec)n=1n=3n=7n=9Fig. 2. Frequency response of the discrete approximationof the operator s0.53.2 Integer Order ControlFor the position control of an inertial mass as shownin Fig.3, the closed loop transfer function between thereference input and the actual inertial mass position isG(s) =ksrJs2 + ksr. (23)For r = 0 the closed loop transfer function will have twopoles on the imaginary axis that ares1,2 = ±i√kJwhich in turn implies that the response of the system willbe oscillatory. On the other hand, setting r = 1 means thatwe are using a pure derivative controller, that is supposedto act on the time derivative of the error not the error itself.The previous analysis shows that using integer orderdifferentiators or integrator is equivalent to jumping onPID plane and just using four point on the PID plane.Fig.4-a shows the PID plane when integer order controlleris used, where entire plane is reduced into 4 possiblecombinations and the rest of the plane is not used.Fig. 3. Position control of an inertial mass3.3 Fraction Order ControlFraction order control allows moving in a continuousfashion in the PID plane, and instead of using finite pointson the plane or jumping from a point to another the entireplane can be used and infinitely many combinations ofdifferentiators and integrators can be used. Fig4-b showsthe entire PID plane that can be used if the controller hasa fraction order Podlunbny et al. (1997).(a) Integer order PID con-troller(b) Fraction order PIDcontrollerFig. 4. Controller DOF region for fraction and Integerorder PID controllerIf an intger order controller is used for the inertial systemshown in Fig.4, the system will be oscillatory or unstableas the only two choices are the pure proportional controlleror the pure derivative. Assuming that the order r is afraction we can achieve better tradeoff between stabilityand robustness, assuming that r=0.5 we get a controllerwith the following structureGc(s) = Ks0.5 . (24)The discrete approximation of the controller isGc(z) = K44.72z9 − 22.36z8 + 4.969z7 − 8.075z6+3.061z5 − 4.947z4 + 2.041z3 − 3.461z2+1.242z − 2.485z9 + 0.5z8 + 0.111z7 + 0.1806z6+0.0684z5 + 0.1106z4 + 0.0456z3+0.0773z2 + 0.02778z + 0.05556(25)where k, is the fraction order controller gain, Fig.5 showsthe response of the inertial system (22) for the fractionalorder controller (24) for different gain values. The purposehere is not to achieve a specific transient response, butto show that using fraction order control we can achievea better tradeoff between stability and robustness, thatcan not be achieved in the case of integer order controlleras jumping between the pure proportional controller topure derivative, makes the system oscillatory or uncon-trollable at all since the derivative actions acts on thetime derivative of the error not the error itself. Indeed,the plant here is not described by a fraction order transferfunction so that we use a fraction order controller thatis more suitable, but the purpose here is to show thebetter tradeoff that can be achieved when we continuouslymove in the PID-plane instead of jumping between fourfinite points representing the possible combinations of theclassical PID-controller.4. PIλDµ-CONTROLLERThe transfer function of the fraction order PID controlleris, Podlunbny (1999)Gc(s) =U(s)E(s)= Kp +KiS−λ +KDSµ (26)where λ and µ are any arbitrary real numbers, selectingλ = 1 and µ = 1 turns the controller into classical PIDcontroller, to show the difference between using a classicalPID-controller and a fractional order one we assume thatwe have a plant that is governed by the following discretetransfer function0 2 4 6 8 1000.20.40.60.811.21.4Time (sec)Position (Radians)  k=0.005k=0.01k=0.015Fig. 5. Fraction order control of an inertial massGp(z) =z3 + 0.9z2 + 0.27z + 0.3468.6z3 − 420z2 + 126.5z − 139.9 (27)selecting the following position on the continous PID planeµ= 0.5 (28)λ= 0.2that represent some location on the continous PID planerather than the classical PID locations that are shown inFig.4, and the controller transfer function becomesGc(s) =U(s)E(s)= Kp +KiS−0.2 +KDS0.5 . (29)This requires a discrete approximation for the fractionorder differentiators0.5 and integrators−0.2. The discreteapproximation of integrator s−0.2 isGs−0.2(z) =0.218z3 + 0.043z2 + 0.0029z + 0.014z3 − 0.2z2 + 0.0134z − 0.066while the approximation of the fraction order differentiatorwas previously computed in (22).Applying both integer and fraction order PID controllerto the discrete plant (27) using a variety of controllergains is shown in Fig.6. The fraction order controller showsbetter transient response characteristics in the sense ofhaving less overshoot and higher response. Indeed, thereason behind selecting these particular PIλDµ points onthe PID plane is not explained or even investigated inthis work, but the purpose is to show that using fractionorder controllers makes all the PID plane available andreachable by the controller, unlike the integer controllerwhere controller does finite jumps between the integerorder differentiator and integrators.5. CONCLUSIONMajority of dynamical systems are described by a fractionorder transfer functions, not only distributed systems suchas flexible shafts and beams, but also lumped systems.This fact doesn’t implies that the real transfer functionshould be derived or worked with, as approximations forthose function are quite enough especially in feedbackcontrol, In other words the for the lumped masses systemthe exact transfer function (12) can be approximatedby a second order transfer function that shows similarresponse. Using fraction order controller makes all thePID plane reachable in the sense of building controllerswith any required set of fraction order differentiators andintegrators, not necessarily the four points on the PID0 2 4 6 8 1002468101214Time (sec)reference  Step referenceResponse with Integer PID Response with fraction PID(a) kp=50 ki=20 kd=00 2 4 6 8 10024681012141618Time (sec)reference  Step referenceResponse with integer PIDResponse with fractional PID(b) kp=20 ki=25 kd=20 2 4 6 8 1002468101214Time (sec)reference  Step referenceResponse with Integer PID Response with fraction PID(c) kp=20 ki=10 kd=5Fig. 6. Fraction and integer order control of a plantdescribed by a fraction order transfer functionplane in the case of integer order control. In other words,using fraction order controllers increase the flexibility ofthe control process, with the ability to achieve bettertradeoff between stability and robustness as it was shownin inertial mass control example. This work shows thatfraction order controllers allows the continuous move in thePID plane that could not be achieved in the case of integerorder controllers, but on the other hand, why and how toselect these particular point on the PID plane was notinvestigated in this work. But considering the controllerwith a fraction order keeps the entire PID-plane usableand reachable, giving the controller much more flexibilityover the integer order ones.ACKNOWLEDGEMENTSThe authors gratefully acknowledge the financial assis-tance provided by Erasmus Mundus University Grantnumber 132878-EM-1-2007-BE-ERA Mundus-ECW.REFERENCESChen, Y.Q. and Moore, K.L. (2002). Discretizationschemes for fractional-order differentiators and integra-tors. J. Am. Chem. Soc., 49, 363–367.Ma, C. and Hori, Y. (2007). Fractional-order con-trol:theory and application in motion control. IEEEIndustrial Electronics Magazine, 1, 6–17.Maione, G. (2006). A rational discrete approximation tothe operator. IEEE Signal Processing Letters, 49, 141–144.Manabe, S. (2002). A suggestion of fractional-order con-troller for flexible spacecraft attitude control. J. Non-linear Dynamics, 29, 251–268.Outaloup, A. (1995). la derivation non entiere:theo-rie,synthese et application. Hermes, Paris, 1.Podlubny, I. (1999). Fractional differential equations. InA. Round (ed.), Frational Differntial Equations, volume198, 125–247. Academic Press, New York, 1st edition.Podlunbny, I., Dorcak, L., and I, K. (1997). On frac-tional derivatives, fractional-order dynamic systems andpiλdµ-controllers. IEEE, 1, 4985–4990.Podlunbny, I. (1999). Fraction-order systems and piλdµ-controllers. IEEE, 44, 208–214.W.J.O’Connar (2007a). Control of flexible mechanicalsystems:wave-based techniques. IEEE, 1, 4192–4202.W.J.O’Connar (2007b). Wave-based analysis and controlof lump-modeled flexible robots. IEEE Transaction onRobotics, 23, 1552–3098.

On fraction order modeling and control of dynamical systems

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On fraction order modeling and control of dynamical systems

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