This paper deals with the concept of (integer-order)memristive systems, which are generalized to non-integerorder case using fractional calculus. We consider thememory effect of the fractional inductor (fractductor),fractional capacitor and fractional memristor. We alsoshow that the memory effect of such devices can be alsoused for an analogue implementation of the fractional-order operator, namely fractional-order integral andfractional-order derivatives. This kind of operator isuseful for realization of the fractional-order controllers.We present theoretical description of such implementationand we proposed the practical realization and did somesimulations and experimental measurements as well.
1. Introduction
Fractional calculus is more than 300 years old idea.These mathematical phenomena allow describe a real ob-ject more accurately than the classical “integer” methods.The real objects are generally fractional [20, 23, 29, 44],however, for many of them the fractionality is very low.A typical example of a non-integer (fractional) order sys-tem is the voltage-current relation of a semi-infinite lossytransmission line [41] or diffusion of the heat through asemi-infinite solid, where heat flow is equal to the half-derivative of the temperature [29]. Besides of the bettermodels of real systems, there is another phenomena inthe fractional calculus, namely memory effect. It is wellknown that the fractional-order systems have an unlim-ited memory (infinite dimensional) while the integer-ordersystems have a limited memory (finite dimensional).
In 1971, professor Leon O. Chua published a paper on
the missing basic circuit element - memristor or memoryresistor. Memristor is a new electrical element which hasbeen predicted and described in 1971 by Leon O. Chuaand for the first time realized by HP laboratory in 2008.Chua proved that memristor behavior could not be dupli-cated by any circuit built using only the other three el-ements (resistor, capacitor, inductor), which is why thememristor is truly fundamental. Memristor is a contrac-tion of memory resistor, because that is exactly its func-tion: to remember its history. The memristor is a two-terminal device whose resistance depends on the magni-tude and polarity of the voltage applied to it and the lengthof time that voltage has been applied. The missing el-ement - the memristor, with memristance M -provides afunctional relation between charge and flux, dφ = Mdq.
Professor Leon O. Chua and Dr. Sung–Mo Kang pub-lished a paper, in 1976, that described a large class of de-vices and systems they called memristive devices and sys-tems [6]. Whereas a memristor has mathematically scalarstate, a system has vector state. The number of state vari-ables is independent of, and usually greater than, the num-ber of terminals. In that paper, Chua applied the model toempirically observed phenomena, including the Hodgkin-Huxley model of the axon and a thermistor at constantambient temperature. He also described memristive sys-tems in terms of energy storage and easily observed elec-trical characteristics. These characteristics match resistiverandom-access memory and phase-change memory, relat-ing the theory to active areas of research. Chua extrap-olated the conceptual symmetry between the resistor, in-ductor, and capacitor, and inferred that the memristor is asimilarly fundamental device. Other scientists had alreadyused fixed nonlinear flux-charge relationships, but Chua’stheory introduces generality. This relation is illustrated inFig. 1.
Figure 1. Connection of four basic electricalelements (Figure is adopted from Ref. [36]).
Thirty-seven years later, on April 30, 2008, StanWilliams and his research group of scientists from HPLabs has finally built real working memristors, thusadding a fourth basic circuit element to electrical circuittheory, one that will join the three better-known ones: thecapacitor, resistor and the inductor. They built a two-terminal titanium dioxide nanoscale device that exhibitedmemristor characteristics [45]. A linear time-invariantmemristor is simply a conventional resistor. Importantthing is that it is impossible to substitute memristor withcombination of the other basic electrical elements andtherefore memristor can provide other new functions [16].
Possible applications of memristive systems [46]:
• new memory without access cycle limitations withnew memory cells for more energy-efficient comput-ers [39] e.g.: 1 bit = 1 memristor;
• new analog computers that can process and associateinformation in a manner similar to that of the humanbrain [35];
• new electronic circuits, e.g. [7, 37]: voltage divider,switcher, compensator, AD – DA converters, etc.;
• new control systems/controllers with memory [8];
In this paper we present the connection between frac-tional calculus (fractional order integral and derivative)and behavior of the memristive systems. As we willsee, the fundamentals of fractional calculus are basedon the memory property of the fractional order inte-gral/derivative and therefore this connection is straightfor-ward. This exceptional property can be used for realiza-tion of the fractional order operator as a basic element forimplementation of the fractional order controllers.
This paper is organized as follows: Section 1 intro-duces memristor and memristive devices. In Section 2 isdescribed the fractional calculus. Section 3 is on analogueelectrical elements which exhibit memristive behavior. InSection 4 are described the fractional-order circuits andproposal for their realization with using the memristivesystem and op-amps. In Section 5 are presented the realmeasurements and simulations. Section 6 concludes thisarticle with some additional remarks.
2. Fractional–order calculus definitions
The idea of fractional calculus has been known sincethe development of the regular calculus, with the firstreference probably being associated with letter betweenLeibniz and L’Hospital in 1695.
Fractional calculus is a generalization of integrationand differentiation to non-integer order fundamental op-erator aDα
t , where a and t are the limits of the operation.The continuous integro-differential operator is defined as
aDαt =
dα
dtα : α > 0,1 : α = 0,∫ t
a(dτ)−α : α < 0.
The three equivalent definitions used for the generalfractional differintegral are the Grunwald-Letnikov (GL)definition, the Riemann-Liouville (RL) and the Caputo’sdefinition [21, 29]. The GL is given as
aDαt f(t) = lim
h→0h−α
[ t−ah ]
∑
j=0
(−1)j
(α
j
)
f(t − jh), (1)
where [.] means the integer part. The RL definition isgiven as
aDαt f(t) =
1Γ(n − α)
dn
dtn
∫ t
a
f(τ)(t − τ)α−n+1
dτ, (2)
for (n − 1 < α < n) and where Γ(.) is the Gamma func-tion. The Caputo’s definition can be written as
aDαt f(t) =
1Γ(α − n)
∫ t
a
f (n)(τ)(t − τ)α−n+1
dτ, (3)
for (n − 1 < α < n). The initial conditions for thefractional order differential equations with the Caputo’sderivatives are in the same form as for the integer-orderdifferential equations.
The Laplace transform method is used for solving engi-neering problems. The formula for the Laplace transformof the RL fractional derivative (2) has the form [29]:
∫ ∞
0
e−st0D
αt f(t) dt = sαF (s) −
n−1∑
k=0
sk0D
α−k−1t f(t),
(4)
for (n − 1 < α ≤ n), where s ≡ jω denotes theLaplace operator. For zero initial conditions [11], Laplace
transform of fractional derivatives (Grunwald-Letnikov,Riemann-Liouville, and Caputo’s), reduces to:
L0Dαt f(t) = sαF (s). (5)
Some others important properties of the fractionalderivatives and integrals we can find out in several works(e.g.: [21, 29], etc.).
For simulation purpose, here we present theOustaloup’s recursive approximation (ORA) algo-rithm [23,24]. The method is based on the approximationof a function of the form:
H(s) = sα, α ∈ R, α ∈ [−1; 1] (6)
for the frequency range selected as (ωb, ωh) by a rationalfunction:
H(s) = Co
N∏
k=−N
s + ω′k
s + ωk(7)
using the following set of synthesis formulas for zeros,poles and the gain:
ω′k = ωb
(ωh
ωb
) k+N+0.5(1−α)2N+1
,
ωk = ωb
(ωh
ωb
) k+N+0.5(1−α)2N+1
,
Co =(
ωh
ωb
)−α2 N∏
k=−N
ωk
ω′k
, (8)
where ωh, ωb are the high and low transitional frequen-cies. An implemetation of this algorithm in Matlab asa function script ora foc() is given in [5].
3. Fractional–order memristive devices
There are a large number of electric and magnetic phe-nomena where the fractional calculus can be used [1, 44].We will consider three of them - capacitor, inductor andmemristor.
Westerlund et al. in 1994 proposed a new linear capac-itor model [43]. It is based on Curie’s empirical law of1889 which states that the current through a capacitor is
I(t) =V0
h1tα,
where h1 and α are constant, V0 is the dc voltage appliedat t = 0, and 0 < α < 1, (α ∈ R).
For a general input voltage V (t) the current is
I(t) = CdαV (t)
dtα≡ C 0D
αt V (t), (9)
where C is capacitance of the capacitor. It is related tothe kind of dielectric. Another constant α (order) is re-lated to the losses of the capacitor. Westerlund provided
in his work the table of various capacitor dielectric withappropriated constant α which has been obtained experi-mentally by measurements.
For a current in the capacitor the voltage is
V (t) =1C
∫ t
0
I(t)dtα ≡ 1C
0D−αt I(t). (10)
Then the impedance of a fractional capacitor is:
Zc(s) =1
Csα=
1ωαC
ej(−α π2 ). (11)
Ideal Bode’s characteristics of the transfer function forreal capacitor (11) are depicted in Fig. 2.
Figure 2. Bode plots of real capacitor.
Westerlund in his work also described behavior of realinductor [44]. For a general current in the inductor thevoltage is
V (t) = LdαI(t)dtα
≡ L 0Dαt I(t), (12)
where L is inductance of the inductor and constant α isrelated to the “proximity effect”. A table of various coilsand their real orders α is described in [34].
Then the impedance of a fractional inductor is:
ZL(s) = Lsα = ωαLejα π2 . (13)
Ideal Bode’s characteristics of the transfer function forreal inductor (13) are depicted in Fig. 3.
As it was already mentioned, Chua in 1971 predicteda new circuit element - called memristor characterized by
a relationship between the charge q(t) and the flux φ(t).It is the fourth basic circuit element [6, 27, 28, 39]. Thevoltage across a charge - controlled memristor is given by
v(t) = M(q(t))i(t), where M(q(t)) = dφ/dq.(14)
Noting from Faraday’s law of induction that magneticflux φ(t) is simply the time integral of voltage (dφ =V (t) dt) and charge q(t) is the time integral of current(dq = I(t) dt), the more convenient form of the current- voltage equation for the memristor is [6]
M(q(t))∫ t
0
I(t)dt =∫ t
0
V (t)dt, (15)
where M(q(t)) is memristance of the memristor. IfM(q(t)) is a constant (M(q(t)) ≡ R(t)), then we obtainOhm’s law R(t) = V (t)/I(t). If M(q(t)) is nontrivial,the equation is not equivalent because q(t) and M(q(t))will vary with time. Furthermore, the memristor is staticif no current is applied. If I(t) = 0, we find V (t) = 0and M(t) is constant. This is the essence of the memoryeffect, which allow us extending the notion of memristivesystems to capacitive and inductive elements in the formof memcapacitors and meminductorswhose properties de-pend on the state and history of the system [17, 40].
Similar to capacitor and inductor, the memristor is alsonot ideal circuit element and we can predict the fractional-order model of such element. Applying the fractional cal-culus to relation (15), we obtain the following general for-
mula for the fractional-order memristive systems:
K 0Dγt I(t) = 0D
βt V (t), (γ, β ∈ R) (16)
where K is the resistance, inductance, capacitance ormemristance, respectively.
Applying the Laplace transform technique (4) to equa-tion (16), we get the following relation
K sγI(s) = sβV (s) (17)
and the resulting impedance of the memristive system(MS) is
ZMS(s) = K sγ−β = K sα, (α ∈ R) (18)
where α is the real order of the memristive system and forthe ideal electrical elements has the following particularvalues, if:
• γ = 0 and β = 0 then α = 0, we obtain resistor andthen K = R [Ω];
• γ = −1 and β = 0 then α = −1, we obtain capacitorand then K = 1/C [F ];
• γ = 0 and β = −1 then α = 1, we obtain inductorand then K = L [H ];
• γ = −1 and β = −1 then α = 0, we obtain memris-tor and then K = M(t) [Ω];
However, as already has been mentioned, the real elec-trical element are not ideal and with the help of fractionalcalculus was shown that the intermediate cases betweenthe known characteristic behaviors of the electrical ele-ments resistor R, capacitor C and inductor L change con-tinuosly [37]. By deduction the memristor M , which hasstorage properties, could be also consider as a real electri-cal element with the fractional order of its mathematicalmodel. The fractional calculus can help us to describedthe memory behavior of the memristor. As we can see inthe equations (1), (2), and (3) kernels of the definitionsconsist of the memory term and consider with the history.It is suitable also for the memristor description and its ap-plications.
General characteristics of the transfer function of a realmemristive system (18) are:
• Magnitude: constant slope of α20dB/dec.;
• Crossover frequency: a function of K;
• Phase: horizontal line of α π2 ;
• Nyquist: straight line at argument α π2 .
The above concepts of memory devices are not neces-sarily limited to resistance – memristor but can in fact begeneralized to capacitative and inductive systems. If x(t)denotes a set of n state variables describing the internalstate of the system, u(t) and y(t) are any two comple-mentary constitutive variables (current, charge, voltage,
or flux) denoting input and output of the system, and g isa generalized response, we can define a general class ofnth-order u controlled dynamical systems called memris-tive systems or devices described by the following equa-tions [7]
dx(t)dt
= f(x, u, t)
y(t) = g(x, u, t)u(t), (19)
where f is a continuous n-dimensional vector functionand we assume unique solution for any initial state x(t)at time t = t0.
General fractional-order differential equation (16) canbe rewritten to its canonical form and then equations (19)become as follow:
0Dαt x(t) = f(x, u, t)y(t) = g(x, u, t)u(t). (20)
4 Analogue fractional-order circuits
We are able to define arbitrary real order α for thememristive system behavior description (18). The am-plitude of this impedance function is A = 20α and thephase angle is Φ = α(π/2) for α ∈ R. Electrical ele-ments (memristive system or fractance) with such prop-erty are sometimes called constant phase element for cer-tain frequency range [1]. So far, the constant phase el-ements (CPE) were approximated by the ladder networkconstructed form RLC elements, tree network, metal-insulator-liquid interface, etc. [3,4,9,10,12–14,20,22,30,38, 41].
Figure 4. Basic connection of twoimpedances with op-amp.
We can use an active operating amplifier (op-amp) andits inverting connection with impedance Z1 in direct con-nection and impedance Z2 in feedback connection. Trans-fer function of circuit depicted in Fig. 4 is:
H(s) =Vo(s)Vin(s)
= −Z2(s)Z1(s)
.
Generally, as electrical element with the impedancesZ1 and Z2 can be used basic electrical elements (resis-tor, capacitor, inductor, memristor) or electrical networks(RC ladder, RC tree, RLC grid, CPE). In this way we canobtain various dividers, filters, integrators, differentiators,
etc. Instead memristive devices or fractance circuit thenew electrical element introduced by G. Bohannan whichis so called ”Fractor” can be used as well [2].
5. Illustrative examples
5.1. Simulation resultsIn Fig. 5 is shown the proposal for analogue implemen-
tation of the fractional-order controller (FOC) with usingthe memristive systems as for example memristor, real ca-pacitor and inductor and op-amps in inverting connection.The memristive systems could be replaced by any CPEor other electrical RLC networks and instead memristorwe can use a usual resistor. Using the suggested circuit ismuch better because of memory property in the FOC.
Figure 5. Analogue fractional - order con-troller built with the memristive systems.
Applying the analogue fractional-order devices de-scribed in Section 4, we are able to realize a new typeof the fractional-order controller (see Fig. 5), which hasthe transfer function:
C(s) =Vo(s)Vin(s)
= −R2
R1
(
−ZM1(s)ZR3(s)
− ZC1(s)ZR4(s)
− ZL1(s)ZR5(s)
)
=R2
R1
(M1
R3sµ +
1C1R4sλ
+L1
R5sδ
)
= Kpsµ + Tis
−λ + Tdsδ. (21)
Usually we set R2 = R1, suppose ideal memristor M1
with µ = 0 and then the controller parameters are:
Kp =M1
R3, Ti =
1C1R4
, Td =L1
R5. (22)
5
For simulation purpose were chosen the following val-ues of electrical elements:
From values of the electrical elements (23), we obtainthe following controller parameters:
Kp = 10, Ti = 100, Td = 1 × 10−3,
µ = 0, λ = 0.94, δ = 0.86. (24)
20
25
30
35
40
45
50
55
60
Mag
nitu
de (
dB)
10−1
100
101
102
103
−90
−45
0
45
Pha
se (
deg)
Bode Diagram
Frequency (rad/sec)
Figure 6. Bode plots of the controller trans-fer function (21) with parameters (24).
In Fig. 6 are depicted simulation results of thefractional-order controller transfer function (21) withparameters (24) obtained via numerical approximationmethod ORA in Matlab.
PIλDµ controller [32], also known as PIλDδ con-troller, was already studied in time domain in [29] andalso in frequency domain in [30]. Investigation and de-tailed analysis of such kind of controller and its partic-ular cases (PDδ, CRONE, Lead-Lag Compensator, TID,etc.) have been done in several additional works (e.g.:[9, 15, 18, 25, 26, 31,33, 47], etc.).
5.2. Experimental measurementHere, we describe a new “element,” a so-called
fractional inductor, or ”Fractductor.” This device hasa fractional-order coupling between flux and current.
Preliminary attempts to construct a fractductor haveproduced a device using magnetorheological fluid as thecore in a transformer-like device. A bode plot (Fig. 7),basic block schematic (Fig. 8), and photo of the experi-mental device (Fig. 9) are shown.
As we can see in Fig. 8, connection of electrical el-ements were done according to suggestions describedin prevoius sections. It is practical realization of the
Figure 9. Photo of fractductor test setupduring collection of data shown in Fig. 7.
fractional-ordermemristive systems which can be used forthe fractional-order controllers implementation as well.Estimated order of the fractductor is α = 0.5 (see Fig. 7).
6. Conclusion
In this brief paper was presented proposal for a newclass of the fractional-order memristive systems, whichare useful for practical implementation of the fractional-order controllers. However this approach gave a good startfor detail analysis and design of the analogue fractional-order controller. The fractional-order controller gives usan insight into the concept of memory of the suggestedfractional operator.
We also proposed a new electrical device, so–calledfractductor, which belongs to class of the analogue gen-eralized fractional-order memristive devices [8]. This de-vice has a fractional-order coupling between flux and cur-rent.
Further work is needed to prove its performance by var-ious simulations and experimental measurements at thecircuit. The results may find wide application in signalprocessing and control systems (e.g.: [8, 19, 42], etc.).
7. Acknowledgment
Ivo Petras was supported in part by the Slovak GrantAgency for Science under grants VEGA: 1/4058/07,1/0365/08, 1/0404/08, and grant APVV-0040-07.
YangQuan Chen was supported in part by Utah StateUniversity New Faculty Research Grant (2002-2003), theTCO Bridging Fund of Utah State University (2005-2006), an NSF SBIR subcontract through Dr. Gary Bo-hannan (2006), and by the National Academy of Sciencesunder the Collaboration in Basic Science and Engineer-ing Program/Twinning Program supported by ContractNo. INT-0002341 from the National Science Foundation(2003-2005).
6
Figure 7. Bode plots of an experimental fractional flux coupling device, the fractductor.
-+
Coil A, 30 turns Coil B, 30 turns
Magnetorheological fluid
Csmall
Rgain
Rout
Magnet Magnet
++- -
Figure 8. Simplified schematic diagram of the fractductor and test circuit.
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