Fractional-Order Memristive Systems

Ivo Petr´asˇ Institute of Control and Informatization of Production Processes BERG Faculty, Technical University of Kosiceˇ B. Nemcovejˇ 3, 042 00 Kosice,ˇ Slovak Republic Tel./Fax: +421-55-602-5194; E-mail: [email protected] YangQuan Chen and Calvin Coopmans Center for Self-Organizing and Intelligent Systems (CSOIS) Electrical and Computer Engineering Department Utah State University, Logan, UT 84322 − 4160, USA Tel.: +1(435)797-0148; Fax: +1(435)797-3054; E-mail: [email protected] http://fractionalcalculus.googlepages.com

Abstract the missing basic circuit element - or memory . Memristor is a new electrical element which has This paper deals with the concept of (integer-order) been predicted and described in 1971 by Leon O. Chua memristive systems, which are generalized to non-integer and for the first time realized by HP laboratory in 2008. order case using fractional calculus. We consider the Chua proved that memristor behavior could not be dupli- memory effect of the fractional (fractductor), cated by any circuit built using only the other three el- fractional and fractional memristor. We also ements (resistor, capacitor, inductor), which is why the show that the memory effect of such devices can be also memristor is truly fundamental. Memristor is a contrac- used for an analogue implementation of the fractional- tion of memory resistor, because that is exactly its func- order operator, namely fractional-order integral and tion: to remember its history. The memristor is a two- fractional-order derivatives. This kind of operator is terminal device whose resistance depends on the magni- useful for realization of the fractional-order controllers. tude and polarity of the applied to it and the length We present theoretical description of such implementation of time that voltage has been applied. The missing el- and we proposed the practical realization and did some ement - the memristor, with memristance M-provides a simulations and experimental measurements as well. functional 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- 1. Introduction vices and systems they called memristive devices and sys- tems [6]. Whereas a memristor has mathematically scalar Fractional calculus is more than 300 years old idea. state, a system has vector state. The number of state vari- These mathematical phenomena allow describe a real ob- ables is independent of, and usually greater than, the num- ject more accurately than the classical “integer” methods. ber of terminals. In that paper, Chua applied the model to The real objects are generally fractional [20, 23, 29, 44], empirically observed phenomena, including the Hodgkin- however, for many of them the fractionality is very low. Huxley model of the axon and a thermistor at constant A typical example of a non-integer (fractional) order sys- ambient temperature. He also described memristive sys- tem is the voltage-current relation of a semi-infinite lossy tems in terms of energy storage and easily observed elec- [41] or diffusion of the heat through a trical characteristics. These characteristics match resistive semi-infinite solid, where heat flow is equal to the half- random-access memory and phase-change memory, relat- derivative of the temperature [29]. Besides of the better ing the theory to active areas of research. Chua extrap- models of real systems, there is another phenomena in olated the conceptual symmetry between the resistor, in- the fractional calculus, namely memory effect. It is well ductor, and capacitor, and inferred that the memristor is a known that the fractional-order systems have an unlim- similarly fundamental device. Other scientists had already ited memory (infinite dimensional) while the integer-order used fixed nonlinear flux-charge relationships, but Chua’s systems have a limited memory (finite dimensional). theory introduces generality. This relation is illustrated in In 1971, professor Leon O. Chua published a paper on Fig. 1.

978-1-4244-2728-4/09/$25.00 ©2009 IEEE This paper is organized as follows: Section 1 intro- duces memristor and memristive devices. In Section 2 is described the fractional calculus. Section 3 is on analogue electrical elements which exhibit memristive behavior. In Section 4 are described the fractional-order circuits and proposal for their realization with using the memristive system and op-amps. In Section 5 are presented the real measurements and simulations. Section 6 concludes this article with some additional remarks.

2. Fractional–order calculus definitions

The idea of fractional calculus has been known since the development of the regular calculus, with the first reference probably being associated with letter between Leibniz and L’Hospital in 1695. Fractional calculus is a generalization of integration and differentiation to non-integer order fundamental op- α erator aDt , where a and t are the limits of the operation. The continuous integro-differential operator is defined as Figure 1. Connection of four basic electrical  elements (Figure is adopted from Ref. [36]). dα  dtα : α>0, α = 1 =0 aDt : α ,  t −α a (dτ) : α<0. Thirty-seven years later, on April 30, 2008, Stan Williams and his research group of scientists from HP The three equivalent definitions used for the general Labs has finally built real working , thus fractional differintegral are the Grunwald-Letnikov (GL) adding a fourth basic circuit element to electrical circuit definition, the Riemann-Liouville (RL) and the Caputo’s theory, one that will join the three better-known ones: the definition [21, 29]. The GL is given as capacitor, resistor and the inductor. They built a two- [ t−a ] h terminal titanium dioxide nanoscale device that exhibited α −α j α aDt f(t) = lim h (−1) f(t − jh), (1) h→0 memristor characteristics [45]. A linear time-invariant j=0 j memristor is simply a conventional resistor. Important thing is that it is impossible to substitute memristor with where [.] means the integer part. The RL definition is combination of the other basic electrical elements and given as therefore memristor can provide other new functions [16]. 1 n t ( ) Possible applications of memristive systems [46]: α ( )= d f τ aDt f t n α−n+1 dτ, (2) Γ(n − α) dt a (t − τ) • new memory without access cycle limitations with new memory cells for more energy-efficient comput- for (n − 1 <α

2 transform of fractional derivatives (Grunwald-Letnikov, in his work the table of various capacitor dielectric with Riemann-Liouville, and Caputo’s), reduces to: appropriated constant α which has been obtained experi- mentally by measurements. { α ( )} = α ( ) L 0Dt f t s F s . (5) For a current in the capacitor the voltage is Some others important properties of the fractional t 1 α 1 −α derivatives and integrals we can find out in several works V (t)= I(t)dt ≡ 0Dt I(t). (10) C 0 C (e.g.: [21, 29], etc.). For simulation purpose, here we present the Then the impedance of a fractional capacitor is: Oustaloup’s recursive approximation (ORA) algo- 1 1 j(−α π ) rithm [23,24]. The method is based on the approximation ( )= = 2 Zc s α α e . (11) of a function of the form: Cs ω C α Ideal Bode’s characteristics of the transfer function for H(s)=s ,α∈ R,α∈ [−1; 1] (6) real capacitor (11) are depicted in Fig. 2. for the frequency range selected as (ωb,ωh) by a rational function:

N  s + ωk H(s)=Co (7) + k k=−N s ω using the following set of synthesis formulas for zeros, poles and the gain:

k+N+0.5(1−α) 2N+1  ωh ωk = ωb , ωb k+N+0.5(1−α) ωh 2N+1 ωk = ωb , ωb α N − 2 ωh ωk Co =  , (8) b ω k=−N ωk where ωh,ωb are the high and low transitional frequen- cies. An implemetation of this algorithm in Matlab as a function script ora foc() is given in [5].

3. Fractional–order memristive devices

There are a large number of electric and magnetic phe- Figure 2. Bode plots of real capacitor. nomena where the fractional calculus can be used [1, 44]. We will consider three of them - capacitor, inductor and Westerlund in his work also described behavior of real memristor. inductor [44]. For a general current in the inductor the Westerlund et al. in 1994 proposed a new linear capac- voltage is itor model [43]. It is based on Curie’s empirical law of 1889 which states that the current through a capacitor is α d I(t) α V (t)=L ≡ L 0D I(t), (12) dtα t ( )= V0 I t α , h1t where L is of the inductor and constant α is related to the “proximity effect”. A table of various coils where h1 and α are constant, V0 is the dc voltage applied and their real orders α is described in [34]. at t =0, and 0 <α<1, (α ∈ R). Then the impedance of a fractional inductor is: For a general input voltage V (t) the current is α α jα π α ZL(s)=Ls = ω Le 2 . (13) d V (t) α I(t)=C ≡ C 0Dt V (t), (9) dtα Ideal Bode’s characteristics of the transfer function for where C is of the capacitor. It is related to real inductor (13) are depicted in Fig. 3. the kind of dielectric. Another constant α (order) is re- As it was already mentioned, Chua in 1971 predicted lated to the losses of the capacitor. Westerlund provided a new circuit element - called memristor characterized by

3 mula for the fractional-order memristive systems: γ β K 0Dt I(t)= 0Dt V (t), (γ,β ∈ R) (16)

where K is the resistance, inductance, capacitance or memristance, respectively. Applying the Laplace transform technique (4) to equa- tion (16), we get the following relation

Ksγ I(s)= sβV (s) (17)

and the resulting impedance of the memristive system (MS) is

γ−β α ZMS(s)=Ks = Ks , (α ∈ R) (18)

where α is the real order of the memristive system and for the ideal electrical elements has the following particular values, if: • γ =0and β =0then α =0, we obtain resistor and then K = R [Ω]; • γ = −1 and β =0then α = −1, we obtain capacitor and then K =1/C [F ]; • γ =0and β = −1 then α =1, we obtain inductor Figure 3. Bode plots of real inductor. and then K = L [H]; • γ = −1 and β = −1 then α =0, we obtain memris- a relationship between the charge q(t) and the flux φ(t). tor and then K = M(t)[Ω]; It is the fourth basic circuit element [6, 27, 28, 39]. The However, as already has been mentioned, the real elec- voltage across a charge - controlled memristor is given by trical element are not ideal and with the help of fractional v(t)=M(q(t))i(t), where M(q(t)) = dφ/dq. calculus was shown that the intermediate cases between (14) the known characteristic behaviors of the electrical ele- ments resistor R, capacitor C and inductor L change con- Noting from Faraday’s law of induction that magnetic tinuosly [37]. By deduction the memristor M, which has flux φ(t) is simply the time integral of voltage (dφ = storage properties, could be also consider as a real electri- V (t) dt) and charge q(t) is the time integral of current cal element with the fractional order of its mathematical (dq = I(t) dt), the more convenient form of the current model. The fractional calculus can help us to described - voltage equation for the memristor is [6] the memory behavior of the memristor. As we can see in the equations (1), (2), and (3) kernels of the definitions t t consist of the memory term and consider with the history. M(q(t)) I(t)dt = V (t)dt, (15) 0 0 It is suitable also for the memristor description and its ap- plications. where M(q(t)) is memristance of the memristor. If General characteristics of the transfer function of a real M(q(t)) is a constant (M(q(t)) ≡ R(t)), then we obtain memristive system (18) are: ’s law R(t)=V (t)/I(t).IfM(q(t)) is nontrivial, • Magnitude: constant slope of α20dB/dec.; the equation is not equivalent because q(t) and M(q(t)) will vary with time. Furthermore, the memristor is static • Crossover frequency: a function of K; if no current is applied. If I(t)=0,wefind V (t)=0 • π ; and M(t) is constant. This is the essence of the memory Phase: horizontal line of α 2 effect, which allow us extending the notion of memristive π • Nyquist: straight line at argument α . systems to capacitive and inductive elements in the form 2 of memcapacitors and meminductors whose properties de- The above concepts of memory devices are not neces- pend on the state and history of the system [17, 40]. sarily limited to resistance – memristor but can in fact be Similar to capacitor and inductor, the memristor is also generalized to capacitative and inductive systems. If x(t) not ideal circuit element and we can predict the fractional- denotes a set of n state variables describing the internal order model of such element. Applying the fractional cal- state of the system, u(t) and y(t) are any two comple- culus to relation (15), we obtain the following general for- mentary constitutive variables (current, charge, voltage,

4 or flux) denoting input and output of the system, and g is etc. Instead memristive devices or fractance circuit the a generalized response, we can define a general class of new electrical element introduced by G. Bohannan which nth-order u controlled dynamical systems called memris- is so called ”Fractor” can be used as well [2]. tive systems or devices described by the following equa- tions [7] 5. Illustrative examples dx(t) = f(x, u, t) 5.1. Simulation results dt In Fig. 5 is shown the proposal for analogue implemen- ( )= ( ) ( ) y t g x, u, t u t , (19) tation of the fractional-order controller (FOC) with using the memristive systems as for example memristor, real ca- where f is a continuous n-dimensional vector function pacitor and inductor and op-amps in inverting connection. and we assume unique solution for any initial state x(t) The memristive systems could be replaced by any CPE at time t = t0. or other electrical RLC networks and instead memristor General fractional-order differential equation (16) can we can use a usual resistor. Using the suggested circuit is be rewritten to its canonical form and then equations (19) much better because of memory property in the FOC. become as follow:

α 0Dt 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 the memristive system behavior description (18). The am- plitude of this impedance function is A =20α and the phase 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 network constructed form RLC elements, tree network, metal- insulator-liquid interface, etc. [3,4,9,10,12–14,20,22,30, 38, 41].

Figure 5. Analogue fractional - order con- troller built with the memristive systems.

Applying the analogue fractional-order devices de- Figure 4. Basic connection of two scribed in Section 4, we are able to realize a new type impedances with op-amp. of the fractional-order controller (see Fig. 5), which has the transfer function: ( ) ( )= Vo s We can use an active operating amplifier (op-amp) and C s Vin(s) its inverting connection with impedance Z 1 in direct con- R2 ZM1 (s) ZC1 (s) ZL1 (s) nection and impedance Z2 in feedback connection. Trans- = − − − − R1 ZR (s) ZR (s) ZR (s) fer function of circuit depicted in Fig. 4 is: 3 4 5 1 = R2 M1 µ + + L1 δ Vo(s) Z2(s) s λ s H(s)= = − . R1 R3 C1R4s R5 Vin(s) Z1(s) µ −λ δ = Kps + Tis + Tds . (21) Generally, as electrical element with the impedances Usually we set R2 = R1, suppose ideal memristor M1 Z1 and Z2 can be used basic electrical elements (resis- with µ =0and then the controller parameters are: tor, capacitor, inductor, memristor) or electrical networks M1 1 L1 (RC ladder, RC tree, RLC grid, CPE). In this way we can Kp = ,Ti = ,Td = . (22) obtain various dividers, filters, integrators, differentiators, R3 C1R4 R5

5 For simulation purpose were chosen the following val- ues of electrical elements:

R3 = R4 =1kΩ,R5 =1Ω; M1 =10kΩ,µ=0; C1 =10µF, λ =0.94; L1 =1mH, δ =0.86. (23)

From values of the electrical elements (23), we obtain the following controller parameters:

−3 Kp =10,Ti = 100,Td =1× 10 , µ =0,λ=0.94,δ=0.86. (24)

Bode Diagram

60

55 50 Figure 9. Photo of fractductor test setup 45 during collection of data shown in Fig. 7. 40

35 Magnitude (dB)

30 25 fractional-order memristive systems which can be used for 20 45 the fractional-order controllers implementation as well. Estimated order of the fractductor is α =0.5 (see Fig. 7).

0 6. Conclusion

Phase (deg) −45 In this brief paper was presented proposal for a new

−90 class of the fractional-order memristive systems, which −1 0 1 2 3 10 10 10 10 10 Frequency (rad/sec) are useful for practical implementation of the fractional- order controllers. However this approach gave a good start Figure 6. Bode plots of the controller trans- for detail analysis and design of the analogue fractional- fer function (21) with parameters (24). order controller. The fractional-order controller gives us an insight into the concept of memory of the suggested fractional operator. In Fig. 6 are depicted simulation results of the We also proposed a new electrical device, so–called fractional-order controller transfer function (21) with fractductor, which belongs to class of the analogue gen- parameters (24) obtained via numerical approximation eralized fractional-order memristive devices [8]. This de- method ORA in Matlab. vice has a fractional-order coupling between flux and cur- λ µ λ δ PI D controller [32], also known as PI D con- rent. troller, was already studied in time domain in [29] and Further work is needed to prove its performance by var- also in frequency domain in [30]. Investigation and de- ious simulations and experimental measurements at the tailed analysis of such kind of controller and its partic- circuit. The results may find wide application in signal δ ular cases (PD , CRONE, Lead-Lag Compensator, TID, processing and control systems (e.g.: [8, 19, 42], etc.). etc.) have been done in several additional works (e.g.: [9, 15, 18, 25, 26, 31,33, 47], etc.). 7. Acknowledgment

5.2. Experimental measurement Ivo Petr´asˇ was supported in part by the Slovak Grant Here, we describe a new “element,” a so-called Agency for Science under grants VEGA: 1/4058/07, fractional inductor, or ”Fractductor.” This device has 1/0365/08, 1/0404/08, and grant APVV-0040-07. a fractional-order coupling between flux and current. YangQuan Chen was supported in part by Utah State Preliminary attempts to construct a fractductor have University New Faculty Research Grant (2002-2003), the produced a device using magnetorheological fluid as the TCO Bridging Fund of Utah State University (2005- core in a -like device. A bode plot (Fig. 7), 2006), an NSF SBIR subcontract through Dr. Gary Bo- basic block schematic (Fig. 8), and photo of the experi- hannan (2006), and by the National Academy of Sciences mental device (Fig. 9) are shown. under the Collaboration in Basic Science and Engineer- As we can see in Fig. 8, connection of electrical el- ing Program/Twinning Program supported by Contract ements were done according to suggestions described No. INT-0002341 from the National Science Foundation in prevoius sections. It is practical realization of the (2003-2005).

6 Figure 7. Bode plots of an experimental fractional flux coupling device, the fractductor.

Magnetorheological fluid Rgain Magnet Magnet - Rout - + + - Csmall +

Coil A, 30 turns Coil B, 30 turns

Figure 8. Simplified schematic diagram of the fractductor and test circuit.

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