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 - memristor or memory resistor. 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 inductor (fractductor), cated by any circuit built using only the other three el- fractional capacitor 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 voltage 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- transmission line [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 memristors, 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<n) and where Γ(.) is the Gamma func- ers [39] e.g.: 1 bit = 1 memristor; tion. The Caputo’s definition can be written as • 1 t (n)( ) new analog computers that can process and associate α ( )= f τ aDt f t α−n+1 dτ, (3) information in a manner similar to that of the human Γ(α − n) a (t − τ) brain [35]; for (n − 1 <α<n). The initial conditions for the • new electronic circuits, e.g. [7, 37]: voltage divider, fractional order differential equations with the Caputo’s switcher, compensator, AD – DA converters, etc.; derivatives are in the same form as for the integer-order • new control systems/controllers with memory [8]; differential equations. The Laplace transform method is used for solving engi- In this paper we present the connection between frac- neering problems. The formula for the Laplace transform tional calculus (fractional order integral and derivative) of the RL fractional derivative (2) has the form [29]: and behavior of the memristive systems. As we will ∞ n−1 see, the fundamentals of fractional calculus are based −st α α k α−k−1 e 0Dt f(t) dt = s F (s) − s 0Dt f(t), on the memory property of the fractional order inte- 0 k=0 gral/derivative and therefore this connection is straightfor- (4) ward. This exceptional property can be used for realiza- tion of the fractional order operator as a basic element for for (n − 1 <α≤ n), where s ≡ jω denotes the implementation of the fractional order controllers. Laplace operator. For zero initial conditions [11], Laplace 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.