Fracmemristor: Fractional-Order Memristor - IEEE Xplore

Received April 10, 2016, accepted April 17, 2016, date of publication April 22, 2016, date of current version May 11, 2016. Digital Object Identifier 10.1109/ACCESS.2016.2557818

Fracmemristor: Fractional-Order Memristor YI-FEI PU1 AND XIAO YUAN2

1 College 2 College

of Computer Science, Sichuan University, Chengdu 610065, China of Electronics and Information Engineering, Sichuan University, Chengdu 610065, China

Corresponding author: Y.-F. Pu ([email protected]) This work was supported in part by the National Natural Science Foundation of China under Grant 61571312, in part by the Foundation Franco-Chinoise Pour La Science Et Ses Applications, and in part by the Science and Technology Support Project of Sichuan Province of China under Grant 2013SZ0071.

ABSTRACT This paper mainly discusses an interesting conceptual framework: the fractional-order memristor (fracmemristor). It is a challenging theoretical problem to determine what the fracmemristor interpolating characteristics between the memristor and the capacitor or inductor are, and where the positions of the fracmemristor in the Chua’s axiomatic element system are. Motivated by this need, in this paper, we introduce an interesting conceptual framework of the fracmemristor, which joins the concepts underlying the fractional-order circuit element and the memristor. We use a state-of-the-art mathematical method, fractional calculus, to analyze the proposed conceptual framework. The term fracmemristor is a portmanteau of the fractional-order memristor. The term fracmemristance refers to the fractional-order impedance of a fracmemristor. First, the relationship between the fracmemristance and the fractance is discussed. Second, the fracmemristances of the purely ideal 1/2-order capacitive fracmemristor and inductive fracmemristor are studied, respectively. The third step is the proposal for the fracmemristances of the purely ideal arbitraryorder capacitive fracmemristor and inductive fracmemristor, respectively. Finally, the fracmemristor is achieved by numerical implementation, and its non-volatility property of memory and nonlinear predictive ability is analyzed in detail experimentally. The predictable characteristics of the fracmemristor are a major advantage when comparing with the classical first-order memristor. INDEX TERMS Fractional calculus, memristor, fractance, fracmemristor, fracmemristance.

I. INTRODUCTION

The memristor was originally envisioned in 1971 by circuit theorist Chua as the missing nonlinear passive two-terminal electrical component relating electric charge and magnetic flux linkage [1]. The memristor was generalized to memristive systems in Chua’s 1976 paper [2]. The memristor has non-volatility property [4]–[9]. In 2008, a team at HP Labs claimed to have found Chua’s missing memristor based on an analysis of a thin film of titanium dioxide [9], [10]. Williams argued that magnetic random access memory, phase change memory and resistive random access memory were memristor technologies [9], [10]. Chua argued for a broader definition that included all 2-terminal non-volatile memory devices based on resistance switching [5]. Chua has more recently argued that the definition could be generalized to cover all forms of two-terminal non-volatile memory devices based on resistance switching effects [4]–[8] although some experimental evidence contradicts this claim, since a non-passive nanobattery effect is observable in resistance switching memory [11]. Meuffels and Schroeder noted 1872

that one of the early memristor papers included a mistaken assumption regarding ionic conduction [12]. Meuffels and Soni furthermore noted that the dynamic state equations set up for a solely current-controlled memristor with the so-called non-volatility property [5]–[8] would allow the violation of Landauer’s principle of the minimum amount of energy required to change ‘‘information’’ states in a system [13], [14]. The concept of a solely current-controlled memristor provides no physical mechanism enabling such a memristor system to erratically change its state just under the influence of white current noise [13], [15]. Nonlinear ionic drift models of the memristor have been proposed by other researchers [16]. As of 2014, the search continues for a model that balances the issues that Strukov’s initial memristor modeling equations do not reflect the actual device physics well [17]. One of the resulting properties of the memristors and memristive systems is the existence of a pinched hysteresis effect [18]. It has been proven that some types of non-crossing pinched hysteresis curves cannot be described by the memristors [19]. There are titanium dioxide

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VOLUME 4, 2016

Y.-F. Pu and X. Yuan: Fracmemristor: Fractional-Order Memristor

memristor [9], [10], [20], [21], polymeric memristor [22], [23], layered memristor [24], ferroelectric memristor [25], and spin memristive systems [26]–[31], respectively. Williams’ solid-state memristors can be combined into devices called crossbar latches, which could replace transistors in future computers [32]. A simple electronic circuit [33] consisting of an LC network and a memristor was used to model experiments on adaptive behavior of unicellular organisms [34]. Versace and Chandler described the modular neural exploring traveling agent model [35], [36]. Application of the memristor crossbar structure in the construction of an analog soft computing system was demonstrated by Merrikh-Bayat and Shouraki [37]. Chua published a tutorial underlining the broad span of complex phenomena and applications [38]. Di Ventra extended the notion of memristive systems to capacitive and inductive elements in the form of memcapacitors and meminductors [14], [39]. In 2011, memristor-based content addressable memory was introduced [40]. Tse demonstrated printed memristive counters based on solution processing, with potential applications as low-cost packaging components [41]. Politecnico showed that a purely passive circuit, employing already-existing components [42]. Some Chaos circuits consisting of a fractional-order Chua’s circuit and a memristor were introduced [43], [44]. Tenreiro Machado studied the generalization of the memristor in the perspective of the fractional-order systems [45]. In 2014, Abdelhouahad proposed the memfractor, which interpolates characteristics between the memristor and the memcapacitor, the meminductor or the second-order memristor [46]. Fractional calculus has become an important novel branch of mathematical analyses [47]–[49]. Fractional calculus is as old as the integer-order calculus, although until recently, its applications were exclusively in mathematics. Scientific study has shown that a fractional-order or fractional dimensional approach is now the best description for many natural phenomena. Many scientific fields such as physics, bioengineering, diffusion processes, viscoelasticity theory, fractal dynamics, fractional control, signal processing, and image processing presently use fractional calculus [47]–[68], which has obtained promising results and ideas demonstrating these fractional mathematical operators can be interesting and useful tools. The application of fractional calculus to analyzing the memristor is an emerging discipline of study in which few studies have been performed [43]–[46]. In the scientific fields of latest signal analysis, signal processing, and circuits and systems, there are many issues on non-linear, non-causal, non-Gaussian, non-stationary, non-minimum phase, non-white additive noise, non-integer-dimensional, and non-integer-order characteristics needed to be analyzed and processed. The classical integer-order signal processing filters and circuit models cannot deal with the aforementioned non-problems efficiently. Fractional calculus has been hybridized with signal processing, circuits and systems, and material science mainly because of its inherent strength VOLUME 4, 2016

of long-term memory, non-locality, and weak singularity. A fractional-order or fractional-dimensional system is now a powerful model for dealing with these non-problems. Following the success in the synthesis of the fractionalorder differentiator and integrator in an analog circuit, the emergence of a promising fractional-order circuit element was called the fractor [50]–[67]. The fractor is essentially a signal processing filter implementing the computation of fractional calculus. The term ‘‘fractance’’ refers to the fractional-order impedance of a fractor. The driving-point impedance function of a fractor is equal to its fractional-order reactance. The fractor has two types, namely, the capacitive fractor and the inductive fractor. The capacitive fractor is the fractional-order capacitor, which is of fractional integral operation. The fractional-order impedance of the capacitive fractor is the capacitive fractance. In a similar way, the inductive fractor is the fractional-order inductor, which is of fractional differential operation. The fractional-order impedance of the inductive fractor is the inductive fractance. As we know, in Chua’s periodic table of all two-terminal circuit elements [1]–[6], the capacitive fractor is lying on the line segment between the capacitor and resistor. The electrical properties of the capacitive fractor fall in between the electrical properties of the capacitor and those of the resistor [50]–[67]. In a similar way, in Chua’s periodic table of all two-terminal circuit elements [1]–[6], the inductive fractor is lying on the line segment between the inductor and resistor. The electrical properties of the inductive fractor fall in between the electrical properties of the inductor and those of the resistor [50]–[67]. From the Chua’s axiomatic element system [1]–[6] and the constitutive relation, according to logical consistency, axiomatic completeness, and formal symmetry, we can assume that there should be a novel corresponding capacitive circuit element and inductive circuit element to the capacitive fractor and inductive fractor, respectively. Therefore, it is natural to ponder a challenging theoretical problem to determine what the fractional-order memristor interpolating characteristics between the memristor and capacitor or inductor are, and where the positions of the fractional-order memristor in the Chua’s axiomatic element system [1]–[6] are. Motivated by this need, in this paper, we proposed to introduce an interesting conceptual framework of the fracmemristor, which joins the concepts underlying the fractional-order circuit element and the meristor. We use a state-of-the-art mathematical method, fractional calculus, to analyze the proposed conceptual framework. In particular, in Chua’s periodic table of all two-terminal circuit elements, the electrical properties of the capacitive fracmemristor should fall in between the electrical properties of the capacitor and those of the memristor. The electrical properties of the inductive fracmemristor should fall in between the electrical properties of the inductor and those of the memristor. The fracmemristor is the fractional-order memristor with predictable function. The predictable characteristics of the fracmemristor are a major advantage when comparing with the classical first-order memristor. 1873


The remainder of this paper is organized as follows: Section 2 recalls some preliminary concepts of the memristor and fractance. Section 3 discusses an interesting conceptual framework of the fracmemristor. In Section 3, first, the relationship between the fracmemristance and fractance is discussed. Second, the fracmemristance of the purely ideal 1/2-order capacitive fracmemristor and inductive fracmemristor is studied, respectively. The third step is the proposal for the fracmemristance of the purely ideal arbitraryorder capacitive fracmemristor and inductive fracmemristor, respectively. In Section 4, the fracmemristor is achieved by numerical implementation, and its non-volatility property of memory and nonlinear predictive ability are analyzed in detail experimentally. In Section 5, the conclusions of this manuscript are presented.

Chua’s axiomatic elements have the element independence β α property. Thus, Dt v(t), Dt i(t) establishes a corresponding

II. RELATED WORK

This section includes a brief introduction to the necessary mathematical background of the memristor and fractance. Chua proposed that there should be a fourth basic element M , which he called the memristor, for memory resistor, completing the set of relations with [1]–[8]: ϕ[q(t)] = M [q(t)]q(t),

(1)

where ϕ and q denotes magnetic flux and quantity of electric charge, respectively. Equation (1) shows that a memristor is as any passive two-terminal circuit element that maintains a functional relation between the time integral of current and the time integral of voltage. The slope of this function is called the memristance, R [q(t)], and is similar to variable resistance [1]–[8]. It follows that: dϕ(q) Ii (t) dq dM (q) = M (q) + q Ii (t) dq = R [q(t)] Ii (t),

Vi (t) =

(2)

where Vi (t) and Ii (t) are the instantaneous value of the input voltage and the input current of a memristor, respectively. Recently, Chua argued for a broader definition that included all 2-terminal non-volatile memory devices based on resistance switching [5]–[8]. Figure 1 shows the periodic table of all two-terminal circuit elements [1]–[8]. In Fig. 1, let us denote the Chua’s axiomatic element and its corresponding electrical characteristics as the symbol C (α,β) , in which α denotes the voltage exponent, β denotes the current exponent. α and β are equal to the order of the time β derivative of v(t) and i(t), respectively. Dαt v(t) and Dt i(t) are collectively referred to as the Chua’s constitutive variables, where D denotes the differential operator. (α, β) is referred to the Chua’s plane of the Chua’s axiomatic element system. C, R, L, M , ML , and MC denote the capacitor, resistor, inductor, memristor, meminductor, and memcapacitor, respectively. The symbol O denotes the other postulated eletments of the Chua’s axiomatic element system. The all 1874

FIGURE 1. Periodic table of all two-terminal circuit elements.

constitutive relation. It follows that [3]–[6]: β

Dαt v(t) − C (α,β) Dt i(t) = 0,

(3)

where α ∈ R and β ∈ R. It follows that [1]–[8], [45], [46]:   C, if α = 0, β = −1     R, if α = 0, β = 0 (4) C (α,β) =  L, if α = −1, β = 0     M , if α = −1, β = −1, where C, R, L, and M denote the electrical characteristics of the capacitor, resistor, inductor, and memristor, respectively. With respect to a capacitive fractor, in Fig. 1, the capacitive fractor is lying on the line segment, S1 , between C and R. The order of a capacitive fractance can be extended to the whole field of negative real numbers. Pu has derived the generic nonlinear relation between the capacitance and resistance of the arbitrary-order capacitive fractance [66], [67]. It follows that: c c F−v = F−(m+p)

= Vi (s)/Ii (s) = c−(m+p) r 1−p s−(m+p) ,

(5)

where v = m + p is a positive real number, m is a posc , c, r, and c(m+p) r 1−p itive integer, and 0 6 p 6 1. F−v denote the driving-point impedance function, capacitance, resistance, and capacitive fractance of a purely ideal v-order capacitive fractor, respectively. The driving-point impedance function of a capacitive fractor is equal to its fractional-order capacitive reactance. Equation (5) represents the fractionalorder capacitive reactance of the purely ideal arbitrary-order capacitive fractor. In addition, with respect to an inductive fractor, in Fig. 1, the inductive fractor is lying on the line segment, S2 , between L and R. The order of an inductive fractance can be extended to the whole field of positive real numbers. Pu has derived the generic nonlinear relation between VOLUME 4, 2016


inductance and resistance of the arbitrary-order inductive fractance [66], [67]. It follows that: l Fvl = Fm+p

= Vi (s)/Ii (s) = l m+p r 1−p sm+p ,

(6)

where v = m + p is a positive real number, m is a positive integer, and 0 6 p 6 1. Fvl , l, r, and l m+p r 1−p denote the driving-point impedance function, inductance, resistance, and inductive fractance of a purely ideal v-order inductive fractor, respectively. The driving-point impedance function of an inductive fractor is equal to its fractional-order inductive reactance. Equation (6) represents the fractional-order inductive reactance of the purely ideal arbitrary-order inductive fractor. III. FRACMEMRISTOR A. RELATIONSHIP BETWEEN FRACMEMRISTANCE AND FRACTANCE

In this subsection, we discuss the relationship between the fracmemristance and fractance. From Fig. 1 and the constitutive relation, according to logical consistency, axiomatic completeness, and formal symmetry, we can see that there should be the corresponding capacitive fracmemristor and inductive fracmemristor to the capacitive fractor and inductive fractor, respectively. The term ‘‘fracmemristor’’ is a portmanteau of ‘‘the fractionalorder memristor.’’ In Fig. 1, the capacitive fracmemristor should be lying on the line segment, S4 , between C and M . The inductive fracmemristor should be lying on the line segment, S3 , between L and M . The term ‘‘fractance’’ refers to the fractional-order impedance of a fractor. The fractionalorder impedance of the capacitive fractor and inductive fractor are the capacitive fractance and inductive fractance, respectively. In a similar way, the term ‘‘fracmemristance’’ refers to the fractional-order impedance of a fracmemristor. The fractional-order impedance of the capacitive fracmemristor and inductive fracmemristor are the capacitive fracmemristance and inductive fracmemristance, respectively. In addition, from Fig. 1, we can further see that the electrical properties of the capacitive fracmemristor should fall in between the electrical properties of the capacitor and those of the memristor. The electrical properties of the inductive fracmemristor should fall in between the electrical properties of the inductor and those of the memristor. The difference between the electrical properties of the memristor and those of the resistor is a major difference between the electrical properties of the fracmemristor and those of the fractor. In addition, equations (1) and (2) show that the memristor definition is based solely on the fundamental circuit variables of current and voltage and their timeintegrals, just like the resistor, capacitor, and inductor. The ideal memristor, for memory resistor, is a special case of generic memristor when R[q(t)] depends only on quantity of electric charge. R[q(t)] (measurement unit of the memristance VOLUME 4, 2016

in Ohm similarly to resistance) is the incremental resistance. Therefore, in a similar way, we can achieve the fracmemristor refer to the aforementioned implementations of the fractor [50]–[67]. As we know, the tree-type [47], [56], two-circuitstype [63], H-type [64], net-grid-type [65] should be four natural implementations of the fractor. What distinguishes these four aforementioned types [47], [56], [63]–[65] from the other approximate implementations [50]–[62] of the fractor is that the formers consist of series of ordinary circuit elements in the form of natural fractal structures exhibiting extreme self-similarity. In contrast to other approximate implementations [50]–[62] of the fractor, the floating point values of the capacitance, inductance, and resistance of these four aforementioned types [47], [56], [63]–[65] of the fractor are not required. In fact, there are zero errors between these four aforementioned types [47], [56], [63]–[65] of fractance with infinite recursive extreme self-similar structures and an ideal fractance. Devices manufactured using the other approximate implementations [50]–[62] of the fractor never represent purely ideal fractance. In a similar way, the aforementioned tree-type, two-circuits-type, H-type, and net-grid-type should also be the natural implementations of the fracmemristor. Because natural implementations usually indicate some fundamental rules of various circuit elements, we study the fracmemristor mainly from the perspective of the fracmemristor of natural implementations. With respect to a purely ideal fractor, the generic electrical characteristics of a purely ideal fractor can be derived from the aforementioned four natural implementations of the fractor [66], [67]. The fractances of the aforementioned four natural implementations of the purely ideal fractor [47], [56], [63]–[65] have the same expression of the electrical properties as (5) and (6). In a similar way, without loss of generality, we can only employ the net-grid-type fracmemristor in this paper to discuss the generic electrical characteristics of a purely ideal fracmemristor. B. FRACMEMRISTANCE OF PURELY IDEAL 1/2-ORDER FRACMEMRISTOR

In this subsection, we discuss the fracmemristance of the purely ideal 1/2-order fracmemristor. The term ‘‘fracmemristance’’ refers to the fractional-order impedance of a fracmemristor. Let us denote the v-order driving-point impedance function (the v-order reactance) of the v-order fracmemristor as the symbol FMv , which is the abbreviation of fracmemristor. For convenience of illustration, in this paper, FMv is also directly referred to as the v-order fracmemristor. Figure 2 shows the structural representation of a purely ideal 1/2-order net-grid-type fracmemristor, FM1/2 . From Fig. 2, we can see that the purely ideal 1/2-order net-grid-type fracmemristor is of extreme self-similar fractal structure with the series connection of infinitely repeated net-grid-type structures, where Za denotes the memristor and Zb denotes the classical passive capacitor or inductor. 1875


First, with regard to a purely ideal 1/2-order capacitive c , let the memristance and capacitance fracmemristor, FM−1/2 of its fracmemristance be equal to R[q(t)] and c, respectively, where q denotes quantity of electric charge. Assuming r[q(s)] is the Laplace transform of R[q(t)]. Suppose the initial energy of the electric element of the purely ideal 1/2-order capacitive fracmemristor is equal to zero. Then, in the Laplace transform domain, Za = r(q) and Zb = 1/cs. From (10), it follows that:

FIGURE 2. A purely ideal 1/2-order net-grid-type fracmemristor.

c = ξ −1/2 s−1/2 , FM−1/2

The number of Za and Zb is equal to two-fold the number of layers. Figure 3 shows its equivalent circuit.

FIGURE 3. Equivalent circuit of purely ideal 1/2-order net-grid-type fracmemristor.

For convenience, in this paper, Za and Zb are also directly referred to as the driving-point impedance functions (the reactances) of a memristor and a capacitor or inductor, respectively. Suppose the currents of Za and Zb are equal to ia (s) and ib (s), respectively. Let the input voltage and input current of FM1/2 be equal to Vi (s) and Ii (s), respectively. From Fig. 3, according to Kirchoff’s current law and Kirchoff’s voltage law, we can obtain the following: ( Za ia + Zb ib = Vi (7) (Za + FM1/2 )ia − (FM1/2 + Zb )ib = 0. According to the Cramer’s rule of linear algebra, it follows that:  (Zb + FM1/2 )Vi  ia (s) =     Za + FM1/2 −(FM1/2 + Zb )    Za Zb (8) (Za + FM1/2 )Vi   i (s) = .  b  Z + FM  −(FM1/2 + Zb )  a 1/2    Za Zb Hence, FM1/2 is equal to: Vi (s) Ii (s) Vi (s) = ia (s) + ib (s) 2Za Zb + FM1/2 (Za + Zb ) = . 2FM1/2 + Za + Zb

FM1/2 =

From (9), it follows that: FM1/2 = Vi (s)/Ii (s) = (Za Zb )1/2 . 1876

where ξ = c/r(q). Equation (11) shows that c1/2 r(q)1/2 denotes the 1/2-order capacitive fracmemristance of the purely ideal 1/2-order capacitive fracmemristor. The 1/2-order capacitive fracmemristor is essentially the −1/2-order fracmemristor. Equation (11) represents the 1/2-order capacitive driving-point impedance function (the 1/2-order capacitive reactance) of the purely ideal 1/2-order capacitive fracmemristor. From (11), we can derive the relation between input voltage Vi (s) and input current Ii (s) of the purely ideal 1/2-order capacitive fracmemristor. It follows that: Vi (s) = ξ −1/2 s−1/2 Ii (s).

(10)

(12)

The inverse Laplace transform of (12) is as follows: n o −1/2 Ii (t)], Vi (t) = c−1/2 L −1 [r(q)]1/2 ∗ [Dt

(13)

where symbol ∗ denotes convolution. From (13), we can see that there is a positive nonlinear correlation between Vi (t) and the 1/2-order fractional integral of Ii (t). From (12), however, it follows that: Ii (s) = ξ 1/2 s1/2 Vi (s).

(14)

The inverse Laplace transform of (14) is as follows: n o 1/2 Ii (t) = c1/2 L −1 [r(q)]−1/2 ∗ [Dt Vi (t)].

(15)

From (15), we can see that there is a positive nonlinear correlation between Ii (t) and the 1/2-order fractional derivative of Vi (t). Second, with regard to a purely ideal 1/2-order inductive l , let the memristance and inductance fracmemristor, FM1/2 of its fracmemristance be equal to r(q) and l, respectively. Suppose the initial energy of the electric element of the purely ideal 1/2-order inductive fracmemristor is equal to zero. Then, in the Laplace transform domain, Za = r(q) and Zb = ls. From (10), it follows that: l FM1/2 = ζ 1/2 s1/2 ,

(9)

(11)

(16)

where ζ = l · r(q). Equation (16) shows that l 1/2 r(q)1/2 denotes the 1/2-order inductive fracmemristance of the purely ideal 1/2-order inductive fracmemristor. The 1/2-order inductive fracmemristor is essentially the 1/2-order fracmemristor. Equation (16) represents the 1/2-order inductive drivingpoint impedance function (the 1/2-order inductive reactance) of the purely ideal 1/2-order inductive fracmemristor. VOLUME 4, 2016


In a similar way, from (16), we can derive the relation between input voltage Vi (s) and input current Ii (s) of the purely ideal 1/2-order inductive fracmemristor. It follows that, respectively: Vi (s) = ζ 1/2 s1/2 Ii (s), Ii (s) = ζ −1/2 s−1/2 Vi (s).

(17) (18)

From (17) and (18), we can see that Vi (t) is positively correlated to the 1/2-order fractional derivative of Ii (t) and Ii (t) is positively correlated to the 1/2-order fractional integral of Vi (t). C. FRACMEMRISTANCE OF PURELY IDEAL ARBITRARY-ORDER FRACMEMRISTOR

In this subsection, we further discuss the fracmemristance of the purely ideal arbitrary-order fracmemristor. The purely ideal arbitrary-order net-grid-type fracmemristor is a recursively nested structure of the aforementioned purely ideal 1/2-order net-grid-type fracmemristor. First, by an extension of aforementioned logic, we can derive the fracmemristance of the purely ideal arbitraryorder capacitive fracmemristor. Suppose Za = r(q) and c Zb = FM−1/2 in Fig. 2. From (10) and (11), it follows that: c F−1/4 = [r(q)]1/2 ξ −1/4 s−1/4 .

(19)

c And then, suppose Za = r(q) and Zb = FM−1/2 n−1 in Fig. 2, where n is a positive integer. From (10), it follows that: c FM−1/2 n = [r(q)]

2n−1 −1 2n−1

ξ

−1 2n

s

−1 2n

.

(20) c FM−1/2

In a similar way, suppose Za = 1/cs and Zb = in Fig. 2. From (10) and (11), it follows that: c FM−3/4

ξ

−1/2 −1/4 −3/4

=c

s

.

(21)

And then, suppose Za = 1/cs and Zb = FM−c 2n−1 −1 /2n−1 ( ) in Fig. 2. From (10), it follows that: c FM−(2 n −1)/2n = c

(

− 2n−1 −1 2n−1

)

−1

ξ 2n s

−(2n −1) 2n

.

(22)

c In a similar way, suppose Za = 1/cs and Zb = FM−1/2 n in Fig. 2. From (10) and (20), it follows that: c FM−(2 n +1)/2n+1 = c

−1 2

[r(q)]

2n−1 −1 2n

−1

ξ 2n+1 s

−(2n +1) 2n+1

c FM−(2 n +1)/2n+1

Suppose Za = 1/cs and Zb = From (10) and (23), it follows that: FM−c (2n+1 +2n +1)/2n+2 =c

−3 4

[r(q)]

2n−1 −1 2n+1

−1

ξ 2n+2 s

.

(23)

in Fig. 2.

c Suppose Za = r(q) and Zb = FM−(2 n −1)/2n+1 in Fig. 2. From (10) and (26), it follows that: 3

c 4 FM−(2 n −1)/2n+2 = [r(q)] c

(

− 2n−1 −1 2n+1

)

−1

ξ 2n+2 s

−(2n −1) 2n+2

.

(27)

c And then, suppose Za = r(q) and Zb = FM−(2 n −1)/2n+k−1

in Fig. 2, where k is a positive integer. From (10), it follows that: −(2n−1 −1) −(2n −1) −1 2k −1 c k c 2n+k−1 ξ 2n+k s 2n+k . 2 FM−(2 = [r(q)] n −1)/2n+k (28) Furthermore, if n and k are non-negative integers, respectively, it follows that: 2n+k − 2n + 1 1 = p 6 1, (29) 6 2 2n+k 1 (2n − 1) 06 =p6 . (30) n+k 2 2 From (25) and (29), if 1/2 6 p 6 1, it follows that: c FM−p =c

(2n −1)(1−p)−(2n −1) 2n −1

[r(q)]

(2n −1)(1−p) 2n −1

s−p

= c−p [r(q)]1−p s−p .

(31)

From (28) and (30), if 0 6 p 6 1/2, it follows that: c FM−p =c

−(2n −1)p 2n −1

[r(q)]

(2n −1)(1−p) 2n −1

= c−p [r(q)]1−p s−p .

s−p (32)

From (31) and (32), we can see that no matter 1/2 6 c has the same analytp 6 1 or 0 6 p 6 1/2, FM−p 1−p ical expression, c−p [r(q)] s−p . Equations (31) and (32) do collectively analytically represent the fractional-order driving-point impedance function of the purely ideal p-order capacitive fracmemristor, where 0 6 p 6 1. Thus, from (31) and (32), if 0 6 p 6 1, it follows that: c FM−p = c−p [r(q)]1−p s−p .

(33)

In addition, as we know, the driving-point impedance function (the capacitive reactance) of a capacitor is as follows: (

− 2n+1 +2n +1 2n+2

)

.

(24)

And then, suppose Za = 1/cs and Zb = FM−c 2n+k−1 −2n +1 /2n+k−1 in Fig. 2, where k is a positive ( ) integer. From (10), it follows that: FM−c (2n+k −2n +1)/2n+k −(2k −1) −(2n+k −2n +1) 2n−1 −1 −1 2n+k = c 2k [r(q)] 2n+k−1 ξ 2n+k s . VOLUME 4, 2016

In a similar way, suppose Za = r(q) and c Zb = FM−(2 n −1)/2n in Fig. 2. From (10) and (22), it follows that: −(2n −1) −(2n−1 −1) −1 1 c 2 c 2n 2n+1 s 2n+1 . ξ (26) FM−(2 = [r(q)] n −1)/2n+1

(25)

c Z−1 (s) = Vi (s)/Ii (s)

= c−1 s−1 ,

(34)

where c is equal to the capacitance of a capacitor. From (25), we can see that with regard to the capacitor, Vi (t) is in direct ratio to the first-order integral of Ii (t) and Ii (t) is in direct ratio to the first-order derivative of Vi (t). A capacitor implements the first-order integral operator. Thus, from (25), the capacitive driving-point impedance function (the capacitive 1877


reactance) of a cascade system of the m-stage first-order integrators is as follows: c Z−m (s) = Vi (s)/Ii (s)

= c−m s−m ,

(35)

where m is a positive integer. Therefore, the order of a purely ideal capacitive fracmemristor can be extended to the whole field of negative real numbers. We can naturally implement a purely ideal arbitrary-order capacitive fracmemristor in a cascade manner. From (33) and (35), the v-order capacitive driving-point impedance function (the v-order reactance) of the purely ideal v-order capacitive fracmemristor is as follows: = Vi (s)/Ii (s) =c

[r(q)]

1−p −(m+p)

s

,

(36)

where v = m + p is a positive real number, m is a positive integer, and 0 6 p 6 1. Equation (36) shows that c(m+p) [r(q)]1−p denotes the v-order capacitive fracmemristance of the purely ideal v-order capacitive fracmemristor. The v-order capacitive fracmemristor is essentially the c is the v-order capacitive −v-order fracmemristor. FM−v driving-point impedance function (the v-order capacitive reactance) of the purely ideal v-order capacitive fracmemristor. From (9), (12), and (36), we can see that the purely ideal v-order capacitive fracmemristor implements a cascade system of a v-order fractional integral operator and a fractional power function of memristance. Comparing with (5) and (36), we can see that the measurement unit and physical dimension of the purely ideal capacitive fracmemristance are the same as those of the purely ideal capacitive fractance [66], [67], because the memristance and resistance have the same measurement unit and physical dimension [1]–[8]. In Fig. 1, the capacitive fracmemristor should be lying on the line segment, S4 , between C and M . In particular, in Chua’s periodic table of all two-terminal circuit elements, the electrical properties of the capacitive fracmemristor should fall in between the electrical properties of the capacitor and those of the memristor. The electrical properties of the inductive fracmemristor should fall in between the electrical properties of the inductor and those of the memristor. The capacitive fracmemristor can be considered in a certain way as an interpolation of the memristor and capacitor. The fractional-order capacitive driving-point impedance function of a capacitive fracmemristor is equal to its fractional-order capacitive reactance. Equation (36) represents the fractional-order capacitive reactance of a purely ideal arbitrary-order capacitive fracmemristor. From (36), we can also see that if the Laplace transform of memristance is equal to an arbitrary power function of s, the v-order capacitive fracmemristor will be turned into a traditional fractor. If v = 0, the v-order capacitive fracmemristor will be turned into a traditional first-order memristor. The capacitive fracmemristor can be converted to a fractor or memristor in some given conditions. 1878

l = [r(q)]1/2 ζ 1/4 s1/4 . FM1/4

(37)

l And then, suppose Za = r(q) and Zb = FM1/2 n−1 in Fig. 2,

where n is a positive integer. From (10), it follows that: l FM1/2 n = [r(q)]

2n−1 −1 2n−1

1

1

ζ 2n s 2n .

(38)

l In a similar way, suppose Za = ls and Zb = FM1/2 in Fig. 2. From (10) and (16), it follows that: l = l 1/2 ζ 1/4 s3/4 . FM3/4

c c FM−v = FM−(m+p) −(m+p)

Second, in a similar way, we can derive the fracmemristance of the purely ideal arbitrary-order inductive l fracmemristor. Suppose Za = r(q) and Zb = FM1/2 in Fig. 2. From (10) and (16), it follows that:

(39)

And then, suppose Za = ls and Zb = FM l 2n−1 −1 /2n−1 ( ) in Fig. 2. From (10), it follows that: l FM(2 n −1)/2n

=l

(2n−1 −1) 2n−1

1

ζ 2n s

(2n −1) 2n

.

(40)

l In a similar way, suppose Za = ls and Zb = FM1/2 n in Fig. 2. From (10) and (38), it follows that: 1

l 2 FM(2 n +1)/2n+1 = l [r(q)]

2n−1 −1 2n

1

ζ 2n+1 s

(2n +1) 2n+1

.

(41)

And then, suppose Za = ls and Zb = FM l 2n+k−1 −2n +1 /2n+k−1 ( ) in Fig. 2, where k is a positive integer. From (10), it follows that: FM(l 2n+k −2n +1)/2n+k (2n+k −2n +1) (2k −1) 2n−1 −1 1 . = l 2k [r(q)] 2n+k−1 ζ 2n+k s 2n+k

(42)

l In a similar way, suppose Za = r(q) and Zb = FM(2 n −1)/2n in Fig. 2. From (10) and (40), it follows that: 1

l 2 FM(2 n −1)/2n+1 = [r(q)] l

(2n−1 −1) 2n

1

ζ 2n+1 s

(2n −1) 2n+1

.

(43)

l And then, suppose Za = r(q) and Zb = FM(2 n −1)/2n+k−1 in

Fig. 2, where k is a positive integer. From (10), it follows that: n−1 −1 ) 1 (2n −1) 2k −1 (2 l k l 2n+k−1 ζ 2n+k s 2n+k . 2 FM(2 = (44) [r(q)] n −1)/2n+k Thus, in a similar way, we can derive the fractionalorder driving-point impedance function of the purely ideal p-order inductive fracmemristor, where 0 6 p 6 1. From (42) and (44), if 0 6 p 6 1, it follows that: FMpl = l p [r(q)]1−p sp .

(45)

In addition, as we know, the driving-point impedance function (the inductive reactance) of an inductor is as follows: Z1l (s) = Vi (s)/Ii (s) = l 1 s1 ,

(46)

where l is equal to the inductance of an inductor. An inductor implements the first-order differential operator. VOLUME 4, 2016


Thus, from (46), the inductive driving-point impedance function (the inductive reactance) of a cascade system of the m-stage first-order differentiators is as follows: Zml (s) = Vi (s)/Ii (s) = l m sm ,

(47)

where m is a positive integer. Therefore, the order of a purely ideal inductive fracmemristance can be extended to the whole field of positive real numbers. We can naturally implement a purely ideal arbitrary-order inductive fracmemristance in a cascade manner. From (45) and (47), the v-order inductive drivingpoint impedance function (the v-order inductive reactance) of the purely ideal v-order inductive fracmemristor is as follows: l FMvl = FMm+p

= Vi (s)/Ii (s) =l

m+p

[r(q)]

x=0

k=0

integer, f (x) is a differintegrable function [47]–[49], [0, x] is the duration of f (x), and s denotes the Laplace operator. When f (x) is causal signal and its fractional primitives must also be zero, we can simplify the Laplace transform for R−L Dvx f (x) 0 R−L v v as L[0 Dx f (x)] = s L[f (x)]. Further, the Laplace transform v of the v-order Caputo differential operator is L[C 0 Dx f (x)] = n−1 P k (k) s f (x) x=0 . When f (x) is causal signal and sv L[f (x)]− k=0

1−p m+p

s

,

(48)

where v = m+p is a positive real number, m is a positive integer, and 0 6 p 6 1. Equation (48) shows that l m+p [r(q)]1−p denotes the v-order inductive fracmemristance of the purely ideal v-order inductive fracmemristor. The v-order inductive fracmemristor is essentially the v-order fracmemristor. FMvl is the v-order inductive driving-point impedance function (the v-order inductive reactance) of the purely ideal v-order inductive fracmemristor. From (9), (12), and (48), we can see that the purely ideal v-order inductive fracmemristor implements a cascade system of a v-order fractional differential operator and a fractional power function of memristance. Comparing with (6) and (48), we can see that the measurement unit and physical dimension of the purely ideal inductive fracmemristance are the same as those of the purely ideal inductive fractance [66], [67], because the memristance and resistance have the same measurement unit and physical dimension [1]–[8]. In Fig. 1, the inductive fracmemristor should be lying on the line segment, S3 , between L and M . In particular, in Chua’s periodic table of all two-terminal circuit elements, the electrical properties of the inductive fracmemristor should fall in between the electrical properties of the inductor and those of the memristor. The inductive fracmemristor can be considered in a certain way as an interpolation of the memristor and inductor. The fractional-order inductive driving-point impedance function of an inductive fracmemristor is equal to its fractional-order inductive reactance. Equation (48) represents the fractional-order inductive reactance of a purely ideal arbitrary-order inductive fracmemristor. From (48), we can also see that if the Laplace transform of memristance is equal to an arbitrary power function of s, the v-order inductive fracmemristor will be turned into a traditional fractor. If v = 0, the v-order inductive fracmemristor will be turned into a traditional first-order memristor. The inductive fracmemristor can be converted to a fractor or memristor in some given conditions. Third, as we know, the commonly used fractional calculus definitions are Grünwald-Letnikov, VOLUME 4, 2016

Riemann-Liouville, and Caputo [47]–[49], respectively. Supv pose that G−L Dvx , R−L Dvx , and C 0 0 0 Dx denote the GrünwaldLetnikov defined, the Riemann-Liouville defined, and the Caputo defined fractional differential operator, respectively. The Laplace transform of the v-order Riemann-Liouville differential operator is L[R−L Dvx f (x)] = sv L[f (x)] − 0 h i n−1 P k R−L v−1−k s 0 Dx , where n − 1 6 v < n is a nonf (x)

its fractional primitives must also be zero, we can simplify the C v v v Laplace transform for C 0 Dx f (x) as L[0 Dx f (x)] = s L[f (x)]. Thus, in this case the three cited definitions of fractional derivatives are equivalent. From (36) and (48), we can see that when the memristance, R[q(t)], is causal signal and its fractional primitives must also be zero, we can use equivalent cited definitions of fractional calculus to the fracmemristor. D. BRANCH-CURRENT ANALYSIS OF PURELY IDEAL ARBITRARY-ORDER FRACMEMRISTOR

In this subsection, from Kirchhoff’s current law and Kirchhoff’s voltage law, the branch-current of the purely ideal arbitrary-order fracmemristor is analyzed. First, with respect to the purely ideal 1/2-order fracmemristor, from (8), it follows that: Zb + FM1/2 ia (s) = , (49) ib (s) Za + FM1/2 where ia (s) and Za = r[q(s)] denote the branch-current and the reactance of the memristor in a purely ideal 1/2-order fracmemristor in the Laplace transform domain, respectively. ib (s) and Zb denote the branch-current and the capacitive or inductive reactance of the corresponding capacitor or inductor in a purely ideal 1/2-order fracmemristor in the Laplace transform domain, respectively. As aforementioned discussion, if ib (s) and Zb represent the branch-current of the capacitor and the capacitive reactance, respectively, FM1/2 represents a purely ideal 1/2-order capacitive fracmemristor. If ib (s) and Zb represent the branch-current of the inductor and the inductive reactance, respectively, FM1/2 represents a purely ideal 1/2-order inductive fracmemristor. Thus, from (9), it follows that: Ii (s) = ia (s) + ib (s),

(50)

where Ii (s) denotes the input current of the purely ideal 1/2-order fracmemristor. Thus, from (49) and (50), it follows that: FM1/2 + Zb Ii (s). (51) ia (s) = 2FM1/2 + Za + Zb 1879


From (10) and (51), it follows that:  !2  1/2 Za  Ii (s). ia (s) = 1 − 1/2 1/2 Za + Zb

(52)

Second, with respect to the purely ideal 1/4-order fracmemristor, from (19) and (37), its equivalent circuit can be shown as given in Fig. 4.

FIGURE 4. Equivalent circuit of purely ideal 1/4-order net-grid-type fracmemristor.

In Fig. 4, Ii (s) denotes the input current of a purely ideal 1/4-order fracmemristor. ia (s) and Za = r[q(s)] denote the branch-current and the reactance of the memristor in a purely ideal 1/4-order fracmemristor in the Laplace transform domain, respectively. As aforementioned discussion, if FM1/2 denotes a purely ideal 1/2-order capacitive fracmemristor, FM1/4 is a corresponding purely ideal 1/4-order capacitive fracmemristor. If FM1/2 denotes a purely ideal 1/2-order inductive fracmemristor, FM1/4 is a corresponding purely ideal 1/4-order inductive fracmemristor. In a similar way, from (10) and Fig. 4, the following can be obtained:     2

1/2

 ia (s) = 1 − 

Za 1/2 Za



1/2 + FM1/2

   Ii (s) !2 

1/2

= 1 −

Za 1/2

Za

 = 1 −

1/4

Za

1/4 1/4

+ Za Zb !2  1/4 Za  1/4

+ Zb

 Ii (s)

Ii (s),

(53)

where Zb denotes the capacitive or inductive reactance of the corresponding capacitor or inductor in FM1/2 . Third, with respect to the purely ideal p-order fracmemristor, from (33) and (45), the following can be obtained: p

FMp = Za1−p Zb ,

(54)

where 0 6 p 6 1. Za = r[q(s)] and Zb denote the reactance of the memristor and the capacitor or inductor in a purely ideal p-order fracmemristor in the Laplace transform domain, respectively. From (54), the equivalent circuit of a purely ideal p/2-order fracmemristor can be shown as given in Fig. 5. In Fig. 5, Ii (s) denotes the input current of a purely ideal p/2-order fracmemristor. ia (s) and Za = r[q(s)] denote the branch-current and the reactance of the memristor in 1880

FIGURE 5. Equivalent circuit of purely ideal p/2-order net-grid-type fracmemristor.

a purely ideal p/2-order fracmemristor in the Laplace transform domain, respectively. As aforementioned discussion, if FMp denotes a purely ideal p-order capacitive fracmemristor, FMp/2 is a corresponding purely ideal p/2-order capacitive fracmemristor. If FMp denotes a purely ideal p-order inductive fracmemristor, FMp/2 is a corresponding purely ideal p/2-order inductive fracmemristor. In a similar way, from (54) and Fig. 5, the following can be obtained:  !2  1/2 Za  Ii (s) ia (s) = 1 − 1/2 1/2 Za + FMp 2    1/2

  = 1 −   = 1 −

Za     Ii (s) 1−p p 1/2 1/2 Za + Za Zb !2  p/2 Za  Ii (s). (55) p/2 p/2 Za + Zb

Moreover, from (35) and (47), we can naturally implement a purely ideal arbitrary-order capacitive or inductive fracmemristor by connecting a purely ideal p/2-order fracmemristor with a m-stage first-order integrators or differentiators in a cascade manner. With regard to a series circuit, the input current for each stage first-order integrator or differentiator is the same as the input current of a purely ideal p/2-order fracmemristor, Ii (s). Thus, in the Laplace transform domain, (55) represents the branch-current of the memristor of the purely ideal (m + p/2)-order, arbitraryorder, fracmemristor. E. ANALOG CIRCUIT REALIZATION OF ARBITRARY-ORDER FRACMEMRISTOR IN ITS NATURAL IMPLEMENTATION

In this subsection, we discuss the analog circuit realization of the arbitrary-order fracmemristor in its natural implementation. As aforementioned discussion, similar to the fractor, the tree-type [47], [56], two-circuits-type [63], H-type [64], net-grid-type [65] should be four natural implementations of the fracmemristor. Therefore, from Fig. 2, we can use the memristor and capacitor or inductor to realize the corresponding arbitrary-order capacitive or inductive fracmemristor in its natural implementation. In order to simplify the analysis, we take the 1/2-order capacitive fracmemristor in the net-grid-type analog circuit as an example for discussion. We can use the memristor and capacitor to achieve the analog VOLUME 4, 2016


circuit realization of a purely ideal fractional-order capacitive fracmemristor. In Fig. 2, we let Za and Zb to be a memristor and a capacitor, respectively. As shown in Fig. 6, from (11) and (36), we construct a 1/2-order capacitive fracmemristor in a series connection of k-layer repeated net-grid-type structures. denotes the memriIn Fig. 6, the symbol of stor [1]–[8] with the same memristance, the symbol of C denotes the capacitor with the same capacitance. The memristor can be obtained by the independent electronic component. Moreover, when the number of layers of the repeated net-grid-type structures increases, the approximation precision of the 1/2-order capacitive fracmemristor also increases. The purely ideal arbitrary-order net-grid-type fracmemristor is a recursively nested structure of the aforementioned purely ideal 1/2-order net-grid-type fracmemristor.

FIGURE 6. Analog circuit of 1/2-order capacitive fracmemristor in series connection of k-layer repeated net-grid-type structures.

Furthermore, for the memristance R[q(t)] can be theoretically an arbitrary variable resistance [1]–[8], its Laplace transform r[q(s)] is also arbitrary. In (36) and (48), the approximate implementation of [r(q)]1−p and L −1 [r(q)]1−p for arbitrary r[q(s)] are difficult and complex. It will be discussed in our future work. IV. EXPERIMENT AND ANALYSIS A. NUMERICAL IMPLEMENTATION OF FRACMEMRISTOR

In this subsection, we achieve the numerical implementation of the fracmemristor before analyzing its electrical characteristics. First, with regard to the capacitive fractor, from (9) and (12), the inverse Laplace transform of (5) is as follows: Vi (t) = c−v r 1− p [D−v t Ii (t)],

(56)

where v = m + p is a positive real number, m is a positive integer, and 0 6 p 6 1. With regard to the inductive fractor, from (9) and (12), the inverse Laplace transform of (6) is as follows: Vi (t) = l v r 1−p [Dvt Ii (t)].

(57)

Without loss of generality, in (36) and (48)-(57), assuming c = 1, l = 1 and r = 1. With regard to the capacitive fracmemristor, from (9) and (12), the inverse Laplace transform of (36) is as follows: n o Vi (t) = L −1 [r(q)]1−p ∗ [D−v (58) t Ii (t)], VOLUME 4, 2016

where r[q(s)] is the Laplace transform of R[q(t)] and symbol ∗ denotes convolution. With regard to the inductive fracmemristor, from (9) and (12), the inverse Laplace transform of (48) is as follows: n o Vi (t) = L −1 [r(q)]1−p ∗ [Dvt Ii (t)]. (59) Equations (58) and (59) show that they are just different in the power of Dt . When the power of Dt is negative, D−v t implements the fractional-order integral. Equation (58) is the expression of the electrical properties of the capacitive fracmemristor. When the power of Dt is positive, Dvt implements the fractional-order differential. Equation (59) is the expression of the electrical properties of the inductive fracmemristor. Equations (58) and (59) also show that the fracmemristor can be considered in a certain way as an interpolation of the memristor and capacitor or inductor. The following comparative experiments based on (58) and (59) show the difference between the electrical properties of the capacitive fracmemristor and those of the inductive fracmemristor. Second, Equations (56) - (59) show that the fractional differential operator Dµ of a one-dimensional digital signal should be implemented, where µ is a real number. Equation (5) shows that for Grünwald-Letnikov definition of fractional calculus, the limit symbol may be removed when N is large enough. Thus, we can numerically implement the fractional differential operator Dµ [68]. It follows that: N −1 kt t −µ N µ X 0(k − µ) µ f (t − ) Dt f (t) ∼ = 0(−µ) 0(k + 1) N

=

k=0 N −1 t −µ N µ X

0(−µ)

k=0

0(k − µ) fk , 0(k + 1)

(60)

where µ is a real number. When µ = v > 0, Dµ achieves the µ-order fractional derivative; When µ = −v < 0, Dµ achieves the µ-order fractional integral; When µ = −1, Dµ achieves the first-order integral. For convenience, assuming N = 50. B. ELECTRICAL CHARACTERISTICS OF FRACMEMRISTOR

In this subsection, we discuss the electrical characteristics of the fracmemristor. We illustrate the interpolated characteristic of the capacitive fracmemristor between a memristor and a capacitor, and that of the inductive fracmemristor between a memristor and an inductor. For convenience of illustration, in (55), we let ia (s) ≈ Ii (s) in the following comparative experiments. In order to illustrate the electrical characteristics of the fracmemristor of the different fractional-orders, we choose different values of µ and v to implement the following comparative experiments. Example 1: In (58) and (59), we suppose that the causal current source Ii (t) applied across the fracmemristor is given by: Ii (t) = t h u(t),

(61) 1881


FIGURE 7. Vi − t of fracmemristor: (a) Capacitive fracmemristor (µ = −v = −0.70); (b) Inductive fracmemristor (µ = v = 0.70).

where h is a positive integer and u(t) is a Heaviside function. Assuming that we have a memristor with memristance: R[q(t)] = q(t) = D−1 t Ii (t) 1 h+1 t u(t). (62) = h+1 And assuming its initial quantity of electric charge q0 + q(0) = 0. The Laplace transform of (62) is as follows: r[(q(s)] = L {R[q(t)]} (63)

Thus, (58) and (59) can be simplified as, respectively:

Vi (t) =

−v−(h+2)(1−p) Ii (t), (h!) Dt 1−p v−(h+2)(1−p) Ii (t). (h!) Dt 1−p

R[q(t)] = q(t) (64) (65)

t h u(t)

Equations (64) and (65) show that when Ii (t) = and R[q(t)] = q(t), the fracmemristor is turned into a traditional fractor. In particular, assuming that h = 0.Thus, (64) and (65) can be further simplified as, respectively: Vi (t) = Vi (t) =

−v−2+2p Dt Ii (t), v−2+2p Dt Ii (t).

(66) (67)

Equations (66) and (67) show that when Ii (t) = u(t) and R[q(t)] = q(t), the fracmemristor is turned into a fractor. Figure 7 shows the waveform of the Vi − t curve of the fracmemristor obtained by solving (56), (57), (62), (66), and (67) when |µ| = v = 0.70. Figure 7 shows that when Ii (t) = u(t) and R[q(t)] = q(t), the 0.70-order capacitive fracmemristor is turned into the 1.30-order capacitive fractor, and the 0.70-order inductive fracmemristor is turned into the 0.10-order inductive fractor. From (36) and (48), we can see that in domain of the Laplace transform, the v-order fracmemristor implements a cascade system of the v-order fractor and a fractional power function of memristance. If the Laplace transform of memristance is equal to an arbitrary power function of s, the v-order fracmemristor will be turned into a traditional fractor. In addition, from (36) and (48), we can also see that if v = 0, 1882

tsin(t) u(t). (68) 2 Assuming that we have a memristor with memristance: Ii (t) =

= (h!)s−h−2 .

Vi (t) =

the v-order fracmemristor will be turned into a traditional first-order memristor. In Fig. 1, the capacitive fracmemristor should be lying on the line segment, S4 , between C and M , and the inductive fracmemristor should be lying on the line segment, S3 , between L and M . The fracmemristor can be converted to a fractor or memristor in some given conditions. Example 2: In (58) and (59), we suppose that the causal current source Ii (t) applied across the fracmemristor is given by:

= D−1 I (t) t i sin(t) tcos(t) = − u(t). 2 2

(69)

And assuming its initial quantity of electric charge q0 + q(0) = 0. From (69), it follows that: r[q (s)] = L {R [q(t)]} 1 = 2 . s2 + 1

(70)

In particular, assuming that v = m+0.5, where m is a positive integer. Thus, p = 0.5. From (70), it follows that: n o n o 1 L −1 [r(q)]1−p = L −1 [r(q)] 2 = [sin(t)]u(t).

(71)

For convenience of illustration, in (71), we only choose its positive root. Figure 8 shows the waveform of the Vi − t curve obtained by solving (56) - (59), (69), and (71) when |µ| = v = 1.5, 4.5. Figure 8 displays the time variations of voltage (Vi − t curve), which have varying amplitude periodic waveforms corresponding to the fractor, memristor, and fracmemristor, respectively. In Fig. 8, in order to illustrate the Vi − t curves of the fracmemristor, memristor, and fractor in the same figure, we divide the experimental values of the fracmemristor by 1000. First, from (2), (56), and (57), we can VOLUME 4, 2016


FIGURE 8. Vi − t of fracmemristor: (a) Capacitive fracmemristor (µ = −v = −1.50); (b) Inductive fracmemristor (µ = v = 1.50), (c) Capacitive fracmemristor (µ = −v = −4.50); (d) Inductive fracmemristor (µ = v = 4.50).

see that because there is not the computation of convolution, the duration of the voltage across the memristor or fractor v should be equal to the duration of Ii (t), D−v t Ii (t), or Dt Ii (t), which is just equal to the duration of the input current of the memristor or fractor. This is the real reason why the curves of the input voltage of the memristor and fractor are cut off in the middle in Fig. 8. Furthermore, from (58) and (59), we can see that in the time domain, the v-order fracmemristor implements the convolution of L −1 [r(q)]1−p and D−v t Ii (t) or Dvt Ii (t). When the input current of the fracmemristor is a periodic signal, the input voltage of the fracmemristor is a varying amplitude periodic signal. Being similar to the memristor, the fracmemristor also has the non-volatility property of memory. Second, Fig. 8 shows that the fracmemristor has nonlinear predictive ability. Because the voltage across the fracmemristor is equal to the convolution of L −1 [r(q)]1−p v and D−v t Ii (t) or Dt Ii (t), its duration should be equal to the summation of the duration of L −1 [r(q)]1−p and that of v D−v t Ii (t) or Dt Ii (t), which is equal to the double duration of the input voltage of the memristor or fractor. Therefore, after the causal current source Ii (t) being power off, the fracmemristor continues to run until its energy is exhausted. From (58) and (59), we can see that the predicted voltage across the fracmemristor is equal to a nonlinear computation of its past current Ii (t), i.e. L −1 [r(q)]1−p ∗[D−v t Ii (t)] input1−p −1 or L ∗[Dvt Ii (t)]. Because [r(q)]1−p has the frac[r(q)] tional power polynomial kernel function of s, it is difficult to directly derive the analytical expression of L −1 [r(q)]1−p VOLUME 4, 2016

for arbitrary r[q(s)] in (58) and (59). We have not illustrated the predicted current across the fracmemristor. Figure 9 shows the waveform of the Vi − Ii curve obtained by solving (56) - (59), (69), and (71) when |µ| = v = 1.5. Figure 9 shows the hysteresis loop (Lissajous curve) in the Vi − Ii plane. In Fig. 9, in order to illustrate the Vi − Ii curves of the fracmemristor, memristor, and fractor in the same figure, we divide the experimental values of the fracmemristor by 1000. With regard to the memristor case, there is a pinched hysteresis loop: double valued Lissajous curve for all time t except when it passes through the pinched point (0, 0). With regard to the fractor case, there is a spiral curve similar to the Achimedean spiral, whose start point is (0, 0). With regard to the fracmemristor case, there is a nonlinear twisted spiral curve, whose start point is (0, 0). Figure 10 shows the waveform of the Vi −q curve obtained by solving (56) - (59), (69), and (71) when |µ| = v = 1.5. Figure 10 shows the hysteresis loop in the Vi − q plane. In Fig. 10, in order to illustrate the Vi − q curves of the fracmemristor, memristor, and fractor in the same figure, we divide the experimental values of the fracmemristor by 1000. p Taking into account that cos(t) = ∓ 1 − sin2 (t), the hysteresis loops of the fractor, memristor, and fracmemristor are the multiple-valued functions of q. In this case, their hysteresis loops are multiple-valued Lissajous curves symmetric with respect to q-axis. 1883


FIGURE 9. Vi − Ii of fracmemristor: (a) Capacitive fracmemristor (µ = −v = −1.50); (b) Inductive fracmemristor (µ = v = 1.50).

FIGURE 10. Vi − q of fracmemristor: (a) Capacitive fracmemristor (µ = −v = −1.50); (b) Inductive fracmemristor (µ = v = 1.50).

FIGURE 11. Vi − t of fracmemristor: (a) Capacitive fracmemristor (µ = −v < 0); (b) Inductive fracmemristor (µ = v > 0).

Figure 11 shows the waveform of the Vi − t curve obtained by solving (58), (59), (69), and (71) when |µ| = v = 0.5, 1.5, 2.5, 3.5. Figure 11 displays the time variations of voltage (Vi − t curves), which have varying amplitude periodic waveforms corresponding to the various fractional-order fracmemristor. Upon decreasing |µ| = v, the Vi − t curve of the fracmemristor changes its shape continuously. The Vi − t curve of a certain fractional-order fracmemristor is equal to a nonlinear continuous interpolation of two Vi − t curves of its left and right proximal fractional-order fracmemristors. Example 3: Because [r(q)]1−p has a fractional power polynomial kernel function of s, it is difficult to directly derive 1884

the analytical expression of L −1 [r(q)]1−p for arbitrary r[q(s)] in (58) and (59). To further illustrate its nonlinear predictive ability, we discuss a special case of the fracmemristor. In particular, in (58) and (59), we suppose that p is given by: 1 p=1− , n

(72)

where n is an positive integer. Assuming that we have a memristor with memristance: r[q (s)] = L {R [q(t)]} = [s−1 Ii (s)]n .

(73) VOLUME 4, 2016


FIGURE 12. Current and voltage of fracmemristor: (a) 0.5-order inductive fracmemristor (without noise); (b) 0.5-order inductive fracmemristor (added white Gaussian noise to the input current of fracmemristor); (c) 1.5-order inductive fracmemristor (without noise); (d) 1.5-order inductive fracmemristor (added white Gaussian noise); (e) 0.5-order capacitive fracmemristor (without noise); (f) 0.5-order capacitive fracmemristor (added white Gaussian noise); (g) 1.5-order capacitive fracmemristor (without noise); (h) 1.5-order capacitive fracmemristor (added white Gaussian noise).

And assuming its initial quantity of electric charge q0 + q(0) = 0. From (73), it follows that: n o n o 1 L −1 [r(q)]1−p = L −1 [r(q)] n = D−1 t Ii (t), VOLUME 4, 2016

(74)

where Ii (t) is an arbitrary causal current source. For convenience of illustration, in (74), we only choose its positive real root. In this example, we suppose that n = 2 and p = 0.5. Figure 12 shows the waveforms of the fracmemristor obtained by solving (58), (59), and (74) when |µ| = v = 0.5, 1.5. 1885


Figure 12 displays the time variations of the current and voltage across the fracmemristor. To further illustrate its nonlinear predictive ability, we use the composite current source mixed by the square wave, sine, and triangular wave signals to experiment. In Fig. 12, in order to illustrate the current and the voltage of the fracmemristor in the same figure, we divide the experimental values of the voltage of the fracmemristor by 100. Furthermore, in order to illustrate the influence of white current noise for the fracmemristor, in Fig. 12(b), 12(d), 12(f), and 12(h), we added the white Gaussian noise to the current across the fracmemristor, whose signal-to-noise ratio is equal to 20. From Fig. 12, we can see that the fracmemristor can be considered in a certain way as an interpolation of the memristor and capacitor or inductor. First, as aforementioned discussion, the v-order fracmem ristor implements the convolution of L −1 [r(q)]1−p and v D−v t Ii (t) or Dt Ii (t).Thus, the fracmemristor depends on the convolution of the current history and its fractional calculus. On the one hand, Fig. 12(b) and 12(d) show that the inductive fracmemristor is unable to protect its memory states and prediction states against unavoidable current noise. In particular, the prediction states of the inductive fracmemristor suffer current noise jamming more serious. According to the theory of fractional calculus, the fractional derivative of the current across the fracmemristor, Dvt Ii (t), enhances its high frequency singular noise. From (48) and (59), we can see that the electrical characteristics of the inductive fractor provide no physical mechanism enabling to the inductive fracmemristor to protect its memory states and prediction states under the influence of current noise. On the other hand, Fig. 12(f) and 12(h) show that the capacitive fracmemristor is able to protect its memory states and prediction states against unavoidable current noise. According to the theory of fractional calculus, the fractional integral of the current across the fracmemristor, D−v t Ii (t), suppresses its high frequency singular noise. From (36) and (58), we can see that the electrical characteristics of the capacitive fractor do provide physical mechanism enabling to the capacitive fracmemristor to protect its memory states and prediction states under the influence of current noise. The concept of the both current and fractional-order controlled fracmemristor provides a physical mechanism enabling a fracmemristor system to erratically change its memory states and prediction states under the influence of white current noise. Second, Fig. 12 shows that the fracmemristor has nonlinear predictive ability. After the causal current source Ii (t) being power off, the fracmemristor continues to run until its energy is exhausted. The predicted voltage across the fracmemristor depends on the nonlinear computation of its early input current. The singularity of the predicted voltage across the fracmemristor reflects the variation trend of the future voltage across the fracmemristor. From (58) and (59), we can see that the voltage across the fracmemristor is equal to the convolution of L −1 [r(q)]1−p v and D−v t Ii (t) or Dt Ii (t), its duration should be equal to the summation of the duration of L −1 [r(q)]1−p and D−v t Ii (t) 1886

or Dvt Ii (t), which is equal to the double duration of the input voltage of the memristor. The predicted voltage across the −1 [r(q)]1−p ∗ [D−v I (t)] or fracmemristor is equal to L i t L −1 [r(q)]1−p ∗ [Dvt Ii (t)]. Because [r(q)]1−p has the fractional power polynomial kernel function of s, it is difficult to directly derive the analytical expression of L −1 [r(q)]1−p for arbitrary r[q(s)] in (58) and (59). We have not illustrated the predicted current across the fracmemristor. V. CONCLUSIONS

The application of fractional calculus to analyzing the memristor is an emerging discipline of study in which few studies have been performed. Fractional calculus has been considered an important novel branch of mathematical analyses. Fractional calculus is as old as the integer-order calculus, although until recently, its application was exclusively in mathematics. Fractional calculus appears to be a promising mathematical method for physical scientists and engineering technicians. Fractional calculus has been hybridized with signal processing, circuits and systems, and material science mainly because of its inherent strength of long-term memory, non-locality and weak singularity. Motivated by this need, in this paper, we proposed to introduce an interesting conceptual framework of the fracmemristor. The term ‘‘fracmemristor’’ is a portmanteau of ‘‘the fractional-order memristor.’’ In particular, in Chua’s periodic table of all two-terminal circuit elements, the electrical properties of the capacitive fracmemristor should fall in between the electrical properties of the capacitor and those of the memristor. The electrical properties of the inductive fracmemristor should fall in between the electrical properties of the inductor and those of the memristor. In this paper, the non-volatility property of memory and nonlinear predictive ability of the fracmemristor are analyzed in detail experimentally. From aforementioned discussion, we can also see that there are many problems else need to be further studied. For example, how to numerically implement L −1 [r(q)]1−p for arbitrary r[q(s)] in (58) and (59), how to numerically implement the predicted current across the fracmemristor, how to achieve the analog circuit realization of the arbitraryorder fracmemristor, how to apply the fracmemristor, and so on. It will be discussed in our future work. REFERENCES [1] L. O. Chua, ‘‘Memristor—The missing circuit element,’’ IEEE Trans. Circuit Theory, vol. CT-18, no. 5, pp. 507–519, Sep. 1971. [2] L. O. Chua and S. M. Kang, ‘‘Memristive devices and systems,’’ Proc. IEEE, vol. 64, no. 2, pp. 209–223, Feb. 1976. [3] L. O. Chua, ‘‘Device modeling via nonlinear circuit elements,’’ IEEE Trans. Circuits Syst., vol. CAS-27, no. 11, pp. 1014–1044, Nov. 1980. [4] L. O. Chua, ‘‘Nonlinear circuit foundations for nanodevices. I. The four-element torus,’’ Proc. IEEE, vol. 91, no. 11, pp. 1830–1859, Nov. 2003. [5] L. O. Chua, ‘‘Resistance switching memories are memristors,’’ Appl. Phys. A, vol. 102, no. 4, pp. 765–783, 2011. [6] L. O. Chua, ‘‘The fourth element,’’ Proc. IEEE, vol. 100, no. 6, pp. 1920–1927, Jun. 2012. [7] T. Prodromakis, C. Toumazou, and L. Chua, ‘‘Two centuries of memristors,’’ Nature Mater., vol. 11, pp. 478–481, May 2012.

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YI-FEI PU received the Ph.D. degree from the College of Electronics and Information Engineering, Sichuan University, in 2006. He is currently a Full Professor, and the Doctoral Supervisor with the College of Computer Science, Sichuan University, and is elected into the Thousand Talents Program of Sichuan Province. He focuses on the application of fractional calculus and fractional partial differential equation to signal analysis and signal processing. He has authored about 20 papers indexed by SCI in journals, such as the IEEE TRANSACTIONS ON IMAGE PROCESSING, the IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, Mathematic Methods in Applied Sciences, and Science in China Series F: Information Sciences, and Science China Information Sciences. He held several research projects, such as the National Nature Science foundation of China and the Returned Overseas Chinese Scholars Project of Education Ministry of China, and holds 11 Chinese inventive patents, as the first or single inventor.

XIAO YUAN received the Ph.D. degree from the School of Electronic Engineering, University of Electronic Science and Technology of China, in 1998. He is an Associate Professor of College of Electronics and Information Engineering with Sichuan University. He focuses on the application of wavelet transform, fractional calculus, and fractional partial differential equation to signal processing and signal analysis. He has authored over 30 papers, which are indexed by SCI and EI or ISTP.

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