On the Recursive Fractional Variable-Order Derivative: Equivalent Switching Strategy, Duality, and Analog Modeling

In this paper, a switching strategy for recursive fractional variable-order derivative is proposed. This strategy can be interpreted as an explanation of order switching mechanism for this particular type of derivative. Additionally, important properties of variable fractional order derivatives, required for prove the main result, are introduced both in a difference equation and a matrix form. Duality between the recursive and standard variable-order derivative is detailed derived. Based on the switching scheme, an analog realization of the recursive variable-order derivative definition is presented. Experimental results obtained for the analog realization are compared to the numerical results.

culus was found a very useful tool for modeling the behavior of many materials and systems, especially those based on the diffusion processes. Ones of such devices that can be modeled more efficiently by fractional calculus are ultracapacitors. Models of these electronic storage devices, which capacity can reach even thousands of Farads, based on fractional order models, were presented in [4,5]. Another system that can be successfully modelled using fractional calculus methods is the heat transfer process in both homogeneous and non-homogeneous media [6,25]. Numerical realization of fractional dynamic systems is more complicated than for integer order case, that is why analog modeling methods for fractional order systems are widely used, e.g., in fractional order chaotic systems [1][2][3].
variable-order derivative definition, which will be useful to develop the main result of the paper. In Sect. 5, a recursive (4th type) definition for variable-order differ-integral is given. Section 6 introduces a matrix form of the recursive type definition of variableorder derivative. The duality between the 1st and 4th type of fractional variable-order derivative definitions is established in Sect. 7. In Sect. 8, the main contribution of this paper, i.e., the identity of the recursive definition to the reductive-switching scheme is presented. Numerical verification of the equivalence between switching-order scheme and 4th type derivative is presented in Sect. 9. Finally, in Sect. 10, an analog realization of the fractional variable-order integral, according to the proposed type of derivative, is presented.

Problem Statement
We have build the electrical circuit (see Fig. 2) that realizes the reductive-switching scheme (more details in Sect. 10). The reductive-switching order occurs when the initial chain of derivatives is reduced according to changing the variable-order (more details are presented in Sect. 8). This switching scheme assumes that changing order is obtained by reducing chain of integrators from the input side. The simple reductiveswitching case (with only one switch, i.e., one change of order) is presented in Fig. 1.

Observation
Let us assume the following, given by formula (1), description of the fractional variable-order derivative (more details about the origin and properties of this formula are given in Sect. 5 The experimental results compared to numerical implementation of formula (1) are presented in Fig. 3.
Observation The formula given by (1) is able to describe with very high accuracy the reductive-switching behavior of the electrical circuit.
Problem formulation The problem that will be solved in this work, is to prove that the formula given by (1) is equivalent to the reductive-switching scheme, and to validate these phenomena using experimental electrical circuits.
In order to prove the above stated problem, we present below the required background from fractional calculus and introduce necessary properties of recursive fractional variable-order derivative.

Fractional Variable-Order Grunwald-Letnikov Type Derivatives
As a base of generalization of the constant fractional order α ∈ R derivative onto variable-order case, the following definition is taken into consideration: where h > 0 is a step time, and n = t/ h . The matrix form of the fractional order derivative is given as follows [16,17]: where , and h = t/k, k is the number of samples. For the case of order changing with time (variable-order case), three types of definitions can be found in the literature [8,33]. The first one is obtained by replacing in (2) a constant-order α by variable-order α(t). In that approach, all coefficients for past samples are obtained for present value of the order, and is given as follows:

Definition 1
The 1st type of fractional variable-order derivative is defined as follows: The second type of definition assumes that coefficients for past samples are obtained for order that was present for these samples. Identity of this definition to the particular (additive) switching scheme was presented in [26]. In this case, the definition has the following form:

Definition 2
The 2nd type of fractional variable-order derivative is defined as follows: The third definition is less intuitive and assumes that coefficients for the newest samples are obtained, respectively, for the oldest orders. For such a case, the following definition applies:

Definition 3
The 3rd type of fractional variable-order derivative is defined as:

Some Properties of the 1st Type of Fractional Variable-Order Derivative
In order to proof the main result of this paper, i.e., the equivalence of the matrix form of the 4th type of variable-order fractional derivative and the reductive-switching scheme, we give the following additional results, which themselves state new contributions in this research area.

Matrix Approach of the 1st Type of Derivative
The matrix form of the 1st type of fractional-variable order derivative is given by [30] ⎛ where 4.2 Proposed Output-Switching Scheme of the 1st Type of Derivative Let us introduce the following output-switching (o-s) scheme presented in Fig. 4 based on the chain of derivatives blocks related by the following switching rule. The switches S i , i = 0, . . . , k, take the following positions . 4 Structure of the multiple output-switching order derivative o-s 0 D α(t) t f (t) (configuration after switch between orders α 0 and α 1 , i.e., in time t ∈ (0, h)) and the output of such a structure we denote o-s 0 D Fig. 4 we have the following result.

Lemma 1
The numerical description of the multiple output-switching scheme, when we switch between orders α 0 , . . . , α k every i h time instant, is the following: ⎛ where and Proof The proof we will start at analysis signals from the input side. The output signal after the α k block has the following form: ⎛ The additional blockα k−1 is connected in whole time despite of time k, that is why this signal is represented by the matrix M(α k−1 , k, k − 1). At the output of this block, we obtain derivative of order α k−1 until time k when the order is equal to α k .
Repeating analogously, we get the output signal from the block of derivativeα 1 in the following form Finally, we obtain the output signal from the block of derivativeα 0 in the following form: ⎛ Combining all this together, we get (7), completing the proof.
Theorem 1 Matrix approach of the 1st type derivative given by (5) is equivalent to the output-switching scheme given by (7), i.e., Proof For simplicity, let us assume the case of one switch between orders, say α 1 and α 2 occurring at time T = τ h, τ ∈ N + , we have the following matrix form based on where and α(t) = α 2 +α 1 for t < T, α 2 for t ≥ T.
The matrix product , obviously corresponds to A W (α, k) given by (6) for The prove of multiple-switching case can be obtained by simple analogy to the proof of one switching case.

Recursive Definition for Variable-Order Differ-Integral
The fractional constant-order difference definition (that can be obtained from Definition 1 for α(t) = const) can be rewritten as follows [27]: where n x (z) being Z-transform of the signal difference of order α of variable x(t). It can be also rewritten as which gives the following relation and can be represented in the time domain as Finally, it can be rewritten in the following form This type of difference is obtained from all values of previous differences. For variableorder case, we can obtain the following definition:

Definition 4
The 4th type of fractional variable-order difference is defined as follows: For a continuous time domain case, the 4th difference definition can be rewritten in the following form [27].

Definition 5
The 4th type of fractional variable-order derivative is defined as follows: Remark 1 For a fractional constant-order α = const, the fractional derivative given by Definition 5 is numerically identical with constant-order fractional derivative given by (2).
Numerical results of the 4th type derivatives, of variable-order α 3 (t) given by It can be seen from Figs. 5 and 6 that the derivative results in the case of 3rd and 4th type definition, for input signal being Heaviside step function, i.e., f (t) = 1(t), are very near to each-other. It follows from the fact that for the constant signal the sequence of samples is insignificant. However, for the linear input signal f (t) = t · 1(t), the difference between these both definitions becomes clearly visible, what is seen in Figs. 7 and 8.
As it can be noticed in Figs. 7 and 8 for the linear input signal the differences between the 4th and others types of derivatives (i.e. 1st, 2nd, and 3rd), are indeed well visible and significant. More plots comparing all the definitions are presented in [27].

Matrix Approach for the 4th Type of Fractional Difference Definition
Let us recall the 4th type of variable-order difference definition in the following form: where and where and v −α r ,i = (−1) i −α r i , for i = 1, . . . , r , i.e., the jth element of q r is Proof It is obtained after consecutive evaluating (10) for each time step l = 0, 1, . . . , k. First, for l = 0, we can write Next, for l = 1: and, generally, for l = r , we have Finally, for l = k: . Combining all this together, completes the proof.

Remark 2 For step time
Remark 3 From direct multiplication of (13), the following matrix product is given by where Example 1 For m = 0 and n = k, from (3), we get the following matrix where q i, j for 1 ≤ j < i ≤ k + 1 are given by (15).

Remark 4 Matrix (3) can be written as
where where Example 2 For n = m, from (3), we get which matches to Q(α r , r ) given by (13) for m = r .

Remark 5
The entries q i, j and q i, j of (3) and (18), respectively, are equal to each other, for i = m + 1, . . . , n + 1 and j = 1, . . . , m, i.e., q i, j = q i, j . Moreover, matrix Q n m does not depend on h.
Example 3 For k = 4, m = 1, and n = 2, using (14) or (3), we get where (q i ) j stands for jth entries of vector q i . More precisely, it is given by Remark 6 A natural extension of the above considerations (by taking the limit h → 0), is the following form of the 4th type of variable-order derivative definition: where Q k 0 is given by (12).

Duality Between the 1st and 4th Type of Fractional Variable-Order Derivatives
In this section, we present the duality between the 1st and 4th type of fractional variable-order derivatives, which will be used to derive the main result of this paper, i.e., the equivalence of the 4th definition and the multiple reductive-switching case. By the duality, we mean the composition property of two types of derivatives, which for the variable-order of the opposite signs yields identity. The duality property, constitutes also an important value itself in the fractional calculus domain.

Lemma 3 The inverse of Q n m is
wherē Proof We will show that the product of matrices Q n m −1 and Q n m given by (22) and (17) whereD and D are the diagonal matrices given in (22) and (17), respectively. Then, we have to show thatQ n m Q n m = I k+1,k+1 , because, obviouslyD D = I k+1,k+1 . Denote by (Q n m ) i the ith row ofQ n m , and by (Q n m ) j the jth column of Q n m . Then, one has to calculate for i = m + 1, . . . , n and j = 1, . . . , m. From (18) and (15), the straightforward calculation gives (q i−1 ) p q p, j for i = m + 1, . . . , n; j = 1, . . . , m.

Example 5 For n = m, from (22) we get
which is indeed the inverse of (19).

Remark 7
Because the structures of the matrices Q k 0 and Q k 0 −1 are different, in the sense that in the first case each entry of the same column, say ith, is multiplied by the same term h −α i−1 , i = 1, . . . , k + 1, and in the second case each entries of the same row, say jth, is multiplied by the same term h α j−1 , j = 1, . . . , k + 1, the following, in general, occurs whereα = {α 0 , . . . , α k }. It means that for a variable-order difference (derivative) the semigroup property does not hold. However, for a constant-order, i.e., α 0 = · · · = α k , and, respectively, α(t) = const, this property holds.
Let us recall the definition of the fractional variable-order difference of the 1st type: where Proof It is sufficient to show that Q k 0 −1 y defines the 1st type of definition given by (29), i.e., it corresponds to the right-hand side of (29). Let us denote by Q k Then, from (28), we get In general, for any 0 ≤ i ≤ k, we get which shows the desired equivalence, i.e., Q k 0 −1 The same can also be proved by comparing matrix Q k 0 −1 with the matrix A W (α, k) defining the 1st derivative definition (see [30]). f (t) (switching from α 1 to α 2 ; configuration before time T ) Remark 8 Obviously, since (30) holds, the following is also true . . .

Theorem 2 The following dualities hold
Proof In both cases, i.e., in the difference and derivative case, it simply follows from the composition of matrices Q k 0 −1 and Q k 0 , which obviously yields identity.

Main Result-Equivalence of Reductive-Switching Order Case
In this section, the equivalence between the reductive-switching scheme and the 4th type of fractional variable-order derivative definition will be presented, for clarity and simplicity, in two parts, namely, in the case of simple (one) switching, and later in the general case-multiple switching.

Simple Reductive-Switching Order Case
The reductive-switching (r-s) order case occurs when the initial chain of derivatives is reduced according to changing the variable-order. The idea, introduced in [30], is depicted in Fig. 9, where the switches S i , i = 1, 2, change their positions depending on an actual value of α(t). If we want to switch from α 1 to α 2 , then, before switching time T , we have: S 1 = a, S 2 = b, and after this time: S 1 = b and S 2 = a. At the instant time T , the derivative block of complementary orderα 1 is disconnected from the front of the derivative block of order α 2 , wherê The numerical scheme for reductive-switching case can be obtained similarly to additive-switching scheme (see details in [30]).

Lemma 5 [30]
For a reductive-switching order case, when the switch from order α 1 to order α 2 occurs at time T = τ h, τ ∈ N + , the numerical scheme has the following form: . . .

Theorem 3
In the case of simple switching (between two orders), matrix description (20) corresponding to the 4th type of fractional variable-oreder derivative is equivalent to the matrix form of simple reductive-switching order derivative scheme given in Lemma 5.
Proof Let us consider the switching order given by (32). In the case of simple reductive-switching scheme based on Lemma 5, we have the following switching description where A(α 2 ) ∈ R (k+1−τ )×τ , and where we have used the already known property, .g., [16]). From the other hand, for the 4th type of derivative and the same switching between orders given by (32), we have the following switching matrix form based on (3): where B(α 2 ) ∈ R (k+1−τ )×τ , C(α 2 ) ∈ R (k+1−τ )×(k+1−τ ) , and D(α 1 ) ∈ R τ ×τ are the suitable block matrices. Assuming for a moment that if we would not change the order, i.e., α 1 = α 2 = α, we would obtain D P = W (α, k) according to Remark 1. From this, we conclude that D(α) = W (α, τ − 1) and C(α) = W (α, k − τ ), and then it gives rise to . Assume again for a moment that we do not switch the order, i.e., we have α 1 = α 2 = α, both in the case of 4th definition and simple-reductive case. Thus, using again Remark 1, we get becauseα 1 = 0, and then W (α 1 , τ − 1) = I τ,τ , from which we conclude that Fig. 10 Structure of multiple reductive-switching order derivative r-s 0 D α(t) t f (t) (configuration before switch between orders α 0 and α 1 ) and then, using (33) Finally, we get D P = r-s P, which ends the proof.

Multiple Reductive-Switching Order Case
The reductive-switching order case occurs when the initial chain of derivatives is reduced according to changing the variable-order (Fig. 10). The switching rule in the multiple-switching case is an analogous extension of the simple-switching case described in Sect. 8.1.

Lemma 6
For a multiple reductive-switching order case, when the switch between orders α 0 , . . . , α k occurs every i h time instant, the numerical scheme has the following form: . . . and The proof of Lemma 6 is a straightforward extension of the proof of Lemma 5 given in [30]. (20) corresponding to the 4th type of fractional-order derivative definition is equivalent to the matrix form of multiple reductive-switching order scheme given by Lemma 6, i.e.,

Theorem 4 Matrix description
Proof In order to proof this theorem, we will use the introduced in Sect. 7 fact about the duality between 1st and 4th type of fractional variable-order derivatives; it means that we have where A W (−α, k) is equal to the matrix product given by (7) expressed for −α. Thus, instead of calculating we perform the following composition which yields the identity, and thereby ends the proof.

Numerical Verification of Equivalence Between Switching-Order scheme and 4th Type Derivative
Numerical verification of reductive-switching scheme is done in Simulink by comparing it to the numerical implementation of the recursive fractional variable-order derivative. It is depicted in Fig. 11, where: Di, i = 1, . . . , 8, are fractional constantorder derivative blocks, Si, i = 1, . . . , 7, are switching blocks, and FVOD is the numerical implementation of the recursive fractional variable-order derivative (for more details see [23]). The effects of simulation according to Fig. 11 are depicted in Fig. 12, where the equivalence between recursive definition and reductive-switching scheme is also verified.

Analog Realization of Switching-Order Scheme
In this section, an analog circuit that corresponds to the proposed numerical definition of 4th type is presented. . . . In this paper, the following method of half-order integrator implementation, introduced in [24], and meticulously investigated in [14,29], will be used. The scheme of this method is presented in Fig. 13. Based on the algorithm described in details in [24], the experimental circuit boards of half-order impedances were constructed. The circuit board that has been used in experimental setup is presented in Fig. 14. The circuit consists of 200 discrete elements with the following values: R 1 = 2.4 k , R 2 = 8.2 k , C 1 = 330 nF, and C 2 = 220 nF.

Analog Realization of the 0.25 Order Impedance
Method mentioned in Sect. 10.1 can be extended to build a fractional order integrator of order 0.25. This can be done by replacing the capacitors in the scheme in Fig. 13 by half-order integrators, which can be 0.5 order domino ladders. This gives an impedance of order α = 0.25, which corresponds to a quarter-order integrator. . . . Fig. 16 A circuit board of the 0.25 order integrator In Fig. 15, the scheme of the approximation of a quarter-order integrator is shown; Z 0.5 is the impedance of half-order domino ladder. As a quarter-order impedance, the circuit board presented in Fig. 16 has been used. The board contains about 5,000 discrete elements and was designed according to the scheme shown in Fig. 15. The main ladder includes 50 branches with the following resistors' values: R 1 = 2.4 k , R 2 = 8.2 k . The half-order integrators have been used in the quarter ladders denoted as Z 0.5 on the scheme.

Experimental Setup
Analog model of switching system, used in experimental setup, corresponds to the switching scheme given in Fig. 9. It contains four TL071 operational amplifiers: two (A 1 and A 3 ) in fractional integrator configuration, and two ( A 2 and A 4 ) in inverting amplifier configuration. All of operational amplifiers were supplied with external symmetrical voltage source ±12 V . The fractional integrators were realized similarly to configuration of analog realization of integer order integrator, but in place of capacitor the fractional impedances Z 1 , Z 2 were used. The fractional order impedances were the domino ladder approximations presented in Sects. 10.1 and 10.2. In such a configuration, obtained output signal is inverted, that is why another operational amplifier in configuration of inverted amplifier is used. The operational amplifier TL071 was specially used, because it has ability to connect special circuit for voltage offset reduction; the scheme of this circuit is described in amplifier data-sheet [7, Fig. 3]. Reduction of this offset voltage is very important, because of integration character of the circuits, and integrated offset could have significant influence in simulation error. We have equipped all used operational amplifiers in Input Offset-Voltage Null Circuit and we tuned it before obtaining experimental results.
As a realization of switches S 1 and S 2 , integrated analog switches DG303 were used. Fractional order impedances with order equal to −0.5 or −0.25, used in exper- . 17 Analog realization of the 4th type of fractional variable-order integral iments, are shown in Figs. 14 and 16, respectively. The experimental circuit (for data acquisition and DG303 switches control) was connected to the dSPACE DS1104 PPC card with a PC. Resistors R allow to adjust a gain of integrators, and impedances Z 1 and Z 2 are used to obtain desired switching-order configuration (according to the reductive-switching scheme presented in Fig. 9). At the beginning, both integrators are connected in the series (switch S 1 is in position a, and switch S 2 is in position b), and after order switch in time T the integrator at front of the chain (based on impedance Z 1 ) is disconnected (switch S 1 is in position b, and switch S 2 is in position a).
In this case, according to the scheme in Fig. 17, the structure has the following parameters: Z 1 and Z 2 are the quarter-order integrators, R = 100 k . The identification results were obtained by numerical minimization of time responses square error for constant orders, time interval t ∈ [0, 1.5], with sampling time T s = 0.001 s, and input signal u(t) = 1(t) being the Heaviside step function. After identification process, the following models for orders −0.5 and −0.25, respectively, in time domain, were obtained:  which gives rise to the following variable-order integrator: or using the duality property: where (for the switching time T = 0.7 s.) Example 7 Integrator with switched orders from α = −1 to α = −0.5.
In this configuration, according to the scheme in Fig. 17: Z 1 and Z 2 are the halforder integrators, R = 100 k . The identification results were obtained by numerical minimization of time responses square error with sampling time T s = 0.001 s, and input signal u(t) = 0.01 · 1(t). After identification process, the following models for orders −1 and −0.5, respectively, in time domain, were obtained:   The switching time was equal to 0.7 s. The experimental results are presented in Fig. 3.
In this example, configuration of the experimental setup is similar to the one already used in Example 6.
The identification results were obtained by numerical minimization of time responses square error with sampling time T s = 0.001 s, and input signal u(t) = 0.1 · t1(t). After identification process, the following models for orders −0.5 and −0.25, respectively, in time domain, were obtained:   Example 9 Integrator with switched orders from α = −1 to α = −0.5 of ramp function input signal.
In this example, configuration of the experimental setup is similar to the one already used in Example 7.
The identification results were obtained by numerical minimization of time responses square error with sampling time T s = 0.001 s, and input signal u(t) = 0.1 · t1(t). After identification process, the following models for orders −1 and −0.5, respectively, in time domain, were obtained:

Conclusions
The paper presented a recursive definition of variable-order differ-integral stated in the form of difference equation, and also in the derived equivalent matrix form. This new definition is also expressed in the form of fractional variable-order derivative. The main result of the paper is a derivation of the identity between the presented recursive definition and the reductive-switching scheme of variable-order derivative. Moreover, in the paper, the equivalence between the 1st type of variable-order derivative definition and the output-switching scheme is also derived and presented. The obtained results allow us to better understand the behavior and peculiarity of this type of definitions of variable-order derivative. Based on this knowledge, it can give rise to more appropriate choice of definitions type, depending on particular application. The paper presents also an analog circuit that corresponds to the proposed numerical definition. Our obtained experimental results show high accuracy of the proposed method of analog modeling. The obtained results can be used, in the future, to improve existing algorithms in control and signal processing areas.