Fractional calculus has been widely used in the areas of viscoelasticity,^[1] diffusion,^[2] random processes,^[3] nonlinear dynamics and chaos,^[4] etc. Particularly, the rational approximation problem of fractional operator has also become a hot spot of research in recent years, where s is the complex frequency variable or the operational variable in the Laplace transform domain and μ is called the operational order whose value is the fraction or the rational number, namely . According to network synthesis, the rational impedance function sequence (the iterating time ) which could be realized with circuit elements, has been used to implement the approximation of the irrational fractional operator function under certain conditions and over a certain frequency range:

In the early 1950s and 1960s, the regular Newton iterating process was proposed by Carlson and Halijak to realize the rational approximation of fractional operator (n denotes an integer equal to or greater than 2).^[8,9] The binomial expansion method was put forward by Steiglitz et al. to implement the rational approximation of the negative half order operator.^[10,11] In the modeling study of constant-argument immittance, Roy made use of the continued fraction expansion to achieve the rational approximation of fractional operator .^[12] From the beginning of this century, order fractance approximation circuit through the Carlson regular Newton iterating process, Carlson iterating process for short, was designed in Ref. [13]. The generalization of the binomial expansion method to obtain the rational approximation impedance function sequence for an arbitrary order fractional operator was presented in Ref. [14]. The rational approximation of fractional operator by variable substitution was also achieved by Krishna et al. based on the study of Roy.^[15,16] Using the continued fraction expansion method to design order fractance approximation circuit was accomplished in Ref. [17]. There are also other methods to implement the rational approximation of fractional operator, such as the elliptic function approximation method,^[18,19] the Padé rational approximation method,^[20,21] and so on.

In many rational approximation methods, the Carlson iterating process about operator has made more and more people attach comprehensive importance to it. Nowadays, when mentioning the study history of fractance and fractance approximation circuit, people would date back to the Carlson iterating process. The impedance function sequence acquired by the Carlson iterating process could approximate to the operator and extend to both high and low frequency uniformly, smoothly and steadily.^[7,13] The K-index^[7] is equal to . That is to say, for every one increase in the iterating time k, the increasing frequency band is almost up to 1.7–1.9 orders of magnitude. It is certainly useful that the Carlson iterating process could realize the rational approximation of fractional operator .

The generalized Carlson iterating process which is to achieve the rational approximation of arbitrary order fractional operator (j is also an integer valued from 1 to ) now has been verified in Refs. [7] and [22]. Performance analysis for the generalized Carlson iterating process, is discussed in this paper. We start with introducing the rational approximation principle about the generalized Carlson iterating process. Next the complexity index of fractance approximation circuit, K-index, P-index, and O-index are introduced to contribute to performance analysis. Then experiments are carried out to acquire these approximation performance indices for n varying from 2 to 5. Finally, we summarize the work of this paper, make an outlook and point out some relative topics in the further research.

Considering the following case in Eq. (1)

The classical Newton iterating formula^[8,23] of solving the arithmetic root is

When , there exists and . and are the classical Newton and Carlson iterating functions which have formerly only been used to accomplish the rational approximation of the fractional operator . and are the corresponding particular cases of and .

Let and , the Newton iterating function and the Carlson iterating function are depicted separately in Figs. 1(a) and 1(b). The both possess basic mathematical properties of the fractance approximation iterating function, such as positive root uniqueness, local convergence, global convergence, and so on.^[7]

Fig. 1.

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	Fig. 1. (color online) Iterating function curves: (a) (linear coordinate); (b) (linear coordinate); (c) (dual logarithmic coordinate); (d) (dual logarithmic coordinate); (e) ; (f) .

In comparison with the Newton iterating process of the second order, the Carlson iterating process is of the third order and has global regularity, namely a strict global monotonic increasing property. So the convergence rate at the root is the largest possible one in the set of predistortion functions and no overshoot or undershoot occurs at the beginning of the process.^[8] Besides, the Carlson iterating function also takes on some other more excellent geometrical features.

2.1. Self-similarity

For different positive real numbers a, these Carlson iterating function curves shown in Figs. 1(b), 1(c), and 1(d) are self-similar. This feature is specially obvious in the dual logarithmic coordinate, as can be seen in Figs. 1(c) and 1(d). This kind of self-similarity can make the Carlson iterating function normalized. Let (χ called the normalized variable) and , we obtain the normalized Carlson iterating function

Therefore, the shapes of these Carlson iterating function curves are the same for all values of positive real number a in the dual logarithmic coordinate.

2.2. Exponential symmetry

Let ( ) and we take the logarithmic operation on each side of Eq. (12), so the exponential form of is

Obviously, there exists , thus the function is odd-symmetric, as shown in Fig. 1(f). The exponential form of the normalized Newton iterating function is illustrated in Fig. 1(e). The exponential symmetry indicates that the frequency band of the impedance function calculated by the Carlson iterating function approximating to fractional operator , is extended to both high and low frequencies uniformly, smoothly, and steadily.^[7]

It has been verified in Ref. [22] that the impedance function calculated by the generalized Carlson iterating function satisfies these following properties:^[7,24,25]

(i) Computational rationality

There exists no irrational computation and only rational computation of complex frequency variable s in the expression of when choosing the rational initial impedance function.

(ii) Positive reality principle

Both numerator polynomial and denominator polynomial should be strict Hurwitz polynomials. In other words, all zeros and poles of the impedance function are located in the negative real axis of the s complex plane or the left-half plane of the s complex plane in conjugate pairs.

(iii) Operational validity

The impedance function takes on the operational characteristics of fractional operator over a certain complex frequency range.

Let

(I) Validity of the amplitude-frequency characteristics

(II) Validity of the phase-frequency characteristics

(III) Validity of the order-frequency characteristics

Take for example, the operational validity is shown in Fig. 2. The solid lines typify the frequency–domain characteristics of the impedance function with the iterating time k increasing. The dotted lines represent the frequency–domain characteristics of the ideal operator . The operational nature of fractional order elements and systems, is precisely characterized by the order–frequency characteristics and the phase–frequency characteristics.

Fig. 2.

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	Fig. 2. (color online) Frequency–domain characteristics curves of the impedance function with and : (a) amplitude–frequency characteristics; (b) phase–frequency characteristics; (c) order–frequency characteristics.

The impedance function meets these properties mentioned above, so it has physical realization and the operational characteristics of fractional operator . In this section, a necessary preliminary to describe the approximating degree of the impedance function to the ideal fractional operator, the performance indices are introduced. We also bring in an index to represent the complexity of the fractance approximation circuit.

3.1. Approximation performance indices: -index, -index, and -index

In order to quantitatively describe the approximation degree of the frequency–domain characteristics to the frequency characteristics , the approximation performance indices are introduced, including the slope index (K-index), the phase frequency index (P-index), and the order frequency index (O-index). Firstly, we explain the phase frequency approximation bandwidth exponent , the order frequency approximation bandwidth exponent and some other related concepts. The phase frequency approximation bandwidth exponent is the difference between the upper limit frequency exponent and the lower limit frequency exponent

In this frequency interval, the phase frequency relative error function satisfies the relationshipwhere i represents the approximation precision grade index.

The order frequency approximation bandwidth exponent is also the difference between the upper limit frequency exponent and the lower limit frequency exponent

In this frequency interval, the order frequency relative error function satisfies the relationship

The phase frequency and order frequency relative error function curves are depicted in Fig. 3 separately. The mathematical meaning of , , , , , and are also illustrated in Fig. 3.

Fig. 3.

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	Fig. 3. (color online) Relative error function curves with , , and : (a) phase-frequency; (b) order-frequency.

The upper limit and lower limit frequency exponent curves with the iterating time k increasing are shown in Fig. 4. and depicted in Fig. 5, are the measure values of magnitude orders of approximation bandwidth. When the iterating time k is greater than a certain positive integer, both curves of and are parallel respectively and have the following simple relationship

(25a)

(25b)

where and identify the slopes, as shown in Fig. 6. In general, the slope index, namely the K-index, identifies the increasing rate of frequency bandwidth exponent with the iterating time k varying and contents the relationship

Fig. 4.

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	Fig. 4. (color online) Upper limit frequency exponent and lower limit frequency exponent with and : (a) phase-frequency; (b) order-frequency.

Fig. 5.

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	Fig. 5. (color online) Approximation bandwidth exponent curves with and : (a) phase-frequency; (b) order-frequency.

Fig. 6.

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	Fig. 6. (color online) Slopes of approximation bandwidth exponent curves with and : (a) phase-frequency; (b) order-frequency.

The geometrical interpretation of P_i and O_i are the intersection points of the vertical coordinate and the extended line segments of and curves respectively, as shown in Fig. 5. To describe the whole approximation performance, the P-index and O-index are denoted by the arithmetic mean of P_i and O_i separately,

The greater the values of K-index, P-index, and O-index, the better the approximation performance of the impedance function to the ideal fractional operator .

3.2. Iterating algorithm of coefficient vectors and complexity of fractance approximation circuit

The coefficient vectors and of numerator polynomial and denominator polynomial are given as follows:

(28a)

(28b)

where the degrees are in descending arrangement.

Let used in Eq. (11) and we can rewrite the rational approximation impedance function sequence of fractional operator in the other form:

Corresponding to this, the iterating formulae of coefficient vectors and are obtained,

(30a)

(30b)

where the operation symbol * denotes convolution and

(31a)

(31b)

The aim of using the zero padding operation is to utilize the matrix operations, which can reduce the amount of calculation. The degrees n_k and d_k of the impedance function are decided by the degrees n₀ and d₀ of the initial impedance, the iterating time k and the operational order μ, then there exists

(32a)

(32b)

(32c)

According to the network synthesis theory, we could use available electrical and electronic elements, such as resistors or capacitors, to design the fractance approximation circuit of impedance function . The degree of the impedance function

indicates how many independent dynamic elements such as capacitors and inductors are used in the network. So we make as a measure index of the complexity of the fractance approximation circuit.

The performance analysis of the generalized Carlson iterating process of fractional operator with arbitrary order has been explored in this paper. In addition, the unit and dimension of the physical quantity are given in this paper, which is useful in the future study. For some special orders, are called in different particular forms, for example, is called Ohm square root second (Ohm root second for short), is Ohm cube root second, is Ohm n-th root second, is Ohm per square root second, is Ohm per cube root second, is Ohm per n-th root second. In further researches, some subjects are to be solved as follows.

i) We attempt to obtain the analytical expressions of zeros and poles of the impedance function, so it is easy to judge the stability and analyze the approximation performance of the circuit network to the fractional operator from zero-pole analytical expressions.

ii) The K-index of the Carlson iterating process approximating to the fractional operator has already been known, how about the K-index with j equal to or greater than 2? Investigating the theoretical expression of the K-index should receive significant attention.

iii) We utilize passive or active elements to design an arbitrary order fractance approximation circuit which is used as a fractional order element to realize arbitrary order fractional calculus. The best situation is to make fractional order capacitors and inductors available in the market, just like normal capacitors and inductors are. Furthermore, a great deal of interest exists in modelling and studying fractional order phenomena and processes in the nature.