W transform and its application in fractional linear systems with rational powers

Fractional linear systems have attracted widespread attention from scholars and researchers due to their excellent performance and potential application prospects. In the analysis and design of fractional linear systems, the solution of fractional linear systems is an important part. So far, the powers of s in the complex ‐ frequency ‐ domain equations obtained by the existing fractional Laplace transform are fractions, which makes it difficult to solve algebraic equations formed by multiple fractional powers. To solve this problem, based on the traditional Laplace transform, a new fractional Laplace transform—W transform is proposed, which can make the power of the equation an integer in the W ‐ domain. The main properties of this transformation are given, and an inversion theorem for W transform is obtained. When this transformation is applied to fractional linear systems with rational powers, the expansion formula will have a large number of terms if the traditional decomposition method is used, which makes the form of time ‐ domain solutions more complex. Therefore, a partial fraction expansion method in the W ‐ domain to simplify the form of time ‐ domain solutions is proposed. On this basis, the general steps of circuit analysis in the W ‐ domain are given. Finally, examples are used to verify the correctness and feasibility of the application.


| INTRODUCTION
With the increase in research on fractional calculus, fractional circuits and systems have been introduced in many fields, including the field of electrical and electronic engineering. System models built using fractional calculus can show different mathematical characteristics than the integer-order model. Considering the applications of fractional circuits and systems, the fractional circuits and systems are studied from different perspectives, such as stability analysis [1,2], timedomain response analysis [3,4] and frequency-domain response analysis [5,6].
In the research of fractional circuits and systems, it is effective to apply the Laplace transform to the fractional derivative. Therefore, the Laplace transform is often used to analyse it in the complex frequency domain, which can greatly simplify the calculation. For commensurate fractional linear systems, if s α is replaced by a variable, the original fractional linear systems are mapped to integer-order systems, and the equations can be fractionally expanded and solved using traditional methods. However, when the system contains multiple fractional powers, there is no universal method for fraction expansion, so the Laplace transform is suitable for solving commensurate or special types of equations [7].
To analyse fractional linear systems, some fractional Laplace transforms have been proposed for fractional calculus based on the traditional Laplace transform. The definition of Yang-Laplace is proposed for the case that the fractional derivative of order α(0 <α < 1) exists, but itself is not differentiable [8]. Yang-Laplace uses E α ð−s α t α Þ instead of e −st in the traditional Laplace transform, the property of displacement is obtained using E α ððx þ yÞ α Þ ¼ E α ðx α ÞE α ðy α Þ. However, the generalised Mittage-Leffler function does not have such a relationship. Therefore, the Yang-Laplace transform can only be used to deal with local commensurate fractional differential equations. The conformable fractional Laplace transform [9] is proposed for conformable fractional calculus. The conformable fractional Laplace transform of conformable fractional derivative is L α fT α f ðtÞg ¼ sF α ðsÞ − f ð0Þ, which is consistent with the differential properties of the traditional Laplace transform, but the conformable fractional Laplace transform can't solve other forms of fractional derivatives. Watugala [10] introduced the Sumudu transform, which is defined as The Sumudu transform is obtained by simple variable substitution and division by a new variable s. On the basis of retaining the advantages of the traditional Laplace transform, the Sumudu transform allows the unit consistency in the differential equations describing physical processes to be maintained. However, the order of the function obtained by transforming the fractional calculus is still fraction, and the solution of incommensurate fractional differential equations is very complicated. Reference [11][12][13] proposed some fractional Laplace transforms with complex offsets and parameters. The added free variable is not meaningful for solving fractional differential equations. If the free variable is fixed to π 2 , the fractional Laplace transform is simplified into the traditional Laplace transform. In addition, for fuzzy fractional calculus, discrete fractional calculus and dynamic objects in three-dimensional space, fuzzy Laplace transform [14], discrete Laplace transform [15] and fractional quaternion Laplace transform [16] are proposed.
There are some common problems in the application of Laplace transform to the fractional domain. For example, the image function of the fractional transfer function in the sdomain is [17].
Where γ kþ1 > γ k ðk ¼ 1; 2; ⋯; maxðm; nÞÞ, b m ; b m−1 ; :::; b 0 and a n−1 ; a n−1 ; :::; a 0 are constant coefficients. The introduction of fractional differential operator makes the transfer function of fractional linear systems no longer a rational function with integer powers of the complex variable s that is the power of s is no longer an integer. This complicates the algebraic analysis of the s-domain in incommensurate situations. It can be seen that in the fractional systems, there is a greater difficulty in using the Laplace transform for analysis.
To solve the above problems, inspired by Refs. [18][19][20][21], a new fractional Laplace transform is proposed, which called W transform. This transformation can effectively analyse fractional systems with rational or irrational powers. In addition, the W transform can be applied to rational fractional linear systems by selecting appropriate parameters. When the order of wis higher, the image function is directly expanded by the traditional decomposition method, then the expansion formula will have a large number of terms, which makes the time-domain form more complex [22]. Therefore, a partial fraction expansion method in the W domain to combine partial terms is proposed, thus simplifying the form of time-domain solutions. So far, using Laplace transform, many scholars have given the analytical solutions of electrical circuits RL, RC and LC defined by Riemann-Liouville, Caputo, Atangana-Baleanu fractional derivatives and so on [23][24][25], but the research objects are simple circuits containing only one power. However, When W transform is used to analyse the rational fractional linear circuit, the complex circuit containing fractional capacitance and fractional inductor with different fractional powers can be solved.
Herein, Section 2 briefly introduces the relevant background knowledge. Section 3 proposes the definition of W transform and inverse transform. Section 4 gives the properties of the W transform. Section 5 gives the partial fraction expansion method for the rational image function in the Wdomain. Section 6 discusses the W transform method of the linear constant coefficient fractional differential equations and fractional state equations with rational powers, and verifies them with examples. Section 7 derives the model of fractional order element in the W-domain, and gives W-domain analytical method of fractional circuit with rational powers. Then examples are used to verify the accuracy of the results. Finally, the conclusions are drawn in Section 8.

| Fractional calculus
The fractional integral is defined as [26].
where J α is the fractional integral operator, α is the integration order and the value is a positive real number, f ðtÞ represents the integrated function, t and a denote the upper and lower bounds of the integral, and ΓðαÞ is Euler's Gamma function, its definition is Using the definition of Laplace transform FðsÞ ¼ ∫ þ∞ 0 − f ðtÞe −st dt and its convolution theorem, the Laplace transform of (1) is obtained as where L is the Laplace transform symbol, and * represents the convolution symbol. Fractional derivative is not defined uniformly in mathematic. The Caputo definition of the fractional derivative is mainly used to study fractional system.
The Caputo fractional derivative is defined as [26].
where D α is fractional differential operator, ⌊ · ⌋ means floor function, ⌈ · ⌉ means ceiling function. The Laplace transform of (3) is It can be seen that the image functions contain the fractional powers of s, which brings difficulties to analyse fractional systems in the s-domain.

| Mittag-Leffler function
The one-parameter Mittag-Leffler function is the simplest Mittag-Leffler function, which is defined as where α ∈ C, z ∈ C, C is a complex set.
The Laplace transform of the one-parameter Mittag-Leffler function is [27].
The two-parameter Mittag-Leffler function is defined as If β ¼ 1, then the two-parameter Mittag-Leffler function degenerates into one-parameter Mittag-Leffler function.
The Laplace transform of the two-parameter Mittag-Leffler function is [27].

| Riemann sheet
When studying multi-valued functions, We hope to find a generalised field, so that the original multi-valued function becomes a single-valued function on this generalised field. This generalised field is Riemann surface [28].
For a multivalued function Pðs α Þ, let w ¼ s 1 v , any Riemann sheet in the s-domain is mapped to a region with an angle of 2π v in the W-domain. for example. The region of the first Riemann When v is a positive integer, the analytic domain of the multivalued function Pðs α Þ can be regarded as Riemann surface with three Riemann sheets, the v Riemann sheets are mapped into a W-plane. For the equation Pðs α Þ ¼ 0, the position of the roots in the W-domain is misalignment, the relationship between the W-plane and the Riemann surface is shown in Figure 1.

converges in a certain region of the W-domain, then the function
is called the W transform of f ðtÞ, recorded as Compared with the traditional Laplace transform, the W transform only replaces s in the Laplace transform with w v , so the s in the Laplace transform can be replaced by w v to obtain the corresponding W-domain image function. Choosing a suitable real number v can make the fractional powers of s become integer powers of w. At the same time, the W transform can only be used to analyse linear circuits, which is the same as the Laplace transform.

Example 1 Solving the W transform of t a−1
ΓðaÞ .

F I G U R E 1 Relationship between W-plane and Riemann surface LIANG AND JIANG
Substituting t a−1 ΓðaÞ into the definition of W transformation, then Let w v t ¼ u, and applying the Euler's Gamma function's definition (1), we get When a ¼ k v , and k is an integer, the image function of t a−1 ΓðaÞ is 1 w k , and the power of w is an integer.

| Inversion of W transform
Definition 2 If f ðtÞ is the original function of FðwÞ, the FðwÞ has no singularities in region B, and the inversion formula is The region B is the shaded region in Figure 2. In the figure, φ 1 is the curve that the straight line ReðsÞ ¼ σ in the s-domain mapped to the first Riemann sheet in the W-domain, φ 2 is a polyline The proof of Definition 2 is given in Appendix.
When v is a positive integer m, the analytic domain of the function can be regarded as Riemann surface with m Riemann sheets, and these m Riemann sheets exist in a Wplane. The inverse transform can be obtained using the residue method. The solution process is similar to the Laplace transform. When v is not a positive integer, because the W transform is invertible, there is a one-to-one correspondence between the original function and the image function, the inverse transform can be obtained according to the W transform of some known functions. Using the following lemma get a very useful formula for inverse transform. Taking this as an example to obtain the inverse transform of partial function.
Lemma 1 [10] if ReðγÞ > 0, ReðβÞ > 0 and |z | <1, existing By combining different parameters according to the needs of the functions, the following derivations can be derived, see the following various corollaries. (6) can be written as where k is an integer. (6) can be written as where k, l are integers. (6) can be written as where k, l, p are integers.
Decomposing cwþd ðw−aÞ 2 þb 2 into the form of two conjugate complex root terms and using (7), we get According to the two-parameter Mittag-Leffler function's definition (4), we can get Because for any natural number l, ða − bjÞ l þ ða þ bjÞ l is a real number and ða − bjÞ l − ða þ bjÞ l is a pure imaginary number, the inverse transform of cwþd ðw−aÞ 2 þb 2 does not contain imaginary numbers.

| PROPERTIES OF THE W TRANSFORM
Many properties of Laplace transform are also true in the W transform. Similar to the proving process of Laplace transform properties, these properties can be proved using the definition of W transform. In particular, when 0 < α < 1, the W transform of Caputo fractional derivative is It can be known from the expression of the differential property that when the initial condition is not zero, to make the power of wan integer, v in W transform should take a suitable positive integer. When the initial condition is zero, as long as αv is a suitable positive integer, the power of w is an integer. Let v ¼ 1 ffi ffi 2 p , applying W transform to both sides of equation, we get Using (11), we can get yðtÞ as

| PARTIAL FRACTION EXPANSION OF RATIONAL IMAGE FUNCTION IN THE W DOMAIN
The powers discussed in the following sections are rational numbers, that is, rational powers, and therefore they can be expressed as the ratio of two positive integers. For example, the rational powers α; β; γ can be expressed as where n α ; n β ; n γ ;n α ;n β ;n γ are positive integers, m is the least common multiple of all rational powers' denominators, n is the greatest common divisor of all rational powers' molecules, or 1 etc., which depends on the specific situation. Obviously, m and n must be co-prime. From a practical point of view, the assumption of rational powers can be made without loss of generality. The number of poles of an integer-order constant coefficient linear differential equation is equal to the order of the differential equation, which does not depend on other coefficients. For fractional systems, if W transform is used to solve linear system equations, and the powers of w in the Wdomain are integers, then fractional differential equations can be solved using conventional methods in the W-domain.
When the W transform is applied to fractional linear system with rational powers, the definition of the W transform (5) can be rewritten as In the case of rational powers, the W transform of Caputo fractional derivative can be written as Obviously, to avoid the fractional powers of w, we need to take n ¼ 1 in the non-zero initial condition.
Differential equations describing fractional linear systems with rational powers can be written as [27].
where a 1 , …, a p , b 1 ,…,b q are real coefficients, α 1 , …, α p , β 1 , …, β q are rational positive real numbers. These rational powers can be expressed as α k ¼n Considering the initial conditions and applying the inverse W transform to both sides of (8), we can obtain Then the image function of fractional linear system with rational powers is From the expression of the rational image function (9) in the W-domain, it can be seen that if the zero initial condition of the equation or the function at the right end of the equation uðtÞ ≠ aδðtÞis to be considered, then n must be taken one to make the powers in the equation all integers. For network function in the W-domain, Equation (9) is simplified as To ensure that the powers of w in the network function n α 1 ; n α 2 ; :::; n α p or n β 1 ; n β 2 ; :::; n β q are co-prime, the value of n is the maximum common divisor of numbers which are the molecules of rational differential powers α 1 ; α 2 ; :::; α p , β 1 ; β 2 ; :::; β q . The m n is not necessarily a positive integer at this time.
Remarkably, from the expressions of (9) and (10), it can be seen that the multiple fractional powers will cause the powers of image function in the W-domain discontinuity, which is different from traditional integer-order linear systems.
For the image function Y ðwÞ, if it is an improper fraction, then Y ðwÞ can be decomposed into the sum of a rational polynomial and a rational proper fraction using polynomial division, that is where, p is a natural number, g 0 ; g 1 ; :::; g p is a real number, GðwÞ ¼ g 0 þ g 1 w þ ::: þ g p w p is a rational polynomial, and NðwÞ DðwÞ is a rational proper fraction. There are two fundamental types that can be derived from the expression of rational polynomial GðwÞ: 1, w k .
The roots of HðwÞ ¼ 0 may be single roots, multiple roots or conjugate complex roots. Therefore, using the traditional integer-order method to expand the rational proper fraction NðwÞ DðwÞ into partial fractions, we can get The conjugate root dwþe ðw−bÞ 2 þc 2 is a combination of two conjugate complex root terms.
There are two fundamental types that can be obtained from the expression of rational proper fraction NðwÞ DðwÞ : 1 w k , 1 ðwþaÞ k (a is a complex number). When the power of w is high and the expansion theorem of partial fraction is used to expand the image function directly, the expansion formula will have a large number of terms, which makes the time-domain form complex. Therefore, to simplify the time-domain expression, we need to fully consider the characteristic that the powers of w are discontinuous and combine the partial fraction.
According to w ¼ s n m , a point in the s-domain corresponds to m points in the W-domain, these points is misalignment, the distances from these points to the origin in the W-domain are the same, and the phase angle difference between adjacent points is the same. Through this feature, it can be found that if HðwÞ ¼ 0 has l roots, the distances between these roots and the origin are same, and the angle between the two adjacent roots is 2π l , then these roots can be combined into a fundamental type of w k w l þb (b is a real number). It corresponds to the conjugate complex root terms in (11) (when l is an odd number, this has a real root term).
If NðwÞ DðwÞ is directly decomposed when there are r poles at the origin in the W-domain, there will appear r terms of a k w k . To simplify expressions in the time-domain, a new decomposition method can be used for NðwÞ DðwÞ , and Equation (11) can be written as Combining the terms in (12), Equation (12) can be rewritten as So we can get a new fundamental type of 1 w l ðw k þaÞ p (a is a complex number).
In summary, the partial fraction expansion of the rational image function in the W-domain can be obtained from the combination of six fundamental types: 1, w k , 1 w k , 1 ðwþaÞ k , w k w l þb and 1 w l ðw k þaÞ p . Where k; l; p are positive integers, a is a complex number, and b is a real number.
From the linearity property of the W transform, it can be known that the expression of original function yðtÞ can be obtained by solving the inverse W transform of these six types.
The original functions of the above six common image functions are shown in Table 1.
W transform only replaces the complex variable s in the Laplace transform with w m n . Therefore, the inverse W transform method of integer-order can be used when the powers of w in the decomposed term is an integer multiple of m n . For example, the original function of b w 2m n þb 2 is sin bt.

| W transform method for linear constant coefficient fractional differential equation
Linear time-invariant fractional systems can be described by. linear constant coefficient fractional differential equations, the expression of which is: where a 1 ,…, a p , b 1 ,…,b q are real coefficients, α 1 ,…, α p , β 1 ,…, β q are rational positive real numbers.
By applying the W transform to both sides of (13), the rational image function is expanded to partial fraction and combining them appropriately by the method in the Section 5, we have Then applying the inverse W transform to FðwÞ term by term, we can obtain the time-domain solution.
Example 5 Considering initial value problem in the case of the inhomogeneous Bagley-Torvik equation [29].
According to the order in the equation, we know that m ¼ 2. Considering the initial value, we should take n ¼ 1. So m n ¼ 2. According to the linearity property and differential property, applying W transform to the both sides of the equation, we obtain.
Using the results in Table 1, we can obtain yðtÞ as The calculated result is consistent with the result in reference [29].

| W transform method for fractional state equation
Linear time-invariant fractional systems can be described using the following state equations.
Simplifying the above formula and using the results in Table 1, we can obtain yðtÞ as This calculation result is consistent with the result in [30], so the method of solving the state equation using the W transform here is correct.

| The form of Kirchhoff's law in W-domain
The expression of Kirchhoff's current law in the time-domain is LIANG AND JIANG -217 ∑i k ðtÞ ¼ 0 By applying W transform to both sides of (16) and using the linearity property of W transform, we can get Similarly, the expression of Kirchhoff's voltage law in the W-domain is ∑U k ðwÞ ¼ 0

| Resistance
The resistance is shown in Figure 3(a). In the associated reference direction, its Volt-ampere relationship is Taking W transform on both sides, the Volt-ampere relationship of the resistance in the W-domain is The W-domain model of the resistance is shown in Figure 3(b).

| Fractional capacitor
The fractional capacitor is shown in Figure 4(a). In the associated reference direction, its Volt-ampere relationship is where α is the fractional order, α ∈ ð0; 1Þ, and C α with unit F=s 1−α denotes the pseudo-capacitance of the capacitor represented in the form of ðα; C α Þ. When the initial condition u C ð0 − Þ of the capacitor is zero, n is the greatest common divisor of all rational powers' molecules. Applying the W transform to both sides of (17), we have where w n α C α is the admittance of the fractional capacitor in the W-domain.
When the initial condition u C ð0 − Þ of the capacitor is not zero, then n ¼ 1, the form of the volt-ampere relationship of the fractional capacitor in the W-domain is where w n α −m C α u C ð0 − Þ is the current of additional current source, which reflects the initial state of the capacitor voltage.
The W-domain model of fractional capacitor is shown in Figure 4(b).
Another volt-ampere relationship of fractional capacitor can be obtained from (18).
Where 1 w nα C α is the impedance of the fractional capacitor in the W-domain, u C ð0 − Þ w m is the voltage of additional voltage source. The W-domain model is shown in Figure 4(c).

| Fractional inductor
The fractional inductor is shown in Figure 5(a). In the associated reference direction, its Volt-ampere relationship is where β is the fractional order, β ∈ ð0; 1�, and L β with unit H=s 1−α denotes the pseudo-inductor of the inductor represented in the form of ðβ; C β Þ. When the initial condition u L ð0 − Þ of the inductor is zero, n is the greatest common divisor of all rational powers' molecules. Applying the W transform on both sides of (19), we have where w n β L β is the impedance of the fractional inductor in the W-domain.
When the initial condition i L ð0 − Þ of the inductor is not zero, then n ¼ 1, the form of the volt-ampere relationship of the fractional inductor in the W-domain is where w n β −m L β i L ð0 − Þ is the voltage of additional voltage source, which reflects the initial state of the inductor current.
The W-domain model of fractional inductor is shown in Figure 5(b).
Another volt-ampere relationship of fractional inductor can be obtained from (20).
Where 1 w n β L β is the admittance of the fractional inductor in the W-domain, i L ð0 − Þ w m is the current of additional current source. The W-domain model is shown in Figure 5(c).

| CONCLUSION
The W transform is proposed and applied to fractional linear systems with rational powers. First, we give the definition of W transform and its inverse transform. The W transform uses e ∓w v t to replace the kernel function e ∓st of the traditional Laplace transform, which can make the fractional calculus with rational powers become integer powers of w in the W domain. Based on these definitions, we derive the properties of the W transform and this transformation can effectively solve fractional differential equations with rational or irrational powers. To simplify the time-domain solutions of differential equations, a partial fraction expansion method in the W domain is presented. On this basis, we solve linear constant coefficient fractional differential equations and fractional state equations.  Finally, the W transform is applied to the analysis of fractional circuits with rational powers, the forms of the fractional-order elements in the W-domain are derived, and the general steps of the W-domain analysis method are given. The W transform proposed here provides a new direction for the analysis of fractional circuits. This paper mainly gives the idea of W transform and applied it to the solution of fractional linear system with rational powers, so further research should be conducted on study the application of W transform in the analysis of fractional linear circuits containing the passive criteria of fractional circuits with rational order elements.