The Z-Transform Method and Delta Type Fractional Difference Operators

TheCaputo-, Riemann-Liouville-, andGrünwald-Letnikov-type difference initial value problems for linear fractional-order systems are discussed. We take under our consideration the possible solutions via the classicalZ-transformmethod. We stress the formula for the image of the discrete Mittag-Leffler matrix function in theZ-transform.We also prove forms of images in theZ-transform of the expressed fractional difference summation and operators. Additionally, the stability problem of the considered systems is studied.


Introduction
Fractional operators are generalizations of the corresponding operators of integer order, as well in continuous as in discrete case.The particular interpretation of the fractional derivative was presented, for example, in [1] or [2].Fractional differences used in models of control systems could be understand as (1) an approximation of continuous operators (see [3]) and (2) a possibility of involving some memory to difference systems, that is, systems in which the current state depends on the full history of systems' states.Systems with fractional derivatives and differences are widely discussed in many papers.However, different authors could use various definitions of operators.Some comparisons between three basic types of fractional difference operators were studied in [4] and in [5] for multistep case.There are many papers, in which the authors usually involve in discrete case the Grünwald-Letnikov-type fractional operator, see, for example, [2,[6][7][8][9][10][11][12][13][14] for case ℎ = 1 and also for general case ℎ > 0. However, the exact formulas for solutions are not used to considered linear initial value problems.The authors mainly use the recurrence method for stating their results.Using the particular type of definitions of fractional difference operators it was possible to state the exact formulas for solutions to initial value problems with the Riemann-Liouville-and the Caputo-type fractional difference operators.The clue point is that solutions to initial value problems with the Grünwald-Letnikov-type operator have the same values as those for systems with the Riemann-Liouville-type operator.
Here we attempt to review methods of solutions for fractional difference systems.By comparing both types of solutions we produce the formula for the Z-transform of particular type of the discrete Mittag-Leffler matrix function.There are few papers where various kinds of discrete Mittag-Leffler functions are discussed.The important paper is [23], where discrete type and -discrete analogues of Mittag-Leffler function are presented.Their relations to fractional differences of the particular case are investigated.The author considers also applications of these functions to numerical analysis and integrable systems.However, the formulas of the considered functions mainly depend on the class of the type of calculus.For example in [19] the authors consider discrete Mittag-Leffler functions connected with nabla Caputo fractional linear difference systems and use there the nabla discrete Laplace transform.In [24] the authors consider a linear nabla (, ℎ)difference equations of noninteger order.In [25] the sequential discrete fractional equations with some particular convention of the fractional difference are solved using the kind of fractional nabla discrete Mittag-Leffler functions.
There is the possible use of the generalized Laplace transformation on time scales, for that see, for example, [26].Such method was used in [16] and [27] under the method called the -transform and the Laplace transform, respectively, for the situation where on the right side of the system there are functions that depend only on time.Additionally, in the dissertation [28] the possible use of the Laplace transform for semilinear systems is mentioned.More classical solution of the problem is also presented in [29], where the authors use the Z-transform to give the conditions for stability of systems with the Caputo-type operator.The conclusions were derived based on the image in the complex domain.The main advantage of the use of the Z-transform is to introduce the natural language for discrete systems; it means to work with sequences instead of discrete functions defined on various domains.
The stability property is the main issue in dynamical systems.As the most of descriptions of behaviour of processes are digital due to computing tools used at computers, it is important to know some conditions for stability of discretetime systems.The results are similar for different operators.In [30,31] the study of the stability problem for discrete fractional systems with the nabla Riemann-Liouville-type difference operator is presented.The authors stress that the Ztransform can be used as a very effective method for stability analysis of systems.In engineering problems more often systems are used with the Grünwald-Letnikov-type fractional difference.One of the attempts to the stability of fractional difference systems is the notion of practical stability, see [6,7].Since the linear systems are defined by some matrices, the conditions connected with eigenvalues of these matrices are presented, for instance, in [11,13,14].
Our paper is rather motivated by the idea of presenting images via Z-transform of fractional difference operators and consequences into stability property.We receive results for stability of linear systems with three operators and with steps ℎ > 0. In the case of the Grünwald-Letnikov-type fractional operator our results are similar to those in [11,13,14].However, steps in the proof of stability condition are based on [31].
The paper is organized as follows.In Section 2, we present the preliminary material needed for further reading.Solutions to the considered linear initial value problems with the Caputo-type fractional difference and the image of the unknown vector function are constructed in Section 3. As the simple consequence of considering two types of solution: one from the classical approach and the second from the Z-transform we can write the images of the one-parameter Mittag-Leffler functions.Sections 4 and 5 deal with images of the unknown vector functions for linear initial value problems with the Riemann-Liouville-and the Grünwald-Letnikov-type fractional differences, respectively.Some comparisons for the Z-transforms of Caputo-type and Riemann-Liouville-type operators are presented in Section 6.In Section 7 we study the stability of the considered systems.Section 8 provides the brief conclusions.Additionally, Table 1 presents basic Z-transform formulas used in the text.
Proof.In fact as we have from (3) that )  , then we easily see equality (6).
For ℎ = 1, (6) can be shortly written as Two fractional ℎ-sums can be composed as follows.
We introduce the two-parameter Mittag-Leffler function defined in different manner and show that both definitions give the same values.Moreover, we use the function for the matrix case and prove by two ways its image in the Ztransform.Let us define the following function: In fact, in the paper we use two of them, namely, In our consideration an important tool is the image of  (,) (, ⋅) with respect to the Z-transform.Proposition 8. Let  (,) (, ⋅) be defined by (8).Then where || > 1 and | − 1|  || 1− > ||.
Proposition 9. Let  ∈ (0, 1] and  <  + 1.Let  be the set of all roots of the following equation: If all elements from  are strictly inside the unit circle, then lim  → ∞  (,) (, ) = 0.
Proof.If all roots of (11) are strictly inside the unit circle, then using theorem of final value for the Z-transform, we easily get the assertion.
By Proposition 9 we get that for some order  there is a "good"  such that all elements from  are inside the unit circle.
Corollary 10.Let  ∈ R. All elements from  (Proposition 9) are inside the unit circle if and only if −2  <  < 0.

Caputo-Type Operator
In this section we recall the definition of the Caputo-type operator and consider initial value problems of fractionalorder systems with this operator.We discuss the problem of solvability of fractional-order systems defined by difference equations with the Caputo-type operator.In [33] versions of solutions of scalar fractional-order difference equation are given.
Proof.This needs the following simple calculations: where | −1 | < 1 to ensure the existence of the summation.
Similarly as in Lemma 6 there exists the transition formula for the Caputo-type operator between the cases for any ℎ > 0 and ℎ = 1.
The next proposition presents the proof of the Ztransform of the one-parameter Mittag-Leffler matrix function, where the idea of transforming a linear equation is used.Let us consider the Caputo difference initial value problem for the linear system that contains only one equation with  = ℎ − , that is, is the scalar one-parameter Mittag-Leffler function.Observe that the Z-transform of this scalar Mittag-Leffler function can be obtained from Proposition 18 by substituting  = 1 and ℎ  =  into (25).

Riemann-Liouville-Type Operator
In this section we recall the definition of the Riemann-Liouville-type operator and consider initial value problems of fractional-order systems with this operator.Similarly as in the case the Caputo-type operator we discuss the problem of solvability of fractional-order systems of difference equations.Family of functions  * , is useful for solving systems with Caputo-type operator.We also formulate the similar family of functions that are used in solutions of systems with Riemann-Liouville-type operator.Let us define the family of functions  , : Z → R parameterized by  ∈ N 0 and by  ∈ (0, 1] with the following values: Proposition 19.Let  , be the function defined by (26).Then for  such that || > 1.
Proof.By simple calculations we have where the summation exists for | −1 | < 1.
Taking particular value of  =  +  − 1 ∈ Z ] and considering the family of functions  , the particular case of Mittag-Leffler function from Definition 7 can be written as , where ] = − 1.Note that in fact  (,) (, ) = ∑  =0    , (), so the right hand side is finite and consequently; values  (,) (, ) always exist.For  = 1 there is a delta exponential function: For the family of functions  , we have similar behaviour for fractional summations as for  * , so we can state the following proposition.The next presented operator is called the Riemann-Liouville-type fractional ℎ-difference operator.The definition of the operator can be found, for example, in [16] (for ℎ = 1) or in [21,22] (for any ℎ > 0).Definition 21.Let  ∈ (0, 1].The Riemann-Liouville-type fractional ℎ-difference operator  Δ  ℎ of order  for a function  : (ℎN)  → R is defined by where  ∈ (ℎN) +(1−)ℎ .
For  ∈ (0, 1] one can get where  ∈ (ℎN)  .Moreover for  = 1 one has The next propositions give useful identities of transforming fractional difference equations into fractional summations.
In [21] the authors show that similarly as in Lemmas 6 and 15 there exists the transition formula for the Riemann-Liouville-type operator between the cases for any ℎ > 0 and ℎ = 1.For the case ℎ = 1, we write /ℎ Δ  := /ℎ Δ  1 .Then the result from Proposition 22 can be stated as where  ∈ N, x() = (ℎ) and with  = ( − 1)ℎ.Then we easily see equality (33).
Then the Z-transform of the Mittag-Leffler function  (,) (ℎ  , ⋅) is given by Proof.Let us consider problem (36a)-(36b) and take the Ztransform on both sides of (36a).Then we get from ( 33

Grünwald-Letnikov-Type Operator
The third type of the operator, which we take under our consideration, is the Grünwald-Letnikov-type fractional ℎdifference operator, see, for example, [2,[6][7][8][9][10][11][12][13][14] for cases ℎ = 1 and also for general case ℎ > 0. From Proposition 30 we have the possibility of stating exact formulas for solutions of initial value problems with the Grünwald-Letnikov-type difference operator by the comparison with parallel problems with the Riemann-Liouville-type operator.And we can use the same formula for the Z-transform of () := ( 0 Δ ℎ )(ℎ + ℎ) as for the Riemann-Liouville-type operator.Then many things could be easily done for systems with the Grünwald-Letnikov-type difference operator.

Table 1 :
Basic Z-transform formulas used in the text.