# TheZ-Transform Method and Delta Type Fractional Difference Operators

Authors: Dorota Mozyrska; Małgorzata Wyrwas
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

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


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## Body

## 1. 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, 6–14] 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.Basic properties of fractional sums and operators were developed firstly in [15] and continued by Atici and Eloe [16, 17], Baleanu and Abdeljawad [18, 19]. Another concept of the fractional sum/difference was introduced in [20–22].Here we attempt to review methods of solutions for fractional difference systems. By comparing both types of solutions we produce the formula for theZ-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 q-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 (q,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 R-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 discrete-time 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 Z-transform 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 viaZ-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 Section2, 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.Table 1
BasicZ-transform formulas used in the text.
y
:
=
y
n
Z
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y
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(
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φ
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α
(
n
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## 2. Preliminaries

Now, we recall the necessary definitions and technical propositions that are used in the sequel therein the paper. Leth>0, a∈R and define hNa:={a,a+h,a+2h,…} and σ(t):=t+h for any t∈(hN)a. Hence t=a+nh for n∈N0. For a function x:(hN)a→R theforwardh-difference operator is defined as (Δhx)(t):=x(σ(t))-x(t)/h, where t∈(hN)a, see [21, 22]. For an arbitrary α∈R and t∈R such that t/h∈R∖{-1,-2,…}, the h-factorial function is defined by th(α):=hαΓ(t/h+1)/Γ(t/h+1-α), where Γ is the Euler gamma function, see [21, 22]. We use the convention that division at a pole yields zero. For h=1 we write shortly t(α) instead of t1(α) and Δ instead of Δ1.Definition 1 (see [22]).
For a functionx:(hN)a→R thefractionalh-sum of orderα>0 is given by Δh-αax(t):=1/Γ(α)∑s=0n‍(t-σ(a+sh))h(α-1)x(a+sh)h, where t=a+(α+n)h, n∈N0 and additionally, Δh0ax(t):=x(t).Fora=0 we write shortly Δh-α instead of 0Δh-α. Note that aΔh-αx:(hN)a+αh→R. From Definition 1 it follows that for t=a+(α+n)h(1)Δh-αaxt=hα∑s=0n‍Γα+n-sΓαΓn-s+1xa+shΔh-αaxt=hα∑s=0n‍n-s+α-1n-sx¯s,
where x¯(s):=x(a+sh). Observe that formula (1) has the form of the convolution of sequences. Let us call for the moment the family of binomial functions defined on Z, parameterized by μ∈R, and given by values: φ~μ(n)=n+μ-1n for n∈N0 and φ~μ(n)=0 for n<0. Then using
(2)φ~αn:=n+α-1n=-1n-αn,
we can write formula (1) in the form of the convolution of φ~α and x¯. In fact
(3)Δh-αaxt=hαφ~α*x¯n,
where “*” denotes a convolution operator; that is, φ~α*x¯(n):=∑s=0nn-s+α-1n-sx¯(s).We recall here that theZ-transform of a sequence {y(n)}n∈N0 is a complex function given by Y(z):=Z[y](z)=∑k=0∞‍y(k)z-k, where z∈C is a complex number for which this series converges absolutely.Note thatZφ~α(z)=∑k=0∞‍k+α-1kz-k=∑k=0∞‍(-1)k-αkz-k=z/z-1α. In the stability analysis it is important to prove to which classes of sequences φ~α belongs. Let ynn∈N0 be a sequence and let y1:=∑s=0∞‍|y(s)| and y∞:=sup⁡n∈N0|y(n)| be the norms defined in the space of sequences. Moreover, let lp(N0), p∈{1,∞}, be the spaces of all sequences satisfying yp<∞.Lemma 2.
Forα<0 and such that α∉Z- the sequence φ~α belongs to l1(N0); that is, ∑s=0∞|φ~α(s)|=φ~α1<∞.Proof.
Letβ:=-α, β>0, and [β]=k be the integer part of β; that is, k-1<β≤k and k∈N0. Then for s∈N0 such that 0≤s≤k we have βs>0 and for s>k we get -1sβs>0. Therefore
(4)φ~α1=∑s=0∞‍φ~αs=∑s=0k‍βs+∑s=k+1∞‍-1sβs=∑s=0k‍βs+∑s=0∞‍-1sβs-∑s=0k‍-1sβs=2∑s=0[k/2]‍β2s+1<∞.
Hence φ~α∈l1(N0).Proposition 3.
Forα∈(0,1) one has φ~α∈l∞(N0).Proof.
Note thatφ~α(0)=1 and
(5)φ~αn=-α-α-1·…·-α-n-1n!φ~αn≤1·2·…·nn!=1,
for n≥1. Hence φ~α∞=1 and consequently, φ~α∈l∞(N0); that is, φ~α∞=sup⁡n∈N0|φ~α(n)|<∞.Proposition 4.
Fort=a+αh+nh∈(hZ)a+αh let one denote y(n):=Δh-αax(t) and x¯(n):=x(a+nh). Then
(6)Zyz=hzz-1αXz,
where X(z):=Zx¯(z).Proof.
In fact as we have from (3) that y(n)=Δh-αax(t)=hαφ~α*x¯(n), then Zy(z)=hαZφ~α(z)X(z). Moreover, Zφ~α(z)=z/z-1α, then we easily see equality (6).Forh=1,   (6) can be shortly written as ZΔ-αax(z)=z/z-1αX(z), where Δ-αax(a+α+n)=:y(n) is treated as a sequence.Two fractionalh-sums can be composed as follows.Proposition 5 (see [32]).
Letx be a real-valued function defined on hNa, where a,h>0. For α,β>0 the following equalities hold: Δh-αa+βhΔh-βax(t)=Δh-α+βax(t)=Δh-βa+αhΔh-αax(t), where t∈hNa+α+βh.In [22] the authors prove the following lemma that gives transition between fractional summation operators for any h>0 and h=1.Lemma 6.
Letx:(hN)a→R and α>0. Then, Δh-αax(t)=hαΔ1-αa/hx~t/h, where t∈hNa+αh and x~(s)=x(sh).In our consideration the crucial role is played by the power rule formula, see [21]: Δh-αaψ(t)=Γ(μ+1)/Γ(μ+α+1)t-a+μhh(μ+α), where ψ(r)=(r-a+μh)h(μ), r∈(hN)a, t∈(hN)a+αh. Then for ψ≡1 we have μ=0. Nextly, taking t=nh+a+αh, n∈N0, we get Δh-αa1(t)=1/Γ(α+1)(t-a)h(α)=Γ(n+α+1)/Γ(α+1)Γ(n+1)hα=n+αnhα. Particularly, we have Z[aΔh-α1](z)=hα/1-1/zα+1=hαz/z-1α+1.Now, we adapt the above formulas to multivariable fractional-order linear case and define Mittag-Leffler matrix functions. In [19] the discrete Mittag-Leffler function is introduced in the following way.Definition 7.
Letα,β,z∈C with Re(α)>0. Thediscrete Mittag-Leffler two-parameter function is defined as
(7)Eα,βλ,z:=∑k=0∞‍λkz+k-1α-1(kα)z+kα-1(β-1)Γαk+β.We introduce thetwo-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 Z-transform. Let us define the following function:
(8)Eα,βλ,n:=∑k=0∞‍λkφ~kα+βn-k.
In fact, in the paper we use two of them, namely, E(α,α)(λ,n)=∑k=0∞λkφ~kα+α(n-k) and E(α)(λ,n):=E(α,1)(λ,n)=∑k=0∞λkφ~kα+1(n-k). For α=1 we have that E(1,1)(λ,n)=E(1)(λ,n)=(1+λ)n.In our consideration an important tool is the image ofE(α,β)(λ,·) with respect to the Z-transform.Proposition 8.
LetE(α,β)(λ,·) be defined by (8). Then
(9)ZEα,βλ,·z=zz-1β1-λzzz-1α-1,
where |z|>1 and z-1αz1-α>|λ|.Proof.
By basic calculations we have(10)ZEα,βλ,·z=∑k=0∞‍∑n=0∞‍λkn-k+kα+β-1n-kz-nZEα,βλ,·z=∑k=0∞‍λkz-k∑s=0∞‍s+kα+β-1sz-sZEα,βλ,·z=∑k=0∞‍λzk∑s=0∞‍-1s-kα-βsz-sZEα,βλ,·z=zz-1β∑k=0∞‍λzkzz-1kαZEα,βλ,·z=zz-1β1-λzzz-1α-1,
where the summation exists for |z|>1 and |z-1|α|z|1-α>|λ|.Proposition 9.
Letα∈(0,1] and β<α+1. Let R be the set of all roots of the following equation:
(11)z-1α=λzα-1.
If all elements from R are strictly inside the unit circle, then limn→∞⁡E(α,β)(λ,n)=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 Proposition9 we get that for some order α there is a “good” λ such that all elements from R are inside the unit circle.Corollary 10.
Letλ∈R. All elements from R (Proposition 9) are inside the unit circle if and only if -2α<λ<0.


## 3. Caputo-Type Operator

In this section we recall the definition of the Caputo-type operator and consider initial value problems of fractional-order 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.Nextly, we define the family of functionsφk,α*:Z→R, parameterized by k∈N0 and by α∈(0,1] with the following values:
(12)φk,α*n:=n-k+kαn-k,forn∈Nk,0,forn<k.Proposition 11.
Letφk,α* be the function defined by (12). Then
(13)Zφk,α*z=1zkzz-1kα+1,
for z such that |z|>1.Proof.
This needs the following simple calculations:(14)Zφk,α*z=∑n=0∞‍φk,α*nz-n=∑n=k∞‍n-k+kαn-kz-n=z-k∑s=0∞‍s+kαsz-s=z-k∑s=0∞‍-1s-kα-1sz-s=z-k1-1z-kα-1=1zkzz-1kα+1,
where |z-1|<1 to ensure the existence of the summation.Note that, for anyα∈(0,1], k,n∈N0, the following relations hold:(1)
φ0,α*(n)=1, φk,α*(k)=1, and φ1,α*(1)=1;(2)
φk,α*(n)=Γ(n-k+kα+1)/Γ(kα+1)Γ(n-k+1) and as the division by pole gives zero, the formula works also for n<k;(3)
φk,α*(n)=1/Γ(kα+1)n-k+kα(kα)=n-k+kαkα;(4)
Z[φ1,α*](z)=1/zz/z-1α+1=1/z-1z/z-1α;(5)
Z[φk,α*](z)=1/zk·z/z-1kα+1=1/(z-1)k·z/z-1k(α-1)+1 for k∈N0;(6)
Forα=1 we have φk,1*(n)=nk and Δ-1n-1k=nk+1. According to item (6) in the properties of functions φk,α* for α=1, we put some comment. Note that ZΔ-1nk(z)=z/z-1·Znk(z)=z/z-1·1/zk·z/z-1k+1=z2/(z-1)k+2 and Znk+1(z)=1/zk+1·z/z-1k+2=z/(z-1)k+2. Therefore, ZΔ-1nk(z)≠Znk+1(z), but taking ZΔ-1n-1k(z) = z/z-1·Zn-1k(z) = z/z-1·1/zk+1·z/z-1k+1 = z/(z-1)k+2 so ZΔ-1n-1k(z) = Znk+1(z).Taking particular value ofz=n+α-1∈Zν and considering the family of functions φk,α* the particular case of the Mittag-Lefler function from Definition 7 can be rewritten as E(α,1)A,n+ν = ∑k=0∞‍Akφk,α*(n) = E(α)(A,n), where ν=α-1. Note that in fact E(α)(A,n) = ∑k=0n‍Akφk,α*(n), so the right hand side is finite and consequently; values E(α)(A,n) always exist. For α=1 there is the delta exponential function: E(1)(A,n)=(I+A)n.The properties of two-indexed functionsφk,s* were given in [34, Proposition 2.5] and generalized for n-indexed functions φk,α* in [34] and in [35] with multiorder (α). In the paper therein, we use functions φk,α* with step h=1 and one order. The next proposition is the particular case for the multi-indexed functions.Proposition 12 (see [35]).
Letα∈(0,1] and ν=α-1. Then for n∈N0 one has
(15)Δ-αφk,α*n+ν=φk+1,α*n.The next step is to present Caputo-typeh-difference operator.Definition 13 (see [32]).
Letα∈(0,1]. TheCaputo-typeh-difference operatorΔh,*αa of order α for a function x:(hN)a→R is defined by
(16)Δh,*αaxt:=Δh-1-αaΔhxt,
where t∈(hN)a+(1-α)h.Note thatΔh,*αax:(hN)a+(1-α)h→R. Moreover, for α=1, the Caputo-type h-difference operator takes the form: Δh,*1ax(t)=Δh0a(Δhx)(t)=(Δhx)(t), where t∈(hN)a. Observe that using function φ~1-α we can write the Caputo-type difference in the following way:
(17)Δh,*αaxt=h-αφ~1-α*Δx¯n,
where t=a+(1-α)h+nh and x¯(n)=x(a+nh).For the Caputo-type fractional difference operator there exists the inverse operator that is the tool in recurrence and direct solving fractional difference equations.Proposition 14 (see [32]).
Letα∈(0,1], h>0, a=(α-1)h, and x be a real-valued function defined on hNa. The following formula holds Δh-αΔh,*αaxt=xt-xa, t∈(hN)αh.Similarly as in Lemma6 there exists the transition formula for the Caputo-type operator between the cases for any h>0 and h=1.Lemma 15 (see [21]).
Letx:(hN)a→R and α>0. Then, Δh,*αax(t)=h-α(a/hΔ1,*αx~)(t/h), where t∈(hN)a+αh and x~(s)=x(sh).From this moment for the caseh=1 we write Δ*αa/h∶=Δ1,*αa/h and similarly, number 1 is omitted for two other operators and fractional summations. Then the result from Proposition 14 can be stated as
(18)Δ-αΔ*αa/hx~n+α=x~n+α-xa,
where n∈N0, x~(s)=x(sh) and a=(α-1)h.Proposition 16.
Fora∈R, α∈(0,1] let one define y(n):=Δh,*αax(t), where t∈(hN)a+(1-α)h and t=a+(1-α)h+nh. Then
(19)Zyz=h-αzz-11-αz-1Xz-zxa,
where X(z)=Z[x¯](z) and x¯(n):=x(a+nh).Proof.
The equality follows fromΔh,*αax(t)=h-αφ~1-α*Δx¯(n). Then using the formula for Z[Δx¯](z)=(z-1)X(z)-zx¯(0)=(z-1)X(z)-zx(a) after simple calculations we get (19).Observe that forα=1 we have Zy(z)=1/h(z-1)X(z)-zx¯(0), which agrees with the transform of difference Δh of x¯. Using the Z-transform of the Caputo-type operator for φk,α*, we can calculate that for k∈N1 the following relation holds:
(20)Δ*α0φk,α*n+1-α=φk-1,α*n,
while for k=0 it becomes zero; that is, Δ*α0φ0,α*(n+1-α)=0.Firstly, let us consider systems with Caputo-type operator forh>0, given by the following form:
(21)Δh,*αaxnh=Axnh+a,n∈N0,
where x:(hZ)a→Rp, α∈(0,1], a=(α-1)h, and A is p×p real matrix. Let us consider the following initial condition: x(a)=x0∈Rp for (21). System (21) with x(a)=x0 can be rewritten in the form
(22a)Δ*ανx~n=A~x~n+ν,n∈N0,(22b)x~ν=x0∈Rp,
where ν=α-1, x~(n+ν)=x(nh+a), and A~=hαA. We also use the notation x¯(n):=x~(n+ν)=x(nh+a), what helps to avoid the problem with domains and allows to see the variable vector as a sequence.Proposition 17 (see [4]).
Letα∈(0,1] and a=(α-1)h. The linear initial value problem
(23a)Δh,*αaxt=Axt+a,t∈hN0(23b)xa=x0,x0∈Rp
has the unique solution given by the formula
(24)xt=EαAhα,thx0=EαAhα,nx0xt=∑k=0∞‍Akhkαφk,α*nx0,
where t∈hNa and t/h=n+ν=n+α-1∈Nα-1.The next proposition presents the proof of theZ-transform of the one-parameter Mittag-Leffler matrix function, where the idea of transforming a linear equation is used.Proposition 18.
Letα∈(0,1] and M(n):=E(α)Ahα,n=∑k=0∞Akhkαφk,α*(n). Then the Z-transform of the one-parameter Mittag-Leffler function E(α)Ahα,· is given by
(25)ZMz=zz-1I-1zzz-1αAhα-1.Proof.
Let us consider problem (22a)-(22b) and take the Z-transform on both sides of (22a). Then for h=1 from (19), we get z/z-11-α((z-1)X(z)-zx(a))=AhαX(z), where X(z)=Z[x¯](z) and x¯(n):=x(a+nh). And after calculations we see that X(z)=z/z-1I-1/zz/z-1αAhα-1x0; hence we get x(t)=x¯(n)=Z-1X(z)(n)=Z-1ZM(z)(n)x0, that is, the solution of the considered initial value problem. And then taking initial conditions as vectors from standard basis from Rp we get the fundamental matrix M, whose transform form should agree with the given.Let us consider the Caputo difference initial value problem for the linear system that contains only one equation withA=λh-α, that is, Δh,*αax(t)=λh-αx(t+a), x(a)=1. Then E(α)λ,n = Z-1z/z-11-λ/zz/z-1α-1(n)=Z-1∑k=0∞‍λk/zkz/z-1kα+1(n) 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 I=1 and Ahα=λ into (25).


## 4. 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 functionsφk,α* 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 φk,α:Z→R parameterized by k∈N0 and by α∈(0,1] with the following values:
(26)φk,αn:=n-k+kα+α-1n-k,forn∈Nk,0,forn<k.Proposition 19.
Letφk,α be the function defined by (26). Then
(27)Zφk,αz=1zkzz-1kα+α,
for z such that |z|>1.Proof.
By simple calculations we have(28)Zφk,αz=∑n=0∞‍φk,αnz-nZφk,αz=∑n=k∞‍n-k+k+1α-1n-kz-nZφk,αz=z-k∑s=0∞‍s+k+1α-1sz-sZφk,αz=z-k∑s=0∞‍-1s-k+1αsz-sZφk,αz=z-k1-1z-kα-α=1zkzz-1(k+1)α,
where the summation exists for |z-1|<1.Note that for anyα∈(0,1] and n,k∈N0 the following relations hold:(1)
φ0,α(0)=1, φk,α(k)=1, and φ1,α(1)=1.(2)
φk,α(n)≥0 and φk,α*(n)≥0.(3)
φk,α(n)=n-(k+1)+(k+1)α((k+1)α-1)/Γ((k+1)α).(4)
Z[φ0,α](z)=z/z-1α, Z[φ1,α](z)=1/zz/z-12α.(5)
Z[φk,α](z)=1/zkz/z-1(k+1)α for k∈N0.(6)
Forα=1, we have φk,1(n)=φk,1*(n)=nk.Taking particular value ofz=n+α-1∈Zν and considering the family of functions φk,α the particular case of Mittag-Leffler function from Definition 7 can be written as E(α,α)A,n+ν = ∑k=0∞‍Akφk,α(n) = E(α,α)(A,n), where ν=α-1. Note that in fact E(α,α)(A,n) = ∑k=0n‍Akφk,α(n), so the right hand side is finite and consequently; values E(α,α)(A,n) always exist. For α=1 there is a delta exponential function: E(1,1)(A,n)=(I+A)n.For the family of functionsφk,α we have similar behaviour for fractional summations as for φk,α* so we can state the following proposition.Proposition 20.
Letα∈(0,1] and ν=α-1. Then for n∈N0, one has
(29)Δ-αφk,αn+ν=φk+1,αn.Proof.
In the proof we use the fact thatZΔ-αφk,α(n+ν)(z) = 1/zz/z-1α·Z[φk,α](z). Then using item 5 from properties of functions φk,α, we see that ZΔ-αφk,α(n+ν)(z) = 1/zk+1z/z-1(k+2)α = Z[φk+1,α](z). Hence taking the inverse transform we get the assertion.The next presented operator is called the Riemann-Liouville-type fractionalh-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.
Letα∈(0,1]. The Riemann-Liouville-type fractional h-difference operator Δhαa of order α for a function x:(hN)a→R is defined by
(30)Δhαaxt:=ΔhΔh-1-αaxt,
where t∈(hN)a+(1-α)h.Forα∈(0,1] one can get
(31)Δh,*αaxt=Δhαxat-xa·t-ah-αΓ1-α=Δhαaxt-xahαt-ah-α,
where t∈(hN)a. Moreover for α=1 one has Δh,*1ax(t)=Δh1ax(t)=Δhx(t).The next propositions give useful identities of transforming fractional difference equations into fractional summations.Proposition 22 (see [33]).
Letα∈(0,1], h>0, a=(α-1)h, and x be a real-valued function defined on hNa. The following formula holds Δh-α0aΔhαxt=xt-xa·h1-α/Γαthα-1=xt-xa·t/hα-1, t∈(hN)αh.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.Lemma 23.
Letx:(hN)a→R and α>0. Then Δhαax(t)=h-αΔ1αa/hx~(t/h), where t∈(hN)a+(1-α)h and x~(s)=x(sh).For the caseh=1, we write Δα∶=a/hΔ1αa/h. Then the result from Proposition 22 can be stated as
(32)Δ-αΔαx~a/hn+α=x~n+α-xa·n+αn+1,
where n∈N, x~(s)=x(sh) and with a=(α-1)h.Proposition 24.
Fora∈R, α∈(0,1], let one define y(n):=Δhαax(t), where t∈(hN)a+(1-α)h and t=a+(1-α)h+nh. Then
(33)Zyz=zhzz-1-αXz-zh-αxa,
where X(z)=Z[x¯](z) and x¯(n):=x(a+nh).Proof.
Leth=1. Then the assertion for h>0 comes from Lemma 23. Let f(n)=Δ-(1-α)νx(n). Then Zyz=ZΔf[z]=(z-1)F(z)-zf(0), where F(z)=Zf(z)=z/z-11-αX(z) and f(0)=x(a)=x¯(0). Then we easily see equality (33).Forα=1 we have the same as for the Caputo-type operator Zy(z)=1/h(z-1)X(z)-zx¯(0), which also agrees with the transform of difference Δh of x¯.Using theZ-transform of the Riemann-Liouville-type operator for k∈N1 we can calculate that
(34)Δα0φk,αn+1-α=φk-1,αn,
while for k=0 we get φ0,α(n)=n+α-1n=φ~α(n) and consequently, Δα0φ0,α(n+1-α) = Δα0φ~α(n+1-α)=0.Now, let us consider systems with the Riemann-Liouville-type operator forh>0, given by the following form:
(35)Δhαaxnh=Axnh+a,n∈N0,
where x:(hZ)a→Rp, α∈(0,1], a=(α-1)h, and A is p×p real matrix and with initial condition x(a)=x0∈Rp. System (35) with x(a)=x0 can be rewritten as follows:
(36a)Δανx~n=A~x~n+ν,n∈N0,(36b)x~ν=x0∈Rp,
where ν=α-1 and x~(n+ν)=x(nh+a), A~=hαA. We also use the notation x¯(n):=x~(n+ν)=x(nh+a), what helps to avoid the problem with domains and allows to see the variable vector as a sequence.Proposition 25 (see [4]).
Letα∈(0,1] and a=(α-1)h. The linear initial value problem,
(37a)Δhαaxt=Axt+a,t∈hN0,(37b)xa=x0,x0∈Rp,
has the unique solution given by the formula
(38)xt=Eα,αAhα,thx0=Eα,αAhα,nx0xt=∑k=0∞‍Akhkαφk,αnx0,
where t∈hN0 and t/h=n+ν=n+α-1∈Nα-1.Proposition 26.
Letα∈(0,1] and M(n):=E(α,α)Ahα,n. Then the Z-transform of the Mittag-Leffler function E(α,α)Ahα,· is given by
(39)Z[M](z)=zz-1αI-1zzz-1αAhα-1.Proof.
Let us consider problem (36a)-(36b) and take the Z-transform on both sides of (36a). Then we get from (33) for h=1, zz/z-1-αX(z)=zx(a)+A~X(z), where X(z)=Z[x¯](z) and x¯(n):=x(a+nh)=x~(ν+n). And after calculations we see that X(z) = z/z-1αI-1/zz/z-1αAhα-1x0; hence x(t) = x¯(n) = Z-1[X(z)](n) = Z-1[Z[M](z)](n)·x0 is the solution of the considered initial value problem. And then taking initial conditions as vectors from standard basis from Rp we get the fundamental matrix M, whose transform Z[M] is given by (39).Letx:hNa→R. Then for A=λh-α we have Δhαax(t)=λh-αx(t+a), x(a)=1. Taking into account the image of the scalar Mittag-Leffler function in the Z-transform we get E(α,α)λ,n=Z-1z/z-1α1-1/zz/z-1αλ-1(n), where I=1 and Ahα=λ are substituted into (39). Hence using the power series expansion the scalar Mittag-Leffler function E(α,α)λ,· can be written as E(α,α)λ,n=Z-1∑k=0∞‍λk/zkz/z-1kα+α(n).


## 5. Grünwald-Letnikov-Type Operator

The third type of the operator, which we take under our consideration, is the Grünwald-Letnikov-type fractionalh-difference operator, see, for example, [2, 6–14] for cases h=1 and also for general case h>0.Definition 27.
Letα∈R. TheGrünwald-Letnikov-typeh-difference operatorΔ~hαaof orderαfor a functionx:(hN)a→R is defined by
(40)Δ~hαaxt:=∑s=0(t-a)/h‍asαxt-sh,
where
(41)asα=-1sαs1hαwithαs=1,fors=0,αα-1⋯α-s+1s!,fors∈N.Proposition 28.
Fora∈R, α∈(0,1], let one define y(n):=Δ~hαax(t), where t∈(hN)a and t=a+nh, n∈N0. Then
(42)Zyz=hzz-1-αXz,
where X(z)=Z[x¯](z) and x¯(n):=x(a+nh).Proof.
Note thatZy(z)=ZΔ~hαax¯(z)=Z[asα*x¯][z]=Zas(α)·X(z)=hz/z-1-αX(z).The following comparison has been proven in [4].Proposition 29.
Leta=(α-1)h. Then ∇hΔh-(1-α)ax(nh)=Δ~hα0y(nh), where y(nh):=x(nh+a), for n∈N0, or x(t)=y(t-a) for t∈(hN)a and ∇hx(nh)=x(nh)-x(nh-h)/h.Proposition 30.
Leta=(α-1)h. Then Δ~hα0y(t+h)=Δhαax(t), where x(t)=y(t-a) for t∈(hN)a.Proof.
Letf(n):=Δ~hα0y(nh+h). Then by Proposition 28 we get Z[f](z) = zhz/z-1-αX(z)-zΔ~hα0y(0) = zhz/z-1-αX(z)-zh-αy(0) = zhz/z-1-α · X(z)-zzh-αx(a) = Zg, where X(z) = Z[x¯](z), x¯(n) = y(nh) = x(a+nh) and g(n):=Δhαax(t), t∈(hN)a+(1-α)h. Hence taking the inverse transform we get the assertion.From Proposition30 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 f(n):=Δ~hα0y(nh+h) as for the Riemann-Liouville-type operator. Then many things could be easily done for systems with the Grünwald-Letnikov-type difference operator.The next proposition is proved in [36] for the case h=1.Proposition 31.
The linear initial value problem,(43)Δ~α0yt+h=Ayt,t∈hN0,y0=y0,y0∈Rp,
has the unique solution given by the formula y(t)=Φ(t)y0 for any t∈(hN)0 and n×n dimensional state transition matrices Φ(t) are determined by the recurrence formula Φ(t+h)=Ahα+IαΦ(t) with Φ(0)=I.Taking into account Propositions25 and 30 we can state the following result.Proposition 32.
The linear initial value problem,(44)Δ~hα0yt+h=Ayt,t∈hN0,y0=y0,y0∈Rp,
has the unique solution given by the formula
(45)yt=E(α,α)Ahα,thy0,
where t∈(hN)0.As a simple conclusion we have that(46)Φnh=Z-1zz-1αI-1zzz-1αAhα-1n.


## 6. Comparisons and Examples

Here we present some comparisons for theZ-transforms of Caputo- and Riemann-Liouville-type operators as the third one used in systems with the left shift gives the same solutions as the Riemann-Liouville-type operator. Then we present the example with mixed operators in two equations. In this chapter we also state the formula in the complex domain of the solution to the initial value problem for semilinear systems like for control systems.Proposition 33.
Lety(n):=Δhαa-Δh,*αax(t), where t=a+(1-α)h+nh. Then
(47)Z[y](z)=h-αzzz-11-α-1x(a).Proof.
It is enough to compare Propositions16 and 24. We can also equivalently take under consideration (31) in the real domain and observe that Z[y](z) = h-αZn-α+1-α(z)x(a) = h-αz∑s=0∞(-1)sα-1sz-s-zx(a) = h-αzz/z-11-α-1x(a).Comparing theZ-transform of two types of the Mittag-Leffler functions we get the following relation.Proposition 34.
Letα∈(0,1], n∈N0. Then
(48)Eα,αA,n=EαA,·*φ~α-1n.Proof.
In fact if we take theZ-transform of the right side, then we have
(49)ZEαA,·*φ~α-1z=ZEαA,·zZφ~α-1z=zz-1I-1zzz-1αAhα-1zz-1α-1=zz-1αI-1zzz-1αAhα-1.
Using formula (39) we get ZE(α)(A,·)*φ~α-1(z)=ZE(α,α)(A,·)(z) and consequently, we reach the assertion.In the following example we consider control systems with the Caputo- and Riemann-Liouville-type operators. We get the same transfer function for initial conditionx(a)=0.Example 35.
Letα∈(0,1], h=1, and a=α-1. Let us consider linear initial value problems of the following forms: (a)
for the Caputo-type operator:(50)Δ*αaxn=Axa+n+Bun,yn=Cxa+n+Dun,(b)
for the Riemann-Liouville-type operator:(51)Δαaxn=Axa+n+Bun,yn=Cxa+n+Dunwith initial condition x(a)=0, and constant matrices A∈Rp×p, B∈Rp×m, C∈Rq×p, D∈Rq×m.
We solve the problems using theZ-transform. Since x(a)=0, then applying the Z-transform to the first equations in both systems, we get the following equation: z/z-11-α(z-1)X(z)=AX(z)+BU(z). Then X(z)=1/z-1·z/z-1α-1·I-1/z-1z/z-1α-1A-1BU(z)=1/z·z/z-1α · I-1/zz/z-1αA-1BU(z), where X=Z[x¯], x¯(n)=x(a+n), and U=Z[u]. Consequently, for the output of the considered systems in the complex domain, we have the following formula: Y(Z)=1/zz/z-1αCI-1/zz/z-1αA-1BU(z)+DU(z), where Y=Z[y]. Then the solution for the output in the real domain is as follows: y(n)=Cf*u(n)+Du(n), where f(n)=E(α,α)(A,n-1)B. We can easily check the value at the beginning that y(0)=Du(0), as E(α,α)(-1)=0.


## 7. Towards Stability Analysis

Let us introduce the stability notions for fractional difference systems involving the Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type operators.Firstly, let us recall that the constant vectorxeq is anequilibrium point of fractional difference system (21) if and only if Δ*αaxeq(t)=ft,xeq=0 (and Δαiaxeq(t)=ft,xeq in the case of the Riemann-Liouville difference systems of the form (35)) for all t∈(hN)0.Assume that the trivial solutionx≡0 is an equilibrium point of fractional difference system (21) (or (35)). Note that there is no loss of generality in doing so because any equilibrium point can be shifted to the origin via a change of variables.Definition 36.
The equilibrium pointxeq=0 of (21) (or (35)) is said to be(a)
stable if, for each ϵ>0, there exists δ=δϵ>0 such that x(a)<δ implies x(a+kh)<ϵ, for all k∈N0,(b)
attractive if there exists δ>0 such that x(a)<δ implies
(52)limk→∞xa+kh=0,(iv)
asymptotically stable if it is stable and attractive. The fractional difference systems (21), (35) are calledstable/asymptotically stable if their equilibrium point xeq=0 is stable/asymptotically stable.Example 37.
Let us consider systems (23a)-(23b) or (37a)-(37b) but with A=0. Then for system (23a)-(23b) we have that x(a+nh)=φ0,α*(n)x0=x0, while for system (37a)-(37b) the solution satisfies the following limn→∞⁡x(a+nh)=limn→∞φ0,α(n)x0=0. Hence, then systems with the Caputo-type operator are stable and systems with the Riemann-Liouville operator are attractive hence asymptotically stable.Proposition 38.
LetR be the set of all roots of the equation
(53)det⁡I-1zzz-1αAhα=0.
Then the following items are satisfied. (a)
If all elements fromR are strictly inside the unit circle, then systems (21) and (35) are asymptotically stable.(b)
If there isz∈R such that |z|>1, both systems (21) and (35) are not stable.Proof.
The main tool in the proof is Proposition9 and the method of decomposition of the matrix of systems into Jordan’s blocks. Some basic tool is similar to those used in [30]. However in the proof we do not use the same tools as in the cited paper. Only the beginning could be read as similar. The Z-transform of x¯(·) is X(z)=z/z-1βI-1/zz/z-1αAhα-1x0=P-1z/z-1βI-1/zz/z-1αJhα-1Px0, where β=1 for systems with the Caputo-type operator and β=α for systems with the Riemann-Liouville-type operator. Moreover the condition from Proposition 9 that β<α+1 is satisfied. Let A=PJP-1, where P is invertible and J=diag(J1,…,Js) and Jl are Jordan’s blocks of order rl,l=1,…,s. Let ki∈N1 be algebraic multiplicities of eigenvalues λi∈Spec⁡(A); let pi∈N1 be their geometric multiplicities, i=1,…,m. The matrix w-1(J)=I-1/zz/z-1αJhα-1 is a diagonal block matrix. The number of blocks corresponding to λi is pi and their form is given by the upper triangular matrix
(54)w-1λiw-2λi…w-rqλi0w-1λi…w-rq+1λi⋮⋮⋱⋮00…w-1λi,
where q=1,…,pi and rq is the size of the block. Because det⁡(w(A))=det⁡(w(J))=∏i=1m‍wki(λi) and X(z) is formed by a linear combination of convolutions of φ~β with the rational functions from (54), R is the set of all poles of (z-1)X(z) as α-β+1>0. (a)
If all elements fromR are strictly inside the unit circle, then limn→∞x¯(n)=limz→1⁡(z-1)X(z)=0.(b)
If there is an element fromR that |z|>1, then limsup⁡n→∞|x(a+n)|=∞ and systems (21) and (35) are not stable.Now we can easily derive some basic corollary.Corollary 39.
If there isλ∈Spec⁡(A), such that |λ|>2/hα, then system (21) is not asymptotically stable. Or equivalently, if system (21) is asymptotically stable, then |λ|≤2/hα for all λ∈Spec⁡(A).Example 40.
Let us consider the family of planar systems with orderα∈(0,1] described by the matrix A=-2α-2α2α-2α. Observe that det⁡A=22α+1≠0 so the considered planar systems have the unique equilibrium point xe=0. Since condition (53) does not depend on the type of an operator, we take into account systems (21) with the Caputo-type difference operator. In our case we get for (i) α=0.8, h=0.5 that elements from R are inside the unit circle and the system is stable, trajectories are presented at Figure 1 but for (ii) α=0.5, h=0.5 we get that the system is unstable, see Figure 2. In this case we can have stability for smaller step h. For the given matrix A and α, the stability of the systems depends also on the length of the step. It is easy to see that λ1=-2α-2αi, λ2=-2α+2αi∈Spec⁡(A) and |λj|=2α+0.5, j=1,2. Then for h=0.5 we have 2/hα=22α and consequently, if h=0.5 and α<0.5, then by Corollary 39 system (21) is not asymptotically stable. Additionally, one can see that for α=0.5 and h=0.5 we get |λj|=2/hα=2 and the equilibrium point is not stable. Therefore, the condition |λ|≤2/hα given for eigenvalues of the matrix describing the system is not sufficient for asymptotic stability of the system.The solution of the Caputo-type fractional-order difference initial value problem for linear system (21) with the initial condition x(0)=0.2, y(0)=0.2, where the matrix A is given in Example 40 and α=0.8, h=0.5. In this case the roots of (53) are strictly inside the unit circle.
(a)
The graph ofx
(b)
The graph ofy
(c)
Phase trajectoryFigure 2
The solution of the Caputo-type fractional-order difference initial value problem for linear system (21) with the initial condition x(0)=0.2, y(0)=0.2, where the matrix A is given in Example 40 and α=0.5, h=0.5. In this case the roots of (53) are outside the unit circle.


## 8. Conclusions

The Caputo-, Riemann-Liouville-, and Grünwald-Letnikov-type fractional-order difference initial value problems for linear systems are discussed. We stress the formula for theZ-transform of the discrete Mittag-Leffler matrix function. Additionally, the stability problem of the considered systems is studied. The next natural step is to consider the semilinear control systems and develop the controllability property via the complex domain.



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