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1 Introduction

Fractional calculus, born at the end of seventeenth century, has provided many hot topicsof research in many fields of science and engineering [4, 19]. The reason of the successof the fractional calculus is its nonlocality. One of the possibility is the considerationof discrete time systems, which have the advantages of simple calculation and smallsingularity [24, 25]. For the details of the most recent advances, one can refer to someexcellent monographs [5, 7,13] and the references cited therein as well.

Despite the wide use of Grünwald–Letnikov fractional difference, it has a major drawback(initial condition independence), which is the restriction on the dynamic behavior oreven the stability. For example, by using the definition given in [20], the state response ofGar is meaningless, which brings many difficulties for analysisand synthesis. To the best of the authors’ knowledge, no results on this property havebeen reported, while it is indeed an essential property to be explored before Grünwald–Letnikov fractional difference can be used as a viable tool. Bearing this in mind, this paperplans to investigate the initial condition independence problem. It should be pointed outthat this work will deepen our understanding on Grünwald–Letnikov fractional differenceand enlarge its applicability.

The outline of the rest paper is organized as follows. Section 2 introduces somebasic knowledge of nabla discrete fractional calculus. Section 3 reveals the initial conditionsindependent property of Grünwald–Letnikov fractional difference and providesthree schemes to empower Grünwald–Letnikov fractional difference equations with availableinitial condition. Section 4 discusses several relevant definitions briefly and presentssome detailed numerical examples to confirm the correctness and effectiveness of theobtained results. Finally, the paper ends with a conclusion in Section 5.

2 Preliminaries

This section recalls some basic definitions and properties on nabla fractional calculus, formore details, we refer the reader to [7, 13].

Definition 1. The nth nabla difference and nth nabla sum for a function Rare defined respectively as

where is the difference/sum order, is the time variable, is the initial instant,is the gamma function.

It can be found that (1) and (2) have a similar form. The distinction lies in that thenabla difference has local memory, and the nabla sum has nonlocal memory. However,to avoid the singularity, i.e.,, the generalized binomial coefficientcan be calculated as alternative expression. For thespecial case of let us make an assumption

Extending the order n in Definition 1 from the positive integers to the positive realnumber, the following definition can be derived.

Definition 2. The αth Grünwald–Letnikov fractional difference/sum for a function is defined as

where and .

When the order like (3). When the order in (4) reduces to the classical tabla sum in (2).When the order becomes the Grünwald–Letnikov fractionalsum, which is also known as the nabla Riemann–Liouville fractional sum [7]. When theorder is not identical to in (1), especially for those since the upper limit of summation in formula (4) is a time-varying variable instead of the constant n. If the number of in the summation iscalled as the memory length, it is worth noting that has a time-varying memorylength, while has a fixed memory length, e.g.,.When the order becomes the Grünwald–Letnikov fractional difference. Due to the perfect match in format of fractional difference and fractional sum,the following relationship follows [23]:

where and

By combining the rising function with the convolution operation,the Grünwald–Letnikov fractional difference/sum can be rewritten as

Where and

Given a function if there exists some such that converges for, then the following equality holds [22]:

where. Notably, the sufficient and necessary condition for is that the multiplicity of pole is less than 1 andall other principal poles of X(s) satisfy js

It can be observed from (5) and (7) that is independent of the initial condition. This property will make it impossible to get from is independent of the initial condition which means that the state or outputfeedback control for the system is meanin gless. For this reason, the objective of this work is to analyze this default propertyand determine a recipe for eliminating this independence.

3 Main results

This section includes two parts, i.e., showing the independence of initial conditions and solving the initial condition independence problem.

3.1 The independent case

Consider the following system with Grünwald–Letnikov definition:

where is the pseudo state, is the input, is the output, and are linear or nonlinear functions.

Defining and using the property in (5), the first equation of (8) can be rewritten as which implies that when is considered cannot be calculated with initial condition x(a). Inspired by [21], a discrete time frequency distributed model is introduced here:

where . It can be checked that the relation from tois equivalent to that of (8). Along this way, different equivalent representation ofcouldgive different equivalent models of (8). For simplicity, it will not be expanded here. To get

the equivalent relation ship, zero initial condition should be configured for the equivalent model like (9), which is different from the Riemann–Liouville case and the Caputo case[21].

In system (8), the fractional difference is not calculated directly. To continue, considerthe fractional difference operation directly

whereis known, is to be calculated, and Since holds for can be calculated by the following systems:

where Note thatcan be accurately calculated without the dependance on initial condition

In the previous discussion, the order is assumed to beIf brings the equivalent description of system(8) as

where zero initial conditions and are needed. Likewise, the large-order case of (10) can be analyzed.

Remark 1. It can be observed that no matter the system output of (8), and the differenceoutput of (10) can be accurately calculated without considering the initial condition. Evenif the order is extended from, the independence of in itial conditions is always the same. This point is actually implied in Definition 2.More specially, the calculation of , only needs the value of

Up till now, the initial conditions independence property of system (8) has been discovered.To further show its influence, let us consider the following linear system governed by

withand can be calcuated as

where If the discrete Mittag-Leffler function is introduced here, then one has

It is worth mentioning that in both (12) and (13), only the input response can be found,and there is no state response. This result is quite coincident with the previous discussion.

Remark 2. One point worth emphasizing is that the exact calculation of in (8). Two methods will be discussed.

The first one is to solve fractional difference equation directly. By using (4) and (6),it follows

where are known, and is to be calculated.(14) can be rewritten as

From this the value of can be calculated sequentially. , which involves the gamma function. When calculating the value of gamma function using MATLAB code gamma(), one has = To avoid the overflow problem, two available ways can be taken. The one is recursive computation by using the property

The other is equivalent transform by using the relation

where the code gammaln() is helpful. Some other treatment can be made before introducing logarithmic function when

The second one is to solve fractional sum equation indirectly. By using (5), (11) canbe rewritten as

where could be calculated like, while the unexpectedcase will not appear any more.

3.2 The dependent case

The system with Caputo fractional difference involves the initial conditions[under Riemann–Liouville definition should consider initial conditions regarding to the Riemann–Liouville fractional difference and Riemann–Liouville fractional sum under Grünwald–Letnikov definition does not depend on initial condition x(a) or any other case. Consequently, the question on how to make fractional-order difference equation connect to initial condition becomes urgent. In this part, four possible schemes are provided.

Scheme 1 Consider the following system:

Where

Using the calculating approach in Remark 2 yields

It can be observed that when calculating , the initial conditions are all adopted. This scheme just hints howto solve the initial conditions independence issue when applying the Grünwald–Letnikovfractional difference.

Defining, the time-delay property of the nabla Laplace transformfollows [22]

from which the state response in frequency domain becomes

from which the state response in frequency domain becomes

Note that (18) coincides with (16) and (17). System (15) will be asymptotically stable if is designed such that the multiplicity of root is less than 1 or all other the principalroots lie in the region j

Scheme 2 Consider the following system:

with

Using the calculating approach in Remark 2 yields

As such, the calculation ofstrongly depends on the initial conditions .

By applying the following time advance property [22]

To sum up, in face of the independence of initial conditions for Grünwald–Letnikovfractional difference, the autonomous system cannot work, whilesystem (15) or (19) can be used as a substitute. The stable condition of the system (19) isthe same with the system (15).

Remark 3. Combing the two methods, a strange but available system follows In this regard,one has the relation ships and, which means that the initial conditions are needed to calculate fork. Different from the methods in (15) and (19), this method is not directlyapplicable to the nonlinear case, such as .

Scheme 3 After wards, let us continue to consider the system .If we assume and take as theinitial condition, the system becomes

with From this the initial condition has beenexerted successfully.

Using the calculating approach in Remark 2 yields

Recalling Definition 2 yields

from whichcan be expressed as

where (20) and are adopted.

It can be found that the initial conditionsare impliedin X(s). The corresponding stability condition becomes that the root of liesin the region . Compared with systems (15) and (19), it is more convenient touse frequency tool in system (21). However, the inconvenience is that this method is notapplicable to the nonlinear case.

Scheme 4 Along this way, consider the following system:

with

Using the calculating approach in Remark 2 yields

The calculation of strongly depends on the initial conditions

Recalling Definition 2 yields

Defining it follows

where (20) and are adopted again.

It can be found that the initial conditions areimplied in X(s). The corresponding stability condition becomes that the root of lies in the region.

Till now, four available schemes have been presented already. Note that the systemsin (15), (19), (21), and (22) only need the initial condition about not the differenceor sum of, which is helpful in practical applications. The nature of them is shifting.Along this way, there are various of methods to introduce the initial condition. The basicprinciple might be to connect the future signal with the past signal and calculate theunknown value with the known value.

Remark 4. The Grünwald–Letnikov fractional difference/sum adopted in [8,9,11,13,14]is actually, which is a special caseof (22) with m = 1. In that case, the initial conditions independence can also be solved.When , which isunexpected. To avoid this confusing phenomenon, the discrete time should beapplied.

4 Further discussion

In this section, the developed results will be discussed for some variants of discreteGrünwald–Letnikov fractional calculus. Afterwards, three detailed numerical examplesare provided to verify the correctness of the proposed theory.

4.1 Extension

In the previous discussion, the basic is assumed. Hereafter, three other definitions will beconsidered.

Definition 3. For , its tempered Grünwald–Letnikov fractional difference/sum is defined by

where

Although a new parameteris introduced, there is no item regarding to the initialcondition in (23).

Definition 4. For Grünwald–Letnikov fractional difference/sum is defined by

Where

By using the nabla Laplace transform, it follows

Although a new parameter is introduced, there is no item regarding to the initialcondition in (23).

Definition 4. Forth Grünwald–Letnikov fractional difference/sum is defined by

where

The elegant formula does not hold, while this definitioncan be rewritten as. Defining

it follows

From this, when is calculated, no more information of is needed. To be more precise, all the fractional differences in Definitions 2,3, and 4 are independent on the initial condition. Fortunately, by using the schemes inSection 3.2, the initial condition dependence can be attained.

Definition 5. ForGrünwald–Letnikov fractional difference/sum is defined by

Where

Notably, the equality strongly implies the possible connectionto the initial condition. Another equality)also shows that the calculation of depends on the history information. Different from the previous definitions, here the memorylength is a constant value instead of a time varying value k 􀀀 a. Along this way,both might be the alternative schemes to generate the initial conditions

4.2 Examples

Example 1. Consider the following five systems:

with. For the purpose ofplotting, denote, for each case. To make the system stable are selected for cases 1–5, respectively.

When the value of is assumed as 1, 􀀀1 in succession, then the resulting result are plottedin Figs. 1(a)–1(d).

It can be observed that with the shifting principle, the initial conditions are available tocalculate in cases 1–5. Moreover, for different can converge to 0as k increases. In this connection, the results in Section 3.2 have been validated.

Example 2. To evaluate the practicability of the proposed result, the following gradientmethods are considered:

In the beginning, the quadratic objective function is considered, where is selected as in case 1 is calculated directly. Similarly, denote is adopted for plotting. Inthis case, the optimization results with different are plotted in Fig. 3(a)–3(f).

These figures clearly show that all the presented cases could find the exact extreme point Since the stability domain of (15) or (19) is difficult to obtain andtherefore cases 2 and 4 are not included in the case of. For cases 1–6, the over shoot appears when Notably, the initial value in case 1 is different from others (seeyellow circle and red circle) since the calculation is, not thesetting value 1.

In what follows, the quartic function shown in Fig. 4 isconsidered. Because the gradientis nonlinear, the exactsolutions of cases 1, 3, 5, and 6 cannot be calculated directly. Therefore, only cases 2and 4 are discussed hereinafter. With the following parameters , and the simulation results can be obtained as Figs. 5(a)–5(b).

Since. Morespecially, is a globalminima point. From Fig. 5(a) it can be found that all the curves converge to the exactextreme points. Likewise, cases 2 and 4 give the similar results. The relationship betweenthe convergent point and the initial condition is clearly shown in Fig. 5(b), which meansthat the initial conditions independence disappears with the special design.

Example 3. Construct the following fractional-order neural network:

With the given conditions, the phase diagram in the three dimensional space is depictedin Fig. 6(a), and the trajectories ofx(k) over time is shown in Fig. 6(b). It couldsee that the proposed initial condition scheme is effective.

Before ending this section, the main contributions of this paper will be summarized tomake it more readability.

• The independence on the initial condition is shown for Grünwald–Letnikov fractional difference equation.

• The equivalent frequency distributed models for a fractional-order system with Grünwald–Letnikov definition are derived.

• The frequency distributed model is also developed for Grünwald–Letnikov frac- tional difference calculation.

• The unavailability of state response for a linear fractional-order system is analyzed.

• The dependent case is considered, which connects the system response with the initial condition.

• The independence problem is also discussed for some variants of Grünwald– Letnikov definition.

5 Conclusions

In this paper, the initial conditions independent property has been studied for the firsttime. Inspired by the meaningless system, the initial valueproblem on Grünwald–Letnikov fractional difference is discussed from the definition, thedistributed frequency model, and the time–domain response. To improve the practicability,four schemes are developed by connecting the initial conditions to the consideredsystem. Apart from this, three illustrative examples are provided to confirm our findings.

It is believed that the proposed principles could greatly enrich the comprehension ofGrünwald–Letnikov fractional difference and facilitate its applications. Future researchefforts will be directed towards the following topics.

(i) Discuss the stability condition of systems further and analyze the dynamic behaviour of its time–domain system response

(ii) Extend the related results to other Grünwald–Letnikov based definitions, i.e., the right-hands, the nonsingular kernel case, time-scales case, etc.

(iii) Apply the initial conditions dependent Grünwald–Letnikov difference to practical applications, such as modeling, controlling, filtering, optimization, etc.

References

1 M.S. Abdelouahab, N.E. Hamri, The Grünwald–Letnikov fractional-order derivative with fixed memory length, Mediterr. J. Math., 13(2):557–572, 2016, https://doi.org/10.1007/s00009-015-0525-3.

2 F.M. Atıcı, S. Chang, J. Jonnalagadda, Grünwald–Letnikov fractional operators: from past to present, Fract. Differ. Calc., 11(1):147–159, 2021, https://doi.org/10.7153/fdc- 2021-11-10.

3 H. Chen, F. Holland, M. Stynes, An analysis of the Grünwald–Letnikov scheme for initial-value problems with weakly singular solutions, Appl. Numer. Math., 139:52–61, 2019, https://doi.org/10.13140/RG.2.2.36154.24001.

4 Y.Q. Chen, A Collection of Fractional Calculus Books, 2020, http://mechatronics. ucmerced.edu/fcbooks/.

5 J.F. Cheng, Fractional Difference Equation Theory, Xiamen Univ. Press, Xiamen, 2011.

6 R. Cioc´, Physical and geometrical interpretation of Grünwald–Letnikov differintegrals: Measurement of path and acceleration, Fract. Calc. Appl. Anal., 19(1):161–172, 2016, https://doi.org/10.1515/fca-2016-0009.

7 C. Goodrich, A.C. Peterson, Discrete Fractional Calculus, Springer, Cham, 2015.

8 W.P. Hunek, Perfect control for right-invertible Grünwald–Letnikov plants – An innovative approach to practical implementation, Fract. Calc. Appl. Anal., 22(2):424–443, 2019, https://doi.org/10.1515/fca-2019-0026.

9 T.Y. Liu, Y.H. Wei, W.D. Yin, Y. Wang, Q. Liang, State estimation for nonlinear discrete–time fractional systems: A Bayesian perspective, Signal Process., 165:250–261, 2019, https://doi.org/10.1016/j.sigpro.2019.06.037.

10. C.L. MacDonald, N. Bhattacharya, B.P. Sprouse, G.A. Silva, Efficient computation of the Grünwald–Letnikov fractional diffusion derivative using adaptive time step memory, J. Comput. Phys., 297:221–236, 2015, https://doi.org/10.1016/j.jcp.2015.04.048.

11 D. Mozyrska, P. Oziablo, M. Wyrwas, Stability of fractional variable order difference systems, Fract. Calc. Appl. Anal., 22(3):807–824, 2019, https://doi.org/10.1515/fca-2019-0044.

12 A.D. Obembe, S.A. Abu-Khamsin, M.E. Hossain, K. Mustapha, Analysis of subdiffusion indisordered and fractured media using a Grünwald–Letnikov fractional calculus model, Comput. Geosci.,22(5):1231–1250, 2018, https://doi.org/10.1007/s10596-018-9749-1.

13 P. Ostalczyk, Discrete Fractional Calculus: Applications in Control and Image Processing, World Scientific, Singapore, New Jersey, 2016, https://doi.org/10.1142/9833.

14 P. Ostalczyk, D. Sankowski, M. Ba¸kała, J. Nowakowski, Fractional-order value identification of the discrete integrator from a noised signal. part i, Fract. Calc. Appl. Anal., 22(1):217–235, 2019, https://doi.org/10.1515/fca-2019-0014.

15 6 A.D. Pano-Azucena, E. Tlelo-Cuautle, J.M. Muñoz-Pacheco, L.G. de la Fraga, FPGA-based implementation of different families of fractional-order chaotic oscillators applying Grünwald– Letnikov method, Commun. Nonlinear Sci. Numer. Simul.,72:516–527, 2019, https://doi.org/10.1016/j.cnsns.2019.01.014.

16 R. Stanisławski, New results in stability analysis for LTI SISO systems modeled by GL- discretized fractional-order transfer functions, Fract. Calc. Appl. Anal., 20(1):243–259, 2017, https://doi.org/10.1515/fca-2017-0013.

17 J.A. Tenreiro Machado, Fractional derivatives: probability interpretation and frequency response of rational approximations, Commun. Nonlinear Sci. Numer. Simul.,14(9-10):3492–3497, 2009, https://doi.org/10.1016/j.cnsns.2009.02.004.

18 M.F. Tolba, L.A. Said, A.H. Madian, A.G. Radwan, FPGA implementation of the fractional order integrator/differentiator: Two approaches and applications, IEEE Trans. Circuits Syst. I, Regul. Pap.,66(4):1484–1495, 2018, https://doi.org/10.1109/TCSI.2018.2885013.

19 A. Tudorache, R. Luca, On a singular Riemann–Liouville fractional boundary value problem with parameters, Nonlinear Anal. Model. Control, 26(1):151–168, 2021, https://doi.org/10.15388/namc.2021.26.21414.

20 Y.H. Wei, Y.Q. Chen, T.Y. Liu, Y. Wang, Lyapunov functions for nabla discrete fractional order systems, ISA Trans., 88:82–90, 2019, https://doi.org/10.1016/j.isatra.2018.12.016.

21 Y.H. Wei, Y.Q. Chen, J.C. Wang, Y. Wang, Analysis and description of the infinite-dimensional nature for nabla discrete fractional order systems, Commun. Nonlinear Sci. Numer. Simul.,72:472–492, 2019, https://doi.org/10.1016/j.cnsns.2018.12.023.

22 Y.H. Wei, Y.Q. Chen, Y. Wang, Y.Q. Chen, Some fundamental properties on the sampling free nabla Laplace transform, in International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2019, https://doi.org/10.1115/DETC2019-97351.

23 Y.H. Wei, W.D. Yin, Y.T. Zhao, Y. Wang, A new insight into the Grünwald–Letnikov discrete fractional calculus, J. Comput. Nonlinear Dyn., 14(4):041008, 2019, https://doi.org/10.1115/1.4042635.

24 G.C. Wu, T. Abdeljawad, J.L. Liu, D. Baleanu, K.T. Wu, Mittag–Leffler stability analysis of fractional discrete-time neural networks via fixed point technique, Nonlinear Anal. Model. Control, 24(6):919–936, 2019, https://doi.org/10.15388/NA.2019.6.5.

25 cb C.B. Zhai, J. Ren, A fractional .-difference equation eigenvalue problem with .(.)-Laplacian operator, Nonlinear Anal. Model. Control, 26(3):482–501, 2021, https://doi.org/10.15388/namc.2021.26.23055.

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