2022
274
Wei Yiheng neudawei@seu.edu.cn
Southeast University, China
Caoa Jinde
Yonsei University, Islandia
Lic Chuang
Hainan University, China
Chend Yangquan
University of California, Merced, CA, China
Abstract:
In this paper, the initial condition independence property of Grünwald–Letnikov fractional difference is revealed for the first time. For example, the solution 
of equation,
cannot be calculated with initial condition 
. First, theinitial condition independence property is carefully investigated in both time domain and frequencydomain. Afterwards, some possible schemes are formulated to make the considered system connectto initial condition. Armed with this information, the concerned property is examined on threemodified Grünwald–Letnikov definitions. Finally, results from illustrative examples demonstratethat the developed schemes are sharp.
Keywords: fractional calculus, independence, initial condition, Grünwald–Letnikov definition, dynamic properties.
Articles
How to empower Grünwald–Letnikov fractional difference equations with available initial condition?*
Caoa Jinde
Lic Chuang
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 of
Gar 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.
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
(1)
(2)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
(3)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
(4)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]:
(5)where
and
By combining the rising function
with the convolution operation,the Grünwald–Letnikov fractional difference/sum can be rewritten as
(6)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.
This section includes two parts, i.e., showing the independence of initial conditions and solving the initial condition independence problem.
Consider the following system with Grünwald–Letnikov definition:
(8)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:
(9)where
. It can be checked that the relation from
to
is equivalent to that of (8). Along this way, different equivalent representation of
couldgive 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
(10)where
is known,
is to be calculated, and
Since
holds for can be calculated by the following systems:
where
Note that
can be accurately calculated without the dependance on initial condition 
In the previous discussion, the order is assumed to be
If 
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
(11)with
and
can be calcuated as
(12)where
If the discrete Mittag-Leffler function
is introduced here, then one has
(13)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
(14)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.
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:
(15)Where 
Using the calculating approach in Remark 2 yields
(16)
(17)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
(18)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:
(19)with 
Using the calculating approach in Remark 2 yields
As such, the calculation of
strongly 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
(21)with 
From this the initial condition has beenexerted successfully.
Using the calculating approach in Remark 2 yields
Recalling Definition 2 yields
from which
can be expressed as
where (20) and
are adopted.
It can be found that the initial conditions
are 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:
(22)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.
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.
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 
(25)
Although a new parameter
is 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
(23)
Although a new parameter 
is introduced, there is no item regarding to the initialcondition in (23).
Definition 4. For
th 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. For
Grünwald–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
Example 1. Consider the following five systems:
(24)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 gradient
is 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 of
x(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.
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.
