# Generalized Fractional-Order Discrete-Time Integrator

Authors: Dorota Mozyrska; Piotr Ostalczyk
Journal: Complexity
 (2017)

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## Abstract

We investigate a generalization of discrete-time integrator. Proposed linear discrete-time integrator is characterised by the variable, fractional order of integration/summation. Graphical illustrations of an analysis of particular vector matrices are presented. In numerical examples, we show relations between the order functions and element responses.


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## Body

## 1. Introduction

In order to build a dynamic system, one should define specifications to be met, apply synthesis techniques, if available, analyse a mathematical model of a system, and simulate the model on a computer to test the effect of various inputs on the behavior of the resulting system. New classes and categories of systems that could be used as new models are still needed. One of the most important tools is an element called “integrator.” In measurements and control applications, an integrator is an element whose output signal is the time integral (in continuous case) or summation (in discrete case) of its input signal. It accumulates the input quantity over a defined time to produce a representative output. For the classical theory, see, for instance, [1, 2]. An integrator may be treated as a fundamental and is commonly used in constructions of more complicated systems via Kelvin’s scheme [2]. This leads to a variety of structures known as realizations. As crucial realizations, those, which reveal such important dynamic properties as stability, controllability, and so forth, are considered. The first-order differential equations can be generalized to the fractional-order ones [3–10]. Hence, we get the fractional-order integrator, which can be used in a modelling of fractional-order dynamic systems.For discrete-time systems, an equivalent element is called a summator or discrete integrator. This dynamic element is described by linear time-invariant first-order difference equation; see [11, 12]. As a generalization of the classical discrete integrator, we can consider discrete summation of fractional order [3, 13–20]. In this paper, we propose a generalization of the fractional-order discrete integrator and call it the variable-, fractional-order discrete-time integrator.Besides applications of integrators in mentioned realizations, another important use is the integration action in the PID controllers; see [1, 21]. The integration action preserves a zero steady state in the closed-loop systems with typical plants. Different types of the variable-, fractional-order elements have been proposed in [22, 23]. For constant orders (fractional or integer orders), all integrators are identical. This property is not valid in the variable-, fractional-order integrators in the mentioned types. Comparing with the model of integrator described in [22], we state here a different and more general model with better motivated initial conditions. Moreover, our investigations of values of coefficients of matrices, that are used for calculations of models, are much more advanced.The proposed integrator may be used in the variable-, fractional-order digital filters [11, 12], described by related variable-, fractional-order difference equations. In the paper, an equivalent but very useful vector matrix description of the variable-, fractional-order integrator is applied. It becomes a great tool in the variable-, fractional-order integrators description. One should mention that to variable-, fractional-order difference equations we cannot apply the Z-transform.The paper is organised as follows. After an introduction to the variable-, discrete-, fractional-order calculus, a description of the variable-, fractional-order discrete integrator is given in Section3. The formula for the variable-, fractional-order integrator response is derived. Our investigations are illustrated by numerical examples.


## 2. Preliminaries

The most important in the evaluation of the variable-, fractional-order backward difference/sum is the kernel function, named after its action theoblivion function. For k,l∈Z and a given order function ν·:Z→R, the function of two discrete variables is defined by its values: aνlk. We assume that order functions have values in the interval 0,1.Definition 1.
Fork,l∈Z and a given order function ν·, one defines the oblivion function, as a discrete function of two variables, by its values aνlk given as(1)aνlk=0fork<01fork=0-1kνlνl-1⋯νl-k+1k!fork>0.It is easy to observe that for opposite values of order function holds the following:(2)a-νlk=0fork<01fork=0νlνl+1⋯νl+k-1k!fork>0.Formula (1) in Definition 1 is equivalent to the following recurrence with respect to k∈N:(3)aνl0=1,aνlk=aνlk-11-νl+1kfork⩾1.In [24], we have proved the following properties for positive values of order function.Proposition 2 (see [24]).
Let the order function have values0<νl<1 for l∈Z. Then, the following properties hold: (a)
For alll∈Z and k∈N1, aνlk<0.(b)
For alll∈Z, the sequence aνlkk∈N1 is increasing.(c)
For each increasing and bounded order functionν· with values in 0,1 and for each l∈N, there is ml such that for k≥ml the sequence aνlkl∈N is increasing; that is, for k≥ml,(4)aνl+1k>aνlk.Particularly for order functions with values in1/2≤νl<1, the border ml=2.(d)
For each increasing and bounded order functionν· with values in 0,1, there is mk∈N1 such that for k≥mk(5)aνk+1k+1>aνkk.In the sequel, we need to prove parallel properties for oblivion function with negative values of order function with values-νl∈-1,0 for l∈Z.Proposition 3.
Let one assume that,∀l∈Z, νl∈0,1. Then the following properties are satisfied: (a)
For alll∈Z and k∈N1, a-νlk>0.(b)
For alll∈Z, the sequence a-νlkk∈N1 is decreasing.(c)
For decreasing and bounded order functionν· with values in 0,1 and for each l∈N, the sequence a-νlkl∈N is decreasing; that is, for k≥N1,(6)a-νl+1k<a-νlk.(d)
For decreasing and bounded order functionν· with values in 0,1 and for each l∈N holds(7)a-νk+1k+1<a-νkk.Proof.
Forl∈Z, we have that a-νl0=1 and for k>0(8)a-νlk=νlνl+1⋯νl+k-1k!.Then, we directly have points (a), (b), and (c). In (b), we need additionally to notice that (νl-1)/k<0, which gives a-νlk+1<a-νlk. We do the next calculations to receive what we claim in point (d). We have the following: (9)a-νk+1k+1-a-νkk=1--νk+1+1k+1a-νk+1k-a-νkk.Moreover, as 0<νk+1<1, then 0<1+(νk+1-1)/(k+1)<1 for k≤1. Knowing that a-νk+1k>0, we receive 1-(-νk+1+1)/(k+1)a-νk+1k<a-νk+1k. Hence,(10)a-νk+1k+1-a-νkk<a-νk+1k-aνkkand then from point (c) we have the thesis.In the next definition, the Grünwald–Letnikov fractional-order backward difference (GL-FOBD) is generalized to the Grünwald–Letnikov variable-, fractional-order backward difference (GL-VFOBD) in part (a) and to the Grünwald–Letnikov variable-, fractional-order backward difference with initialization (GL-VFOBDwI) in part (b). For definition and properties of the Grünwald–Letnikov fractional-order backward difference (GL-FOBD) for constant order, we refer to [10, 18, 25, 26].Definition 4.
Letf be a discrete-variable bounded real valued function.(a)
The Grünwald–Letnikov variable-, fractional-order backward difference with initialization (GL-VFOBDwI) with an order functionν:Z→R+∪0 is defined as an infinite sum, provided that the series is convergent:(11)Δ-∞GLνkfk=∑i=0∞aνkifk-i=1aνk1aνk2⋯fkfk-1fk-2⋮.(b)
The Grünwald–Letnikov variable-, fractional-order backward difference (GL-VFOBD) with an order functionν:Z→R+∪0 started at k0 is defined as a finite sum(12)Δk0GLνkfk=∑i=k0kaνki-k0fk+k0-i=∑i=0k-k0aνkifk-i=1aνk1⋯aνkk-k0fkfk-1⋮fk0+1fk0.Fork0=0, the GL-VFOBD becomes a discrete convolution: Δ0GLνkfk=(a∗f)k=(f∗a)k, where ak≔aνkk. In particular case of constant order function, we have the following (for k>k0 and k0 can be finite or k0=-∞):(i)
Δk0GL0fk=fk;(ii)
Δk0GL1fk=fk-fk-1;(iii)
Δk0GL-1fk=∑i=0k-k0fk-i‍, if only the summation exists.Next, one assumes thatνk=0 for k⩽k0-1. Equality (11) is defined for any k⩾k0>-∞, so it is also valid for any k-1,k-2,…,k0+1,k0,k0-1,….Collecting all such equalities like (11) in one vector matrix form, one obtains(13)Δ-∞GLνkfk=A-∞k0,kνkfkfk0-1,where(14)fk=fkfk-1⋮fk0+1fk0,fk0-1=fk0-1fk0-2⋮,Δ-∞GLνkfk=Δ-∞GLνkfk⋮Δ-∞GLνk0fk0,(15)A-∞k0,kνk≔Ak0kνkI-∞k0-1νk.Inside matrix (15) we extract the following parts:(16)Ak0kνk≔1aνk1⋯aνkk-k001⋯aνk-1k-k0-1⋮⋮⋮00⋯1,(17)I-∞k0-1νk≔aνkk-k0+1aνkk-k0+2⋯aνk-1k-k0aνk-1k-k0+1⋯⋮⋮aν01aν02⋯.Forν=1 and k0=0, appropriate matrices have the following forms:(18)Ak0k-1=111⋯11011⋯11001⋯11⋮⋮⋮⋮⋮000⋯11000⋯01,(19)Ak0k1=1-10⋯0001-1⋯00001⋯00⋮⋮⋮⋮⋮000⋯1-1000⋯01.Moreover, it is worth noticing thatAk0k0=1k-k0+1, which is the k-k0+1×k-k0+1-dimensional identity matrix.We give now the series of properties of finite dimensional matricesAk0kνk and Ak0k-νk and their inverses, for order function with values νk∈0,1.Proposition 5 (see [24]).
Assume that for allk≥k0, νk∈0,1 and νk=0 for k≤k0-1. Then, all elements of the inverse matrix Ak0kνk-1 are nonnegative.Proposition 6 (see [24]).
For an order functionνk>0 and a constant order νc>0, the following equality holds: Akνk0Akνc0=A0kνk+νc.By direct calculations, one can check thatA0kνcA0kνk≠A0kνk+νc. Proposition 6 confirms the fact that for νk=ν=const, we have Akν0Akνc0=A0kν+νc. As a consequence of the last equality, one has the following Ak0kν=Ak0k-ν-1. By direct calculations, we can check that(20)Ak0kνk-1≠Ak0k-νk.The crucial matrix in the VFODI response is Ak0k-νk. We prove the following in [22].Proposition 7 (see [22]).
For order functions with values0≤νk⩽1 for k≥k0, all elements of the matrix Ak0k-νk are nonnegative.Proposition 8 (see [22]).
For two order functions with values0<ν1k<ν2k⩽1, for k≥k0, all elements of the matrix aij=Ak0k-ν1k are less than or equal to these of the matrix bij=Ak0k-ν2k; that is, aij≤bij.


## 3. Variable-, Fractional-Order Difference Integrator (VFODI)

### 3.1. Description of the VFODI

Letu be the given function u:Z↦R and f:R2↦R be given bounded function of two variables. Let ¯f(k)≔f(yk-1,uk),k∈Z. The variable-, fractional-order discrete-time difference equation for the pair of orders ν,μ is described by the following fractional-order difference equation:(21)Δ-∞GLνkyk=Δk0GL-μkf¯k,k≥k0with initial conditions(22)yk0-1=yk0-1yk0-2⋮and values uk=0 for k<k0.From definitions of fractional operators, (21) has the following recurrence solution:(23)yk=-∑i=1∞aνkiyk-i+∑i=0k-k0a-μkifyk-1-i,uk-i,k≥k0.In the simplest situations, we present solutions given by (23) in the form (i)
forν≡0,μ≡0,yk=fyk-1,uk and this looks like classical difference equation;(ii)
forν≡1,μ≡0,yk=yk-1+fyk-1,uk and this is the case of the so-called nabla operator equation;(iii)
forν≡0,μ≡1,yk=∑i=1k-k0fyk-1-i,uk-i‍;(iv)
forν≡1,μ≡1,yk=yk-1+∑i=1k-k0fyk-1-i,uk-i‍.In the next definition of variable-, fractional-order difference integrator (VFODI), we assume thatfyk-1,uk=b0uk. Let us introduce the following notation:(24)yk=ykyk-1⋮yk0+1yk0,uk=ukuk-1⋮uk0+1uk0,Yk=ykyk-1⋮,fork∈Z.If k≥k0, then we can write Y(k) in the following way Yk=ykyk0-1.The variable-, fractional-order difference integrator (VFODI) is described by the following fractional-order difference matrix-vector equation:(25)A-∞kνk·Yk=b0Ak0k-μkukwith initial condition Yk0-1=yk0-1. On the right side of (25), we use finite matrix Ak0k-μk as on the left side there is a rectangular (finite rows and infinite columns) matrix operator. It is because we claim as in the classical cases that values uk=0 for k<k0. Then, even if we write infinite operator matrix on the right side, it will give the same action as multiplication by an infinite number of zeros of u·.Proposition 9.
Equation (25) with initial condition Yk0-1=yk0-1 has the solution given by the following description:(26)yk=Ak0kνk-1·b0Ak0k-μkuk-I-∞k0-1νkyk0-1.Proof.
The proof follows from the fact that taking into account the initial conditions vectorYk0-1 and the partition of the matrices A-∞k0,kνk, we have from (25)(27)Ak0kνkI-∞k0-1νkykyk0-1=b0Ak0k-μkuk.
It is possible as matricesAk0kνk and Ak0k-μk are always nonsingular.For example, fork=1 and zero initial conditions vector Yk0-1=0 with k0=0, we have that (28)y1=y1y0=b0A1ν1-1A1-μ1u1u0,y1=b0u1+u0ν1+μ1u0.Hence, y0=b0u0 and y(1)=b0(u1+u0ν1+μ1).Proposition 10.
The VFODI of form (25) with fractional orders satisfying the condition that for each k≥k0, νk+μk=1 and zero initial conditions vector Yk0-1=0 is a classical first-order summator (discrete-time integrator).Proof.
By the assumption-μk=νk-1, then(29)Ak0kνkyk=b0Ak0kνk-1uk.Based on investigations in [24], we know that (29) is equivalent to(30)Ak0kνkyk=b0Ak0kνkAk0k-1uk.Matrix Ak0kνk is always nonsingular as an upper triangular matrix with ones on the main diagonal. Hence, from (30) one immediately gets yk=b0Ak0k-1uk and equivalently(31)Ak0k1yk=b0uk.Matrix Ak0k1 has the form (19). From the first row of (31) with (19), one obtains yk-yk-1=b0uk or equivalently yk=b0∑i=0kui‍.Example 11.
In the numerical example, usefulness of the VFODI is presented. In the first one, the discrete unit step responses of the VFODI for assumed fractional-order function are presented. In the second example, some particular application of the VFODI is shown.
Let us consider two fractional-order functions given by their values fork∈N:(32)νk=0.5-0.5e-0.01k,μk=-0.5+0.5e-0.05k.In Figure 1, plots of order functions νk, black line, and μk, red line, are presented. As a consequence in Figures 2(a) and 2(b), images and 3D matrix values are presented.
In Figures3(a) and 3(b), images and 3D matrix values of Ak0kνk-1 are presented, where p1,p2 are pixels positions and cp1,p2 are the color representations.
The image and values of matrixAk0kνk-1Ak0k-μk are given in Figures 4(a) and 4(b).Figure 1
Plots of order functions defined by formula (32).Figure 2
Image of the matrix and 3D plot of values of the matrixAk0kνk-1.
(a)
Image of the matrix
(b)
3D plot of values of the matrixFigure 3
Image of the matrix and 3D plot of values of the matrixAk0k-μk.
(a)
Image of the matrix
(b)
3D plot of values of the matrixFigure 4
Image of the matrix and 3D plot of values of the matrixAk0kνk-1Ak0k-μk.
(a)
Image of the matrix
(b)
3D plot of values of the matrix


### 3.2. Particular Forms of VFODI

We consider here two special cases of (25), the first one for μk≡0 and the second one for νk≡0.(i)
Let us considerμk≡0. Then, (25) takes form(33)A-∞kνk·Yk=b0ukand the VFODI response, with initial conditionYk0-1=yk0-1, is described by the formula proved in Proposition 9:(34)yk=b0Ak0kνk-1·uk-Ak0kνk-1·I-∞k0-1νkyk0-1.One should emphasise that the initial condition vector is infinite dimensional. This is characteristic for systems with “memory” of the state.Example 12.
In the following numerical example, we examine the discrete unit step responses of the first form of the VFODI. We present plots of order functions and solutions of VFODI, given by (34) with b0=0.001,k0=0. (a)
Let us consider order function given by(35)ν1k=1-e-0.01k.The order functionν1 is plotted in Figure 5(a). The graph of the unit step response is given in Figure 5(b). The considered order function is characterised by two time intervals separated by a time instant k=400. In the first one, the order function increases monotonically from 0 to 1. Hence, the summation force is weaker but growing. In the second interval, the function is almost constant. This means that the VFODI acts as a classical integrator. The VFODI response presented in Figure 5(b) is similar to the response of the ideal summator with lag.(b)
Now we consider decreasing order function given by(36)ν2k=1+e-0.01k.The graph of the order function (36) is given in Figure 6(a). Values of order function begin at 2 and monotonically tend to 1. This means that summation force successively declines to the classical summator. The response is shown in Figure 6(b).Figure 5
Plots of the order functionν1 given by (35) and the unit step response of the 1st form of VFODI represented by formula (34) with b0=0.001,k0=0.
(a)
The plot of the order function
(b)
The unit step responseFigure 6
Plots of the order functionν2 given by (36) and the unit step response of the 1st form of VFODI represented by formula (34) with b0=0.001,k0=0.
(a)
The plot of the order function
(b)
The unit step responseExample 13.
The VFOIE possesses also the property related to the classical integrator. For zero input signal and nonzero initial conditions, it preserves a nonzero output. In this example, we split our consideration to two different initial conditions:(37)y-11=1000⋮,y-12=0010⋮.We change in the item order functions, considering increasing and decreasing order functions. (c)
For the order functionν1k=1-e-0.01k, we obtain the VFOIE homogenous responses presented in Figures 7(a) and 7(b), respectively. Indeed responses differ a little from each other. The second response reaches slightly lower steady state. This statement confirms that the oblivion function decreases due to the initial condition placed in the past. For the order function ν2k=1+e-0.01k, we obtain the VFOIE homogenous responses presented in Figures 8(a) and 8(b), respectively.(d)
In this part, we examine the homogenous response toy-11 with a periodic order function given by ν3k=1+sin⁡πk/40. The plots of order functions and the simulated solutions are presented in Figures 9(a), 9(b), and 9(c), respectively. Values of both responses are tending to zero.(ii)
Let us takeνk≡0. Then, (25) takes the form(38)Yk=b0Ak0k-μkukand the VFODI response, with initial conditionYk0-1=yk0-1, is described by the formula proved in Proposition 9:(39)yk=b0Ak0k-μkuk-I-∞k0-1νkyk0-1.Figure 7
The VFOI homogenous responses toy-11 and y-12 with b0=0.001,k0=0 in formula (34) and the order ν1k=1-e-0.01k.
(a)
Plot of response toy-11
(b)
Plot of response toy-12Figure 8
The VFOI homogenous responses toy-11 and y-12 with b0=0.001,k0=0 in formula (34) and the order ν2k=1+e-0.01k.
(a)
Plot of response toy-11
(b)
Plot of response toy-12Figure 9
Plots of sinusoidal order functionν3k=1+sin⁡πk/40 and the VFOI homogenous response to y-11 and y-12 represented by formula (34) with b0=0.001,k0=0.
(a)
Sinusoidal order function
(b)
The VFOI homogenous response toy-11
(c)
The VFOI homogenous response toy-12


## 3.1. Description of the VFODI

Letu be the given function u:Z↦R and f:R2↦R be given bounded function of two variables. Let ¯f(k)≔f(yk-1,uk),k∈Z. The variable-, fractional-order discrete-time difference equation for the pair of orders ν,μ is described by the following fractional-order difference equation:(21)Δ-∞GLνkyk=Δk0GL-μkf¯k,k≥k0with initial conditions(22)yk0-1=yk0-1yk0-2⋮and values uk=0 for k<k0.From definitions of fractional operators, (21) has the following recurrence solution:(23)yk=-∑i=1∞aνkiyk-i+∑i=0k-k0a-μkifyk-1-i,uk-i,k≥k0.In the simplest situations, we present solutions given by (23) in the form (i)
forν≡0,μ≡0,yk=fyk-1,uk and this looks like classical difference equation;(ii)
forν≡1,μ≡0,yk=yk-1+fyk-1,uk and this is the case of the so-called nabla operator equation;(iii)
forν≡0,μ≡1,yk=∑i=1k-k0fyk-1-i,uk-i‍;(iv)
forν≡1,μ≡1,yk=yk-1+∑i=1k-k0fyk-1-i,uk-i‍.In the next definition of variable-, fractional-order difference integrator (VFODI), we assume thatfyk-1,uk=b0uk. Let us introduce the following notation:(24)yk=ykyk-1⋮yk0+1yk0,uk=ukuk-1⋮uk0+1uk0,Yk=ykyk-1⋮,fork∈Z.If k≥k0, then we can write Y(k) in the following way Yk=ykyk0-1.The variable-, fractional-order difference integrator (VFODI) is described by the following fractional-order difference matrix-vector equation:(25)A-∞kνk·Yk=b0Ak0k-μkukwith initial condition Yk0-1=yk0-1. On the right side of (25), we use finite matrix Ak0k-μk as on the left side there is a rectangular (finite rows and infinite columns) matrix operator. It is because we claim as in the classical cases that values uk=0 for k<k0. Then, even if we write infinite operator matrix on the right side, it will give the same action as multiplication by an infinite number of zeros of u·.Proposition 9.
Equation (25) with initial condition Yk0-1=yk0-1 has the solution given by the following description:(26)yk=Ak0kνk-1·b0Ak0k-μkuk-I-∞k0-1νkyk0-1.Proof.
The proof follows from the fact that taking into account the initial conditions vectorYk0-1 and the partition of the matrices A-∞k0,kνk, we have from (25)(27)Ak0kνkI-∞k0-1νkykyk0-1=b0Ak0k-μkuk.
It is possible as matricesAk0kνk and Ak0k-μk are always nonsingular.For example, fork=1 and zero initial conditions vector Yk0-1=0 with k0=0, we have that (28)y1=y1y0=b0A1ν1-1A1-μ1u1u0,y1=b0u1+u0ν1+μ1u0.Hence, y0=b0u0 and y(1)=b0(u1+u0ν1+μ1).Proposition 10.
The VFODI of form (25) with fractional orders satisfying the condition that for each k≥k0, νk+μk=1 and zero initial conditions vector Yk0-1=0 is a classical first-order summator (discrete-time integrator).Proof.
By the assumption-μk=νk-1, then(29)Ak0kνkyk=b0Ak0kνk-1uk.Based on investigations in [24], we know that (29) is equivalent to(30)Ak0kνkyk=b0Ak0kνkAk0k-1uk.Matrix Ak0kνk is always nonsingular as an upper triangular matrix with ones on the main diagonal. Hence, from (30) one immediately gets yk=b0Ak0k-1uk and equivalently(31)Ak0k1yk=b0uk.Matrix Ak0k1 has the form (19). From the first row of (31) with (19), one obtains yk-yk-1=b0uk or equivalently yk=b0∑i=0kui‍.Example 11.
In the numerical example, usefulness of the VFODI is presented. In the first one, the discrete unit step responses of the VFODI for assumed fractional-order function are presented. In the second example, some particular application of the VFODI is shown.
Let us consider two fractional-order functions given by their values fork∈N:(32)νk=0.5-0.5e-0.01k,μk=-0.5+0.5e-0.05k.In Figure 1, plots of order functions νk, black line, and μk, red line, are presented. As a consequence in Figures 2(a) and 2(b), images and 3D matrix values are presented.
In Figures3(a) and 3(b), images and 3D matrix values of Ak0kνk-1 are presented, where p1,p2 are pixels positions and cp1,p2 are the color representations.
The image and values of matrixAk0kνk-1Ak0k-μk are given in Figures 4(a) and 4(b).Figure 1
Plots of order functions defined by formula (32).Figure 2
Image of the matrix and 3D plot of values of the matrixAk0kνk-1.
(a)
Image of the matrix
(b)
3D plot of values of the matrixFigure 3
Image of the matrix and 3D plot of values of the matrixAk0k-μk.
(a)
Image of the matrix
(b)
3D plot of values of the matrixFigure 4
Image of the matrix and 3D plot of values of the matrixAk0kνk-1Ak0k-μk.
(a)
Image of the matrix
(b)
3D plot of values of the matrix


## 3.2. Particular Forms of VFODI

We consider here two special cases of (25), the first one for μk≡0 and the second one for νk≡0.(i)
Let us considerμk≡0. Then, (25) takes form(33)A-∞kνk·Yk=b0ukand the VFODI response, with initial conditionYk0-1=yk0-1, is described by the formula proved in Proposition 9:(34)yk=b0Ak0kνk-1·uk-Ak0kνk-1·I-∞k0-1νkyk0-1.One should emphasise that the initial condition vector is infinite dimensional. This is characteristic for systems with “memory” of the state.Example 12.
In the following numerical example, we examine the discrete unit step responses of the first form of the VFODI. We present plots of order functions and solutions of VFODI, given by (34) with b0=0.001,k0=0. (a)
Let us consider order function given by(35)ν1k=1-e-0.01k.The order functionν1 is plotted in Figure 5(a). The graph of the unit step response is given in Figure 5(b). The considered order function is characterised by two time intervals separated by a time instant k=400. In the first one, the order function increases monotonically from 0 to 1. Hence, the summation force is weaker but growing. In the second interval, the function is almost constant. This means that the VFODI acts as a classical integrator. The VFODI response presented in Figure 5(b) is similar to the response of the ideal summator with lag.(b)
Now we consider decreasing order function given by(36)ν2k=1+e-0.01k.The graph of the order function (36) is given in Figure 6(a). Values of order function begin at 2 and monotonically tend to 1. This means that summation force successively declines to the classical summator. The response is shown in Figure 6(b).Figure 5
Plots of the order functionν1 given by (35) and the unit step response of the 1st form of VFODI represented by formula (34) with b0=0.001,k0=0.
(a)
The plot of the order function
(b)
The unit step responseFigure 6
Plots of the order functionν2 given by (36) and the unit step response of the 1st form of VFODI represented by formula (34) with b0=0.001,k0=0.
(a)
The plot of the order function
(b)
The unit step responseExample 13.
The VFOIE possesses also the property related to the classical integrator. For zero input signal and nonzero initial conditions, it preserves a nonzero output. In this example, we split our consideration to two different initial conditions:(37)y-11=1000⋮,y-12=0010⋮.We change in the item order functions, considering increasing and decreasing order functions. (c)
For the order functionν1k=1-e-0.01k, we obtain the VFOIE homogenous responses presented in Figures 7(a) and 7(b), respectively. Indeed responses differ a little from each other. The second response reaches slightly lower steady state. This statement confirms that the oblivion function decreases due to the initial condition placed in the past. For the order function ν2k=1+e-0.01k, we obtain the VFOIE homogenous responses presented in Figures 8(a) and 8(b), respectively.(d)
In this part, we examine the homogenous response toy-11 with a periodic order function given by ν3k=1+sin⁡πk/40. The plots of order functions and the simulated solutions are presented in Figures 9(a), 9(b), and 9(c), respectively. Values of both responses are tending to zero.(ii)
Let us takeνk≡0. Then, (25) takes the form(38)Yk=b0Ak0k-μkukand the VFODI response, with initial conditionYk0-1=yk0-1, is described by the formula proved in Proposition 9:(39)yk=b0Ak0k-μkuk-I-∞k0-1νkyk0-1.Figure 7
The VFOI homogenous responses toy-11 and y-12 with b0=0.001,k0=0 in formula (34) and the order ν1k=1-e-0.01k.
(a)
Plot of response toy-11
(b)
Plot of response toy-12Figure 8
The VFOI homogenous responses toy-11 and y-12 with b0=0.001,k0=0 in formula (34) and the order ν2k=1+e-0.01k.
(a)
Plot of response toy-11
(b)
Plot of response toy-12Figure 9
Plots of sinusoidal order functionν3k=1+sin⁡πk/40 and the VFOI homogenous response to y-11 and y-12 represented by formula (34) with b0=0.001,k0=0.
(a)
Sinusoidal order function
(b)
The VFOI homogenous response toy-11
(c)
The VFOI homogenous response toy-12


## 4. Final Conclusions

The form of the variable-, fractional-order difference integrator (VFODI) is characterised by the two independent fractional-order functions. Both order functions are assumed to be nonnegative. There is no restriction concerning their equality. There is an immense choice of the fractional-order selection. One of the promising choices appears to be a relation of the order function with the input and output signalsνuk and μuk or νyk and μyk. In the closed-loop systems with VFO PID controller, the order functions can be related to the closed-loop error signal.



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