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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.

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, [

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 [

Besides applications of integrators in mentioned realizations, another important use is the integration action in the PID controllers; see [

The proposed integrator may be used in the variable-, fractional-order digital filters [

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 Section

The most important in the evaluation of the variable-, fractional-order backward difference/sum is the kernel function, named after its action the

For

It is easy to observe that for opposite values of order function holds the following:

Formula (

In [

Let the order function have values

For all

For all

For each increasing and bounded order function

Particularly for order functions with values in

For each increasing and bounded order function

In the sequel, we need to prove parallel properties for oblivion function with negative values of order function with values

Let one assume that,

For all

For all

For decreasing and bounded order function

For decreasing and bounded order function

For

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 [

Let

The Grünwald–Letnikov variable-, fractional-order backward difference with initialization (GL-VFOBDwI) with an order function

The Grünwald–Letnikov variable-, fractional-order backward difference (GL-VFOBD) with an order function

For

Next, one assumes that

Collecting all such equalities like (

Inside matrix (

For

Moreover, it is worth noticing that

We give now the series of properties of finite dimensional matrices

Assume that for all

For an order function

By direct calculations, one can check that

For order functions with values

For two order functions with values

Let

From definitions of fractional operators, (

for

for

for

for

In the next definition of variable-, fractional-order difference integrator (VFODI), we assume that

The variable-, fractional-order difference integrator (VFODI) is described by the following fractional-order difference matrix-vector equation:

Equation (

The proof follows from the fact that taking into account the initial conditions vector

It is possible as matrices

For example, for

The VFODI of form (

By the assumption

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 for

In Figures

The image and values of matrix

Plots of order functions defined by formula (

Image of the matrix and 3D plot of values of the matrix

Image of the matrix

3D plot of values of the matrix

Image of the matrix and 3D plot of values of the matrix

Image of the matrix

3D plot of values of the matrix

Image of the matrix and 3D plot of values of the matrix

Image of the matrix

3D plot of values of the matrix

We consider here two special cases of (

Let us consider

and the VFODI response, with initial condition

One should emphasise that the initial condition vector is infinite dimensional. This is characteristic for systems with “memory” of the state.

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 (

Let us consider order function given by

The order function

Now we consider decreasing order function given by

The graph of the order function (

Plots of the order function

The plot of the order function

The unit step response

Plots of the order function

The plot of the order function

The unit step response

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:

For the order function

In this part, we examine the homogenous response to

Let us take

and the VFODI response, with initial condition

The VFOI homogenous responses to

Plot of response to

Plot of response to

The VFOI homogenous responses to

Plot of response to

Plot of response to

Plots of sinusoidal order function

Sinusoidal order function

The VFOI homogenous response to

The VFOI homogenous response to

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 signals

The authors declare that they have no conflicts of interest.

This research was partially supported by the Bialystok University of Technology Grant S/WI/1/2016 (Dorota Mozyrska) and the Lodz University of Technology Grant 501∖12-24-1-5437 (Piotr Ostalczyk) and funded from the sources for research by Ministry of Science and Higher Education.