Systems with Fractional Variable-Order Difference Operator of Convolution Type and Its Stability

In the paper the Grunwald–Letnikov-type linear fractional variable order discrete-time systems are studied. The conditions for stability and instability are formulated. The regions of the systems stability are determined accordingly to locus of eigenvalues of a matrix associated to the considered system. DOI:  http://dx.doi.org/10.5755/j01.eie.24.5.21846


I. INTRODUCTION AND PRELIMINARIES
Recently, theory of fractional dynamical systems has become one of the essential tool for modelling in various technical disciplines, particularly in electrotechnics, chemistry, electrochemistry or viscoelasticity, see for instance [1]- [3].The systems with the fractional variable orders for continuous-time as well for discrete-time have been studied in [4]- [8].In the paper we deal with discretetime systems described by the Grünwald-Letnikov operator of convolution type, and with step 0 h  .This allow us to introduce the continuous case by using the limit with h tending to zero.We formulate and prove the conditions for the stability of the considered discrete-time systems.We determined the regions of location of eigenvalues of matrices associated to the systems in order to guarantee the asymptotic stability of the considered systems.These regions were presented as the images of some map associated with the GL-VFOBD-h.Some of the frames of these regions are illustrated in the example.
Definition 1 ([4]).For , k l   and a given order function ( ) v  we define the oblivion function, as a discrete function of two variables, by its values [ ( )] ( ) where 1 x  with an order function : {0} x be a continuous bounded real valued function.The Grünwald-Letnikov variable-, fractional-order differential operator of function with an order function : {0} where > 0 h , 0 t  .The sign     denotes the floor function.

II. DIFFERENCE SYSTEMS WITH FRACTIONAL VARIABLE ORDER
Let us consider the system with a variable-order the following form By ( 3) system ( 5) can be rewritten in the following recurrence way where 1 k  . where where ) (z . Proof.Note that ( 5) can be equivalently rewritten as where 0 k  .Then taking the one-side Z-transform of system (10), see more in [9], one gets the following equation in Z-domain where In order to get the thesis one needs to take inverse Ztransform of (12).
First let us formulate a condition for instability of system (5). where and = arg( )    , then system (5) is unstable.
Proof.Let  be given by (9) and Then by ( 13 . In practice it is enough to calculate the right side of (13) for some steps.Now let us give the necessary and sufficient condition for the stability of system (5).
Proof.Observe that if all roots of the equation are inside the unit circle, then system (5) (or equivalently, ( 6)) is asymptotically stable, see for instance [9].Note that guarantees system (5) (or equivalently, ( 6) or (10)) to be asymptotically stable.
Example 8. We consider in the example the system with three different order functions.The considered system has and matrix We present for systems and their orders the stability regions.In the same figures we illustrate by crosses eigenvalues of .
A  In Fig. 1 and Fig. 2 we present the frames of regions of stability for order functions given by       .Then, (9) can be rewritten in the following recurrence way where where ( ) z  is defined by ( 9).The sufficient condition for the asymptotic stability of equation ( 23) is as follows: Proposition 10.

IV. CONCLUSIONS
In the paper the stability of the Grünwald-Letnikov-type linear fractional-, variable-order discrete-time systems with step 0  h was studied.We investigate systems described by the Grünwald-Letnikov operator of convolution type, and with step 0  h .The convolution-type operator allow us to use Z-transform and formulate and prove the conditions for the stability of the considered discrete-time systems.We determined the regions of location of eigenvalues of matrices associated to the systems in order to guarantee the asymptotic stability of the considered systems.Summarizing we have produced conditions and descriptions for the stability of the Grünwald-Letnikov-type linear fractional-, variable-order discrete-time systems with step 0  h .

Formula ( 1 )
in Definition 1 can be equivalently expressed by the following recurrence with respect to : k   Manuscript received 4 January, 2018; accepted 7 April, 2018.The work was supported by Polish funds of National Science Center, granted on the basis of decision DEC-2016/23/B/ST7/03686.

Proposition 4 .
System (5) with initial value (0) n x   has the unique solution given by

1
) we get that  does not belong to the set { ( ) :| | 1} z z z   .Hence the eigenvalue  of the matrix A lies outside of the image of unit circle in the mapping : is unstable.Since values of coefficients [ ( )] ( ) k a k  are tending very fast to zero, one can approximate series in condition (8) by taking the right hand side for some steps.Observe that for > unit circle if and only if all eigenvalues of A belong to fact that all eigenvalues of A are inside the set restricted by the figure from[0,1] .In considered situations we receive asymptotically stable system. In Fig.3and Fig.4we present the frames of regions of stability for order functions given by

1
27)then (23) is asymptotically stable.Proof.We are interested in the set of roots of the equation 23) is asymptotically stable.Using the properties of order function and Proposition 10 one gets the following result: