# Explicit criteria for stability of fractional *h*-difference two-dimensional systems

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## Abstract

In the paper the explicit conditions for stability of linear fractional order *h*-difference systems with the Grünwald–Letnikov-type operator are presented. The state variables of the considered systems are taken from the plane. As the tool the \({\mathcal {Z}}\)-transform, which can be considered as an effective method for the stability analysis of linear systems, is used. The main result gives the sufficient and necessary condition for the asymptotic stability of the considered system according to the entries of the given matrix associated with the system.

## Keywords

Stability Fractional operator \({\mathcal {Z}}\)-Transform## 1 Introduction

The stability analysis is one of the most essential problems in dynamical systems as in control theory. Recently, it has been investigated for the fractional systems in some papers, see for example [1, 2, 3] for continuous-time case and [4, 5, 6] for the discrete-time case. Contrary to the continuous case, the stability theory of fractional difference equations is less developed. In [7, 8, 9] \({\mathcal {Z}}\)-transform is used as an effective method for stability analysis of linear discrete-time fractional order systems. The explicit stability conditions for a linear fractional difference system with the Caputo-type operator of order \(\alpha \in (0,1)\) are presented in [10]. The alternative (less convenient for practical purposes compared to [10]) stability conditions for the Caputo difference systems are presented in [11]. Additionally, in [10] the discussion concerning the stability behaviour of systems with the Riemann–Liouville-type difference operator is given. In [10] the stability conditions are considered as a direct extension of the classical results given in [12] for the difference systems.

Our main goal is to formulate the explicit stability conditions for the two–dimensional *h*-difference systems with the Grünwald–Letnikov-type operator since in many applications one needs explicit criteria on the entries of the matrix associated with the considered system. The formulated alternative stability conditions could be considered as an extension of Theorem 2.37 in [12] to the fractional systems.

The main results are stated in Theorem 1, where there are given the exact conditions on elements of the matrix of the considered system. The proposed method is not pretended to be better than the other’s method available in the literature, but it is stated as exact conditions on values of the matrix. This method is restricted to two-dimensional system as it is particularly often that some models are needed only for two-dimensional systems.

It should be also stressed that the Grünwald–Letnikov-type fractional *h*-difference operator is used in papers connected with applications of fractional differences in circuit systems, see for example [13].

The paper is organized as follows. In Sect. 2 the basic definitions of *h*-difference fractional order operator are given. Section 3 provides the properties of the \({\mathcal {Z}}\)-transform acting on fractional operators, especially on the Grünwald–Letnikov-type operator. In Sect. 4 the problem of stability of linear multi-parameter fractional difference control systems with the Grünwald–Letnikov *h*-difference operator is considered. Finally, Sect. 5 provides brief conclusions.

## 2 Preliminaries

Firstly, we recall some necessary definitions and notations used in the sequel therein the paper. Let \(a\in {\mathbb {R}}\). Then \((h{\mathbb {N}})_{a}:=\{a,a+h,a+2h,\ldots \}\). Let *x* denote a real function defined on \((h\mathbb {N})_{a}\), i.e. \(x:(h{\mathbb {N}})_{a}\rightarrow {\mathbb {R}}\). Let us recall the definition of the Grünwald–Letnikov-type difference operators, see for example [14, 15, 16] for cases \(h=1\) and extended for general case \(h>0\) in [17]. Here we present basic results for the case when \(h>1\).

### **Definition 1**

*Grünwald–Letnikov-type h-difference operator*\( _{a}{\varDelta }_{h}^{\alpha }\)

*of order*\(\alpha \)

*for a function*\( x:(h{\mathbb {N}})_{a}\rightarrow {\mathbb {R}}\) is defined by

The Grünwald–Letnikov-type *h*-difference operator can be extended to vector valued sequences in the componentwise manner, i.e. for \(x=\left( x_{1},x_{2}\right) : (h{\mathbb {N}})_{a}\rightarrow {\mathbb {R}}^{2}\) we have \(_{a}\varDelta _{h}^{\alpha } x=\left( _{a}\varDelta _{h}^{\alpha } x_{1}, _{a} \varDelta _{h}^{\alpha } x_{2}\right) \). If \(a=0\), then we will write: \({\varDelta }_{h}^{\alpha }:=_{0} {\varDelta }_{h}^{\alpha }\).

*Y*(

*z*) and a region of convergence, find the signal

*y*(

*n*) whose \({\mathcal {Z}}\)-transform is

*Y*(

*z*) and has the specified region of convergence. The presented \({\mathcal {Z}}\)-transform involves, by definition, only the values of

*y*(

*n*) for \(n \ge 0\). Similarly as in the case of the classical \({\mathcal {Z}}\)-transform, the sequence \(\{y(n)\}_{n\in {\mathbb {N}}_{0}}\) can be obtain from the function

*Y*(

*z*) by a process called the inverse \({\mathcal {Z}}\)-transform. This process is symbolically denoted as \(y(n)={\mathcal {Z}}^{-1}[Y(z)](n)\).

### **Proposition 1**

## 3 Systems

*h*-difference operator given by

*A*is a \(2\times 2\) matrix, with the initial condition

*A*and \(\det A\) is the determinant of

*A*. Moreover, let

*fractional characteristic function of matrix*

*A*

*for order*\(\alpha \)

*and step*\(h>0\). And the equation

*fractional characteristic equation of matrix*

*A*

*for order*\(\alpha \)em and step \(h>0\). Using the results given for the Grünwald–Letnikov difference systems of the form 3 presented in [7] we can state the following proposition:

### **Proposition 2**

*I*is the identity matrix and \(n\in {\mathbb {N}}_1\).

## 4 Stability

*equilibrium point*of the fractional difference system (3) if and only if

*A*is nonzero, then system (3) has only one equilibrium point \({x}^\mathrm{eq}=0\).

### **Definition 2**

- (a)
*stable*if, for each \(\epsilon >0\), there exists \(\delta =\delta \left( \epsilon \right) >0\) such that \(\left\| x_0\right\| < \delta \) implies \(\Vert x(nh)\Vert < \epsilon \), for all \(n\in {\mathbb {N}}_{0}\). - (b)
*attractive*if there exists \(\delta >0\) such that \(\left\| x_0\right\| < \delta \) implies \(\lim \limits _{n\rightarrow \infty } x(nh)=0.\) - (c)
*asymptotically stable*if it is stable and attractive.

*stable*/

*asymptotically stable*if their equilibrium points \({x}^\mathrm{eq}=0\) are stable/asymptotically stable.

### **Proposition 3**

### **Proposition 4**

*A*.

### *Proof*

The result is based on those presented in [9]. Here we only use them for matrix \(h^{\alpha }A\).

The main result that connects entries of the matrix *A* and the stability of the system is stated in the following theorem.

### **Theorem 1**

*R*are strictly inside the unit circle if and only if one of the set of conditions holds:

- 1)
\( \left\{ \begin{array}{l} -2^{\alpha +1}<h^{\alpha } \mathrm{tr}\, A<0 \\ 0<\det A\le \frac{1}{4}\mathrm{tr}^{2}\,A\\ 4^{\alpha }+(2h)^{\alpha } \mathrm{tr}\, A+h^{2\alpha }\det A>0 \end{array}\right. \),

- 2)
\( \left\{ \begin{array}{l} \mathrm{tr}\, A\ne 0 \\ \frac{1}{4}\mathrm{tr}^{2}\, A<\det A<\left( 2\left| \sin \frac{\psi -\alpha \frac{\pi }{2}}{2-\alpha } \right| \right) ^{2\alpha } \end{array}\right. \), where \(\psi =\arctan \frac{\sqrt{4\det A-\mathrm{tr}^2\,A}}{\mathrm{tr}\,A}\) or \(\psi =\pi +\arctan \frac{\sqrt{4\det A-\mathrm{tr}^2\,A}}{\mathrm{tr}\,A}\),

- 3)
\( \left\{ \begin{array}{l} \mathrm{tr}\, A=0\\ 0<\det A<\left( 2\left| \cos \frac{\pi }{2(2-\alpha )} \right| \right) ^{2\alpha } \end{array}\right. \).

### *Proof*

The proof is based on the cases that for real roots of \(p_{(A,\alpha ,h)}(\lambda )=0\) elements of *R* are strictly inside the unit circle if and only if \(\lambda = z\left( \frac{z-1}{hz}\right) ^{\alpha }\) are from the interval \((-2^{\alpha },0)\). Hence we need to find the solution of the systems of inequalities: \(p_1^{2}-4p_2\ge 0\), \(-2^\alpha < \frac{-p_1\pm \sqrt{p_1^{2}-4p_2}}{2}<0\), with \(p_1=- h^{\alpha } tr A, p_2=h^{2\alpha } det A\) that gives the set 1). The parts 2) and 3) are the version of Proposition 4.

The interesting and less difficult statement we receive for the order \(\alpha =\frac{1}{2}\) and \(h=1\). Then

### **Corollary 1**

*R*are strictly inside the unit circle if and only if one of the set of conditions holds:

- 1)
\( \left\{ \begin{array}{l} -2\sqrt{2}<\mathrm{tr}\, A<0 \\ 0<\det A\le \frac{1}{4}\mathrm{tr}^{2}\,A\\ 2+\sqrt{2} \mathrm{tr}\,A+\det A>0 \end{array}\right. \),

- 2)
\( \left\{ \begin{array}{l} \mathrm{tr}\, A\ne 0 \\ \frac{1}{4}\mathrm{tr}^{2}\, A<\det A<2\left| \sin \left( 2\psi /3-\frac{\pi }{6}\right) \right| \end{array}\right. \), where \(\psi =\arctan \frac{\sqrt{4\det A-\mathrm{tr}^{2}\,A}}{\mathrm{tr}\,A}\) or \(\psi =\pi +\arctan \frac{\sqrt{4\det A-\mathrm{tr}^{2}\,A}}{\mathrm{tr}\,A}\),

- 3)
\( \left\{ \begin{array}{l} \mathrm{tr}\, A=0\\ 0<\det A<1 \end{array}\right. \).

### *Remark 1*

It is known that for \(\alpha =1\) the definition of the fractional operator on the right hand side of Eq. (3) \(\left( \varDelta ^{\alpha } x\right) (n+1)=\sum _{s=0}^{n+1}c_s^{(1)}x(n+1-s)=x(n+1)-x(n)\), as \(c_0^{(1)}=1\), \(c_1^{(1)}=-1\) and \(c_s^{(1)}=0\) for \(s>1\). Moreover, it is easy to notice that conditions 1), for real case, from Proposition 1 coincide with those proposed in the book [12] for classical difference equation, i.e. \(x(n+1)=(I+A)x(n)\).

### *Example 1*

## 5 Conclusion

The paper describes sufficient and necessary conditions for the asymptotic stability of fractional difference two-dimensional systems with the Grünwald–Letnikov operator. These conditions depend on the entries of the given matrix associated with the considered system.

## Notes

### Acknowledgments

The work was supported by Bialystok University of Technology grant S/WI/1/2016.

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