Stability and l_∞ Performance Analysis for Nabla Discrete Fractional Linear Positive System with Time-Varying Delays

Abstract

In this paper, the stability and $l_{\infty }$-gain problem are investigated for the Nabla discrete fractional linear positive systems with bounded time-varying delays. First, a sufficient condition and a necessary condition are presented to ensure the system’s positivity. Then, based on the system’s positivity property, an asymptotically stable condition is established. Furthermore, it is demonstrated that the $l_{\infty }$-gain of such systems is determined by the system matrices and is independent of the magnitude of delays. Finally, numerical examples are provided to demonstrate the validity of the obtained results.

1. Introduction

Fractional calculus [1] refers to the integration and differentiation of non-integer orders. It possesses many unique and significant properties, such as memory hereditary [2] and infinite dimensionality [3]. Therefore, fractional calculus finds wide applications in various fields of science, including heat conduction [4], viscoelastic materials [5], thermal diffusion [6], capacitance [7], and so on.

Currently, research on fractional systems mainly focuses on continuous systems. However, due to the fact that discrete fractional calculus not only has good non-local properties but also saves computational resources and avoids performance loss caused by discretization, more and more attention has been paid to discrete fractional systems [8]. Currently, numerous researchers have proposed various definitions of fractional difference, with the Nabla difference being the most prevalent. A dynamic system that is modeled using the Nabla difference is referred to as a Nabla discrete fractional order system [9,10]. The infinite-dimensional properties of the Nabla discrete fractional system were examined in [11]. An explicit formulation of the state response for discrete fractional-order systems was presented in [12]. The stability concerns related to the Nabla discrete fractional system were addressed in [13,14]. More recently, Refs. [15,16] investigated the modeling and simulation aspects of the Nabla discrete fractional dynamic system.

On the other hand, physical systems encountered in the real world are characterized by non-negative quantities, such as the population of species in biological contexts, product pricing in economic frameworks, and matter density in physical sciences. These systems are commonly designated as positive systems, indicating that their state variables and outputs remain non-negative as long as the initial conditions and inputs are also non-negative. The preservation of non-negativity is a critical task of positive systems. Prior research has established non-negativity conditions for continuous fractional systems, as referenced in [17], and for discrete fractional systems, as detailed in [18,19]. Additionally, time delay is a significant factor influencing system performance. The non-negativity characteristics have been examined for continuous-time fractional linear systems with time-varying delays, as well as for discrete-time fractional linear systems with h delays, as discussed in [20] and [21], respectively. However, the conditions required to ensure the non-negativity of the Nabla discrete fractional linear system with time-varying delays have yet to be explored. Consequently, this paper aims to investigate the non-negativity conditions relevant to the Nabla discrete fractional linear system with bounded time-varying delays.

Stability analysis is one of the important problems in the dynamical system theory. Based on the Lyapunov function method, the stability and stabilization problems for continuous fractional linear positive systems were solved in [22]. The asymptotic stability conditions of continuous fractional linear positive systems with delays were established in [20,23]. Some conditions for practical stability of discrete fractional linear positive systems were established in [24,25]. The robust stability of discrete fractional linear positive systems was investigated in [26]. However, there are few results in the analysis of the stability of Nabla discrete fractional linear positive systems.

Moreover, the $L_{\infty }$-gain serves as a crucial performance indicator for positive systems. A necessary and sufficient condition for the $L_{\infty }$-gain in integer-order positive systems was established through linear programming in [27]. The $L_{\infty }$-gain issue for continuous fractional positive systems was explored in [28], while the same problem for these systems with bounded time-varying delays was examined in [20]. A notable observation is that the $L_{\infty }$-gain of continuous fractional positive systems is unaffected by the size of the delays or the fractional order of the system. This paper aims to investigate whether this characteristic also applies to discrete fractional systems.

Based on above introduction, this paper primarily explores the problems of non-negativity, stability, and $l_{\infty }$-gain of the Nabla discrete fractional linear positive systems with bounded time-varying delays. It applies the concept of Nabla fractional difference to establish conditions for non-negativity. By taking advantage of the monotonic characteristics of positive systems, this paper examines the asymptotic stability of the system. Lastly, it will analyze the $l_{\infty }$-gain problems of these systems using the comparison principle.

The remainder of this paper is structured as follows. Section 2 presents essential definitions and lemmas. Section 3 establishes the conditions necessary for characterizing non-negativity. Section 4 examines the concept of asymptotic stability. Section 5 addresses issues related to $l_{\infty }$-gain. In Section 6, we illustrate the effectiveness of these results through examples. In Section 7, conclusions are provided.

Notations: $\mathbb{R}$ denotes the set of real numbers, ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space, ${\mathbb{R}}_{++}^n$ denotes the set $\{x\in {\mathbb{R}}^n,x > 0\}$, $\mathbb{Z}$ stands for the set of integers, and ${\mathbb{Z}}_{+}$ denotes the set of positive integers. For any $a,b\in \mathbb{R}$, define, respectively, two sets ${\mathbb{N}}_{a+1}: = \{a + 1,a + 2,a + 3,\dots \}$, ${\mathbb{N}}_a^b: = \{a + 1,a + 2,a + 3,\dots ,b\}$. ${\mathbf{1}}_n$ denotes the column vector with all entries equal to 1. I denotes the identity matrix with appropriate dimensions. The ∞-norm of a column vector $x\in {\mathbb{R}}^n$ is defined as ${\parallel x\parallel }_{\infty }: = {max}_{1\le i\le n}\left|x_i\right|$. ${\left[A\right]}_{ij}$ denotes $(i,j)$-th entry of a matrix A. $A \ge 0$ denotes matrix A is non-negative. The M-matrix M is defined as its off-diagonal elements are all non-positive and its inverse $M^{-1} \ge 0$. The Metzler matrix is defined as all its off-diagonal elements are non-negative. The Hurwitz matrix is defined as all its eigenvalues have negative real part. $\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha }{\beta }}\right): = {\displaystyle \frac{\mathrm{\Gamma }(\alpha + 1)}{\mathrm{\Gamma }(\beta + 1)\mathrm{\Gamma }(\alpha - \beta + 1)}}$, where $\alpha$, $\beta \in \mathbb{R}$. $\mathrm{\Gamma }\left(z\right): = {\int }_0^{+\infty }{e}^{-t}{t}^{z-1}dt$ is the Gamma function. The $l_{\infty }$-norm of a vector-valued function $\omega :{\mathbb{N}}_{a+1}\to {\mathbb{R}}^n$ is defined as

$${\parallel \omega \parallel }_{l_{\infty }}:=ess\underset{k\in {\mathbb{N}}_{a+1}}{sup}{\parallel \omega \left(k\right)\parallel }_{\infty },$$

$\omega \left(k\right)\in l_{\infty }({\mathbb{N}}_{a+1},{\mathbb{R}}^n)$ denotes the space of vector-valued function $\omega \left(k\right)$ defined on ${\mathbb{N}}_{a+1}$ with finite $l_{\infty }$-norm.

2. Preliminaries

Here, we will introduce some basic definitions and lemmas on the Nabla discrete fractional calculus.

Definition 1 

([29]). For a function $f:{\mathbb{N}}_{a+1-n}\to \mathbb{R}$, its n-th order Nabla backward difference is defined as

$${\nabla }^{n}f\left(k\right)=\sum _{j=0}^n{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)f(k-j),$$

where $n\in {\mathbb{Z}}_{+}$, $k\in {\mathbb{N}}_{a+1}$, and $a\in \mathbb{R}$.

Definition 2 

([29]). For a function $f:{\mathbb{N}}_{a+1}\to \mathbb{R}$, its α-th order Nabla fractional sum is defined as

$${}_a\nabla _k^{-\alpha }f\left(k\right)=\sum _{j=0}^{k-a-1}{(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{-\alpha }{j}}\right)f(k-j),$$

where  $\alpha \in (n-1,n)$, $n\in {\mathbb{Z}}_{+}$, $k\in {\mathbb{N}}_{a+1}$, and  $a\in \mathbb{R}$.

Definition 3 

([29]). For a function $f:{\mathbb{N}}_{a+1-n}\to \mathbb{R}$, its Caputo and Riemann–Liouville Nabla fractional differences are defined, respectively, as

$${}_a{}^C\nabla _k^{\alpha }f\left(k\right)={}_a\nabla _k^{-(n-\alpha )}{\nabla }^{n}f\left(k\right)={}_a\nabla _k^{\alpha -n}{\nabla }^{n}f\left(k\right),$$

and

$$\begin{array}{c}\hfill {}_a{}^{RL}\nabla _k^{\alpha }f\left(k\right)={\nabla }^n{}_a\nabla _k^{-(n-\alpha )}f\left(k\right)={\nabla }^n{}_a\nabla _k^{\alpha -n}f\left(k\right),\end{array}$$

where  $\alpha \in (n-1,n)$, $n\in {\mathbb{Z}}_{+}$, $k\in {\mathbb{N}}_{a+1}$, and $a\in \mathbb{R}$.

In this paper, we mainly adopt the Nabla fractional difference in the Caputo sense.

The continuous Mittag–Leffler function is an important tool for analyzing fractional differential equations, which is a generalization of the exponential function. Similarly, the discrete Mittag–Leffler function is also a crucial tool for analyzing discrete fractional difference equations. Its definition is as follows.

Definition 4 

([29]). The one- and two-parameter discrete Mittag–Leffler functions are defined as follows:

$${\mathcal{F}}_{\alpha ,\beta }(\lambda ,k,a)=\sum _{j=0}^{+\infty }{\displaystyle \frac{{\lambda }^j}{\mathrm{\Gamma }(j\alpha +\beta )}}{(k-a)}^{\overline{j\alpha +\beta -1}},$$

and

$${\mathcal{F}}_{\alpha }(\lambda ,k,a)=\sum _{j=0}^{+\infty }{\displaystyle \frac{{\lambda }^j}{\mathrm{\Gamma }(j\alpha +1)}}{(k-a)}^{\overline{j\alpha }},$$

where  $\alpha >0$, $\beta \in \mathbb{R}$, $k\in {\mathbb{N}}_a$, $a\in \mathbb{R}$, and  $p^{\overline{q}}={\displaystyle \frac{\mathrm{\Gamma }(p+q)}{\mathrm{\Gamma }\left(p\right)}}$.

The $\mathcal{N}$-transform is an important method for analyzing discrete fractional systems. In the following, we will present the definition of the $\mathcal{N}$-transform and its properties relevant to this paper.

Definition 5 

([29]). For a function $f:{\mathbb{N}}_{a+1}\to \mathbb{R}$, its $\mathcal{N}$-transform is defined as

$${\mathcal{N}}_a\left\{f\right\}\left(s\right)=\sum _{k=1}^{+\infty }{(1-s)}^{k-1}f(k+a).$$

Lemma 1 

([29]). Let $\alpha \in (n-1,n)$ and $n\in {\mathbb{Z}}_{+}$. If the $\mathcal{N}$-transform of $f:{\mathbb{N}}_{a+1-n}\to \mathbb{R}$ converges for $|s - 1| < \rho$ ($\rho > 0$), then

$${\mathcal{N}}_a\left\{{}_a{}^C\nabla _k^{\alpha }f\right\}\left(s\right)=s^{\alpha }{\mathcal{N}}_a\left\{f\right\}\left(s\right)-\sum _{i=0}^{n-1}{s}^{\alpha -i-1}{\nabla }^i{f\left(k\right)|}_{k=a},$$

for $|s - 1| < \rho$.

Lemma 2 

([29] Convolution Theorem). Let $f,g:{\mathbb{N}}_{a+1}\to \mathbb{R}$, then

$${\mathcal{N}}_a\{f\ast g\}\left(s\right)={\mathcal{N}}_a\left\{f\right\}\left(s\right){\mathcal{N}}_a\left\{g\right\}\left(s\right)$$

where ∗ denotes the convolution operation, i.e.,

$$f\ast g=\sum _{j=a+1}^{k}f(k-j+a+1)g\left(j\right).$$

Lemma 3 

([11] Final Value Theorem). Let $f:{\mathbb{N}}_{a+1}\to \mathbb{R}$, $a\in \mathbb{R}$, if $F\left(s\right)={\mathcal{N}}_a\left\{f\right\}\left(s\right)$ and the poles of $sF\left(s\right)$ satisfy $|s - 1| > 1$, then,

$$\underset{k\to +\infty }{lim}f\left(k\right)=\underset{s\to 0}{lim}sF\left(s\right).$$

In order to investigate the non-negativity of Nabla discrete fractional linear systems with time-varying delays, the subsequent fundamental lemmas are introduced.

Lemma 4 

([11]). Consider the Nabla discrete fractional difference equation

$${}_a{}^C\nabla _k^{\alpha }x\left(k\right)=\lambda x\left(k\right)+f\left(k\right),x\left(a\right)=x_0,k\in {\mathbb{N}}_{a+1},$$

then it has a unique solution

$$x\left(k\right)=x_0{\mathcal{F}}_{\alpha }(\lambda ,k,a)+\sum _{s=a+1}^k{\mathcal{F}}_{\alpha ,\alpha }(\lambda ,k,a)f(k-s+a+1),$$

where $\alpha \in (0,1)$, $k\in {\mathbb{N}}_{a+1}$, $a\in \mathbb{R}$, $x\left(k\right)\in \mathbb{R}$, and $f\left(k\right)\in \mathbb{R}$.

Lemma 5 

([30]). For any $\alpha \in (0,1)$, $m_j\left(\alpha \right)={(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha -1}{j}}\right)$ is the j-th coefficient of fractional $(1-\alpha )$-th sum, then one has $m_j\left(\alpha \right)>0$ and $m_j\left(\alpha \right)-m_{j+1}\left(\alpha \right)>0$.

3. Positivity Analysis

In this section, we will provide the non-negativity conditions of Nabla discrete fractional linear systems.

Consider the following Nabla discrete fractional linear system with time-varying delays.

$$\begin{array}{c}\left\{\begin{array}{c}{}_a{}^C\nabla _k^{\alpha }x\left(k\right)=Ax\left(k\right)+Bx(k-d\left(k\right))+E\omega \left(k\right),\hfill \\ y\left(k\right)=Cx\left(k\right)+Dx(k-\tau \left(k\right))+F\omega \left(k\right),k\in {\mathbb{N}}_{a+1},\hfill \\ x\left(s\right)=\varphi \left(s\right),s\in {\mathbb{N}}_{a-max\{d,\tau \}}^a,\hfill \end{array}\right.\hfill \end{array}$$

(1)

where $a\in \mathbb{R}$ and $\alpha \in (0,1)$. $x\left(k\right)\in {\mathbb{R}}^n$, $\omega \left(k\right)\in {\mathbb{R}}^p$, and $y\left(k\right)\in {\mathbb{R}}^q$ represent the system state, input, and output, respectively. A, B, E, C, D, and F are known constant real matrices with appropriate dimensions. $\varphi \left(s\right)$ is the initial condition. $d\left(k\right),\tau \left(k\right):{\mathbb{N}}_{a+1}\to \mathbb{Z}$ are time-varying delays and satisfy $0\le d\left(k\right)\le d$, $0\le \tau \left(k\right)\le \tau$ ($d,\tau \in \mathbb{Z}$).

In order to achieve non-negativity conditions of system (1), we firstly give the definition of positivity.

Definition 6. 

System (1) is positive, if for any initial condition $\varphi \left(s\right)\ge 0(s\in {\mathbb{N}}_{a-max\{d,\tau \}}^a)$ and input $\omega \left(k\right)\ge 0(k\in {\mathbb{N}}_{a+1})$, its solution $x(k,\varphi )\ge 0$ and output $y\left(k\right)\ge 0$ for all $k\in {\mathbb{N}}_{a+1}$.

In the following, a criterion will be given to determine the positivity of system (1).

Theorem 1. 

System (1) is positive, if $I-A$ (for $d\left(k\right)\not\equiv 0$) or $I-(A+B)$ (for $d\left(k\right)\equiv 0$) is M-matrix, $B\ge 0$, $E\ge 0$, $C\ge 0$, $D\ge 0$, and $F\ge 0$.

Proof. 

We will show that for any initial condition $\varphi \left(s\right)\ge 0(s\in {\mathbb{N}}_{a-max\{d,\tau \}}^a)$ and input $\omega \left(k\right)\ge 0$, then state variable $x\left(k\right)\ge 0$ and output $y\left(k\right)\ge 0$ hold for all $k\in {\mathbb{N}}_{a+1}$.

Based on the Nabla fractional difference definition in the Caputo sense, one has

$$\begin{array}{cc}\hfill {}_a{}^C\nabla _k^{\alpha }x\left(k\right)& ={}_a\nabla _k^{\alpha -1}\nabla ^{1}x\left(k\right)\hfill \\ \hfill & =\sum _{j=0}^{k-a-1}{m}_j\left(\alpha \right)[x(k-j)-x(k-j-1)]\hfill \\ \hfill & =x\left(k\right)+\sum _{j=0}^{k-a-2}[m_{j+1}\left(\alpha \right)-m_j\left(\alpha \right)]x(k-j-1)-m_{k-a-1}\left(\alpha \right)x\left(a\right),\hfill \end{array}$$

where $m_j\left(\alpha \right)={(-1)}^j\left({\displaystyle \genfrac{}{}{0pt}{}{\alpha -1}{j}}\right)$ is the j-th coefficient of $(1-\alpha )$-th fractional sum of x. Then the state equation of system (1) can be re-written as

$$\begin{array}{cc}\hfill & x\left(k\right)+\sum _{j=0}^{k-a-2}[m_{j+1}\left(\alpha \right)-m_j\left(\alpha \right)]x(k-j-1)-m_{k-a-1}\left(\alpha \right)x\left(a\right)\hfill \\ \hfill =& Ax\left(k\right)+Bx(k-d(k\left)\right)+E\omega \left(k\right).\hfill \end{array}$$

(2)

For the case of the delay $d\left(k\right)\not\equiv 0$, Equation (2) can be rewritten as

$$x\left(k\right)={(I-A)}^{-1}\mathrm{\Psi }\left(k\right),$$

(3)

where $\mathrm{\Psi }\left(k\right)={\sum }_{j=0}^{k-a-2}[m_j\left(\alpha \right)-m_{j+1}\left(\alpha \right)]x(k-j-1)+m_{k-a-1}\left(\alpha \right)x\left(a\right)+Bx(k-d\left(k\right))+E\omega \left(k\right)$. Using mathematical induction method, we will show that $\mathrm{\Psi }\left(k\right)\ge 0$ holds for all $k\in {\mathbb{N}}_{a+1}$.

First, for $k=a+1$, we have

$$\mathrm{\Psi }(a+1)=x\left(a\right)+Bx(a+1-d(a+1\left)\right)+E\omega (a+1).$$

Since $x\left(a\right)\ge 0$, $x(a+1-d(a+1\left)\right)\ge 0$, $\omega (a+1)\ge 0$, $B\ge 0$, and $E\ge 0$, then $\mathrm{\Psi }(a+1)\ge 0$.

Next, assume that $\mathrm{\Psi }\left(k\right)\ge 0$ for $k\le n(n\in {\mathbb{N}}_{a+1})$.

Then, for the case of $k=n+1$, one has

$$\begin{array}{cc}\hfill \mathrm{\Psi }(n+1)=& \sum _{j=0}^{n-a-1}[m_j\left(\alpha \right)-m_{j+1}\left(\alpha \right)]x(n-j)+m_{n-a}\left(\alpha \right)x\left(a\right)\hfill \\ & +Bx(n+1-d(n+1\left)\right)+E\omega (n+1).\hfill \end{array}$$

Since $\mathrm{\Psi }\left(k\right)\ge 0$ and $(I-A)$ is an M-matrix, then $x\left(k\right)={(I-A)}^{-1}\mathrm{\Psi }\left(k\right)\ge 0$ for all $k\in {\mathbb{N}}_{a+1}^n$. From Lemma 5, one can know $m_j\left(\alpha \right)-m_{j+1}\left(\alpha \right)>0$ and $m_j\left(\alpha \right)>0$ for all $j\ge 0$. And since $x\left(a\right)\ge 0$, $x(n+1-d(n+1\left)\right)\ge 0$, $\omega (n+1)\ge 0$, $B\ge 0$, and $E\ge 0$, one can derive that $\mathrm{\Psi }(n+1)\ge 0$. Therefore, one can obtain $\mathrm{\Psi }\left(k\right)\ge 0$ for all $k\in {\mathbb{N}}_{a+1}$.

And, since $I-A$ is an M-matrix, then ${(I-A)}^{-1}\ge 0$. Thus, system equality (3) implies that $x\left(k\right)\ge 0$ for all $k\in {\mathbb{N}}_{a+1}$.

For the case of delay $d\left(k\right)\equiv 0$, Equation (2) can be written as

$$x\left(k\right)={(I-(A+B))}^{-1}\stackrel{-}{\mathrm{\Psi }}\left(k\right),$$

(4)

where

$$\begin{array}{c}\hfill \stackrel{-}{\mathrm{\Psi }}\left(k\right)=\sum _{j=0}^{k-a-2}[m_j\left(\alpha \right)-m_{j+1}\left(\alpha \right)]x(k-j-1)+m_{k-a-1}\left(\alpha \right)x\left(a\right)+E\omega \left(k\right).\end{array}$$

As the case of $d\left(k\right)\equiv 0$, using the mathematical induction method again, we can also prove that $\stackrel{-}{\mathrm{\Psi }}\left(k\right)\ge 0$ for all $k\in {\mathbb{N}}_{a+1}$. Since $I-(A+B)$ is an M-matrix, then ${(I-(A+B))}^{-1}\ge 0$. Thus, Equation (4) implies that $x\left(k\right)\ge 0$.

Finally, since $\varphi \left(s\right)\ge 0(s\in {\mathbb{N}}_{a-max\{d,\tau \}}^a)$, $x\left(k\right)\ge 0(k\in {\mathbb{N}}_{a+1})$, and system matrices $C\ge 0$, $D\ge 0$, we have $y\left(k\right)\ge 0(k\in {\mathbb{N}}_{a-max\{d,\tau \}})$. This completes the whole proof. □

Next, a necessary condition of the positive system (1) is given as follows.

Theorem 2. 

System (1) is positive, then ${(I-A)}^{-1}\ge 0$, ${(I-A)}^{-1}B\ge 0$, ${(I-A)}^{-1}E\ge 0$, $C\ge 0$, $D\ge 0$, $F\ge 0$.

Proof. 

Assume that system (1) is positive. We will first show that ${(I-A)}^{-1}\ge 0$. Let $\omega (a+1)=0$, $x(a+1-d(a+1\left)\right)=0$, one has

$${}_a{}^C\nabla _{a+1}^{\alpha }x(a+1)=x(a+1)-x\left(a\right)=Ax(a+1).$$

The above equality can be rewritten as

$$x(a+1)={(I-A)}^{-1}x\left(a\right).$$

Since the initial condition $x\left(a\right)\ge 0$ can be arbitrary, and $x(a+1)$ exists, the inverse of $(I-A)$ exists. Therefore, ${(I-A)}^{-1}\ge 0$ for $x(a+1)\ge 0$, and $x\left(a\right)\ge 0$ is arbitrary.

In the following, we will prove that ${(I-A)}^{-1}B\ge 0$. Choose $x\left(a\right)=0$, $\omega (a+1)=0$, $x(a+1-d(a+1))=e_j$ ($e_j$ denotes the unit vector in ${\mathbb{R}}^n$ with the j-coordinate equals to 1). Then, $x(a+1)={(I-A)}^{-1}Bx(a+1-d(a+1))={(I-A)}^{-1}Be\left(j\right)$, since $x(a+1)\ge 0$, ${(I-A)}^{-1}B\ge 0$.

Finally, we will show that ${(I-A)}^{-1}E\ge 0$. By choosing $x\left(a\right)=0$ and $x(a+1-d(a+1\left)\right)=0$, we have $x(a+1)={(I-A)}^{-1}E\omega (a+1)$. Since $\omega \left(a\right)\ge 0$ is arbitrary and $x(a+1)\ge 0$, then ${(I-A)}^{-1}E\ge 0$.

Similarly, one can derive $C\ge 0$, $D\ge 0$, $F\ge 0$. □

4. Stability Analysis

In this section, we will investigate the stability problems of the Nabla discrete fractional linear positive system (1) with input $\omega \left(k\right)=0$, which has the following form.

$$\begin{array}{c}\left\{\begin{array}{c}{}_a{}^C\nabla _k^{\alpha }x\left(k\right)=Ax\left(k\right)+Bx(k-d\left(k\right)),\hfill \\ x\left(s\right)=\varphi \left(s\right),s\in {\mathbb{N}}_{a-d}^a.\hfill \end{array}\right.\hfill \end{array}$$

(5)

Before discussing the stable problem, it is essential to establish certain supplementary definitions and lemmas that will facilitate the stability analysis.

Definition 7 

([30]). A matrix $A\in {\mathbb{R}}^{n\times n}$ is called fractional Schur if and only if all the eigenvalues ${\lambda }_i\left(A\right)$ of matrix A satisfy ${\lambda }_i\left(A\right)\in \{s^{\alpha }:|s^{\alpha }-1|>1\},i=1,2,\dots ,l(l\le n)$, where $\alpha \in (0,1)$.

Lemma 6 

([30]). Let $A\in {\mathbb{R}}^{n\times n}$, $\alpha \in (0,1)$, the Nabla discrete fractional linear system ${}_a{}^C\nabla _k^{\alpha }x\left(k\right)=Ax\left(k\right)$ is asymptotically stable if and only if the matrix A is fractional Schur.

Lemma 7 

([30]). Let $A\in {\mathbb{R}}^{n\times n}$, $\alpha \in (0,1)$, the Nabla discrete fractional linear system ${}_a{}^C\nabla _k^{\alpha }x\left(k\right)=Ax\left(k\right)$ is positive if and only if $I-A$ is an M-matrix.

Lemma 8 

([31]). Let $A\in {\mathbb{R}}^{n\times n}$ be Metzler. Then, the following conditions are equivalent:

(i) 

A is Hurwitz;

(ii) 

A is non-singular and $A^{-1}\le 0$;

(iii) 

There exists a vector $\lambda \in {\mathbb{R}}_{++}^n$ such that ${\lambda }^{T}A<0$;

(iv) 

There exists a vector $\eta \in {\mathbb{R}}_{++}^n$ such that $A_{\eta }<0$.

Based on the above lemmas, a necessary condition for the asymptotic stability of the positive system ${}_a{}^C\nabla _k^{\alpha }x\left(k\right)=Ax\left(k\right)$ with $0<\alpha <1$ can be obtained as follows.

Theorem 3. 

The Nabla discrete fractional linear positive system

$${}_a{}^C\nabla _k^{\alpha }x\left(k\right)=Ax\left(k\right),k\in {\mathbb{N}}_{a+1}$$

(6)

with $0<\alpha <1$, is asymptotically stable, then A is Hurwitz.

Proof. 

Since system (6) is positive, it follows from Lemma 7 that $I-A$ is an M-matrix, which implies $A-I$ is Metzler and ${(A-I)}^{-1}\le 0$. Then, from Lemma 8, one can derive that the matrix $A-I$ is Hurwitz.

Let $\lambda (A-I)$ and $\lambda \left(A\right)$ denote the eigenvalues of matrices $A-I$ and A, respectively, then, one has

$$Re\left(\lambda \right(A-I\left)\right)<0andRe\left(\lambda \right(A\left)\right)<1.$$

(7)

Since system (6) is asymptotically stable, it follows from Lemma 6 that A is fractional Schur, that is,

$$\lambda \left(A\right)\in \{s^{\alpha }:|s^{\alpha }-1|>1\}.$$

(8)

Note that A is Metzler, then the largest eigenvalue ${\lambda }^{\ast }\left(A\right)$ of A is real. Thus, from (7) and (8), one can derive that A is Hurwitz. □

Comparison Lemma is a crucial tool for analyzing positive systems. The Comparison Lemma for the Nabla discrete fractional linear system (5) is presented as follows.

Lemma 9. 

Assume that system (5) is positive. Let $x(k,{\varphi }_1)$ and $x(k,{\varphi }_2)$ represent the system trajectories under initial conditions ${\varphi }_1\left(s\right)$ and ${\varphi }_2\left(s\right)$, respectively. Then, $x(k,{\varphi }_1)\le x(k,{\varphi }_2)$ holds when ${\varphi }_1\left(s\right)\le {\varphi }_2\left(s\right)$ for $s\in {\mathbb{N}}_{a-d}^a$.

Proof. 

Let $e\left(k\right)=x(k,{\varphi }_2)-x(k,{\varphi }_1)$; one has error system

$${}_a{}^C\nabla _k^{\alpha }e\left(k\right)=Ae\left(k\right)+Be(k-d\left(k\right)).$$

(9)

Since system (5) is positive, then it follows from Theorem 2 that ${(I-A)}^{-1}\ge 0$ and ${(I-A)}^{-1}B\ge 0$, and since the initial condition ${\varphi }_2\left(s\right)-{\varphi }_1\left(s\right)\ge 0$, $s\in {\mathbb{N}}_{a-d}^a$, it follows from Theorem 1 that the error system (9) is positive. Thus, $e\left(k\right)\ge 0$ for all $k\in {\mathbb{N}}_{a+1}$, that is $x(k,{\varphi }_1)\le x(k,{\varphi }_2)$ for all $k\in {\mathbb{N}}_{a+1}$. □

Next, we consider the following system

$$\begin{array}{c}\left\{\begin{array}{c}{}_a{}^C\nabla _k^{\alpha }\overline{x}\left(k\right)=A\overline{x}\left(k\right)+B\overline{x}(k-d),k\in {\mathbb{N}}_{a+1}\hfill \\ \overline{x}\left(s\right)=\varphi \left(s\right),s\in {\mathbb{N}}_{a-d}^a,\hfill \end{array}\right.\hfill \end{array}$$

(10)

where the system matrices A and B and initial condition $\overline{x}\left(s\right)=\varphi \left(s\right)$ are defined same as in system (5), constant d is the supermum of time delay $d\left(k\right)$, and $k\in {\mathbb{N}}_{a+1}$.

In the following, the relationship between the trajectories of systems (5) and (10) will be given.

Lemma 10. 

Consider systems (5) and (10), assume that $I-A$ is an M-matrix, $B\ge 0$, $A+B$ is Hurwitz, the initial condition $\varphi \left(s\right)=\lambda (s\in {\mathbb{N}}_{a-d}^a)$ and λ satisfies that $(A+B)\lambda <0$, then one has

(i) 

$\overline{x}\left(k\right)\le \lambda$ for all $k\in {\mathbb{N}}_{a+1}$;

(ii) 

$\overline{x}\left(k_2\right)\le \overline{x}\left(k_1\right)$ for all $k_2\ge k_1$, $k_1,k_2\in {\mathbb{N}}_{a+1}$;

(iii) 

$\underset{k\to +\infty }{lim}\overline{x}\left(k\right)=0$;

(iv) 

$x\left(k\right)\le \overline{x}\left(k\right)$ for all $k\in {\mathbb{N}}_{a+1}$.

where $x\left(k\right)$ and $\overline{x}\left(k\right)$ represent the state trajectories of systems (5) and (10) under the initial condition $\varphi \left(s\right)=\lambda (s\in {\mathbb{N}}_{a-d}^a)$, respectively.

Proof. 

(i) Define $e\left(k\right)=\lambda -\overline{x}\left(k\right)$, then, from system (10), one has

$${}_a{}^C\nabla _k^{\alpha }e\left(k\right)=Ae\left(k\right)+Be(k-d)-(A+B)\lambda .$$

(11)

Since the initial condition $e\left(s\right)=\lambda -\lambda =0(s\in {\mathbb{N}}_{a-d}^a)$, $-(A+B)\lambda \ge 0$, $I-A$ is an M-matrix, and $B\ge 0$, then it follows from Theorem 1 that the error system (11) is positive, thus $e\left(k\right)\ge 0$ for all $k\in {\mathbb{N}}_{a+1}$, that is, $\overline{x}\left(k\right)\le \lambda$ holds for all $k\in {\mathbb{N}}_{a+1}$.

(ii) For any $c\in {\mathbb{Z}}_{+}$, define $\overline{e}\left(k\right)=\overline{x}\left(k\right)-\overline{x}(k+c)$ for all $k\in {\mathbb{N}}_{a+1}$, then, based on system system (10), one has

$${}_a{}^C\nabla _k^{\alpha }\overline{e}\left(k\right)=A\overline{e}\left(k\right)+B\overline{e}(k-d).$$

(12)

Since the initial condition $\overline{e}\left(s\right)=\overline{x}\left(k\right)-\overline{x}(k+c)=\lambda -\overline{x}(k+c)\ge 0(s\in {\mathbb{N}}_{a-d}^a)$, $I-A$ is an M-matrix, $B\ge 0$. From Theorem 1, it is known that the error system (12) is positive. Thus, $\overline{x}\left(k\right)\ge \overline{x}(k+c)$ holds for all $k\in {\mathbb{N}}_{a+1}$, which implies $\overline{x}\left(k_2\right)\le \overline{x}\left(k_1\right)$ for all $k_2\ge k_1$ $(k_1,k_2\in {\mathbb{N}}_{a+1})$.

(iii) From conditions (i) and (ii), one can deduce that $\overline{x}\left(k\right)$ is monotonically non-increasing and bounded. Therefore, $\underset{k\to +\infty }{lim}\overline{x}\left(k\right)$ exists. By considering the linearity and positivity of system (10), one can infer that ${}_a{}^C\nabla _k^{\alpha }\overline{x}\left(k\right)$ is bounded.

Let $\underset{k\to +\infty }{lim}\overline{x}\left(k\right)={\overline{x}}_0$, it follows from Lemma 3 that

$$\underset{s\to 0}{lim}s\overline{X}\left(s\right)=\underset{k\to +\infty }{lim}\overline{x}\left(k\right)={\overline{x}}_0,$$

and

$$\underset{s\to 0}{lim}s{\mathcal{N}}_a{\{}_a^C{\nabla }_k^{\alpha }\overline{x}\left(k\right)\}=\underset{k\to +\infty }{lim}{}_a{}^C\nabla _k^{\alpha }\overline{x}\left(k\right),$$

where $\overline{X}\left(s\right)={\mathcal{N}}_{a}x\left(k\right)$. From Lemma 1, one has ${\mathcal{N}}_a\left\{{}_a{}^C\nabla _k^{\alpha }\overline{x}\left(k\right)\right\}=s^{\alpha }\overline{X}\left(s\right)-s^{\alpha -1}\overline{x}\left(a\right)$. Then,

$$\begin{array}{cc}\hfill (A+B){\overline{x}}_0& =\underset{s\to 0}{lim}s{\mathcal{N}}_a{\{}_a^C{\nabla }_k^{\alpha }\overline{x}\left(k\right)\}\hfill \\ \hfill & =\underset{s\to 0}{lim}{s}^{\alpha }[s\overline{x}\left(s\right)-\overline{x}\left(a\right)]=0.\hfill \end{array}$$

Therefore, $(A+B){\overline{x}}_0=0$. Since $A+B$ is Hurwitz, it is invertible. Thus, ${\overline{x}}_0=0$.

(iv) Define $\widehat{e}\left(k\right)=\overline{x}\left(k\right)-x\left(k\right)$ and consider systems (5) and (10), one has

$${}_a{}^C\nabla _k^{\alpha }\widehat{e}\left(k\right)=A\widehat{e}\left(k\right)+B\widehat{e}(k-d\left(k\right))+B(\overline{x}(k-d)-\overline{x}(k-d\left(k\right))).$$

(13)

Since the initial condition $\widehat{e}\left(s\right)=\overline{x}\left(s\right)-x\left(s\right)=\lambda -\lambda =0(s\in {\mathbb{N}}_{a-d}^a)$, $\overline{x}(k-d)-\overline{x}(k-d\left(k\right))\ge 0$ and $I-A$ is an M-matrix, $B\ge 0$. Then, from Theorem 1, one can deduce that the error system (13) is positive. Thus, $x\left(k\right)\le \overline{x}\left(k\right)$ for all $k\in {\mathbb{N}}_{a+1}$. □

Based on the above analysis, we can now derive a sufficient and necessary condition for the asymptotic stability of system (5).

Theorem 4. 

Consider system (5) and assume that $I-A$ and $I-(A+B)$ are M-matrix and $B\ge 0$. Then, system (5) is asymptotically stable if and only if $A+B$ is Hurwitz.

Proof. 

Conditions $I-A$ being an M-matrix and $B\ge 0$ imply that system (5) is positive.

(Sufficiency) Let $A+B$ be Hurwitz. Firstly, we prove that system (5) is stable. For any $\epsilon >0$, there exists a vector $\lambda \in {\mathbb{R}}_{++}^n$ and an initial condition $\varphi \left(s\right)$ such that $0\le \varphi \left(s\right)\le \lambda \le \epsilon {\mathbf{1}}_n$ and $(A+B)\lambda <0$. From Lemmas 9 and 10, one has $x(k,\varphi )\le x(k,\lambda )\le \overline{x}(k,\lambda )\le \lambda \le \epsilon {\mathbf{1}}_n$. Therefore, ${\parallel x(k,\varphi )\parallel }_{\infty }\le \epsilon$.

In the following, we will prove the convergence of positive system (5). For any initial condition $\varphi \left(s\right)\ge 0$, there exists a vector $\lambda \in {\mathbb{R}}_{++}^n$ such that $0\le \varphi \left(s\right)\le \lambda$ and $(A+B)\lambda <0$. Based on Lemmas 9 and 10, one has $x(k,\varphi )\le x(k,\lambda )\le \overline{x}(k,\lambda )$ and $\underset{k\to +\infty }{lim}\overline{x}(k,\lambda )=0$. Thus, $\underset{k\to +\infty }{lim}x(k,\varphi )=0$.

(Necessity) Consider system (5) is positive and asymptotically stable. Let $d\left(k\right)=0$, then ${}_a{}^C\nabla _k^{\alpha }x\left(k\right)=(A+B)x\left(k\right)$ is positive and asymptotically stable. It follows from Theorem 3 that $A+B$ is Hurwitz. □

Remark 1. 

According to Theorem 4, one can know that the stability of Nabla discrete fractional positive systems does not depend on the fractional order α or the upper bound d of delay $d\left(k\right)$. This finding aligns with the results for integer positive systems, as discussed in [32].

5. l_∞-Gain Analysis

In this section, we focus on the $l_{\infty }$-gain analysis of Nabla discrete fractional linear positive systems (1). To begin, we make the following assumptions.

Assumption 1. 

Assume that the Nabla discrete fractional linear system (1) satisfies the following conditions:

(i) 

System matrices $I-A$ and $I-(A+B)$ are M-matrix, $B\ge 0$, $E\ge 0$, $C\ge 0$, $D\ge 0$, and $F\ge 0$;

(ii) 

Matrix $A+B$ is Hurwitz;

(iii) 

The initial condition $\varphi \left(s\right)=0$ for all $s\in {\mathbb{N}}_{a-max\{d,\tau \}}^a$.

This assumption mainly explains that the system being analyzed in this section is positive and that the initial conditions are set to zero.

Now, the $l_{\infty }$-gain of a system is defined as follows.

Definition 8 

([30]). Consider a linear system $\Sigma :\omega \left(k\right)\to y\left(k\right)$, where $\omega \left(k\right)\in l_{\infty }({\mathbb{N}}_{a+1},{\mathbb{R}}^p)$, $y\left(k\right)\in l_{\infty }({\mathbb{N}}_{a+1},{\mathbb{R}}^q)$ represent the input and output, respectively. Then, the $l_{\infty }$-gain of system Σ is defined by

$${∥\Sigma ∥}_{l_{\infty }}:=\underset{{∥\omega ∥}_{l_{\infty }}=1}{sup}{∥y∥}_{l_{\infty }}=\underset{{∥\omega ∥}_{l_{\infty }}\ne 0}{sup}{\displaystyle \frac{{∥y∥}_{l_{\infty }}}{{∥\omega ∥}_{l_{\infty }}}}.$$

The following comparison principles are an important tool for analyzing system $l_{\infty }$ performance.

Lemma 11. 

Consider system (1) under Assumption 1. Let $y_1\left(k\right)$ and $y_2\left(k\right)$ be the outputs of system (1) under inputs ${\omega }_1\left(k\right)$ and ${\omega }_2\left(k\right)$, respectively. Then, for any $k\in {\mathbb{N}}_{a+1}$, if ${\omega }_1\left(k\right)\le {\omega }_2\left(k\right)$, it implies that output $y_1\left(k\right)\le y_2\left(k\right)$.

Proof. 

Let $\omega \left(k\right)={\omega }_2\left(k\right)-{\omega }_1\left(k\right)$ and $y\left(k\right)=y_2\left(k\right)-y_1\left(k\right)$. If the Nabla discrete fractional system (1) is linear and positive, then $y\left(k\right)\ge 0$ when $\omega \left(k\right)\ge 0$ for any $k\in {\mathbb{N}}_{a+1}$, indicating that $y_1\left(k\right)\le y_2\left(k\right)$ when ${\omega }_1\left(k\right)\le {\omega }_2\left(k\right)$ for any $k\in {\mathbb{N}}_{a+1}$. □

Based on Definition 8 and Lemma 11, we only need to study the output performance of the Nabla discrete fractional system (1) under input $\omega \left(k\right)={\mathbf{1}}_p$,

$$\begin{array}{c}\left\{\begin{array}{c}{}_a{}^C\nabla _k^{\alpha }x\left(k\right)=Ax\left(k\right)+Bx(k-d\left(k\right))+E{\mathbf{1}}_p,\hfill \\ y\left(k\right)=Cx\left(k\right)+Dx(k-\tau \left(k\right))+F{\mathbf{1}}_p.\hfill \end{array}\right.\hfill \end{array}$$

(14)

Meanwhile, we consider system (14) with constant delays $d\left(k\right)=d$, $\tau \left(k\right)=\tau$ and $d\left(k\right)=0$, $\tau \left(k\right)=0$, respectively, as follows

$$\begin{array}{c}\left\{\begin{array}{c}{}_a{}^C\nabla _k^{\alpha }\tilde{x}\left(k\right)=A\tilde{x}\left(k\right)+B\tilde{x}(k-d)+E{\mathbf{1}}_p,\hfill \\ \tilde{y}\left(k\right)=C\tilde{x}\left(k\right)+D\tilde{x}(k-\tau )+F{\mathbf{1}}_p,\hfill \end{array}\right.\hfill \end{array}$$

(15)

and

$$\begin{array}{c}\left\{\begin{array}{c}{}_a{}^C\nabla _k^{\alpha }\underset{\sim }{x}\left(k\right)=(A+B)\underset{\sim }{x}\left(k\right)+E{\mathbf{1}}_p,\hfill \\ \underset{\sim }{y}\left(k\right)=(C+D)\underset{\sim }{x}\left(k\right)+F{\mathbf{1}}_p.\hfill \end{array}\right.\hfill \end{array}$$

(16)

The following lemmas describe the relationships among the trajectories of systems (14)–(16).

Lemma 12. 

Under Assumption 1, the trajectory of system (15) satisfies

(i) 

$\tilde{x}\left(k_2\right)\ge \tilde{x}\left(k_1\right)$ for any $k_2>k_1>a$, $k_1,k_2\in {\mathbb{N}}_{a+1}$;

(ii) 

$\underset{k\to +\infty }{lim}\tilde{x}\left(k\right)=-{(A+B)}^{-1}E{\mathbf{1}}_p$.

Proof. 

(i) Given any integer $c\in {\mathbb{Z}}_{+}$, define error variate $\tilde{e}\left(k\right)=\tilde{x}(k+c)-\tilde{x}\left(k\right)$, then $\tilde{e}\left(k\right)$ satisfies that

$${}_a{}^C\nabla _k^{\alpha }\tilde{e}\left(k\right)=A\tilde{e}\left(k\right)+B\tilde{e}(k-d).$$

(17)

Note that $\tilde{e}\left(s\right)=\tilde{x}(s+c)-\tilde{x}\left(s\right)=0$ for all $s\in {\mathbb{N}}_{a-max\{d,\tau \}}^a$ since $\tilde{x}(s+c)=0$ and $\tilde{x}\left(s\right)=0$. Due to $I-A$ is an M-matrix and $B\ge 0$, from Theorem 1, we can deduce that system (17) is positive, that is, $\tilde{e}\left(k\right)\ge 0$ for all $k\in {\mathbb{N}}_{a+1}$. Thus, $\tilde{x}(k+c)\ge \tilde{x}\left(k\right)$, which means that $\tilde{x}\left(k_2\right)\ge \tilde{x}\left(k_1\right)$.

($ii$) Let ${\tilde{x}}_0=-{(A+B)}^{-1}E{\mathbf{1}}_p$ and define error signal $\widehat{e}\left(k\right)={\tilde{x}}_0-\tilde{x}\left(k\right)$, then $\widehat{e}\left(k\right)$ satisfies that

$$\begin{array}{c}\left\{\begin{array}{c}{}_a{}^C\nabla _k^{\alpha }\widehat{e}\left(k\right)=A\widehat{e}\left(k\right)+B\widehat{e}(k-d),\hfill \\ \widehat{e}\left(s\right)={\tilde{x}}_0,s\in {\mathbb{N}}_{a-d}^a.\hfill \end{array}\right.\hfill \end{array}$$

(18)

Since $I-A$ is an M-matrix and $B\ge 0$, according to Theorem 1, we deduce that system (18) is positive. Since $A+B$ is Hurwitz, it follows from Theorem 2 that system (18) is asymptotically stable, that is, $\underset{k\to +\infty }{lim}\widehat{e}\left(k\right)=\underset{k\to +\infty }{lim}({\tilde{x}}_0-\tilde{x}\left(k\right))=0$. Thus, $\underset{k\to +\infty }{lim}\tilde{x}\left(k\right)=-{(A+B)}^{-1}E{\mathbf{1}}_p$. □

Lemma 13. 

Under Assumption 1, the trajectory $\underset{\sim }{x}\left(k\right)$ of Nabla discrete system (16) satisfies

(i) 

$\underset{\sim }{x}\left(k_2\right)\ge \underset{\sim }{x}\left(k_1\right)$ for any $k_2>k_1>a$, $k_1,k_2\in {\mathbb{N}}_{a+1}$;

(ii) 

$\underset{k\to +\infty }{lim}\underset{\sim }{x}\left(k\right)=-{(A+B)}^{-1}E{\mathbf{1}}_p$.

Proof. 

The lemma can be proven using the proof method of Lemma 12, which is omitted here. □

Lemma 14. 

Under Assumption 1, the trajectories and outputs of system (14)–(16) satisfy that

$$\left(i\right)\tilde{x}\left(k\right)\le x\left(k\right)\le \underset{\sim }{x}\left(k\right)forallk\in {\mathbb{N}}_{a+1};$$

$$\left(ii\right)\tilde{y}\left(k\right)\le y\left(k\right)\le \underset{\sim }{y}\left(k\right)forallk\in {\mathbb{N}}_{a+1}.$$

Proof. 

(i) Define error variable $e\left(k\right)=x\left(k\right)-\tilde{x}\left(k\right)$, from systems (14) and (15), one has

$$\begin{array}{c}\left\{\begin{array}{c}{}_a{}^C\nabla _k^{\alpha }e\left(k\right)=Ae\left(k\right)+Be(k-d\left(k\right))+B[\tilde{x}(k-d\left(k\right))-\tilde{x}(k-d)],\hfill \\ e\left(s\right)=0,s\in {\mathbb{N}}_{a-max\{d,\tau \}}^a.\hfill \end{array}\right.\hfill \end{array}$$

(19)

From Lemma 12, one can deduce that $\tilde{x}(k-d\left(k\right))\ge \tilde{x}(k-d)$, that is, $\tilde{x}(k-d\left(k\right))-\tilde{x}(k-d)\ge 0$. Since $I-A$ is an M-matrix and $B\ge 0$, by Theorem 1, one can conclude that system (19) is positive, that is, $e\left(k\right)=x\left(k\right)-\tilde{x}\left(k\right)\ge 0$. Thus, $\tilde{x}\left(k\right)\le x\left(k\right)$ holds for all $k\in {\mathbb{N}}_{a+1}$.

Similarly, we can deduce that $x\left(k\right)\le \underset{\sim }{x}\left(k\right)$ holds for all $k\in {\mathbb{N}}_{a+1}$.

($ii$) From Lemma 13 and condition (i), one has

$$\begin{array}{cc}\hfill y\left(k\right)& =Cx\left(k\right)+Dx(k-\tau \left(k\right))+F{\mathbf{1}}_p\hfill \\ \hfill & \ge C\tilde{x}\left(k\right)+D\tilde{x}(k-\tau \left(k\right))+F{\mathbf{1}}_p\hfill \\ \hfill & \ge C\tilde{x}\left(k\right)+D\tilde{x}(k-\tau )+F{\mathbf{1}}_p=\tilde{y}\left(k\right).\hfill \end{array}$$

Thus, $y\left(k\right)\ge \tilde{y}\left(k\right)$. Following the similar manner, we can deduce that $y\left(k\right)\le \underset{\sim }{y}\left(k\right)$. □

To date, we can provide specific calculations of the $l_{\infty }$-gain of Nabla discrete fractional linear system (1).

Theorem 5. 

Under Assumption 1, the $l_{\infty }$-gain of Nabla discrete fractional linear system (1) can be expressed as ${\parallel F-(C+D)(A+B)}^{-1}{E\parallel }_{\infty }$.

Proof. 

For any input $\omega \left(k\right)\ge 0$ with ${\parallel \omega \parallel }_{l_{\infty }}=1$, it follows that $\omega \left(k\right)\le {\mathbf{1}}_p$. Utilizing Lemmas 12 and 13, we can deduce that

$$\underset{k\to +\infty }{lim}\tilde{x}\left(k\right)=\underset{k\to +\infty }{lim}\underset{\sim }{x}\left(k\right)=-{(A+B)}^{-1}E{\mathbf{1}}_p.$$

Consequently,

$$\underset{k\to +\infty }{lim}\tilde{y}\left(k\right)=\underset{k\to \infty }{lim}\underset{\sim }{y}\left(k\right)=[F-(C+D){(A+B)}^{-1}E]{\mathbf{1}}_p.$$

According to Lemma 14, we have

$$\underset{k\to +\infty }{lim}y\left(k\right)=[F-(C+D){(A+B)}^{-1}E]{\mathbf{1}}_p.$$

From Definition 8, it follows that

$$\underset{{∥\omega ∥}_{l_{\infty }}=1}{sup}{∥y∥}_{l_{\infty }}\ge {\parallel [F-(C+D){(A+B)}^{-1}E]\parallel }_{\infty }.$$

Conversely, by Lemma 14, we find that

$$\begin{array}{cc}\hfill y\left(k\right)& \le \underset{\sim }{y}\left(k\right)\hfill \\ \hfill & =(C+D)\underset{\sim }{x}\left(k\right)+F{\mathbf{1}}_p\hfill \\ \hfill & \le [F-(C+D){(A+B)}^{-1}E]{\mathbf{1}}_p.\hfill \end{array}$$

From Definition 8, we also have

$$\underset{{∥\omega ∥}_{l_{\infty }}=1}{sup}{∥y∥}_{l_{\infty }}\le {\parallel [F-(C+D){(A+B)}^{-1}E]\parallel }_{\infty }.$$

Thus, we can conclude that

$$\underset{{∥\omega ∥}_{l_{\infty }}=1}{sup}{∥y∥}_{l_{\infty }}={\parallel [F-(C+D){(A+B)}^{-1}E]\parallel }_{\infty }.$$

This concludes the proof. □

Remark 2. 

According to Theorem 5, one can know that the $l_{\infty }-$ gain of the Nabla discrete fractional positive system does not depend on the fractional order α or the upper bound d of delay $d\left(k\right)$. When $\alpha =1$, this conclusion is consistent with the result for integer-order positive systems provided in [33].

6. Numerical Examples

Example 1. 

Consider the Nabla discrete fractional linear system (1) with the following system matrices

$$\begin{array}{ccc}\hfill A& =& \left[\begin{array}{ccc}-4& 0.5& 0.5\\ 1& -4& 2\\ 0.25& 0.75& -3.5\end{array}\right],B=\left[\begin{array}{ccc}0.2& 0.5& 0.6\\ 0.5& 0.3& 0.1\\ 0.2& 0.4& 0.3\end{array}\right],C=\left[\begin{array}{ccc}0.5& 0.3& 0.7\\ 1.0& 0.3& 0.1\end{array}\right],\hfill \\ \hfill D& =& \left[\begin{array}{ccc}0.2& 0.7& 0.4\\ 0& 0.1& 0.5\end{array}\right],E=\left[\begin{array}{cc}0.3& 0.1\\ 0.2& 0.1\\ 0& 0.5\end{array}\right],F=\left[\begin{array}{cc}0.1& 0.2\\ 0.3& 0.2\end{array}\right].\hfill \end{array}$$

We can see that $I-A$ and $I-(A+B)$ are M-matrices, $B\ge 0$, $E\ge 0$, $C\ge 0$, $D\ge 0$, and $F\ge 0$. Thus, it follows from Theorem 1 that this system is positive.

Simple calculation demonstrates that the eigenvalues of $A+B$ are ${\lambda }_1=-0.3808$, ${\lambda }_2=-4.4628$, and ${\lambda }_3=-5.0563$; this means that matrix $A+B$ is Hurwitz, from Theorem 2, one can deduce that the system is asymptotically stable.

Given an initial condition $\varphi \left(s\right)=\left[1.2|cos(10s\left)\right|,1.2|cos(5s\left)\right|,0.5|cos(7s\left)\right|\right]$ ($s\in {\mathbb{N}}_{a-2}^a$), time delays $d\left(k\right)=\lceil sin\left(0.5\pi \right)\rceil$ and $\tau \left(k\right)=\lceil sin\left(0.1\pi k\right)\rceil +1$. The various state trajectories of the system (1) with fractional orders $\alpha =0.2$, $\alpha =0.5$, and $\alpha =0.8$ are depicted in Figure 1. These state trajectories demonstrate that they all asymptotically approach zero. This observation clearly indicates that the stability of Nabla discrete fractional positive systems is independent of the fractional order, which is consistent with the result presented in Theorem 4. It is important to note that this conclusion does not apply to the Nabla discrete fractional system without non-negativity constraints. Reference [34] has shown that the stability of the general Nabla discrete fractional system is dependent on the fractional order in the absence of such constraints.

On the other hand, we can computer the limit of output trajectory is $[F-(C+D){(A+B)}^{-1}E]{\mathbf{1}}_2={[1.1568,1.0706]}^T$ as $k\to \infty$. Similarly, the various output trajectories of the Nabla discrete fractional positive system with fractional orders $\alpha =0.2$, $\alpha =0.5$, and $\alpha =0.8$ are depicted in Figure 2. All these output trajectories asymptotically approach the same fixed value. This observation demonstrates that the $l_{\infty }$-gain of the Nabla discrete fractional positive system is independent of the fractional order $\alpha$, which coincides with the conclusion presented in Theorem 5.

Example 2. 

Consider the Nabla discrete fractional linear system (1) with the following system matrices

$$\begin{array}{ccc}\hfill A& =& \left[\begin{array}{ccc}0.1& 0.5& 0.5\\ 0.2& -0.3& 2\\ 0.5& 0.3& -0.4\end{array}\right],B=\left[\begin{array}{ccc}0.2& 0.1& 0.6\\ 0.5& 0.2& 0.1\\ 0.2& 0.4& 0.2\end{array}\right],C=\left[\begin{array}{ccc}0.5& 0.3& 0.7\\ 1.0& 0.3& 0.1\end{array}\right],\hfill \\ \hfill D& =& \left[\begin{array}{ccc}0.2& 0.7& 0.4\\ 0& 0.1& 0.5\end{array}\right],E=\left[\begin{array}{cc}0.3& 0.1\\ 0.2& 0.1\\ 0& 0.5\end{array}\right],F=\left[\begin{array}{cc}0.1& 0.2\\ 0.3& 0.2\end{array}\right].\hfill \end{array}$$

We can compute that

$${(I-A)}^{-1}=\left[\begin{array}{ccc}11.8447& 8.2524& 16.0194\\ 12.4272& 9.8058& 18.4466\\ 6.8932& 5.0485& 10.3883\end{array}\right]\ge 0.$$

since $B\ge 0$, $E\ge 0$, then ${(I-A)}^{-1}B\ge 0$, ${(I-A)}^{-1}E\ge 0$.

Given the same initial conditions and time delays as in Example 1, the state trajectory of the Nabla discrete fractional system described in Equation (1) is depicted in Figure 3. This figure indicates that the system is not positive. This observation demonstrates that the conditions for the validity of Theorem 2, namely, ${(I-A)}^{-1}B\ge 0$, ${(I-A)}^{-1}B\ge 0$, and ${(I-A)}^{-1}E\ge 0$ are necessary but not sufficient conditions.

7. Conclusions

This paper investigates the asymptotic stability and $l_{\infty }$-gain problem of Nabla discrete fractional linear positive systems with bounded time-varying delays. Firstly, we derive both a sufficient condition and a necessary condition to ensure the positivity of these systems. Then, by employing a comparison principle, we present a necessary and sufficient criterion for the asymptotic stability of Nabla discrete fractional linear positive systems. Additionally, we demonstrate that the $l_{\infty }$-gain of the system is determined by the system matrices. Finally, numerical examples are provided to validate the aforementioned results.