Wirtinger-based fractional summation inequality for stability analysis of nabla discrete fractional-order time-delay systems

Abstract

This paper centers on the stability analysis of nabla discrete fractional-order systems with time delay. The innovation in this study lies in the exploration of novel fractional Wirtinger summation inequalities, aiming to mitigate the conservatism inherent in discrete-time systems with time delay. The introduced fractional Wirtinger summation inequality encompasses both integer-order Wirtinger summation inequality and integer-order Jensen summation inequality, effectively bridging the gap between integer-order and fractional summation inequalities. Additionally, the paper introduces certain properties of fractional difference, skillfully integrating integer-order difference with fractional difference into a cohesive framework. Furthermore, to analyze the stability of the models under consideration, the paper presents several new lemmas utilizing the generalized nabla discrete fractional Lyapunov direct method and proposes multiple time-delay-dependent stability criteria. Finally, a series of numerical examples is presented to illustrate the correctness and practicality of the obtained results.

1 Introduction

In recent years, fractional calculus has garnered substantial attention across diverse fields owing to its distinctive characteristics, such as non-locality, long-term memory, and hereditary properties. As a result, it has found applications in a myriad of potential domains [1,2,3,4]. In a recent review [2], the authors delved into the real-world applications of fractional calculus, unveiling extensive prospects within this burgeoning field.

As a crucial indicator in control systems, stability has garnered widespread attention due to the frequent manifestation of delay phenomena in practical situations. Consequently, the introduction of time delay into models can lead to stability issues, and substantial efforts have been dedicated to the advancement of stability analysis for linear time-delay fractional-order systems [5,6,7,8,9,10,11,12]. Undoubtedly, the Lyapunov method stands out as a powerful tool for studying stability.

In recent years, various integral inequality techniques have been commonly employed to mitigate conservatism and enhance the stability of continuous-time delayed systems [13,14,15,16,17,18,19]. Scholars addressing continuous integer-order delayed systems have proposed techniques like the Wirtinger-type integral inequality method [14], the free-weighting matrix method [15], and the time-delay interval partition method [16]. Jensen inequality [13] is particularly useful for problems involving multiple integral terms in functional derivatives. Seuret and Gouaisbaut [14] subsequently introduced the Wirtinger integral inequality and established that Jensen inequality is a special case of it. Recently, Zeng et al. [15] enhanced the Wirtinger integral inequality using a free-matrix approach, demonstrating its degeneration into the Wirtinger inequality under specific conditions. In continuous fractional-order delayed systems, where integral inequalities play an important role in studying stability, an increasing number of researchers are dedicated to constructing fractional integral inequalities to reduce conservatism [17,18,19]. Yang et al. [19] proposed the Caputo–Wirtinger integral inequality, and numerical experiments have showcased its effectiveness in mitigating conservatism for time-varying delay systems.

Given the close approximation of solutions between difference and continuous equations, discretizing practical systems into difference equations is paramount for further research. In recent years, various summation inequality techniques have been widely employed to tackle the issue of conservatism in reducing the stability of discrete-time delayed systems [20,21,22]. In the realm of discrete integer-order delayed systems, Zhang et al. [20] employed the Abel lemma to derive a novel finite-sum inequality, exploring the stability of discrete integer-order systems with time delay. Concurrently, Seuret et al. employed an alternative method to establish the Wirtinger summation inequality, showcasing its inclusion of Jensen summation inequality as a special case. They extended their study to investigate the stability of discrete integer-order systems with time-varying delays, incorporating information on time-varying delays into the obtained stability conditions [21]. Subsequently, Zhang et al. refined the results presented in [21] by leveraging the reciprocally convex lemma and the Wirtinger summation inequality, achieving relatively lower conservatism [22]. In recent years, the stability of discrete fractional-order systems with time delay has garnered significant attention from scholars. Huang et al. [23] utilized Burton’s technique to explore the stability of fractional delta discrete-time neural networks with time delay. They pointed out the challenge of finding a suitable Lyapunov function for fractional differential or difference systems with pure delay. Additionally, Wei et al. [12] employed Z-transform and matrix eigenvalue analysis to establish the LMI stability criterion for delta discrete fractional-order systems with time delay.

Besides, the quality of the results obtained by Lyapunov–Krasovskii functional method not only depends on the type of functional, but also relates to the method of handling functional summation term. When taking the difference of a discrete fractional Lyapunov–Krasovskii functional $V(r,\omega _r) =\sum _{i=-b+1}^{-a}\sum _{j=r+i+1}^{r}{}_{a}^{C}\text {}\nabla _{j}^{\alpha }\omega ^{T}(j)R{}_{a}^{C}\text {}\nabla _{j}^{\alpha }\omega (j)$, quadratic summation term $-\sum _{i=r-b+1}^{r-a}{}_{a}^{C}\text {}\nabla _{i}^{\alpha }\omega ^{T}(i)R{}_{a}^{C}\text {}\nabla _{i}^{\alpha }\omega (i)$ will arise. If the upper bound estimation of this term can be effectively handled, stability criteria with less conservatism for discrete fractional-order systems can be established.

Motivated by this, in this paper, we aim to construct a new class of fractional Wirtinger summation inequalities to address the conservatism issue in the stability of discrete fractional-order delay systems. Therefore, this paper will introduce several new nabla fractional Wirtinger summation inequalities and employ them in the stability analysis of nabla discrete fractional-order delayed systems, resulting in reduced conservatism. Additionally, the paper will present some properties of fractional difference and effectively integrates integer-order difference with fractional difference, highlighting that fractional difference serves as a generalization of classical integer-order difference. Finally, the paper will introduce several generalized Lyapunov stability theories as theoretical tools for studying the stability of nabla discrete fractional-order systems with time delay, accompanied by the provision of several time-delay-dependent stability criteria.

The main contributions of this paper can be summarized as follows:

• The transitional properties from discrete fractional difference to integer-order difference are presented, which fully shows that fractional difference is a generalization of classical integer-order difference.

• Several new fractional Wirtinger summation inequalities are proposed, which include integer-order Wirtinger summation inequality and integer-order Jensen summation inequality.

• Several generalized Lyapunov stability theories are proposed as theoretical tools for studying the stability of nabla discrete fractional-order delayed systems.

• A new fractional Lyapunov–Krasovskii functional, which incorporates a double summation inequality, is constructed to obtain novel stability criteria for time-delay systems.

• A time-delay-dependent stability criterion is proposed, and an upper bound on the time delay is obtained.

Notations: $_{\ell }^{}\text {}\nabla _{r}^{\alpha }$ and $\nabla ^n$ denote nabla fractional difference operator and n-order nabla difference operator, respectively. ${\mathbb {R}}$ and ${\mathbb {Z}}$ stand for the set of real numbers and the set of integers, respectively. ${\mathbb {Z}}_{+}$ and ${\mathbb {Z}}_{-}$ represent the set of positive and negative integer numbers, respectively. ${\mathbb {R}}^n$ stands for the n-dimensional euclidean space. ${\mathbb {R}}^{n \times m}$ stands for the set of $n \times m$ dimensional real matrices. $S^T$ represents the transpose of a matrix S. $S\succ 0~(S\prec 0)$ denotes S is a symmetric and positive (negative) definite matrice. $He\left\{ S \right\} =S+S^T$. ${\mathscr {N}}_{\ell }(\cdot )$ represents the nabla Laplace transform operator. ${\mathbb {N}}_{\ell }=\left\{ \ell , \ell +1, \ell +2, \dots \right\}$ and ${\mathbb {N}}_{\ell _1}^{\ell _2}=\left\{ \ell _1, \ell _1+1, \ell _1+2, \dots , \ell _2 \right\}$ denote discrete sequences. $\lambda _{\min }(S)$ represents the minimum eigenvalue of matrix S. $r^{{\bar{\varrho }} }\triangleq \frac{\varGamma (r+\varrho )}{\varGamma (r)}$, where, $r\notin {\mathbb {Z}}_{-}, \varGamma (\cdot )$ represents the gamma function.

2 Preliminaries

Definition 1

(Goodrich and Peterson [1]) The m-order nabla difference of a function $\omega : {\mathbb {N}}_{\ell +1-m}\rightarrow {\mathbb {R}}$ is written by

$$\begin{aligned} \nabla ^{m}\omega (r)=\sum _{\imath =0}^{m}(-1)^{\imath }\left( {\begin{array}{c}m\\ \imath \end{array}}\right) \omega (r-\imath ), \end{aligned}$$

(1)

where $\left( {\begin{array}{c}m\\ \imath \end{array}}\right) \triangleq \frac{\varGamma (m+1)}{\varGamma (\imath +1)\varGamma (m-\imath +1)}$, $m\in {\mathbb {Z}}_{+}$, $\ell \in {\mathbb {R}}$ and $r\in {\mathbb {N}}_{\ell +1}$.

With the defined nabla difference, it yields

$$\begin{aligned} \nabla \left\{ \psi (r)\phi (r)\right\} =\phi (r)\nabla \psi (r)+\psi (r-1)\nabla \phi (r), \end{aligned}$$

(2)

then, the formula of summation by parts is as follows

$$\begin{aligned} \sum _{\imath =\ell +1}^{r}\psi (\imath -1)\nabla \phi (\imath )=\psi (\imath )\phi (\imath )|_{\ell }^{r}-\sum _{\imath =\ell +1}^{r}\phi (\imath )\nabla \psi (\imath ). \end{aligned}$$

(3)

Definition 2

(Goodrich and Peterson [1]) The $\gamma$th difference$\setminus$sum of a function $\omega : {\mathbb {N}}_{\ell +1}\rightarrow {\mathbb {R}}$ is written as follows

$$\begin{aligned} _{\ell }^{G}\text {}\nabla _{r}^{\gamma }\omega (r)=\sum _{\imath =\ell +1}^{r}\frac{(r-\imath +1)^{\overline{-\gamma -1}}}{\varGamma (-\gamma )}\omega (\imath ), \end{aligned}$$

(4)

where $\gamma \in {\mathbb {R}}{\setminus {\mathbb {Z}}_+}$, $\varrho \in {\mathbb {R}}$, $\ell \in {\mathbb {R}}$, $r\in {\mathbb {N}}_{\ell +1}$.

Definition 3

(Goodrich and Peterson [1]) The $\alpha$th nabla fractional difference of $\omega : {\mathbb {N}}_{\ell +1-m}\rightarrow {\mathbb {R}}$ in the Caputo and Riemann–Liouville senses, are defined as follows

$$\begin{aligned}&_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\omega (r)={} _{\ell }^{G}\text {}\nabla _{r}^{\alpha -m }{}\nabla ^{m}\omega (r), \end{aligned}$$

(5)

$$\begin{aligned}&_{\ell }^{R}\text {}\nabla _{r}^{\alpha }\omega (r)=\nabla ^{m}{} _{\ell }^{G}\text {}\nabla _{r}^{\alpha -m }\omega (r), \end{aligned}$$

(6)

where $\alpha \in (m-1,m)$, $m\in {\mathbb {Z}}_{+}$, $r\in {\mathbb {N}}_{\ell +1}$ and $\ell \in {\mathbb {R}}$.

Definition 4

(Goodrich and Peterson [1]) The nabla Laplace transform of a function $\omega : {\mathbb {N}}_{\ell +1}\rightarrow {\mathbb {R}}$, $\ell \in {\mathbb {R}}$ is expressed as follows

$$\begin{aligned} {\mathscr {N}}_{\ell }\left\{ \omega (r) \right\} \triangleq \sum _{r=1}^{+\infty }(1-s)^{r-1}\omega (r+\ell ), \end{aligned}$$

(7)

where s makes the above infinite series converge.

Definition 5

(Goodrich and Peterson [1]) The nabla Mittag–Leffler function is expressed as follows

$$\begin{aligned} {\mathcal {F}}_{\alpha ,\gamma }\left( \lambda ,r,\ell \right) = \sum _{\imath =0}^{+\infty }\lambda ^{\imath }\frac{(r-\ell )^{\overline{\imath \alpha +\gamma -1}}}{\varGamma (\imath \alpha +\gamma )}, \end{aligned}$$

(8)

where $\alpha >0$, $\gamma \in {\mathbb {R}}$, $r\in {\mathbb {N}}_{\ell +1}$ and $\ell \in {\mathbb {R}}$.

Especially, when $\gamma =1$, it degenerates into the single parameter nabla Mittag–Leffler function as follows

$$\begin{aligned} {\mathcal {F}}_{\alpha }\left( \lambda ,r,\ell \right) ={\mathcal {F}}_{\alpha ,1 }\left( \lambda ,r,\ell \right) = \sum _{\imath =0}^{+\infty }\lambda ^{\imath }\frac{(r-\ell )^{\overline{\imath \alpha }}}{\varGamma (\imath \alpha +1)}. \end{aligned}$$

(9)

Additionally, the nabla discrete Laplace transform of Mittag–Leffler function (see Ref. [1]) is given as follows

$$\begin{aligned} {\mathscr {N}}_{\ell }\left\{ {\mathcal {F}}_{\alpha ,\gamma }\left( \lambda ,r,\ell \right) \right\} =\frac{s^{\alpha -\gamma }}{s^{\alpha }-\lambda }, \end{aligned}$$

(10)

for $\left| 1-s \right| < 1$, $\left| s^{\alpha }\right| > \left| \lambda \right|$. Ref. [24] also expresses that

$$\begin{aligned} {\mathscr {N}}_{\ell }\left\{ _{\ell }^{C}\text {}\nabla _{r}^{\alpha }\omega (r) \right\}= & {} s^{\alpha }{\mathscr {N}}_{\ell }\left\{ \omega (r) \right\} \nonumber \\ {}{} & {} -\sum _{\imath =0}^{m-1}s^{\alpha -\imath -1}\nabla ^{\imath }\omega (r)|_r=\ell . \end{aligned}$$

(11)

Lemma 1

(Ostalczyk [25]) Let $\omega : {\mathbb {N}}_{\ell +1-m}\rightarrow {\mathbb {R}}$, $\alpha \in (m-1,m)$ and $r\in {\mathbb {N}}_{\ell +1}$, the relationship of two fractional difference in the Caputo and Riemann–Liouville senses is given as follows

$$\begin{aligned} _{\ell }^{R}\text {}\nabla _{r}^{\alpha }\omega (r)={}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\omega (r)+\sum _{\imath =0}^{m-1}\frac{(r-\ell )^{\overline{\imath -\alpha }}}{\varGamma (\imath -\alpha +1)}{}\nabla ^{\imath }\omega (\ell ). \nonumber \\ \end{aligned}$$

(12)

Lemma 2

(Jiang et al. [26]) Let $\omega : {\mathbb {N}}_{\ell _1}^{\ell _2} \rightarrow {\mathbb {R}}^n$ be a sequence of discrete-time variables, for any symmetric matrice $S \in {\mathbb {R}}^{n \times n}\succ 0$, it holds that

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}\omega ^{T}(\imath )S\omega (\imath ) \nonumber \\&\quad \ge \frac{1}{\ell _2-\ell _1}\Big ( \sum _{\imath =\ell _1+1}^{\ell _2}\omega (\imath )\Big )^TS\Big ( \sum _{\imath =\ell _1+1}^{\ell _2}\omega (\imath )\Big ). \end{aligned}$$

(13)

Lemma 3

(Wei et al. [27]) Let $f: {\mathbb {N}}_{\ell +1}\rightarrow {\mathbb {R}}$ be a bounded function and assume $\lim _{r\rightarrow +\infty }f(r)$ exists, it holds that $\lim _{r\rightarrow +\infty }f(r)=\lim _{s\rightarrow 0}sF(s)$ with $F(s)={\mathscr {N}}_{\ell }\left\{ f(r) \right\}$.

Lemma 4

(Wei et al. [9]) Assume $f(\ell )=g(\ell )$ and $_{\ell }^{C}\text {}\nabla _{r}^{\alpha }f (r)\ge {_{\ell }^{C}\text {}\nabla _{r}^{\alpha }g (r)}$ where $\alpha \in (0,1)$, then $f(r)\ge g(r)$.

3 Main results

In this section, compatibility properties between discrete fractional difference and discrete integer-order difference are firstly established, highlighting that discrete fractional difference extends the domain of discrete integer-order difference. Secondly, several new discrete fractional summation inequalities are introduced, presenting a new perspective for the investigation of discrete fractional-order time-delay systems. Additionally, in the exploration of the stability of nabla discrete fractional-order time-delay systems, this section extends several fractional Lyapunov stability theories. Finally, a Lyapunov–Krasovskii function, incorporating fractional difference, is formulated, providing time-delay-dependent LMI-form stability criteria for nabla discrete fractional-order time-delay systems.

3.1 Properties of fractional difference

We first discuss the properties of the lower limit and upper limit of fractional difference under the Caputo and Riemann–Liouville senses, respectively.

Lemma 5

For the fractional difference with order $\gamma \in (m-1,m)$ and $r\in {\mathbb {N}}_{\ell +1}$, the following equalities hold

$$\begin{aligned} \lim _{\gamma \rightarrow (m-1)^+}{}_{\ell }^{C}\text {}\nabla _{r}^{\gamma }\omega (r)&={}\nabla ^{m-1}\omega (r)-{}\nabla ^{m-1}\omega (\ell ), \end{aligned}$$

(14)

$$\begin{aligned} \lim _{\gamma \rightarrow m^-}{}_{\ell }^{C}\text {}\nabla _{r}^{\gamma }\omega (r)&={}\nabla ^{m}\omega (r), \end{aligned}$$

(15)

$$\begin{aligned} \lim _{\gamma \rightarrow (m-1)^+}{}_{\ell }^{R}\text {}\nabla _{r}^{\gamma }\omega (r)&={}\nabla ^{m-1}\omega (r), \end{aligned}$$

(16)

$$\begin{aligned} \lim _{\gamma \rightarrow m^-}{}_{\ell }^{R}\text {}\nabla _{r}^{\gamma }\omega (r)&={}\nabla ^{m}\omega (r). \end{aligned}$$

(17)

Proof

(i) In the light of Definition 3, one has

$$\begin{aligned}&\lim _{\gamma \rightarrow (m-1)^+}{}_{\ell }^{C}\text {}\nabla _{r}^{\gamma }\omega (r)\nonumber \\&\quad =\lim _{\gamma \rightarrow (m-1)^+}\sum _{\imath =\ell +1}^{r}\frac{(r-\imath +1)^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}{}\nabla ^{m}\omega (\imath )\nonumber \\&\quad =\lim _{\gamma \rightarrow (m-1)^+}\sum _{\imath =\ell +1}^{r}\frac{1}{\varGamma (m-\gamma )}\frac{\varGamma (r-\imath +m-\gamma )}{\varGamma (r-\imath +1)}\nabla ^{m}\omega (\imath )\nonumber \\&\quad =\sum _{\imath =\ell +1}^{r}\nabla ^{m}\omega (\imath )\nonumber \\&\quad =\nabla ^{m-1}\omega (r)-\nabla ^{m-1}\omega (\ell ), \end{aligned}$$

(18)

which completes the proof of formula (14).

(ii) Let $\phi (\imath )\triangleq \nabla ^{m-1}\omega (\imath )$ and $\psi (\imath -1)\triangleq \frac{(r-\imath +1)^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}$. According to formula (3), it follows

$$\begin{aligned} {}_{\ell }^{C}\text {}\nabla _{r}^{\gamma }\omega (r)&=\sum _{\imath =\ell +1}^{r}\psi (\imath -1)\nabla \phi (\imath )\nonumber \\&=\psi (\imath )\phi (\imath )|_{\ell }^{r}-\sum _{\imath =\ell +1}^{r}\phi (\imath )\nabla \psi (\imath )\nonumber \\&=\frac{0^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}\phi (r)-\frac{(r-\ell )^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}\phi (\ell )\nonumber \\&\quad +\sum _{\imath =\ell +1}^{r}\frac{(r-\imath +1)^{\overline{m-\gamma -2}}}{\varGamma (m-\gamma -1)}\phi (\imath )\nonumber \\&=\phi (r)-\frac{(r-\ell )^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}\phi (\ell )\nonumber \\&\quad +\sum _{\imath =\ell +1}^{r-1}\frac{(r-\imath +1)^{\overline{m-\gamma -2}}}{\varGamma (m-\gamma -1)}\phi (\imath ), \end{aligned}$$

(19)

where $\frac{0^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}=0, \frac{1^{\overline{m-\gamma -2}}}{\varGamma (m-\gamma -1)}=1$, $\nabla \frac{(r-\imath )^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}= -\frac{(r-\imath +1)^{\overline{m-\gamma -2}}}{\varGamma (m-\gamma -1)}$. Then, calculating the terms of (19) yields

$$\begin{aligned}&\lim _{\gamma \rightarrow m^-}\frac{(r-\ell )^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}\nonumber \\&\quad =\lim _{\gamma \rightarrow m^-}\frac{1}{\varGamma (m-\gamma )}\frac{\varGamma (r-\ell +m-\gamma -1)}{\varGamma (r-\ell )}\nonumber \\&\quad =\lim _{\gamma \rightarrow m^-}\frac{ (r-\ell -2+m-\gamma )(r-\ell -3+m-\gamma )\cdots (m-\gamma )}{\varGamma (r-\ell )}\nonumber \\&\quad =0, \end{aligned}$$

(20)

and

$$\begin{aligned}&\lim _{\gamma \rightarrow m^-}\sum _{\imath =\ell +1}^{r-1}\frac{(r-\imath +1)^{\overline{m-\gamma -2}}}{\varGamma (m-\gamma -1)}\phi (\imath )\nonumber \\&\quad =\lim _{\gamma \rightarrow m^-}\frac{2^{\overline{m-\gamma -2}}}{\varGamma (m-\gamma -1)}\phi (r-1)\nonumber \\&\qquad +\lim _{\gamma \rightarrow m^-}\sum _{\imath =\ell +1}^{r-2}\frac{(r-\imath +1)^{\overline{m-\gamma -2}}}{\varGamma (m-\gamma -1)}\phi (\imath )\nonumber \\&\quad =\lim _{\gamma \rightarrow m^-}\frac{1}{\varGamma (m-\gamma -1)}\frac{\varGamma (m-\gamma )}{\varGamma (2)}\phi (r-1)\nonumber \\&\qquad +\lim _{\gamma \rightarrow m^-}\sum _{\imath =\ell +1}^{r-2}\frac{1}{\varGamma (m-\gamma -1)}\frac{\varGamma (r-\imath +m-\gamma -1)}{\varGamma (r-\imath +1)}\phi (\imath )\nonumber \\&\quad =-\phi (r-1)\nonumber \\&\qquad +\lim _{\gamma \rightarrow m^-}\sum _{\imath =\ell +1}^{r-2}\frac{1}{\varGamma (r-\imath +1)}\Big [(r-\imath -1+m-\gamma -1)\nonumber \\&\qquad \times (r-\imath -2+m-\gamma -1)\cdots (m-\gamma )(m-\gamma -1)\Big ]\phi (\imath )\nonumber \\&\quad =-\phi (r-1). \end{aligned}$$

(21)

Substituting formulae (20)–(21) into (19) yields

$$\begin{aligned} \lim _{\gamma \rightarrow m^-}{} _{\ell }^{C}\text {}\nabla _{r}^{\gamma }\omega (r)&=\phi (r)-\phi (r-1)\nonumber \\&=\nabla \phi (r)=\nabla ^{m}\omega (r). \end{aligned}$$

(22)

This completes the proof of (15).

(iii) On the basis of Definition 3, one has

$$\begin{aligned}&\lim _{\gamma \rightarrow (m-1)^+}{}_{\ell }^{R}\text {}\nabla _{r}^{\gamma }\omega (r)\nonumber \\&\quad =\lim _{\gamma \rightarrow (m-1)^+}{}\nabla ^{m}\sum _{\imath =\ell +1}^{r}\frac{(r-\imath +1)^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}\omega (\imath )\nonumber \\&\quad =\nabla ^{m}\lim _{\gamma \rightarrow (m-1)^+}\sum _{\imath =\ell +1}^{r}\frac{1}{\varGamma (m-\gamma )}\frac{\varGamma (r-\imath +m-\gamma )}{\varGamma (r-\imath +1)}\omega (\imath )\nonumber \\&\quad =\nabla ^{m-1}\Big [\sum _{\imath =\ell +1}^{r}\omega (\imath )-\sum _{\imath =\ell +1}^{r-1}\omega (\imath )\Big ]\nonumber \\&\quad =\nabla ^{m-1}\omega (r), \end{aligned}$$

(23)

which the proof of (16) is thus completed.

(iv) It is easy to calculate that

$$\begin{aligned}&\lim _{\gamma \rightarrow m^-}{}_{\ell }^{R}\text {}\nabla _{r}^{\gamma }\omega (r)\nonumber \\&\quad =\lim _{\gamma \rightarrow m^-}{}\nabla ^{m}\sum _{\imath =\ell +1}^{r}\frac{(r-\imath +1)^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}\omega (\imath )\nonumber \\&\quad =\lim _{\gamma \rightarrow m^-}{}\nabla ^{m-1}\Big [\sum _{\imath =\ell +1}^{r}\frac{(r-\imath +1)^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}\omega (\imath )\nonumber \\&\qquad -\sum _{\imath =\ell +1}^{r-1}\frac{(r-\imath )^{\overline{m-\gamma -1}}}{\varGamma (m-\gamma )}\omega (\imath )\Big ]\nonumber \\&\quad ={}\nabla ^{m-1}\Big [\omega (r)+\lim _{\gamma \rightarrow m^-}\sum _{\imath =\ell +1}^{r-1}\frac{(r-\imath +1)^{\overline{m-\gamma -2}}}{\varGamma (m-\gamma -1)}\omega (\imath )\Big ]\nonumber \\&\quad ={}\nabla ^{m-1}[\omega (r)-\omega (r-1)]={}\nabla ^{m}\omega (r), \end{aligned}$$

(24)

where $\lim _{\gamma \rightarrow m^-}\sum _{\imath =\ell +1}^{r-1}\frac{(r-\imath +1)^{\overline{m-\gamma -2}}}{\varGamma (m-\gamma -1)}\omega (\imath )=-\omega (r-1)$. This completes the proof of (17). $\square$

Remark 1

From the above discussion, the Riemann–Liouville difference is a reasonable generalization of classical integer-order difference.

Remark 2

The properties presented in Lemma 5 in this paper are similar to the properties of Caputo derivative and Riemann–Liouville derivative given in Ref. [28].

3.2 Fractional Wirtinger summation inequalities

This section, a novel Wirtinger-based fractional summation inequalities is proposed as follows:

Lemma 6

Let $\omega : {\mathbb {N}}_{\ell _1}^{\ell _2} \rightarrow {\mathbb {R}}^n$ be a sequence of discrete-time variables. For a symmetric matrice $R\succ 0$, it holds that

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega ^{T}(\imath )R{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\nonumber \\&\quad \ge \frac{1}{\ell _2-\ell _1} \begin{bmatrix} \varPsi _1 \\ \varPsi _2 \end{bmatrix}^{T} \begin{bmatrix} R &{} 0\\ 0 &{} 3(\frac{\ell _2-\ell _1+1}{\ell _2-\ell _1-1})R \end{bmatrix} \begin{bmatrix} \varPsi _1 \\ \varPsi _2 \end{bmatrix}, \end{aligned}$$

(25)

where $\alpha \in (m-1,m)$, $\varPsi _1=\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\ell }^{\alpha }\omega (\imath )$ and $\varPsi _2=\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )-\frac{2}{\ell _2-\ell _1+1}\sum _{\imath =\ell _1+1}^{\ell _2}\sum _{j=\ell _1+1}^{\imath }{}_{\ell _1}^{}\text {}\nabla _{j}^{\alpha }\omega (j)$.

Proof

Set the auxiliary function $x(\imath )$ as follows

$$\begin{aligned} x(\imath )&\triangleq {}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )-\frac{1}{\ell _2-\ell _1}\sum _{j=\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\\&\qquad \nabla _{j}^{\alpha }\omega (j) -3\frac{2\imath -\ell _2-\ell _1-1}{(\ell _2-\ell _1)(\ell _2-\ell _1-1)}\Big (\sum _{j=\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{j}^{\alpha }\omega (j)\\&\qquad -\frac{2}{\ell _2-\ell _1+1}\sum _{j=\ell _1+1}^{\ell _2}\sum _{k=\ell _1+1}^{j}{}_{\ell _1}^{}\text {}\nabla _{k}^{\alpha }\omega (k)\Big )\\&\quad ={}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )-\frac{1}{\ell _2-\ell _1}\varPsi _1-3\frac{2\imath -\ell _2-\ell _1-1}{(\ell _2-\ell _1)(\ell _2-\ell _1-1)}\varPsi _2.\\ \end{aligned}$$

Noting that

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2} 1=\ell _2-\ell _1, \end{aligned}$$

(26)

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}(2\imath -\ell _2-\ell _1-1)=0,\end{aligned}$$

(27)

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}(2\imath -\ell _2-\ell _1-1)^2\nonumber \\&\quad =\frac{(\ell _2-\ell _1)(\ell _2-\ell _1-1)(\ell _2-\ell _1+1)}{3}, \end{aligned}$$

(28)

and

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}(2\imath -\ell _2-\ell _1-1){}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\nonumber \\&\quad =(\ell _1-\ell _2+1){}_{\ell _1}^{}\text {}\nabla _{\ell _1+1}^{\alpha }\omega (\ell _1+1)\nonumber \\&\qquad +\sum _{\imath =\ell _1+2}^{\ell _2}(2\imath -\ell _2-\ell _1-1){}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\nonumber \\&\quad =(\ell _1-\ell _2+1){}_{\ell _1}^{}\text {}\nabla _{\ell _1+1}^{\alpha }\omega (\ell _1+1)+\sum _{\imath =\ell _1+2}^{\ell _2}\psi (\imath -1)\nabla \phi (\imath )\nonumber \\&\quad =(\ell _1-\ell _2+1){}_{\ell _1}^{}\text {}\nabla _{\ell _1+1}^{\alpha }\omega (\ell _1+1)\nonumber \\&\qquad +\psi (\imath )\phi (\imath )|_{\ell _1+1}^{\ell _2}-2\sum _{\imath =\ell _1+2}^{\ell _2}\phi (\imath )\nonumber \\&\quad =(\ell _1-\ell _2+1){}_{\ell _1}^{}\text {}\nabla _{\ell _1+1}^{\alpha }\omega (\ell _1+1)-2\sum _{\imath =a+2}^{\ell _2}\sum _{j=\ell _1+1}^{\imath }{}_{\ell _1}^{}\text {}\nabla _{j}^{\alpha }\omega (j)\nonumber \\&\qquad +(\ell _2-\ell _1+1)\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\nonumber \\&\qquad -(\ell _1-\ell _2+3){}_{\ell _1}^{}\text {}\nabla _{\ell _1+1}^{\alpha }\omega (a+1)\nonumber \\&\quad =(\ell _2-\ell _1+1)\Big [-\frac{2}{\ell _2-\ell _1+1}{}_{\ell _1}^{}\text {}\nabla _{\ell _1+1}^{\alpha }\omega (\ell _1+1)\nonumber \\&\qquad +\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )-\frac{2}{\ell _2-\ell _1+1}\sum _{\imath =\ell _1+2}^{\ell _2}\sum _{j=\ell _1+1}^{\imath }{}_{\ell _1}^{}\text {}\nabla _{j}^{\alpha }\omega (j)\Big ]\nonumber \\&\quad =(\ell _2-\ell _1+1)\Big [\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\nonumber \\&\qquad -\frac{2}{\ell _2-\ell _1+1}\sum _{\imath =\ell _1+1}^{\ell _2}\sum _{j=\ell _1+1}^{\imath }{}_{\ell _1}^{}\text {}\nabla _{j}^{\alpha }\omega (j)\Big ]\nonumber \\&\quad =(\ell _2-\ell _1+1)\varPsi _2, \end{aligned}$$

(29)

where $\psi (\imath -1)\triangleq (2\imath -\ell _2-\ell _1-1)$, $\phi (\imath )\triangleq \sum _{j=\ell _1+1}^{\imath } {}_{\ell _1}^{}\text {}\nabla _{j}^{\alpha }\omega (j)$. Then

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}x^{T}(\imath )Rx(\imath )\nonumber \\&\quad = \sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega ^{T}(\imath )R{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )+\frac{1}{(\ell _2-\ell _1)^2}\Big (\sum _{\imath =\ell _1+1}^{\ell _2} 1\Big )\nonumber \\&\qquad \times \varPsi _1^TR\varPsi _1+\frac{9}{(\ell _2-\ell _1)^2(\ell _2-\ell _1-1)^2}\nonumber \\&\qquad \times \Big (\sum _{\imath =\ell _1+1}^{\ell _2} (2\imath -\ell _2-\ell _1-1)^2\Big )\varPsi _2^TR\varPsi _2\nonumber \\&\qquad -\frac{2}{\ell _2-\ell _1}\varPsi _1^TR\Big (\sum _{\imath =\ell _1+1}^{\ell _2} {}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\Big )\nonumber \\&\qquad -\frac{6}{(\ell _2-\ell _1)(\ell _2-\ell _1-1)}\varPsi _2^TR\nonumber \\&\qquad \times \Big (\sum _{\imath =\ell _1+1}^{\ell _2}(2\imath -\ell _2-\ell _1-1) {}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\Big )\nonumber \\&\qquad +\frac{6}{(\ell _2-\ell _1)^2(\ell _2-\ell _1-1)}\nonumber \\&\qquad \times \Big (\sum _{\imath =\ell _1+1}^{\ell _2}2\imath -\ell _2-\ell _1-1\Big )\varPsi _1^TR\varPsi _2\nonumber \\&\quad =\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega ^{T}(\imath )R{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )+\frac{1}{(\ell _2-\ell _1)}\varPsi _1^TR\varPsi _1\nonumber \\&\qquad +\frac{3(\ell _2-\ell _1+1)}{(\ell _2-\ell _1)(\ell _2-\ell _1-1)}\varPsi _2^TR\varPsi _2-\frac{2}{\ell _2-\ell _1}\varPsi _1^TR\varPsi _1\nonumber \\&\qquad -\frac{6(\ell _2-\ell _1+1)}{(\ell _2-\ell _1)(\ell _2-\ell _1-1)}\varPsi _2^TR\varPsi _2\nonumber \\&\quad =\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega ^{T}(\imath )R{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )-\frac{1}{\ell _2-\ell _1}\varPsi _1^TR\varPsi _1\nonumber \\&\qquad -\frac{3(\ell _2-\ell _1+1)}{(\ell _2-\ell _1)(\ell _2-\ell _1-1)}\varPsi _2^TR\varPsi _2. \end{aligned}$$

(30)

Therefore

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}{}_{a}^{}\text {}\nabla _{\imath }^{\alpha }\omega ^{T}(\imath )R{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\nonumber \\&\quad \ge \frac{1}{\ell _2-\ell _1}\varPsi _1^TR\varPsi _1+\frac{3(\ell _2-\ell _1+1)}{(\ell _2-\ell _1)(\ell _2-\ell _1-1)}\varPsi _2^TR\varPsi _2\nonumber \\&\quad =\frac{1}{\ell _2-\ell _1} \begin{bmatrix} \varPsi _1 \\ \varPsi _2 \end{bmatrix}^{T} \begin{bmatrix} R &{} 0\\ 0 &{} 3(\frac{\ell _2-\ell _1+1}{\ell _2-\ell _1-1})R \end{bmatrix} \begin{bmatrix} \varPsi _1 \\ \varPsi _2 \end{bmatrix}. \end{aligned}$$

(31)

The proof of (25) is thus completed. $\square$

Corollary 1

Let $\omega : {\mathbb {N}}_{\ell _1}^{\ell _2} \rightarrow {\mathbb {R}}^n$ be a sequence of discrete-time variables. For a symmetric matrice $R\succ 0$, it holds that

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega ^{T}(\imath )R{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath ) \nonumber \\&\quad \ge \frac{1}{\ell _2-\ell _1} \begin{bmatrix} \varPsi _1 \\ \varPsi _2 \end{bmatrix}^{T} \begin{bmatrix} R &{} 0\\ 0 &{} 3R \end{bmatrix} \begin{bmatrix} \varPsi _1 \\ \varPsi _2 \end{bmatrix}, \end{aligned}$$

(32)

where $\alpha \in (m-1,m)$, $\varPsi _1$ and $\varPsi _2$ are defined in Lemma 6.

Proof

Because of $\frac{\ell _2-\ell _1+1}{\ell _2-\ell _1-1}>1$, it follows directly from Lemma 6 to Corollary 1. $\square$

Corollary 2

(Fractional Jensen summation inequality) Let $\omega : {\mathbb {N}}_{\ell _1}^{\ell _2} \rightarrow {\mathbb {R}}^n$ be a sequence of discrete-time variables. For a symmetric matrice $R\succ 0$, it holds that

$$\begin{aligned} \sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega ^{T}(\imath )R{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath ) \ge \frac{1}{\ell _2-\ell _1}\varPsi _1^TR\varPsi _1, \end{aligned}$$

(33)

where $\alpha \in (m-1,m)$, $\varPsi _1$ and $\varPsi _2$ are defined in Lemma 6.

Proof

Either Lemma 6 or Corollary 1 leads directly to Corollary 2. $\square$

Remark 3

When $\alpha$ tend to $1^{-}$ in Corollary 2, one has

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}\nabla \omega ^{T}(\imath )R\nabla \omega (\imath )\nonumber \\&\quad \ge \frac{1}{\ell _2-\ell _1}\Big (\sum _{\imath =\ell _1+1}^{\ell _2}\nabla \omega (\imath )\Big )^TR\Big (\sum _{\imath =\ell _1+1}^{\ell _2}\nabla \omega (\imath )\Big ), \end{aligned}$$

(34)

which is the discretized form of Jensen inequality in Ref. [13].

Remark 4

Due to the addition of a positive definite term at the right end of the inequality, Corollary 1 and Lemma 6 have lower conservatism than the fractional Jensen summation inequality.

Remark 5

Note that for $\lim _{\alpha \rightarrow 1^-}$, it can be shown from Lemma 6 that

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}\Big (\nabla \omega ^{T}(\imath )\Big )R\nabla \omega (\imath ) =\sum _{\imath =\ell _1}^{\ell _2-1}\Big (\varDelta \omega ^{T}(\imath )\Big )R\varDelta \omega (\imath )\nonumber \\&\quad \ge \frac{1}{\ell _2-\ell _1} \begin{bmatrix} {\tilde{\varPsi }}_1 \\ {\tilde{\varPsi }}_2 \end{bmatrix}^{T} \begin{bmatrix} R &{} 0\\ 0 &{} 3(\frac{\ell _2-\ell _1+1}{\ell _2-\ell _1-1})R \end{bmatrix} \begin{bmatrix} {\tilde{\varPsi }}_1 \\ {\tilde{\varPsi }}_2 \end{bmatrix}, \end{aligned}$$

(35)

where

$$\begin{aligned} {\tilde{\varPsi }}_1&=\omega (\ell _2)-\omega (\ell _1)\\ {\tilde{\varPsi }}_2&=\sum _{\imath =\ell _1+1}^{\ell _2}{}\nabla \omega (\imath )-\frac{2}{\ell _2-\ell _1+1}\sum _{\imath =\ell _1+1}^{\ell _2}\sum _{j=\ell _1+1}^{\imath }\nabla \omega (j)\\&=\omega (\ell _2)-\omega (\ell _1)-\frac{2}{\ell _2-\ell _1+1}\sum _{\imath =\ell _1+1}^{\ell _2}\Big (\omega (\imath )-\omega (\ell _1)\Big )\\&=\omega (\ell _2)-\omega (\ell _1)\\&\quad -\frac{2}{\ell _2-\ell _1+1}\Big [\sum _{\imath =\ell _1}^{\ell _2}\omega (\imath )-(\ell _2-\ell _1+1)\omega (\ell _1)\Big ]\\&=\omega (\ell _2)+\omega (\ell _1)-\frac{2}{\ell _2-\ell _1+1}\sum _{\imath =\ell _1}^{\ell _2}\omega (\imath ).\\ \end{aligned}$$

Obviously, the fractional summation inequality derived from Lemma 6 is a generalization of the integer-order summation inequality of Lemma 2 of Ref. [21]. Meanwhile, Lemma 6 is also a generalization of the discretized form of Corollary 5 in Ref. [14].

Remark 6

After taking the limit on the right end of Lemma 6, it is completely consistent with Lemma 2 in Ref. [21], which not only effectively combines fractional difference with integer difference, but also has extremely important significance for the study of discrete fractional systems.

Remark 7

An alternative proof of Lemma 6 is as follows.

Proof

Let $m_0$ be a scalar, $\psi _\imath =2\imath -\ell _2-\ell _1-1$. On one hand,

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}\Big [{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )+m_0\psi _\imath \varPsi _2\Big ]^TR\Big [{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )+m_0\psi _\imath \varPsi _2\Big ]\nonumber \\&\quad =\sum _{\imath =\ell _1+1}^{\ell _2}\Big [{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )^TR{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )+2m_0\varPsi _2^TR\psi _\imath {}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\nonumber \\&\qquad +m_0^2\psi _\imath ^2\varPsi _2^TR\varPsi _2 \Big ] \nonumber \\&\quad =\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )^TR{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )+2m_0(\ell _2-\ell _1+1)\varPsi _2^TR\varPsi _2\nonumber \\&\qquad +m_0^2\frac{(\ell _2-\ell _1)(\ell _2-\ell _1-1)(\ell _2-\ell _1+1)}{3}\varPsi _2^TR\varPsi _2\nonumber \\&\quad =\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )^TR{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\nonumber \\&\qquad +\frac{(\ell _2-\ell _1+1)\Big [(m_0(\ell _2-\ell _1)(\ell _2-\ell _1-1)+3)^2-9\Big ]}{3(\ell _2-\ell _1)(\ell _2-\ell _1-1)}\nonumber \\&\qquad \times \varPsi _2^TR\varPsi _2. \end{aligned}$$

(36)

On the other hand, using Lemma 2, we obtain

$$\begin{aligned}{} & {} \sum _{\imath =\ell _1+1}^{\ell _2}\Big [{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )+m_0\psi _\imath \varPsi _2\Big ]^TR\Big [{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )+m_0\psi _\imath \varPsi _2\Big ] \nonumber \\{} & {} \quad \ge \frac{1}{\ell _2-\ell _1}\Big (\varPsi _1+m_0 \sum _{\imath =\ell _1+1}^{\ell _2}\psi _\imath \varPsi _2\Big )^T R\nonumber \\{} & {} \qquad \times \Big (\varPsi _1+m_0 \sum _{\imath =\ell _1+1}^{\ell _2}\psi _\imath \varPsi _2\Big )\nonumber \\{} & {} \quad =\frac{1}{\ell _2-\ell _1}\varPsi _1^TR\varPsi _1, \end{aligned}$$

(37)

where $\sum _{\imath =\ell _1+1}^{\ell _2}\psi _\imath =0$. Taking $m_0=\frac{-3}{(\ell _2-\ell _1) (\ell _2-\ell _1-1)}$, and combining (36) and (37), gives

$$\begin{aligned}&\sum _{\imath =\ell _1+1}^{\ell _2}{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega ^{T}(\imath )R{}_{\ell _1}^{}\text {}\nabla _{\imath }^{\alpha }\omega (\imath ) \nonumber \\&\quad \ge \frac{1}{\ell _2-\ell _1}\varPsi _1^TR\varPsi _1\nonumber \\&\qquad +\frac{(\ell _2-\ell _1+1)\Big [9-(m_0(\ell _2-\ell _1)(\ell _2-\ell _1-1)+3)^2\Big ]}{3(\ell _2-\ell _1)(\ell _2-\ell _1-1)}\nonumber \\&\qquad \times \varPsi _2^TR\varPsi _2\nonumber \\&\quad = \frac{1}{\ell _2-\ell _1}\varPsi _1^TR\varPsi _1+\frac{3(\ell _2-\ell _1+1)}{(\ell _2-\ell _1)(\ell _2-\ell _1-1)}\varPsi _2^TR\varPsi _2\nonumber \\&\quad =\frac{1}{\ell _2-\ell _1} \begin{bmatrix} \varPsi _1 \\ \varPsi _2 \end{bmatrix}^{T} \begin{bmatrix} R &{} 0\\ 0 &{} 3(\frac{\ell _2-\ell _1+1}{\ell _2-\ell _1-1})R \end{bmatrix} \begin{bmatrix} \varPsi _1 \\ \varPsi _2 \end{bmatrix}. \end{aligned}$$

(38)

This completes the proof of Lemma 6. $\square$

3.3 Stability analysis of nabla discrete fractional-order systems with time delay

To derive the subsequent findings in this paper, two novel criteria will be presented for ensuring the asymptotic stability of nabla discrete fractional-order nonlinear systems. Now, contemplate the nabla discrete fractional-order nonlinear system as delineated below:

$$\begin{aligned} {}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\omega (r)=f(r, \omega ), \end{aligned}$$

(39)

where $\alpha \in (0,1)$, $r\in {\mathbb {N}}_{\ell +1}$, $f: [\ell +1, \infty ]\times \varLambda \rightarrow {\mathbb {R}}^n$ is locally Lipschitz on $\omega$ and $\varLambda \subset {\mathbb {R}}^n$ is a state space containing the equilibrium point $\omega _e=0$.

Definition 6

(Wei et al. [9]) The system (39) is called discrete-time Mittag–Leffler stable if

$$\begin{aligned} \left\| \omega (r) \right\| \le \left\{ \mu (\omega (\ell )){\mathcal {F}}_{\alpha }\left( \lambda ,r,\ell \right) \right\} ^c, \end{aligned}$$

(40)

where $\alpha \in (0,1)$, $c>0$, $\lambda <0$, $\mu (\omega )\ge 0$ is locally Lipschitz on $\omega$ with Lipschitz constant $\mu _0$. If inequality (40) is satisfied for any initial state $\omega (\ell )$, then system (39) is globally Mittag–Leffler stable at $\omega _e=0$.

Lemma 7

Supposing the Lyapunov function $V(r,\omega (r)): [\ell +1, \infty ]\times \varLambda \rightarrow {\mathbb {R}}$ is locally Lipschitz on $\omega$ satisfying

(41)

$$\begin{aligned}&{}_{\ell }^{C} \nabla _{r}^{\alpha }V(r,\omega (r))\le -\lambda _3\left\| \omega (r) \right\| ^{bc}+\lambda _4(r-\ell )^{\overline{-\alpha }}, \end{aligned}$$

(42)

where $\alpha \in (0,1)$, $r\in {\mathbb {N}}_{\ell +1}$, $\varLambda \subset {\mathbb {R}}^n$ is state space containing the $\omega _e=0$, and are any positive numbers. Then, the system (39) is Mittag–Leffler stable at $\omega _e=0$.

Proof

Similar to the proofs of Theorem 5 in Ref. [9] and Lemma 4 in Ref. [19], there exists a nonnegative function d(r) that satisfies the following equation

$$\begin{aligned} {}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }V(r,\omega (r))+d(r)&=-\lambda _3\lambda _2^{-1}V(r,\omega (r))\nonumber \\&\quad +\lambda _4(r-\ell )^{\overline{-\alpha }}. \end{aligned}$$

(43)

The following equation is obtained by taking the nabla Laplace transform to (43)

$$\begin{aligned}&s^{\alpha }V(s)-s^{\alpha -1}V(\ell , \omega (\ell ))+D(s)\nonumber \\ {}&\quad =-\lambda _3\lambda _2^{-1}V(s) +\lambda _4\varGamma (1-\alpha )s^{\alpha -1}, \end{aligned}$$

(44)

where ${\mathscr {N}}_{\ell }\left\{ (r-\ell )^{\overline{-\alpha }} \right\} =\varGamma (1-\alpha )s^{\alpha -1}$ (see Ref. [1]), $D(s)={\mathscr {N}}_{\ell }\left\{ d(r) \right\}$ and $V(s)= {\mathscr {N}}_{\ell }\left\{ V(r,\omega (r)) \right\}$. By organizing the above equation, we can solve the V(s) as follows

$$\begin{aligned} V(s)= & {} \frac{\Big (V(\ell , \omega (\ell ))+\lambda _4\varGamma (1-\alpha )\Big )s^{\alpha -1}}{s^\alpha +\lambda _3\lambda _2^{-1}}\nonumber \\ {}{} & {} - \frac{D(s)}{s^\alpha +\lambda _3\lambda _2^{-1}}. \end{aligned}$$

(45)

On account of the Lipschitz condition and nabla Laplace inverse transform, we can know that

$$\begin{aligned} V(r,\omega (r))&=\Big (V(\ell , \omega (\ell ))+\lambda _4\varGamma (1-\alpha )\Big ){\mathcal {F}}_{\alpha }\left( -\lambda _3\lambda _2^{-1},r,\ell \right) \nonumber \\&\quad -d(r)*{\mathcal {F}}_{\alpha ,\alpha }\left( -\lambda _3\lambda _2^{-1},r,\ell \right) \end{aligned}$$

(46)

is the unique solution of (43). Since ${\mathcal {F}}_{\alpha ,\alpha }\left( -\lambda _3\lambda _2^{-1},r,\ell \right)$ is non-negative, then

$$\begin{aligned} V(r,\omega (r))\le \Big (V(\ell , \omega (\ell ))+\lambda _4\varGamma (1-\alpha )\Big ){\mathcal {F}}_{\alpha }\left( -\lambda _3\lambda _2^{-1},r,\ell \right) . \nonumber \\ \end{aligned}$$

(47)

Combining (41) and (47), one has

(48)

Let $\mu =\frac{V(\ell , \omega (\ell ))+\lambda _4\varGamma (1-\alpha )}{\lambda _1}$. Because $V(r,\omega (r))$ is locally Lipschitz on $\omega$, it follows that $\mu =\frac{V(\ell , \omega (\ell ))+\lambda _4\varGamma (1-\alpha )}{\lambda _1}$ is also Lipschitz on $\omega$ and $\mu >0$, which implies the system (39) is Mittag–Leffler stable at $\omega _e=0$. The proof is completed. $\square$

Remark 8

If $h(r)=(r-\ell )^{\overline{-\alpha }}$ is taken from Theorem 2 in Ref. [11], using the final value theorem of nabla Laplace transform of Lemma 3 gives

$$\begin{aligned} \lim _{r\rightarrow +\infty }(r-\ell )^{\overline{-\alpha }}=\lim _{s\rightarrow 0}s\varGamma (1-\alpha )s^{\alpha -1}=0, \end{aligned}$$

(49)

this satisfies the conditions (22) and (23) of Theorem 2 in Ref. [11], for more details, see the proof of Theorem 2 in Ref. [11]. Therefore, Lemma 7 can also be obtained by using Theorem 2 in Ref. [11].

Lemma 8

Supposing Lyapunov function $V(r,\omega (r))$ satisfies

$$\begin{aligned}&\lambda _1(\left\| \omega (r) \right\| )\le V(r,\omega (r))\le \lambda _2(\left\| \omega (r) \right\| ), \end{aligned}$$

(50)

$$\begin{aligned}&\quad {}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }V(r,\omega (r))\le -\lambda _3(\left\| \omega (r) \right\| )+\lambda _4(r-\ell )^{\overline{-\alpha }}, \end{aligned}$$

(51)

where $\alpha \in (0,1)$, $r\in {\mathbb {N}}_{\ell +1}$, $\lambda _i(i=1,2,3)$ are discrete class$-{\mathcal {K}}$ and $\lambda _4$ is any positive constant. Then, the system (39) is asymptotically stable at $\omega _e=0$.

Proof

Combining (50) and (51), it holds that

$$\begin{aligned} {}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }V(r,\omega (r))&\le -\lambda _3\Big (\lambda _2^{-1}(V(r,\omega (r)))\Big )\nonumber \\&\quad +\lambda _4(r-\ell )^{\overline{-\alpha }}. \end{aligned}$$

(52)

Then the limit for both sides of the above inequality is

$$\begin{aligned} \lim _{r\rightarrow +\infty }{}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }V(r,\omega (r))\le 0, \end{aligned}$$

(53)

it follows that there exists a constant ${\mathcal {K}}>0$ such that the inequality ${}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }V(r,\omega (r))\le 0$ holds for all $r>{\mathcal {K}}$. Using Lemma 4 gives $V(r,\omega (r))\le V(\ell ,\omega (\ell ))$ for all $r>{\mathcal {K}}$.

Case 1 Assuming the existence of a positive constant $r_1\ge {\mathcal {K}}$, such that $V(r_1,\omega (r_1))=0$, then $\omega (r_1)=0$. Thus $\omega (r)\equiv 0, \forall r\ge r_1$.

Case 2 Assuming the existence of a positive number $\varepsilon$ such that $V(r,\omega (r))\ge \varepsilon$ holds for arbitrary $r\ge 0$. Whereupon, using a similar proof of Theorem 4 in [9], one has

$$\begin{aligned} {}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }V(r,\omega (r))\le -\mu V(r,\omega (r))+\lambda _4(r-\ell )^{\overline{-\alpha }},\nonumber \\ \end{aligned}$$

(54)

where $\mu =\frac{\lambda _3\big (\lambda _2^{-1} (\varepsilon )\big )}{V(\ell ,\omega (\ell ))}$. Then, based on the proof of Lemma 7, we obtain $\lim _{r \rightarrow \infty }V(r,\omega (r))=0$, which contradicts the assumption.

To sum up, we have $\lim _{r \rightarrow \infty }V(r,\omega (r))=0$. Subsequently, by formula (50), we obtain $\lim _{r \rightarrow \infty }\omega (r)=0$. This completes the proof of Lemma 8. $\square$

In the following paper, we consider the system described as follows

$$\begin{aligned} \left\{ \begin{aligned}&_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\omega (r)=A\omega (r)+B\omega (r-\sigma ), \quad \forall r\in {\mathbb {N}}_{\ell +1},\\ {}&\omega (s)=\phi (s),\qquad \qquad \qquad \qquad \quad \forall s\in {\mathbb {N}}_{-\sigma }^\ell , \end{aligned} \right. \nonumber \\ \end{aligned}$$

(55)

where $\alpha \in (0,1)$, $\ell \in {\mathbb {R}}$, $\omega (r)\in {\mathbb {R}}^n$ is the state vector with $n\in {\mathbb {Z}}_{+}$, $\phi$ is the initial condition and A, B are constant matrices.

Based on the nabla fractional Wirtinger summation inequalities, the following stability theorem is provided.

Theorem 1

Given $\sigma >1$, the system (55) is asymptotically stable if there exist matrices $S\in {\mathbb {R}}^{2n\times 2n}\succ 0, P\in {\mathbb {R}}^{2n\times 2n}\succ 0$ and $R\in {\mathbb {R}}^{n\times n}\succ 0$ such that

$$\begin{aligned} \varPhi \prec 0, \end{aligned}$$

(56)

where

$$\begin{aligned} \varPhi&=He\left\{ \varPi _1^TS\varPi _2 \right\} +\varPi _3^TP\varPi _3-\varPi _4^TP\varPi _4 + \sigma ^2\varPi _0^TR\varPi _0\\&\quad -e_4^TRe_4-3\Big (\frac{\sigma +1}{\sigma -1}\Big )\varPi _5^TR\varPi _5,\\ \varPi _0&=Ae_1+Be_2, \varPi _1=[e_1^T, e_2^T]^T, \varPi _2=[\varPi _0^T, e_3^T]^T, \\ \varPi _3&=[e_1^T, \varPi _0^T]^T, \varPi _4=[e_2^T, e_3^T]^T, \varPi _5=e_4-2e_5,\\ \eta _1(r)&=\big [\omega ^T(r), \omega ^T(r-\sigma )\big ]^T, \\ \eta _2(r)&=\Big [\omega ^T(r), {}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\omega ^T(r) \Big ]^T,\\ \varTheta (r)&=\Big [\omega ^T(r), \omega ^T(r-\sigma ), {}_{\ell }^{C}\text {}\nabla _{r-\sigma }^{\alpha }\omega ^T(r-\sigma ),\\&\quad \sum _{\imath =r-\sigma +1}^{r}{}_{\ell }^{C}\text {}\nabla _{\imath }^{\alpha }\omega ^T(\imath ),\\&\quad \frac{1}{\sigma +1}\sum _{\imath =r-\sigma +1}^{r}\sum _{j=r-\sigma +1}^{\imath }{}_{\ell }^{C}\text {}\nabla _{j}^{\alpha }\omega ^T(j) \Big ]^T, \\ e_\imath&=\big [O_{n\times (\imath -1)n}, I_n, O_{n\times (5-\imath )n}\big ],\quad \imath =1, 2, 3, 4, 5. \\ \end{aligned}$$

Proof

Let the Lyapunov–Krasovskii functional as follows

$$\begin{aligned} V(\omega _r)=\sum _{\imath =1}^{3}V_\imath (\omega _r), \end{aligned}$$

(57)

where

$$\begin{aligned} \begin{aligned} V_1(\omega _r)&={}_{\ell }^{G}\text {}\nabla _{r}^{1-\alpha }\Big (\eta _1^T(r)S\eta _1(r)\Big ),\\ V_2(\omega _r)&=\sum _{\imath =r-\sigma +1}^{r}\eta _2^T(\imath )P\eta _2(\imath ),\\ V_3(\omega _r)&=\sigma \sum _{\imath =-\sigma +1}^{0}\sum _{j=r+\imath +1}^{r}{}_{\ell }^{C}\text {}\nabla _{j}^{\alpha }\omega ^T(j)R{}_{\ell }^{C}\text {}\nabla _{j}^{\alpha }\omega (j).\\ \end{aligned} \end{aligned}$$

By the Caputo fractional difference of $V(\omega _r)$ and taking the limit gives

$$\begin{aligned}&\lim _{\gamma \rightarrow 1^-}{}_{\ell }^{C}\text {}\nabla _{r}^{\gamma }V(\omega _r)\nonumber \\&\quad ={}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\Big (\eta _1^T(r)S\eta _1(r)\Big )+\frac{\eta _1^T(\ell )S\eta _1(\ell )}{\varGamma (1-\alpha )} (r-\ell )^{\overline{-\alpha }}\nonumber \\&\qquad +\sum _{\imath =2}^{3}\nabla V_\imath (\omega _r). \end{aligned}$$

(58)

Firstly, we calculate the first item above to get

$$\begin{aligned}&{}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\Big (\eta _1^T(r)S\eta _1(r)\Big )\nonumber \\&\quad \le \Big ({}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\eta _1^T(r)\Big )S\eta _1(r)+\eta _1^T(r)S{}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\eta _1(r)\nonumber \\&\quad =\varTheta ^T(r)He\left\{ \varPi _1^TS\varPi _2 \right\} \varTheta (r). \end{aligned}$$

(59)

Next, the computation of $\nabla V_2(\omega _r)$ and $V_3(\omega _r)$ yield

$$\begin{aligned} \nabla V_2(\omega _r)&=\eta _2^T(r)P\eta _2(r)-\eta _2^T(r-\sigma )P\eta _2(r-\sigma )\nonumber \\&=\varTheta ^T(r)\Big [\varPi _3^TP\varPi _3-\varPi _4^TP\varPi _4 \Big ]\varTheta (r). \end{aligned}$$

(60)

And according to Lemma 6, one has

$$\begin{aligned}&\nabla V_3(\omega _r)\nonumber \\&\quad =\sigma ^2{}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\omega ^T(r)R{}_{\ell }^{C}\text {}\nabla _{r}^{\alpha }\omega (r)\nonumber \\&\qquad -\sigma \sum _{\imath =r-\sigma +1}^{r}{}_{\ell }^{C}\text {}\nabla _{\imath }^{\alpha }\omega ^T(\imath )R{}_{\ell }^{C}\text {}\nabla _{\imath }^{\alpha }\omega (\imath )\nonumber \\&\quad \le \varTheta ^T(r)\Big [\sigma ^2\varPi _0^TR\varPi _0-e_4^TRe_4\nonumber \\&\qquad -3\Big (\frac{\sigma +1}{\sigma -1}\Big )\varPi _5^TR\varPi _5 \Big ]\varTheta (r). \end{aligned}$$

(61)

Substituting the formulae (59)−(61) into (58), it follows that

$$\begin{aligned}&\lim _{\gamma \rightarrow 1-}{}_{\ell }^{C}\text {}\nabla _{r}^{\gamma }V(\omega _r)\nonumber \\&\quad \le \varTheta ^T(r)\Big [He\left\{ \varPi _1^TS\varPi _2 \right\} +\varPi _3^TP\varPi _3-\varPi _4^TP\varPi _4 \nonumber \\&\qquad + \sigma ^2\varPi _0^TR\varPi _0-e_4^TRe_4-3\Big (\frac{\sigma +1}{\sigma -1}\Big )\varPi _5^TR\varPi _5 \Big ] \varTheta (r)\nonumber \\&\qquad +\frac{\eta _1^T(\ell )S\eta _1(\ell )}{\varGamma (1-\alpha )} (r-\ell )^{\overline{-\alpha }}\nonumber \\&\quad =\varTheta ^T(r)\varPhi \varTheta (r)+\frac{\eta _1^T(\ell )S\eta _1(\ell )}{\varGamma (1-\alpha )} (r-\ell )^{\overline{-\alpha }}\nonumber \\&\quad \le \lambda _{\max } (\varPhi )\varTheta ^T(r)\varTheta (r)+\frac{\eta _1^T(\ell )S\eta _1(\ell )}{\varGamma (1-\alpha )} (r-\ell )^{\overline{-\alpha }}. \end{aligned}$$

(62)

In the light of Lemma 8, the system (55) is asymptotically stable at $\omega _e=0$. The proof is completed. $\square$

Corollary 3

Given $\sigma >1$, the system (55) is asymptotically stable if there exist matrices $S\in {\mathbb {R}}^{2n\times 2n}\succ 0, P\in {\mathbb {R}}^{2n\times 2n}\succ 0$ and $R\in {\mathbb {R}}^{n\times n}\succ 0$ such that

$$\begin{aligned} \varPhi&=He\left\{ \varPi _1^TS\varPi _2 \right\} +\varPi _3^TP\varPi _3-\varPi _4^TP\varPi _4 + \sigma ^2\varPi _0^TR\varPi _0\nonumber \\&\quad -e_4^TRe_4-3\varPi _5^TR\varPi _5\nonumber \\&\quad \prec 0, \end{aligned}$$

(63)

where the above matrices are defined in Theorem 1.

4 Numerical simulation examples

Example 1

Consider the system (55) with

$$\begin{aligned} A=\begin{bmatrix} -0.8&{} 0\\ 0.05&{}-0.91 \end{bmatrix},\quad B=\begin{bmatrix} -0.1&{}0 \\ -0.1&{}-0.1 \end{bmatrix}, \end{aligned}$$

(64)

$\sigma =20$, $\alpha =0.85$, $\omega (\ell )= [1.5, -0.8]^T$ and $\ell = 3$.

The feasible solution for LMI (56) in Theorem 1 can be obtained by Matlab’s LMI toolbox as follows:

$$\begin{aligned}{} & {} \begin{array}{cccc} S=\left( \begin{array}{cccc} 1.5080 &{} 0.0316 &{} -0.0195 &{} 0.0009\\ 0.0316 &{} 1.3839 &{} -0.0041 &{} -0.0068\\ -0.0195 &{} -0.0041 &{} 0.4240 &{} 0.0009\\ 0.0009 &{} -0.0068 &{} 0.0009 &{} 0.4235 \end{array}\right) ,&\end{array} \\{} & {} \begin{array}{cccc} P=\left( \begin{array}{cccc} 0.6655 &{} 0.0074 &{} 0.2645 &{} -0.0045\\ 0.0074 &{} 0.6894 &{} 0.0036 &{} 0.2577\\ 0.2645 &{} 0.0036 &{} 0.7016 &{} -0.0008\\ -0.0045 &{} 0.2577 &{} -0.0008 &{} 0.6851 \end{array}\right) , \end{array} \\{} & {} \begin{array}{cccc} R=\left( \begin{array}{cc} 0.0046 &{} 0.0003\\ 0.0003 &{} 0.0035 \end{array}\right) . \end{array} \end{aligned}$$

Therefore, the system (64) is asymptotically stable according to Theorem 1. Figure 1 shows the solution of the system (64) gradually converges to the origin, thus confirming the correctness and effectiveness of Theorem 1.

Fig. 1

The state trajectories of the system (64)

Example 2

Consider the system (55) with

$$\begin{aligned} A= & {} \begin{bmatrix} -4&{} -0.8 &{} 0.1\\ 0.7&{} -0.5 &{} 0.9\\ 0.2&{} -0.8 &{}-1.5 \end{bmatrix}, B=\begin{bmatrix} 0.2&{} -0.8 &{} -0.1\\ -1.5&{} -0.1 &{} 0.1\\ -0.1&{} -0.1 &{}0.1 \end{bmatrix}, \nonumber \\ \end{aligned}$$

(65)

$\sigma =15$, $\alpha =0.9$, $\omega (\ell )= [1.3, -0.7, 1.6]^T$ and $\ell =2$.

Similarly, the feasible solution of LMI (56) within Theorem 1 employing the LMI toolbox of Matlab is as follows:

$$\begin{aligned}{} & {} \begin{array}{cccc} S_1=\left( \begin{array}{cccccc} 2.5144 &{} 0.5425 &{} -0.2095 \\ 0.5425 &{} 2.0872 &{} 1.0826 \\ -0.2095 &{} 1.0826 &{} 2.6534 \end{array}\right) ,&\end{array} \\{} & {} \begin{array}{cccc} S_2=\left( \begin{array}{cccccc} -0.1411 &{} 0.3928 &{} 0.1199\\ 0.2219 &{} 0.1785 &{} 0.0084\\ -0.3300 &{} 0.0344 &{} -0.0327 \end{array}\right) ,&\end{array} \\{} & {} \begin{array}{cccc} S_3=\left( \begin{array}{cccccc} -0.1411 &{} 0.2219 &{} -0.3300 \\ 0.3928 &{} 0.1785 &{} 0.0344 \\ 0.1199 &{} 0.0084 &{} -0.0327 \end{array}\right) ,&\end{array} \\{} & {} \begin{array}{cccc} S_4=\left( \begin{array}{cccccc} 0.8850 &{} 0.0842 &{} -0.0273\\ 0.0842 &{} 0.4982 &{} 0.1112\\ -0.0273 &{} 0.1112 &{} 0.6200 \end{array}\right) ,&\end{array} \\{} & {} \begin{array}{cccccc} P_1=\left( \begin{array}{cccccc} 3.9119 &{} 0.6485 &{} -0.2787 \\ 0.6485 &{} 0.9721 &{} -0.0514 \\ -0.2787 &{} -0.0514 &{} 1.1060 \end{array}\right) , \end{array} \\{} & {} \begin{array}{cccccc} P_2=\left( \begin{array}{cccccc} 0.8048 &{} -0.0641 &{} -0.1097\\ 0.2142 &{} 0.3727 &{} 0.3263\\ -0.0914 &{} -0.1855 &{} 0.3504 \end{array}\right) , \end{array} \\{} & {} \begin{array}{cccccc} P_3=\left( \begin{array}{cccccc} 0.8048 &{} 0.2142 &{} -0.0914 \\ -0.0641 &{} 0.3727 &{} -0.1855 \\ -0.1097 &{} 0.3263 &{} 0.3504 \end{array}\right) , \end{array} \\{} & {} \begin{array}{cccccc} P_4=\left( \begin{array}{cccccc} 0.6570 &{} 0.1152 &{} -0.0315\\ 0.1152 &{} 0.6519 &{} 0.1960\\ -0.0315 &{} 0.1960 &{} 0.8244 \end{array}\right) , \end{array} \\{} & {} \begin{array}{cccc} S=\left( \begin{array}{cccccc} S_1 &{} S_2 \\ S_3 &{} S_4 \end{array}\right) ,&\end{array} \begin{array}{cccc} P=\left( \begin{array}{cccccc} P_1 &{} P_2 \\ P_3 &{} P_4 \end{array}\right) ,&\end{array} \\{} & {} \begin{array}{ccc} R=\left( \begin{array}{ccc} 0.0031 &{} 0.0014 &{} -0.0003\\ 0.0014 &{} 0.0043 &{} 0.0043\\ -0.0003 &{} 0.0043 &{} 0.0102 \end{array}\right) . \end{array} \end{aligned}$$

Hence, the system (65) is asymptotically stable in line with Theorem 1. Figure 2 illustrates the solution of the system (65) steadily converges towards the origin, thus validating the accuracy and efficacy of Theorem 1.

Fig. 2

The state trajectories of the system (65)

Table 1 Upper bound on $\sigma$ obtained for Examples 1 and 2

Table 1 represents the maximum upper bound for the time delay $\sigma$ in Examples 1 and 2.

Remark 9

The range of the time-delay values is (0,1) according to Theorems 2 and 4 in Ref. [12], but the numerical results suggest that the range of the time-delay values can be much larger. Therefore, the stability criteria proposed in this paper for discrete fractional-order time-delay systems are relatively less conservative and effective. By combining the stability criteria obtained in this paper with Ref. [12], the range of the time-delay values can be arbitrary.

Example 3

Consider a simple fractional-order RLC circuit with time delay, where inductance L and capacitance C are both fractional-order components. Assuming the V(t) is an input voltage, the current I(t) is the state variable, the ${\bar{R}}$ is resistance, and the output voltage $V_C(t)$ is the output. Assuming there is a time delay in the current transmission process, then according to the Kirchhoff’s voltage law, the motion equation of the such circuit system can be expressed as

$$\begin{aligned} L{}_{0}^{C}\text {D}_{t}^{\alpha }I(t)+{\bar{R}}I(t-\sigma )+\frac{1}{C} \int _{0}^{t} (t-s)^{\beta -1}I(s)ds=V(t). \end{aligned}$$

(66)

Let

$$\begin{aligned}&p_1(t)=I(t),&p_2(t)=V_C(t)=\frac{1}{C} \int _{0}^{t} (t-s)^{\beta -1}I(s)ds, \end{aligned}$$

then the formula (66) can be transformed into the state equation form as follows:

$$\begin{aligned} \begin{bmatrix} {}_{0}^{C}\text {D}_{t}^{\alpha }p_1(t)\\ {}_{0}^{C}\text {D}_{t}^{\beta }p_2(t) \end{bmatrix}&= \begin{bmatrix} 0&{} -\frac{1}{L} \\ \frac{\varGamma (\beta )}{C} &{}0 \end{bmatrix} \begin{bmatrix} p_1(t)\\ p_2(t) \end{bmatrix}+ \begin{bmatrix} -\frac{{\bar{R}}}{L}&{} 0 \\ 0 &{}0 \end{bmatrix} \begin{bmatrix} p_1(t-\sigma )\\ p_2(t-\sigma ) \end{bmatrix}\nonumber \\&\quad +\begin{bmatrix} \frac{V(t)}{L} \\ 0 \end{bmatrix}. \end{aligned}$$

(67)

Assuming that the equilibrium point of system (67) is $p^*$, and by $\omega (t)=p(t)-p^*$, then the Eq. (67) can be transformed into

$$\begin{aligned} {}_{0}^{C}\text {D}_{t}^{\gamma }\omega (t)=A\omega (t)+B\omega (t-\sigma ), \end{aligned}$$

(68)

where ${}_{0}^{C}\text {D}_{t}^{\gamma }=({}_{0}^{C}\text {D}_{t}^{\alpha }, {}_{0}^{C}\text {D}_{t}^{\beta })^T$, $\omega (t)=(\omega _1(t),\omega _2(t))^T$,

$$\begin{aligned} A=\begin{bmatrix} 0&{} -\frac{1}{L} \\ \frac{\varGamma (\beta )}{C} &{}0 \end{bmatrix},\quad B= \begin{bmatrix} -\frac{{\bar{R}}}{L}&{} 0 \\ 0 &{}0 \end{bmatrix}. \end{aligned}$$

The corresponding discretization form is

$$\begin{aligned} {} _{\ell }^{C}\text {}\nabla _{r}^{\gamma }\omega (r)=A\omega (r)+B\omega (r-\sigma ). \end{aligned}$$

(69)

In the following, we consider the stabilitation problem for the system (69) by empolying the state feedback control $U(r)=K_1\omega (r)+K_2\omega (r-\sigma )$, in which $K_1\in {\mathbb {R}}^{2\times 2}$ and $K_2\in {\mathbb {R}}^{2\times 2}$ are the given controller gain. Then the following closed-loop system can be obtained

$$\begin{aligned} {}_{\ell }^{C}\text {}\nabla _{r}^{\gamma }\omega (r)=(A+K_1)\omega (r)+ (B+K_2)\omega (r-\sigma ). \end{aligned}$$

(70)

Taking $\alpha =\beta =0.92$, $L=0.2$, ${\bar{R}}=1$, $C=0.5$, $\sigma =20$, $\omega (\ell )= [0.5, -0.5]^T$, $\ell =5$, and

$$\begin{aligned} K_1= \begin{bmatrix} -4&{}0 \\ 0&{}-4 \end{bmatrix},\quad K_2= \begin{bmatrix} 0&{} 1 \\ 1 &{}-2 \end{bmatrix}. \end{aligned}$$

The feasible solution for LMI (56) in Theorem 1 can be obtained by Matlab’s LMI toolbox as follows:

$$\begin{aligned}{} & {} \begin{array}{cccc} S=\left( \begin{array}{cccc} 5.1609 &{} -2.9222 &{} 1.7683 &{} -3.2971\\ -2.9222 &{} 15.7019 &{} 2.8505 &{} 1.8326\\ 1.7683 &{} 2.8505 &{} 3.3386 &{} -2.7284\\ -3.2971 &{} 1.8326 &{} -2.7284 &{} 8.6407 \end{array}\right) ,&\end{array} \\{} & {} \begin{array}{cccc} P=\left( \begin{array}{cccc} 33.0366 &{} -13.1143 &{} 2.8347 &{} -6.1658\\ -13.1143 &{} 28.0199 &{} -0.7934 &{} 5.0254\\ 2.8347 &{} -0.7934 &{} 0.6940 &{} -1.0754\\ -6.1658 &{} 5.0254 &{} -1.0754 &{} 5.9837 \end{array}\right) , \end{array} \\{} & {} \begin{array}{cccc} R=10^{-3}\times \left( \begin{array}{cc} 0.0595 &{} -0.1065\\ -0.1065 &{} 0.9380 \end{array}\right) . \end{array} \end{aligned}$$

Therefore, the system (70) is asymptotically stable according to Theorem 1. Figure 3 shows the solution of the system (70) gradually converges to the origin, thus confirming the correctness and effectiveness of Theorem 1.

Fig. 3

The state trajectories of the system (70)

5 Conclusions

In addressing the issue of conservatism in the stability analysis of nabla discrete fractional-order time-delay systems, this study has employed newly constructed fractional Wirtinger summation inequalities. Firstly, with the Lyapunov–Krasovskii functional incorporating double fractional summation terms, the process of taking fractional derivatives has become intricate. To overcome this, transitional properties from fractional difference to integer-order difference have been introduced. Secondly, the study has devised several novel fractional Wirtinger summation inequalities and has demonstrated that as the order approaches $1^-$, it degenerates into the integer-order Wirtinger summation inequality. This feature has proven pivotal in mitigating conservatism in the study of nabla discrete fractional-order time-delay systems. Subsequently, to investigate the stability of the considered systems, several generalized Lyapunov stability theories have been proposed. Finally, the correctness and effectiveness of the obtained results have been validated through numerical simulation experiments.

In future research, we will work on improving the proposed fractional Wirtinger summation inequality to further mitigate conservatism in discrete fractional-order systems with time delay.