A new result on stability analysis for discrete system with interval time-varying delays

Abstract

This paper is concerned with the stability criteria for a discrete-time linear system with interval time-varying delays. By using the time delay division we construct an augmented Lyapunov–Krasovskii functional for two delay subintervals. Moreover, we use a new summation inequality to estimate the derivatives of LKFs more accurately and derive less conservative criteria. Finally, we present two numerical examples to demonstrate that the obtained results are less conservative.

1 Introduction

During the transmission and processing of signal or data, time delay inevitably occurs in such practical systems as neural networks and networked control systems. It is well known that the presence of time delay often affects the dynamic behavior of the system, resulting in deterioration or even instability in system performance. Therefore, the research on the stability of time delay systems has been the focus of many scholars [1,2,3,4,5].

One essential research focus on a system with time delay is how to reduce the conservatism of the stability condition. In view of this, many researchers choose to approach this from two perspectives, selecting an appropriate Lyapunov–Krasovskii functional (LKF) and making a more accurate estimation of its derivative. With regard to the former aspect, many significant results have been achieved such as constructing an amplified LKF [6], increasing its integral multiplicity [7], or dividing the time delay interval [8]. With regard to the latter, the Jensen inequality [9, 10] is the first powerful tool used to estimate the derivative of LKF in the stability theory of time-delay systems. Later on, the Wirtinger-based integral inequality [11, 12], the free-matrix-based integral inequality [13,14,15], and a series of integral inequalities [16] have been proposed. These inequalities are widely applied in time delay systems [17,18,19,20,21,22,23,24,25,26,27,28]. Of course, there are other methods or combinations of different methods, such as the free-weighting matrix approach proposed in the literature [29], the basic idea is introducing some free weighting matrix through the Newton–Leibniz formula and/or the equation of state of the system, and the less conservative stability results can also be obtained.

It is worth noting that not only continuous-time systems, but also discrete-time systems play an important role and have received extensive attention in the literature [30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47]. In [12], by using the Wirtinger-based inequality an improved stability condition is obtained. However, the allowable delay upper bound can be further lifted. This inspires our research in this paper.

Based on the above discussions, by means of a new summation inequality and time delay division, less conservative criteria than some existing ones are obtained, and two examples are presented to demonstrate the effectiveness and superiority of the proposed approach. The main contributions of this paper are as follows:

• (i) A new summation inequality is obtained by the Schmidt orthogonalization method, and the corresponding result is less conservative.

• (ii) The proof of Corollary 1 in [15] can be simplified by the substitution of variables, which is demonstrated by Lemma 2.1 in this paper.

Notation: Throughout this paper, $\Re ^{n}$ denotes the n-dimensional Euclidean space, and $\Re ^{n \times m}$ is the set of all $n \times m$ real matrices; I is used to denote the identity matrix of proper dimension; $\operatorname{diag}\{\cdot \cdot \cdot \}$ denotes block diagonal matrix, and $\langle f,g\rangle$ denotes the inner product of f and g; $P>0$ (≥0) means that P is a real symmetric and positive definite (semipositive definite) matrix; the symmetric term in a matrix is denoted by ∗, and $\operatorname{sym}[A]=A+A^{T}$; $N^{+}(N)$ is the set of all positive (nonnegative) integers.

2 Preliminaries

Consider the following discrete time-varying delay system:

$$ \textstyle\begin{cases} x(k+1)=Ax(k)+Bx(k-d(k)), \quad k\in N, \\ x(k) =\phi (k), \quad k=-d _{2}, -d_{2}+1,\ldots, 0, \end{cases} $$

(1)

where $x(k) = {({x_{1}}(k),{x_{2}}(k),\ldots,{x_{n}}(k))^{T}} \in {\Re ^{n}}$ is the state vector, $\phi (k)\in {\Re ^{n}}$ is the initial condition, $n\in N^{+}$, A and $B \in {\Re ^{n \times n}}$ are constant matrices, and $d(k)$ is the discrete time-varying delay satisfying $d_{1}\leq d(k)\leq d_{2}$, where $d_{1}$ and $d_{2}$ are nonnegative integers.

Lemma 2.1

Let $x(i)$ be a sequence of discrete-time variables in $[a,a+n]\rightarrow \Re ^{n}$. For symmetric matrices $R\in {\Re ^{n\times n}}$ and $Z_{11},Z_{22},Z_{33} \in {\Re ^{4n\times 4n}}$ and for any matrices $Z_{12},Z_{13},Z_{23} \in {\Re ^{4n\times 4n}}$ and $N_{1},N_{2},N_{3} \in {\Re ^{4n\times n}}$ satisfying

$$\Xi =\left(\begin{array}{cccc}{Z}_{11}& Z_{12}& Z_{13}& N_1\\ \ast & Z_{22}& Z_{23}& N_2\\ \ast & \ast & Z_{33}& N_3\\ \ast & \ast & \ast & R\end{array}\right)\ge 0,$$

we have the following inequality:

$$ -\sum_{i = a}^{a+n-1} \eta ^{{T}}(i) R \eta (i) \leq \omega ^{ {T}} \varOmega \omega , $$

(2)

where

$$\begin{array}{c}\eta (i)=x(i+1)-x(i),\hfill \\ \omega ={(x^T(a+n),x^T(a),\frac{1}{n+1}\sum _{i=a}^{a+n}{x}^T(i),\frac{2}{(n+1)(n+2)}\sum _{i=a}^{a+n}\sum _{j=i}^{a+n}{x}^T(j))}^T,\hfill \\ \Omega =n(Z_{11}+\frac{1}{3}{Z}_{22}+\frac{1}{5}{Z}_{33})+sym[N_1{\Pi }_1+N_2{\Pi }_2+N_3{\Pi }_3],\hfill \\ {\Pi }_1={\epsilon }_1-{\epsilon }_2,{\Pi }_2={\epsilon }_1+{\epsilon }_2-2{\epsilon }_3,{\Pi }_3={\epsilon }_1-{\epsilon }_2+6{\epsilon }_3-6{\epsilon }_4,\hfill \\ {\epsilon }_1=\left(\begin{array}{cccc}I& 0& 0& 0\end{array}\right),{\epsilon }_2=\left(\begin{array}{cccc}0& I& 0& 0\end{array}\right),\hfill \\ {\epsilon }_3=\left(\begin{array}{cccc}0& 0& I& 0\end{array}\right),{\epsilon }_4=\left(\begin{array}{cccc}0& 0& 0& I\end{array}\right).\hfill \end{array}$$

Proof

Define

$$ f_{0}(i)=1,\quad\quad f_{1}(i)=i,\quad\quad f_{2}(i)=i^{2}. $$

By the Schmidt orthogonalization method, where $\langle f,g\rangle =\sum_{i = 1}^{n} f(i)g(i)$, we obtain the following orthogonal polynomials $g_{k}(i)$, $k={0,1,2}$:

$$ g_{0}(i)=1,\quad\quad g_{1}(i)=i-\frac{n+1}{2},\quad \quad g_{2}(i)=i^{2}-(n+1)i+ \frac{(n+1)(n+2)}{6}. $$

Then converting the coefficient of constant term to 1 or −1 by multiplying by a positive number, we have

$$ h_{0}(i)=1,\quad\quad h_{1}(i)=-1+\frac{2}{ n+1}i, \quad\quad h_{2}(i)=1-\frac{6}{ n+2}i+\frac{6}{(n+1)(n+2)}i^{2}, $$

and $h_{k}(i)$, $k={0,1,2}$, satisfy

$$ \langle h_{0},h_{0}\rangle =n,\quad\quad \langle h_{1},h_{1}\rangle =n \frac{n-1}{3(n+1)},\quad\quad \langle h_{2},h _{2}\rangle =n \frac{(n-1)(n-2)}{5(n+1)(n+2)}. $$

Let $\varsigma (i)=(h_{0}(i)u^{T}, h_{1}(i)u^{T}, h_{2}(i)u ^{T}, \xi ^{T}(i))^{T}$, where $u= (y^{T}(n+1), y^{T}(1), \frac{1}{n+1} \sum_{i = 1}^{n+1} y^{T}(i), \frac{2}{(n+1)(n+2)} \sum_{i = 1}^{n+1}\sum_{j = i}^{n+1} y^{T}(j) )^{T}$, $y(i)$ is a sequence of discrete-time variable in $[1,1+n]\rightarrow \Re ^{n}$, and $\xi (i)=y(i+1)-y(i)$. Noticing that $\sum_{i = 1}^{n+1} iy(i)=\sum_{i = 1}^{n+1}\sum_{j = i}^{n+1} y(j)$, we can obtain

$$ \sum_{i = 1}^{n } h_{k}(i)\xi (i)= \varPi _{k+1}u, \quad k={0,1,2}. $$

It is easy to see that $\varsigma ^{T}(i) \varXi \varsigma (i)\geq 0$. Then summing both sides of the inequality before from 1 to n yields

$$ -\sum_{i = 1}^{n} \xi ^{{T}}(i) R \xi (i) \leq u^{{T}} \tilde{\varOmega } u, $$

where $\tilde{\varOmega }=n(Z_{11}+\frac{n-1}{3(n+1)}Z_{22}+ \frac{(n-1)(n-2)}{5(n+1)(n+2)}Z_{33}) +\operatorname{sym}[N_{1}\varPi _{1}+N_{2}\varPi _{2}+N _{3}\varPi _{3}]$.

Let $x(i)=y(i-a+1)$ be a sequence of discrete-time variables in $[a,a+n]\rightarrow \Re ^{n}$ satisfying

$$ -\sum_{i = a}^{a+n-1} \eta ^{{T}}(i) R \eta (i)=-\sum_{i = 1}^{n} \xi ^{{T}}(i) R \xi (i) \leq u^{{T}} \tilde{\varOmega } u = \omega ^{{T}} \tilde{\varOmega } \omega . $$

Define ${\varOmega }=\tilde{\varOmega }+\frac{2n}{3(n+1)}Z_{22}+\frac{6n ^{2}}{5(n+1)(n+2)}Z_{33}$. Then Ω is affinely dependent on n, and we obtain (2). This completes the proof. □

Remark 1

Define $f_{3}(i)=i^{3}$. Then $h_{3}(i)=-1+\lambda \frac{12}{ n+3}i-\frac{30}{(n+2)(n+3)}i^{2}+ \frac{20}{(n+1)(n+2)(n+3)}i^{3}$, where $\lambda = \frac{6n^{2}+15n+11}{6(n+1)(n+2)}$, and it is not the fourth Legendre polynomial anymore [16]. Thus deriving a general form of the discrete inequality (2) is still an interesting and challenging task.

Remark 2

By the substitution of variables demonstrated in Lemma 2.1, the proof of Corollary 1 in [15] can be simplified. By means of the new summation inequality the results calculated based on the criteria given in this paper are less conservative than those reported in the existing literature from Tables 1 and 2.

Table 1 Allowable delay upper bound $d_{2}$ for various $d_{1}$ for Example 1

Table 2 Allowable delay upper bound $d_{2}$ for various $d_{1}$ for Example 2

3 Main results

For reducing the conservatism, we further present a delay partitioning method, where the time delay interval is divided into two segments: $[d_{1}, d_{2}]=[d_{1}, d]\cup [d, d_{2}]$.

Theorem 3.1

For given positive integers $d_{1}$ and $d_{2}$ satisfying $d_{1} \leq d(k)\leq d_{2}$, system (1) is asymptotically stable if there exist matrices $P>0$ ($\in {\Re ^{4n\times 4n}}$), $Q_{i}>0$ ($\in {\Re ^{n \times n}}$), symmetric matrices $R\in {\Re ^{n\times n}}$ and $Z_{j11},Z_{j22},Z_{j33} \in {\Re ^{4n\times 4n}}$, any matrices $Z_{j12},Z_{j13},Z_{j23} \in {\Re ^{4n\times 4n}}$ and $N_{ji}\in {\Re ^{4n\times n}}$ ($i=1,2,3$; $j=1,2,\ldots,7$) such that the following LMIs hold:

$$\begin{aligned}& \begin{aligned}[b] \varPsi _{1} \bigl(d(k) \bigr)&=D_{1}+D_{3}+D_{4}+D_{5}+D_{6}+{G_{1} \bigl(d (k) \bigr)}+{G_{3} \bigl(d (k) \bigr)}+ {G_{4} \bigl(d (k) \bigr)}+{{H_{1}}}+{{H_{2}}} \\ &< 0, \quad d(k)\in [d_{1}, d]\cap N,\end{aligned} \end{aligned}$$

(3)

$$\begin{aligned}& \begin{aligned}[b] \varPsi _{2} \bigl(d(k) \bigr) &=D_{2}+D_{3}+D_{4}+D_{7}+D_{8}+{G_{2} \bigl(d (k) \bigr)}+{G_{5} \bigl(d (k) \bigr)}+{G_{6} \bigl(d (k) \bigr)}+{{H_{1}}}+{{H_{3}}} \\ &< 0, \quad d(k)\in [d, d_{2}] \cap N, \end{aligned} \end{aligned}$$

(4)

$${\Xi }_j=\left(\begin{array}{cccc}{Z}_{j11}& Z_{j12}& Z_{j13}& N_{j1}\\ \ast & Z_{j22}& Z_{j23}& N_{j2}\\ \ast & \ast & Z_{j33}& N_{j3}\\ \ast & \ast & \ast & R_i\end{array}\right)\ge 0,\mathit{\text{and}}i=\{\begin{array}{cc}1\hfill & \mathit{\text{if }}j=1,\hfill \\ 2\hfill & \mathit{\text{if }}j=2,3,5,\hfill \\ 3\hfill & \mathit{\text{if }}j=4,6,7,\hfill \end{array}$$

(5)

where

$$\begin{aligned}& D_{1}={E^{T}_{2}}P{E_{2}}-{E^{T}_{1}}P{E_{1}}, \\& D_{2}={E^{T}_{4}}P{E_{4}}-{E^{T}_{3}}P{E_{3}}, \\& D_{3}={e^{T}_{1}}Q_{1}{e_{1}}-{e^{T}_{2}}(Q_{1}-Q_{2}){e_{2}}-{e^{T} _{4}}(Q_{2}-Q_{3}){e_{4}}-{e^{T}_{5}}Q_{3}{e_{5}}, \\& D_{4}=d_{1}(e_{0}-e_{1})^{T}R_{1}(e_{0}-e_{1})+{ \alpha _{1}}\operatorname{sym}[N_{11} \varPi _{1}+N_{12}\varPi _{2}+N_{13}\varPi _{3}]{\alpha _{1}^{T}}, \\& \begin{aligned} D_{5}={} &(d-d_{1}) (e_{0}-e_{1})^{T}R_{2}(e_{0}-e_{1})+{ \alpha _{2}}\operatorname{sym}[N _{21}\varPi _{1}+N_{22}\varPi _{2}+N_{23}\varPi _{3}]{\alpha _{2}^{T}} \\ &{}+{\alpha _{3}}\operatorname{sym}[N_{31}\varPi _{1}+N_{32}\varPi _{2}+N_{33}\varPi _{3}]{\alpha _{3}^{T}}, \end{aligned} \\& D_{6}= (d_{2}-d) (e_{0}-e_{1})^{T}R_{3}(e_{0}-e_{1})+{ \alpha _{4}}\operatorname{sym}[N _{41}\varPi _{1}+N_{42}\varPi _{2}+N_{43}\varPi _{3}]{\alpha _{4}^{T}}, \\& D_{7}=(d-d_{1}) (e_{0}-e_{1})^{T}R_{2}(e_{0}-e_{1})+{ \alpha _{5}}\operatorname{sym}[N _{51}\varPi _{1}+N_{52}\varPi _{2}+N_{53}\varPi _{3}]{\alpha _{5}^{T}}, \\& \begin{aligned} D_{8}={} &(d_{2}-d) (e_{0}-e_{1})^{T}R_{3}(e_{0}-e_{1})+{ \alpha _{6}}\operatorname{sym}[N _{61}\varPi _{1}+N_{62}\varPi _{2}+N_{63}\varPi _{3}]{\alpha _{6}^{T}} \\ &{}+{\alpha _{7}}\operatorname{sym}[N_{71}\varPi _{1}+N_{72}\varPi _{2}+N_{73}\varPi _{3}]{\alpha _{7}^{T}}, \end{aligned} \\& G_{1} \bigl(d(k) \bigr)=\operatorname{sym} \bigl[(E_{2}-E_{1})^{T}PF_{1} \bigl(d(k) \bigr) \bigr], \\& G_{2} \bigl(d(k) \bigr)=\operatorname{sym} \bigl[(E_{4}-E_{3})^{T}PF_{2} \bigl(d(k) \bigr) \bigr], \\& G_{3} \bigl(d(k) \bigr)={\alpha _{2}} \biggl[ \bigl(d(k)-d_{1} \bigr) \biggl(Z_{211}+\frac{1}{3}Z_{222}+ \frac{1}{5}Z_{233} \biggr) \biggr]{\alpha _{2}^{T}}, \\& G_{4} \bigl(d(k) \bigr)={\alpha _{3}} \biggl[ \bigl(d-d(k) \bigr) \biggl(Z_{311}+\frac{1}{3}Z_{322}+ \frac{1}{5}Z_{333} \biggr) \biggr]{\alpha _{3}^{T}}, \\& G_{5} \bigl(d(k) \bigr)={\alpha _{6}} \biggl[ \bigl(d(k)-d \bigr) \biggl(Z_{611}+\frac{1}{3}Z_{622}+ \frac{1}{5}Z_{633} \biggr) \biggr]{\alpha _{6}^{T}}, \\& G_{6} \bigl(d(k) \bigr)={\alpha _{7}} \biggl[ \bigl(d_{2}-d(k) \bigr) \biggl(Z_{711}+\frac{1}{3}Z_{722}+ \frac{1}{5}Z_{733} \biggr) \biggr]{\alpha _{7}^{T}}, \\& F_{1} \bigl(d(k) \bigr)= \bigl(0,0, \bigl(d(k)-d_{1}+1 \bigr)e_{7}^{T}+ \bigl(d-d(k)+1 \bigr)e_{8}^{T} ,0 \bigr)^{T}, \\& F_{2} \bigl(d(k) \bigr)= \bigl(0,0, \bigl(d(k)-d+1 \bigr)e_{8}^{T} + \bigl(d_{2}-d(k)+1 \bigr)e_{9}^{T},0 \bigr)^{T}, \\& \begin{aligned} E_{1}= {}&\biggl(e_{1}^{T},(d_{1}+1)e_{6}^{T}-e_{1}^{T},(d_{2}-d+1)e_{9}^{T}-e _{2}^{T}-e_{3}^{T}-e_{4}^{T}, \\ &{} \frac{1}{2}(d_{1}+1) (d_{1}+2)e_{10}^{T}-(d _{1}+1)e_{1}^{T} \biggr)^{T}, \end{aligned} \\& \begin{aligned} E_{2}= {}&\biggl(e_{0}^{T},(d_{1}+1)e_{6}^{T}-e_{2}^{T},(d_{2}-d+1)e_{9}^{T}-e _{3}^{T}-e_{4}^{T}-e_{5}^{T}, \\ &{} \frac{1}{2}(d_{1}+1) (d_{1}+2)e_{10}^{T}-(d _{1}+1)e_{6}^{T} \biggr)^{T}, \end{aligned} \\& \begin{aligned} E_{3}= {}&\biggl(e_{1}^{T},(d_{1}+1)e_{6}^{T}-e_{1}^{T},(d-d_{1}+1)e_{7}^{T}-e _{2}^{T}-e_{3}^{T}-e_{4}^{T}, \\ &{} \frac{1}{2}(d_{1}+1) (d_{1}+2)e_{10}^{T}-(d _{1}+1)e_{1}^{T} \biggr)^{T}, \end{aligned} \\& \begin{aligned} E_{4}= {}&\biggl(e_{0}^{T},(d_{1}+1)e_{6}^{T}-e_{2}^{T},(d-d_{1}+1)e_{7}^{T}-e _{3}^{T}-e_{4}^{T}-e_{5}^{T}, \\ &{} \frac{1}{2}(d_{1}+1) (d_{1}+2)e_{10}^{T}-(d _{1}+1)e_{6}^{T} \biggr)^{T}, \end{aligned} \\& H_{1}={\alpha _{1}} \biggl[{d_{1} \biggl(Z_{111}+\frac{1}{3}Z_{122}+\frac{1}{5}Z_{133} \biggr)} \biggr] {\alpha _{1}^{T}}, \\& H_{2}={\alpha _{4}} \biggl[(d_{2}-d) \biggl(Z_{411}+\frac{1}{3}Z_{422}+\frac{1}{5}Z _{433} \biggr) \biggr]{\alpha _{4}^{T}}, \\& H_{3}={\alpha _{5}} \biggl[(d-d_{1}) \biggl(Z_{511}+\frac{1}{3}Z_{522}+\frac{1}{5}Z _{533} \biggr) \biggr]{\alpha _{5}^{T}}, \end{aligned}$$

and

$$\begin{aligned}& \alpha _{1}= \bigl(e^{T}_{1},e^{T}_{2},e^{T}_{6},e^{T}_{10} \bigr),\quad\quad \alpha _{2}= \bigl(e^{T}_{2},e^{T}_{3},e^{T}_{7},e^{T}_{11} \bigr),\quad \quad \alpha _{3}= \bigl(e ^{T}_{3},e^{T}_{4},e^{T}_{8},e^{T}_{12} \bigr), \\& \alpha _{4}= \bigl(e^{T}_{4},e^{T}_{5},e^{T}_{9},e^{T}_{13} \bigr),\quad \quad \alpha _{5}= \bigl(e^{T}_{2},e^{T}_{4},e^{T}_{7},e^{T}_{11} \bigr),\quad \quad \alpha _{6}= \bigl(e ^{T}_{4},e^{T}_{3},e^{T}_{8},e^{T}_{12} \bigr), \\& \alpha _{7}= \bigl(e^{T}_{3},e^{T}_{5},e^{T}_{9},e^{T}_{13} \bigr),\quad \quad e_{i}=(0_{n \times (i-1)n},I,0_{n\times (13-i)n}), \quad i=1,2,\ldots,13, \\& e_{0}=Ae_{1}+Be_{3}. \end{aligned}$$

Proof

Construct a Lyapunov functional as follows:

$$ V(k) = \sum_{i = 1}^{5} {{V_{i}}(k)}, $$

where

$$\begin{aligned}& {V_{1}}(k) = {\theta ^{T}}(k)P{\theta (k)}, \\ & {V_{2}}(k) = \sum_{i =k-d_{1}}^{k-1} x^{T}(i)Q_{1}x(i)+\sum_{i =k-d}^{k-d_{1}-1} x^{T}(i)Q_{2}x(i) +\sum_{i =k-d _{2}}^{k-d-1} x^{T}(i)Q_{3}x(i), \\ & {V_{3}}(k) = \sum_{i = -d_{1}}^{-1} \sum_{j = k+i}^{k-1} \eta ^{{T}}(j) R_{1} \eta (j), \\ & {V_{4}}(k) = \sum_{i = -d }^{-d_{1}-1} \sum_{j = k+i} ^{k-1} \eta ^{{T}}(j) R_{2} \eta (j), \\ & {V_{5}}(k) = \sum_{i = -d_{2}}^{-d -1} \sum_{j = k+i} ^{k-1} \eta ^{{T}}(j) R_{3} \eta (j), \end{aligned}$$

and

$$ \theta (k)= \Biggl(x^{T}(k), \sum_{i =k-d_{1}}^{k-1} x^{T}(i), \sum_{i =k-d_{2}}^{k-d_{1}-1} x^{T}(i), \sum_{i = k-d _{1}}^{k-1}\sum _{j = i}^{k-1} x^{T}(j) \Biggr)^{T}. $$

For simplicity, we first define the following vectors:

$$\begin{aligned}& \begin{aligned} {\xi }_{1}^{T}(k)={}& \bigl( {x^{T}}(k)\quad {x^{T}}(k-d_{1})\quad {x^{T}} \bigl(k-d(k) \bigr) \quad {x^{T}}(k-d) \quad {x^{T}}(k-d_{2}) \\ & U^{T}_{1} \quad U^{T}_{2} \quad U^{T}_{3}\quad U^{T}_{4} \quad W^{T}_{1} \quad W^{T}_{2} \quad W^{T}_{3}\quad W^{T}_{4} \bigr), \end{aligned} \\& \begin{aligned} {\xi }_{2}^{T}(k)={}& \bigl( {x^{T}}(k)\quad {x^{T}}(k-d_{1})\quad {x^{T}} \bigl(k-d(k) \bigr) \quad {x^{T}}(k-d) \quad {x^{T}}(k-d_{2}) \\ & U^{T}_{1} \quad U^{T}_{5} \quad U^{T}_{6}\quad U^{T}_{7} \quad W^{T}_{1} \quad W^{T}_{5} \quad W^{T}_{6}\quad W^{T}_{7} \bigr), \end{aligned} \end{aligned}$$

where

$$\begin{aligned}& U_{1}=\frac{1}{d_{1}+1}\sum_{i =k-d_{1}}^{k} x(i), \\ & U_{2}=\frac{1}{d(k)-d_{1}+1}\sum_{i =k-d(k)}^{k-d_{1}} x(i), \\ & U_{3}=\frac{1}{d-d(k)+1}\sum_{i =k-d }^{k-d(k)} x(i), \\ & U_{4}=\frac{1}{d_{2}-d +1}\sum_{i =k-d_{2}}^{k-d} x(i), \\ & U_{5}=\frac{1}{d -d_{1}+1}\sum_{i =k-d }^{k-d_{1}} x(i), \\ & U_{6}=\frac{1}{d(k)-d+1}\sum_{i =k-d(k) }^{k-d } x(i), \\ & U_{7}=\frac{1}{d_{2}-d(k)+1}\sum_{i =k-d_{2}}^{k-d(k)} x(i), \\ & W_{1}=\frac{2}{(d_{1}+1)(d_{1}+2)} \sum_{i = k-d_{1}}^{k} \sum_{j = i}^{k} x(j), \\ & W_{2}=\frac{2}{(d(k)-d_{1}+1)(d(k)-d_{1}+2)}\sum_{i =k-d(k)} ^{k-d_{1}}\sum_{j = i}^{k-d_{1}} x(j), \\ & W_{3}=\frac{2}{(d -d(k)+1)(d -d(k)+2)}\sum_{i =k-d }^{k-d(k)} \sum_{j = i}^{k-d(k)} x(j), \\ & W_{4}=\frac{2}{(d_{2}-d+1)(d_{2}-d+2)} \sum_{i = k-d_{2}}^{k-d} \sum_{j = i}^{k-d} x(j), \\ & W_{5}=\frac{2}{(d -d_{1}+1)(d -d_{1}+2)}\sum_{i =k-d }^{k-d _{1}} \sum_{j = i}^{k-d_{1}} x(j), \\ & W_{6}=\frac{2}{(d(k)-d+1)(d(k)-d+2)}\sum_{i =k-d(k) }^{k-d } \sum_{j = i}^{k-d } x(j), \\ & W_{7}=\frac{2}{(d_{2}-d(k)+1)(d_{2}-d(k)+2)}\sum_{i =k-d_{2}} ^{k-d(k)}\sum_{j = i}^{k-d(k)} x(j). \end{aligned}$$

Case (i): $d(k)\in [d_{1}, d]\cap N$. The forward difference of $V_{i}(k)$ ($i=1,2,\ldots,5$) is given by

$$\begin{aligned}& \begin{aligned}[b] \triangle V_{1}(k)= {}&{\theta ^{T}}(k+1)P{\theta (k+1)}-{\theta ^{T}}(k)P {\theta (k)} \\ ={} &{\xi }_{1}^{T}(k) \bigl\{ {E^{T}_{2}}P{E_{2}}-{E^{T}_{1}}P{E_{1}}+ \operatorname{sym} \bigl[(E _{2}-E_{1})^{T}PF_{1} \bigl(d(k) \bigr) \bigr] \bigr\} {\xi }_{1}(k) \\ ={} &{\xi }_{1}^{T}(k) \bigl[D_{1}+G_{1} \bigl(d(k) \bigr) \bigr]{\xi }_{1}(k), \end{aligned} \end{aligned}$$

(6)

$$\begin{aligned}& \begin{aligned}[b] \triangle V_{2}(k)= {}& x^{T}(k)Q_{1}x(k)-x^{T}(k-d_{1})Q_{1}x(k-d_{1})+x ^{T}(k-d_{1})Q_{2}x(k-d_{1}) \\ & {} -x^{T}(k-d )Q_{2}x(k-d ) +x^{T}(k-d )Q_{3}x(k-d)-x^{T}(k-d_{2})Q_{3}x(k-d _{2}) \\ = {}&{\xi }_{i}^{T}(k) \bigl[{e^{T}_{1}}Q_{1}{e_{1}}-{e^{T}_{2}}(Q_{1}-Q_{2}) {e_{2}}-{e^{T}_{4}}(Q_{2}-Q_{3}){e_{4}}-{e^{T}_{5}}Q_{3}{e_{5}} \bigr]{\xi } _{i}(k) \\ = {}&{\xi }_{i}^{T}(k)D_{3}{\xi }_{i}(k), \quad i=1,2. \end{aligned} \end{aligned}$$

(7)

Using Lemma 2.1, we can obtain

$$\begin{aligned}& \begin{aligned}[b] \triangle V_{3}(k)={} &\sum _{i=-d_{1}}^{-1}\sum_{j = k+i+1} ^{k} \eta ^{{T}}(j) R_{1} \eta (j) -\sum _{i = -d_{1}}^{-1} \sum_{j = k+i}^{k-1} \eta ^{{T}}(j) R_{1} \eta (j) \\ = {}&d_{1}\eta ^{{T}}(k) R_{1} \eta (k)-\sum _{i =k -d_{1}}^{k-1} \eta ^{{T}}(i) R_{1} \eta (i) \\ \leq{} &{\xi }_{i}^{T}(k) \bigl[d_{1}(e_{0}-e_{1})^{T}R_{1}(e_{0}-e_{1}) \bigr] {\xi }_{i}(k) \\ & {}+{\xi }_{i}^{T}(k) \bigl(e^{T}_{1},e^{T}_{2},e^{T}_{6},e^{T} _{10} \bigr) \biggl\{ d_{1} \biggl(Z_{111}+ \frac{1}{3}Z_{122}+\frac{1}{5}Z_{133} \biggr) \\ & {} +\operatorname{sym}[N_{11}\varPi _{1}+N_{12} \varPi _{2}+N_{13}\varPi _{3}] \biggr\} \bigl(e^{T}_{1},e^{T} _{2},e^{T}_{6},e^{T}_{10} \bigr)^{T}{\xi }_{i}(k) \\ = {}&{\xi }_{i}^{T}(k)[D_{4}+H_{1}]{\xi }_{i}(k), \quad i=1,2, \end{aligned} \end{aligned}$$

(8)

$$\begin{aligned}& \begin{aligned}[b] \triangle V_{4}(k)= {}&\sum _{i = -d }^{-d_{1}-1}\sum_{j = k+i+1}^{k} \eta ^{{T}}(j) R_{2} \eta (j) -\sum _{i = -d }^{-d_{1}-1}\sum_{j = k+i}^{k-1} \eta ^{{T}}(j) R_{2} \eta (j) \\ = {}&(d -d_{1})\eta ^{{T}}(k) R_{2} \eta (k)-\sum _{i =k -d(k)}^{k-d _{1} -1} \eta ^{{T}}(i) R_{2} \eta (i) -\sum_{i =k-d }^{k-d(k)-1} \eta ^{{T}}(i) R_{2} \eta (i) \\ \leq {}& {\xi }_{1}^{T}(k) \bigl[(d -d_{1}) (e_{0}-e_{1})^{T}R_{2}(e_{0}-e_{1}) \bigr] {\xi }_{1}(k) +{\xi }_{1}^{T}(k) \bigl(e^{T}_{2},e^{T}_{3},e^{T}_{7},e^{T} _{11} \bigr) \biggl\{ \bigl(d(k)-d_{1} \bigr)\hspace{-20pt} \\ & {} \times \biggl(Z_{211}+\frac{1}{3}Z_{222}+ \frac{1}{5}Z_{233} \biggr) +\operatorname{sym}[N_{21} \varPi _{1}+N_{22}\varPi _{2}+N_{23} \varPi _{3}] \biggr\} \\ & {}\times\bigl(e^{T}_{2},e^{T}_{3},e^{T}_{7},e ^{T}_{11} \bigr)^{T}{\xi }_{1}(k) \\ & {} +{\xi }_{1}^{T}(k) \bigl(e^{T}_{3},e^{T}_{4},e^{T}_{8},e^{T}_{12} \bigr) \biggl\{ \bigl(d-d(k) \bigr) \biggl(Z _{311}+ \frac{1}{3}Z_{322}+ \frac{1}{5}Z_{333} \biggr) \\ & {} +\operatorname{sym}[N_{31}\varPi _{1}+N_{32} \varPi _{2}+N_{33}\varPi _{3}] \biggr\} \bigl(e^{T}_{3},e^{T} _{4},e^{T}_{8},e^{T}_{12} \bigr)^{T}{\xi }_{1}(k) \\ = {}&{\xi }_{1}^{T}(k) \bigl[D_{5}+G_{3} \bigl(d (k) \bigr)+G_{4} \bigl(d (k) \bigr) \bigr]{\xi }_{1}(k), \end{aligned} \end{aligned}$$

(9)

$$\begin{aligned}& \begin{aligned}[b] \triangle V_{5}(k)={} &\sum _{i = -d_{2}}^{-d -1}\sum_{j = k+i+1}^{k} \eta ^{{T}}(j) R_{3} \eta (j) -\sum _{i = -d_{2}}^{-d -1}\sum_{j = k+i}^{k-1} \eta ^{{T}}(j) R_{3} \eta (j) \\ \leq{} & {\xi }_{1}^{T}(k) \bigl[(d_{2}-d ) (e_{0}-e_{1})^{T}R_{3}(e_{0}-e _{1}) \bigr]{\xi }_{1}(k)+{\xi }_{1}^{T}(k) \bigl(e^{T}_{4},e^{T}_{5},e^{T}_{9},e ^{T}_{13} \bigr) \biggl\{ (d_{2}-d )\hspace{-20pt} \\ & {} \times \biggl(Z_{411}+\frac{1}{3}Z_{422}+ \frac{1}{5}Z_{433} \biggr) +\operatorname{sym}[N_{41} \varPi _{1}+N_{42}\varPi _{2}+N_{43} \varPi _{3}] \biggr\} \\ & {} \times\bigl(e^{T}_{4},e^{T}_{5},e^{T}_{9},e ^{T}_{13} \bigr)^{T}{\xi }_{1}(k) \\ ={} &{\xi }_{1}^{T}(k)[D_{6}+H_{2}]{\xi }_{1}(k). \end{aligned} \end{aligned}$$

(10)

From (6)–(10) we have $\triangle V(k) \leq {\xi }_{1}^{T}(k)\varPsi _{1}(d(k)) {\xi }_{1}(k)$. According to (3), we can obtain $\triangle V(k) <0$ for any ${\xi }_{1}(k)\neq 0$ and $d(k)\in [d_{1}, d]\cap N$.

Case (ii): $d(k)\in [d, d_{2}]\cap N$. Similarly, we have

$$\begin{aligned}& \begin{aligned}[b] \triangle V_{1}(k)={} &{\theta ^{T}}(k+1)P{\theta (k+1)}-{\theta ^{T}}(k)P {\theta (k)} \\ ={} &{\xi }_{2}^{T}(k) \bigl\{ {E^{T}_{4}}P{E_{4}}-{E^{T}_{3}}P{E_{3}}+ \operatorname{sym} \bigl[(E _{4}-E_{3})^{T}PF_{2} \bigl(d(k) \bigr) \bigr] \bigr\} {\xi }_{2}(k) \\ = {}&{\xi }_{2}^{T}(k) \bigl[D_{2}+G_{2} \bigl(d(k) \bigr) \bigr]{\xi }_{2}(k), \end{aligned} \end{aligned}$$

(11)

$$\begin{aligned}& \begin{aligned}[b] \triangle V_{4}(k)= {}&\sum _{i = -d }^{-d_{1}-1}\sum_{j = k+i+1}^{k} \eta ^{{T}}(j) R_{2} \eta (j) -\sum _{i = -d }^{-d_{1}-1}\sum_{j = k+i}^{k-1} \eta ^{{T}}(j) R_{2} \eta (j) \\ \leq {}& {\xi }_{2}^{T}(k) \bigl[(d-d_{1} ) (e_{0}-e_{1})^{T}R_{2}(e_{0}-e _{1}) \bigr]{\xi }_{2}(k)+{\xi }_{2}^{T}(k) \bigl(e^{T}_{2},e^{T}_{4},e^{T}_{7},e ^{T}_{11} \bigr) \biggl\{ (d-d_{1} )\hspace{-20pt} \\ & {} \times \biggl(Z_{511}+\frac{1}{3}Z_{522}+ \frac{1}{5}Z_{533} \biggr)+\operatorname{sym}[N_{51} \varPi _{1}+N_{52}\varPi _{2}+N_{53} \varPi _{3}] \biggr\} \\ & {} \times \bigl(e^{T}_{2},e^{T}_{4},e^{T}_{7},e ^{T}_{11} \bigr)^{T}{\xi }_{2}(k) \\ = {}&{\xi }_{2}^{T}(k)[D_{7}+H_{3}]{ \xi }_{2}(k), \end{aligned} \end{aligned}$$

(12)

$$\begin{aligned}& \begin{aligned}[b] \triangle V_{5}(k)= {}&\sum _{i = -d_{2}}^{-d -1}\sum_{j = k+i+1}^{k} \eta ^{{T}}(j) R_{3} \eta (j) -\sum _{i = -d_{2}}^{-d -1}\sum_{j = k+i}^{k-1} \eta ^{{T}}(j) R_{3} \eta (j) \\ ={} &(d_{2} -d )\eta ^{{T}}(k) R_{3} \eta (k)- \sum_{i =k -d(k)} ^{k-d -1} \eta ^{{T}}(i) R_{3} \eta (i) -\sum_{i =k-d_{2} }^{k-d(k)-1} \eta ^{{T}}(i) R_{3} \eta (i) \\ \leq {}& {\xi }_{2}^{T}(k) \bigl[(d_{2} -d ) (e_{0}-e_{1})^{T}R_{3}(e_{0}-e _{1}) \bigr]{\xi }_{2}(k) +{\xi }_{2}^{T}(k) \bigl(e^{T}_{4},e^{T}_{3},e^{T}_{8},e ^{T}_{12} \bigr) \biggl\{ \bigl(d(k)-d \bigr)\hspace{-30pt} \\ & {} \times \biggl(Z_{611}+\frac{1}{3}Z_{622}+ \frac{1}{5}Z_{633} \biggr)+\operatorname{sym}[N_{61} \varPi _{1}+N_{62}\varPi _{2}+N_{63} \varPi _{3}] \biggr\} \\ & {} \times \bigl(e^{T}_{4},e^{T}_{3},e^{T}_{8},e ^{T}_{12} \bigr)^{T}{\xi }_{2}(k) \\ & {} +{\xi }_{2}^{T}(k) \bigl(e^{T}_{3},e^{T}_{5},e^{T}_{9},e^{T}_{13} \bigr) \biggl\{ \bigl(d _{2}-d(k) \bigr) \biggl(Z_{711}+ \frac{1}{3}Z_{722}+\frac{1}{5}Z_{733} \biggr) \\ & {} +\operatorname{sym}[N_{71}\varPi _{1}+N_{72} \varPi _{2}+N_{73}\varPi _{3}] \biggr\} \bigl(e^{T}_{3},e^{T} _{5},e^{T}_{9},e^{T}_{13} \bigr)^{T}{\xi }_{2}(k) \\ = {}&{\xi }_{2}^{T}(k) \bigl[D_{8}+G_{5} \bigl(d (k) \bigr)+G_{6} \bigl(d (k) \bigr) \bigr]{\xi }_{2}(k), \end{aligned} \end{aligned}$$

(13)

and $\triangle V(k) \leq {\xi }_{2}^{T}(k)\varPsi _{2}(d(k)){\xi }_{2}(k)$, which under condition (4) indicates that $\triangle V(k)$ is negative definite.

Now by combining Case (i) and Case (ii), $\triangle V(k)$ is negative definite, which guarantees that system (1) is asymptotically stable. This completes the proof. □

Remark 3

To show the advantage of time delay division for the stability condition without using time delay division, we omitted the proof process and only display the results in Table 1.

4 Numerical illustrations

In this section, we present some numerical examples that often appear in the literature to demonstrate the effectiveness and sophistication of the proposed approach.

Example 1

Consider system (1) with the following parameters:

$$A=\left(\begin{array}{cc}0.8& 0\\ 0.05& 0.9\end{array}\right),B=\left(\begin{array}{cc}-0.1& 0\\ -0.2& -0.1\end{array}\right).$$

Example 2

Consider system (1), where

$$A=\left(\begin{array}{cc}0.7& 0.1\\ 0.05& 0.7\end{array}\right),B=\left(\begin{array}{cc}-0.1& 0.1\\ -0.1& -0.2\end{array}\right).$$

The allowable delay upper bound $d_{2}$ can be found for given $d_{1}$ or vice versa. The simulation results are listed in Tables 1 and 2.

Remark 4

In [12] and this paper, the stability conditions are obtained by using the Wirtinger-based integral inequality and the free-matrix-based integral inequality, respectively. We can see from Table 1 that the latter is superior to the former in reducing the conservatism when time delay division is not used, whereas the conservatism of both is the same when time delay division is used. Therefore, on the basis of some processing techniques, such as the time delay division mentioned before, our future research is discussing the relationship between the tightness of inequalities and the conservatism of criteria.

Remark 5

From Table 2 we can see that with the lower bounds $d_{1}=2,5,7,10,20$, the corresponding $d_{2}=13,14,16,19,29$ obtained in [40], $d_{2}=14,17,18,21,31$ obtained by Theorem 1 in [12], and $d_{2}=15,17,18,21,31$ derived by Theorem 3.1 in this paper. Thus the results obtained from Theorem 3.1 in this paper are significantly better than those obtained from the existing methods.

5 Conclusion

In this paper, the problem of stability criteria of discrete systems with interval time-varying delays has been investigated. By means of the discrete form of free-matrix-based integral inequality and the time delay division, less conservative criteria than some existing ones are derived due to a tighter estimation of the new inequality. Finally, some illustrative examples are presented to demonstrate the effectiveness of the obtained method.