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On delay-interval-dependent robust stability of LPD discrete-time system with mixed time-varying delays and nonlinear uncertainties

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Abstract

In this work, a delay-interval-dependent robust stability problem for linear parameter dependent (LPD) discrete-time system with discrete and distributed time-varying delays and nonlinear uncertainties is addressed. By adopting combinations of mixed model transformation, various inequality expressions, utilization of zero equations, and new parameter dependent Lyapunov–Krasovskii functional, a new delay-interval-dependent asymptotic stability criterion is derived in the form of linear matrix inequalities. Moreover, an improved delay-interval-dependent asymptotic stability criterion of discrete-time linear system with discrete time-varying delay and nonlinear uncertainties is also achieved. Some numerical examples are also illustrated to exhibit the effectiveness with less conservatism of the proposed stability criterion.

Introduction

During the past decades, various stability problems of discrete-time systems with delays have drawn many researchers’ attention, for delays in systems may lead to instability or bad performances [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. Discrete-time systems with time-varying delays have been well recognized to be typical examples of applications to engineering [3]. In recent years, various stability analyses of both discrete- and continuous-time systems with respect to uncertainty about time-invariant parameters have been among many challenging problems. An important aspect of uncertainty in a linear system is a linear parameter-dependent (LPD) system in which the uncertain state matrices in the polytope are mathematically expressed by a convex combination of known matrices. Most stability criteria have been obtained via Lyapunov–Krasovskii functional approaches, in which parameter-dependent Lyapunov functions have been employed [1, 8, 13, 14, 25, 28]. Stability analysis for discrete-time systems with interval time-varying delay and nonlinear uncertainties was studied in [21]. The robust stability criteria of LPD continuous-time system with delays were presented in [28]. Some results have been obtained for delay-interval-dependent robust stability of discrete-time systems with discrete and distributed time-varying delays [27]. However, few results have been obtained for robust stability for LPD discrete-time systems with interval time-varying delays and nonlinear perturbations.

Motivated by the above statement, the problem of stability for LPD discrete-time systems with mixed interval time-varying delays and nonlinear uncertainties is studied. Through the construction of a parameter-dependent Lyapunov–Krasovskii functional, model transformation, various forms of inequalities and utilization of zero equations, a delay-interval-dependent robust stability criterion is presented in terms of linear matrix inequalities (LMIs) whose feasibility can be easily checked. Then, based on the derived delay-interval-dependent robust stability criterion, an improved delay-range-dependent stability criterion is proposed for a discrete-time system with interval discrete time-varying delay and nonlinear uncertainties. Moreover, we include some numerical examples to exhibit the effectiveness with less conservatism of the proposed stability criterion.

Problem formulation

Notations, definition and lemmas used throughout this paper are given as follows. By \({{Z}}^{+}\) we denote the set of nonnegative integers; \({{R}}^{n}\) denotes the n-dimensional space with the vector norm \(\|\cdot \|\); by \(\|\xi \|\) we denote the Euclidean vector norm of \(\xi \in {R}^{n}\), that is, \(\|\xi \|^{2}=\xi ^{T}\xi \); \({R}^{n\times r}\) denotes the space of all \((n\times r)\)-dimensional real matrices; \(A^{T}\) denotes the transpose of a matrix A, while I denotes the identity matrix. Matrix A is called semi-positive definite (\(A \geq 0\)) if \(x^{T}Ax \geq 0\), for all \(x \in {R}^{n}\); A is positive definite (\(A > 0\)) if \(x^{T}Ax > 0\) for all \(x\neq 0\). Matrix B is called semi-negative definite (\(B \leq 0\)) if \(x^{T}Bx \leq 0\), for all \(x \in {R}^{n}\); B is negative definite (\(B < 0\)) if \(x^{T}Bx < 0\) for all \(x\neq 0\); \(A > B \) means \(A- B >0\); \(A \geq B \) means \(A- B \geq 0\); \(\lambda (A)\) denotes the set of all eigenvalues of A; \(\lambda _{\max }(A) = \max \{\mathop{\mathrm{Re}}\lambda: \lambda \in \lambda (A)\}\); \(\lambda _{\min }(A) = \min \{ \mathop{\mathrm{Re}}\lambda: \lambda \in \lambda (A)\}\); \(\lambda _{\max }(A(\varphi )) = \max \{\lambda _{\max }(A_{i}):i=1,2,\dots,N \}\); \(\lambda _{\min }(A(\varphi )) = \min \{\lambda _{\min }(A_{i}):i=1,2, \dots,N\}\), and represents the elements below the main diagonal of a symmetric matrix.

Consider the LPD discrete-time system with mixed interval time-varying delays and nonlinear perturbations of the form

$$\begin{aligned} & {\xi }(k+1) = A(\varphi )\xi (k)+ B(\varphi )\xi \bigl(k-h(k) \bigr)+C(\varphi )\sum_{i=1}^{\infty }\delta (i) \xi (k-i) \\ &\phantom{{\xi }(k+1) =}{}+f_{1}\bigl(k,\xi (k)\bigr)+f_{2}\bigl(k,\xi \bigl(k-h(k) \bigr)\bigr), \end{aligned}$$
(1)
$$\begin{aligned} &\xi (s) = \phi (s),\quad { s \in \{-h_{2},-h_{2}+1,\ldots,0\}}, \end{aligned}$$
(2)

where \(\xi (k) \in {R}^{n}\) is the state variable for \(k\in {{Z}}^{+}\) and \(\phi (s)\) is the initial value at s; \(A(\varphi )\), \(B(\varphi )\), \(C(\varphi ) \in {R}^{n\times n}\) stand for uncertain matrices defined by

$$\begin{aligned} &A(\varphi )=\sum_{i=1}^{N}\varphi _{i}A_{i}, \qquad B(\varphi )= \sum _{i=1}^{N}\varphi _{i}B_{i},\qquad C(\varphi )= \sum_{i=1}^{N}\varphi _{i}C_{i}, \\ &\sum_{i=1}^{N}\varphi _{i}=1,\quad \varphi _{i}\geq 0, A_{i}, B _{i},C_{i} \in {R}^{n \times n}, i=1,\dots,N. \end{aligned}$$

The nonlinear perturbations \(f_{1}(k,\xi (k))\) and \(f_{2}(k,\xi (k-h(k)))\) are bounded in magnitude:

$$\begin{aligned} &f_{1}^{T}\bigl(k,\xi (k)\bigr)f_{1}\bigl(k,\xi (k)\bigr)\leq \gamma ^{2} \xi ^{T}(k) \xi (k), \end{aligned}$$
(3)
$$\begin{aligned} &f_{2}^{T}\bigl(k,\xi \bigl(k-h(k)\bigr)\bigr)f_{2} \bigl(k,\xi \bigl(k-h(k)\bigr)\bigr)\leq \beta ^{2}x^{T} \bigl(k-h(k)\bigr) \xi \bigl(k-h(k)\bigr), \end{aligned}$$
(4)

where γ and β are given positive real constants. The discrete time-varying delay \(h(k)\) satisfies

$$ 0< h_{1}\leq h(k)\leq h_{2}, $$

where \(h_{1}\) and \(h_{2}\) are known positive integers. Moreover, there exists a constant scalar w such that the function \(\delta (\cdot )\) satisfies

$$\begin{aligned} \sum_{i=1}^{\infty }\delta (i)=w< \infty. \end{aligned}$$
(5)

Definition 2.1

([25])

System (1) is said to be robustly stable if there exists a positive-definite function \(V(k,\xi (k)):{Z}^{+}\times {R}^{n} \rightarrow {R}\) such that

$$ \Delta V\bigl(k,\xi (k)\bigr)=V\bigl(k+1,\xi (k+1)\bigr)-V\bigl(k,\xi (k) \bigr)< 0, $$

along any trajectory of solution of system (1) with (2).

Lemma 2.2

([5])

For any constant matrix \(W\in {R}^{n\times n}, W=W^{T}>0\), two integers \(r_{M}\) and \(r_{m}\) satisfying \(r_{M}\geq r_{m}\), vector function \(\xi: [r_{m},r_{M}]\rightarrow {{R}}^{n}\), the following inequality holds:

$$\begin{aligned} \Biggl( \sum_{i=r_{m}}^{r_{M}}\xi (i) \Biggr)^{T}W \Biggl(\sum_{i=r_{m}}^{r_{M}} \xi (i) \Biggr) \leq (r_{M}-r_{m}+1) \sum _{i=r_{m}}^{r_{M}}\xi ^{T}(i)W\xi (i). \end{aligned}$$

Lemma 2.3

([26])

Let \(Q\in {R}^{n\times n}\) be a positive-definite matrix, \(\xi _{i} \in {R}^{n},i=1,2,\ldots\) . If the sums concerned are well defined, then

$$\begin{aligned} \Biggl[\sum_{i=k-M_{1}}^{k-M_{2}-1}\sum _{j=i}^{k-M_{2}-1} \xi _{j} \Biggr]^{T} Q \Biggl[\sum_{i=k-M_{1}}^{k-M_{2}-1} \sum_{j=i}^{k-M_{2}-1} \xi _{j} \Biggr] \leq \frac{(M_{1}-M_{2})^{2}}{2}\sum_{i=k-M_{1}}^{k-M_{2}-1} \sum_{j=i}^{k-M_{2}-1}\xi _{j}^{T} Q \xi _{j}. \end{aligned}$$

Lemma 2.4

([30])

Let \(M\in {R}^{n\times n}\) be a positive-definite matrix, \(\xi _{i} \in {R}^{n}\) and \(a_{i}\geq 0\), \(i=1,2,\ldots\) . If the sums concerned are well defined, then

$$\begin{aligned} &\Biggl[\sum_{i=k-M_{2}}^{k-1} \xi _{i} \Biggr]^{T} M \Biggl[\sum _{i=k-M_{2}} ^{k-1} \xi _{i} \Biggr] \leq M_{2}\sum_{i=k-M_{2}}^{k-1}\xi _{i}^{T} M \xi _{i}, \\ &\Biggl[ \sum_{i=-M_{2}}^{-1}\sum _{i=k+j}^{k-1} \xi _{j} \Biggr]^{T} M \Biggl[ \sum_{i=-M_{2}}^{-1} \sum_{i=k+j}^{k-1} \xi _{j} \Biggr] \leq \frac{(M_{2})^{2}}{2} \sum_{i=-M_{2}}^{-1} \sum_{i=j+k}^{k-1}\xi _{j}^{T} M \xi _{j}, \\ &\Biggl[ \sum_{i=1}^{\infty } a_{i} \xi _{i} \Biggr]^{T} M \Biggl[ \sum _{i=1}^{\infty } a_{i}\xi _{i} \Biggr] \leq \Biggl[ \sum_{i=1}^{\infty } a_{i} \Biggr]\sum_{i=1}^{\infty } a_{i}\xi _{i}^{T} M \xi _{i}. \end{aligned}$$

Main results

Referring to system (1), a new delay-range-dependent robust stability approach is presented here. The Lyapunov–Krasovskii functional method is combined with the LMI technique. Let us introduce the following notations to be used later on:

$$\begin{aligned} &L_{w}(\varphi )=\sum_{i=1}^{N} \varphi _{i}L^{w}_{i},\qquad M(\varphi )=\sum _{i=1}^{N}\varphi _{i}M_{i},\qquad Z_{1}(\varphi )=\sum_{i=1}^{N} \varphi _{i}Z^{1}_{i}, \\ &Z_{2}(\varphi )=\sum_{i=1}^{N} \varphi _{i}Z^{2}_{i},\qquad P_{k}( \varphi )=\sum_{i=1}^{N}\varphi _{i}P^{k}_{i},\qquad Q_{s}(\varphi )= \sum_{i=1}^{N}\varphi _{i}Q^{s}_{i}, \\ &M_{r}(\varphi )=\sum_{i=1}^{N} \varphi _{i}M^{r}_{i},\qquad N_{r}( \varphi )=\sum_{i=1}^{N}\varphi _{i}N^{r}_{i}, \\ &\sum_{i=1}^{N}\varphi _{i}=1,\quad \varphi _{i}\geq 0, J, W, L ^{w}_{i},M_{i},M^{r}_{i},N^{r}_{i},Z^{1}_{i},Z^{2}_{i},P^{k}_{i},Q ^{s}_{i}\in R^{n \times n}, \\ &w=1,2,3, r=1,2,3,4, k=1,2,\dots,18, s=1,2,\dots,19, i=1,2,\dots,N, \\ &\prod _{i,j} = \bigl[ \varSigma _{i,j}^{m,n} \bigr]_{27\times 27}, \end{aligned}$$
(6)

where \(\varSigma _{i,j}^{m,n}={\varSigma _{i,j}^{n,m}}^{T}, m,n=1, 2, 3, \dots, 27, i,j=1, 2, 3,\dots, N\),

$$\begin{aligned} &\varSigma ^{1,1}_{i,j}={G^{1}_{i}}+{G^{2}_{i}}+{G^{1}_{i}}^{T}+{G^{2} _{i}}^{T}-{L^{1}_{i}}^{T}A^{1}_{j}+{L^{1}_{i}}^{T}B^{1}_{j}-{L^{1} _{i}}^{T}+{A^{1}_{i}}^{T}{L^{1}_{j}} \\ &\phantom{\varSigma ^{1,1}_{i,j}=}{} +{B^{1}_{i}}^{T}{L^{1}_{j}}-L^{1}_{i}+P^{2}_{i}+P^{3}_{i}+P^{4} _{i}+P^{5}_{i}+P^{8}_{i}+P^{9}_{i}+{h_{1}^{2}}Q^{1}_{i} \\ &\phantom{\varSigma ^{1,1}_{i,j}=}{} +r^{2}{h_{1}^{2}}Q^{4}_{i}+{h_{2}^{2}}Q^{7}_{i}+r^{2}{h_{2}^{2}}Q ^{10}_{i}+\rho ^{2}Q^{13}_{i}+r^{2} \rho ^{2}Q^{16}_{i}+h_{1}M^{1}_{i} \\ &\phantom{\varSigma ^{1,1}_{i,j}=}{} +N^{1}_{i}+{N^{1}_{i}}^{T}+h_{2}M^{2}_{i}+N^{2}_{i}+{N^{2}_{i}} ^{T}+rh_{1}M^{3}_{i}+N^{3}_{i}+{N^{3}_{i}}^{T} \\ &\phantom{\varSigma ^{1,1}_{i,j}=}{} +rh_{2}M^{4}_{i}+N^{4}_{i}+{N^{4}_{i}}^{T}-h_{1}^{2}P^{17}_{i}+ \epsilon _{1}\varphi ^{2}I+wP^{19}_{i}, \\ &\varSigma ^{1,2}_{i,j}=P^{1}_{i}+{G^{1}_{i}}^{T}+{G^{2}_{i}}^{T}-{L^{1} _{i}}^{T}+{A^{1}_{i}}^{T}{L^{2}_{j}}+{B^{1}_{i}}^{T}{L^{2}_{j}}-L^{2} _{i}+{h_{1}^{2}}Q^{2}_{i} \\ &\phantom{\varSigma ^{1,2}_{i,j}=}{} +r^{2}{h_{1}^{2}}Q^{5}_{i}+{h_{2}^{2}}Q^{8}_{i}+r^{2}{h_{2}^{2}}Q ^{11}_{i}+\rho ^{2}Q^{14}_{i}+r^{2} \rho ^{2}Q^{17}_{i}, \\ &\varSigma ^{1,3}_{i,j}=-N^{1}_{i},\qquad \varSigma ^{1,4}_{i,j}=-N^{2}_{i},\qquad \varSigma ^{1,5}_{i,j}=-G^{1}_{i}+{L^{1}_{i}}^{T}B^{2}_{j},\qquad \varSigma ^{1,6}_{i,j}=-N^{3}_{i}, \\ &\varSigma ^{1,7}_{i,j}=-N^{4}_{i},\qquad \varSigma ^{1,8}_{i,j}=-G^{2}_{i}+ {L^{1}_{i}}^{T}A^{2}_{j}+h_{1}P^{17}_{i},\qquad \varSigma ^{1,21}_{i,j}=-G^{1} _{i}-{L^{1}_{i}}^{T}B^{1}_{j}, \\ &\varSigma ^{1,22}_{i,j}=-G^{2}_{i}+{L^{1}_{i}}^{T}A^{2}_{j}+{A^{1}_{i}} ^{T}L^{3}_{j}+{B^{1}_{i}}^{T}L^{3}_{j}-L^{3}_{i}, \\ &\varSigma ^{1,25}_{i,j}={L^{1}_{i}}^{T},\qquad \varSigma ^{1,26}_{i,j}={L ^{1}_{i}}^{T},\qquad \varSigma ^{1,27}_{i,j}={L^{1}_{i}}^{T}C_{j}, \\ &\varSigma ^{2,2}_{i,j}=P^{1}_{i}-{L^{2}_{i}}^{T}-L^{2}_{i}+{h_{1}^{2}}Q ^{3}_{i}+r^{2}{h_{1}^{2}}Q^{6}_{i}+{h_{2}^{2}}Q^{9}_{i}+r^{2}{h_{2} ^{2}}Q^{12}_{i}+\rho ^{2}Q^{15}_{i} \\ &\phantom{\varSigma ^{2,2}_{i,j}=}{} +r^{2}\rho ^{2}Q^{18}_{i}+{h_{1}}P^{10}_{i}+{h_{2}}P^{11}_{i}+r {h_{1}}P^{12}_{i}+r{h_{2}}P^{13}_{i}+h_{1}^{2}P^{14}_{i} \\ &\phantom{\varSigma ^{2,2}_{i,j}=}{} +h_{2}^{2}P^{15}_{i}+\rho ^{2}P^{16}_{i}+\frac{h_{1}^{4}}{4}P ^{17}_{i}+\frac{({h_{2}-h_{1}})^{2}}{4}P^{18}_{i}+h_{2}^{2}Z^{1}_{i}+(rh _{2})^{2}Z^{2}_{i}, \\ &\varSigma ^{2,5}_{i,j}=-G^{1}_{i}+{L^{2}_{i}}^{T}B^{2}_{j},\qquad \varSigma ^{2,8}_{i,j}=-G^{2}_{i}+{L^{2}_{i}}^{T}A^{2}_{j},\qquad \varSigma ^{2,21} _{i,j}=-G^{1}_{i}+{L^{2}_{i}}^{T}B^{1}_{j}, \\ &\varSigma ^{2,22}_{i,j}=-G^{2}_{i}+{L^{2}_{i}}^{T}A^{2}_{j}-L^{3}_{i},\qquad \varSigma ^{2,25}_{i,j}={L^{2}_{i}}^{T},\qquad \varSigma ^{2,26}_{i,j}= {L^{2}_{i}}^{T}, \\ &\varSigma ^{2,27}_{i,j}={L^{2}_{i}}^{T}C_{j},\qquad \varSigma ^{3,3}_{i,j}=-P ^{2}_{i}+P^{6}_{i}-P^{18}_{i},\qquad \varSigma ^{3,23}_{i,j}=P^{18}_{i}, \\ &\varSigma ^{4,4}_{i,j}=-P^{4}_{i}-P^{6}_{i},\qquad \varSigma ^{5,5}_{i,j}=-P ^{8}_{i}-P^{18}_{i}+ \epsilon _{2}\beta ^{2}I,\qquad \varSigma ^{5,22}_{i,j}= {B^{2}_{i}}^{T}L^{3}_{j}, \\ &\varSigma ^{5,24}_{i,j}=P^{18}_{i}, \qquad \varSigma ^{6,6}_{i,j}=-P^{3}_{i}+P ^{7}_{i},\qquad \varSigma ^{7,7}_{i,j}=-P^{5}_{i}-P^{7}_{i}, \\ &\varSigma ^{8,8}_{i,j}=-P^{9}_{i}-P^{17}_{i},\qquad \varSigma ^{8,22}_{i,j}= {A^{2}_{i}}^{T}L^{3}_{j},\qquad \varSigma ^{9,9}_{i,j}=-Q^{1}_{i}, \varSigma ^{9,15}_{i,j}=-Q^{2}_{i}, \\ &\varSigma ^{10,10}_{i,j}=-Q^{4}_{i},\qquad \varSigma ^{10,16}_{i,j}=-Q^{5} _{i},\qquad \varSigma ^{11,11}_{i,j}=-Q^{7}_{i},\qquad \varSigma ^{11,17}_{i,j}=-Q ^{8}_{i}, \\ &\varSigma ^{12,12}_{i,j}=-Q^{10}_{i},\qquad \varSigma ^{12,18}_{i,j}=-Q^{11} _{i},\qquad \varSigma ^{13,13}_{i,j}=-Q^{13}_{i},\qquad \varSigma ^{13,19}_{i,j}=-Q ^{14}_{i}, \\ &\varSigma ^{14,14}_{i,j}=-Q^{16}_{i},\qquad \varSigma ^{14,20}_{i,j}=-Q^{17} _{i},\qquad \varSigma ^{15,15}_{i,j}=-Q^{3}_{i}-P^{14}_{i},\qquad \varSigma ^{16,16} _{i,j}=-Q^{6}_{i}, \\ &\varSigma ^{17,17}_{i,j}=-Q^{9}_{i}-P^{15}_{i},\qquad \varSigma ^{18,18}_{i,j}=-Q ^{12}_{i},\qquad \varSigma ^{19,19}_{i,j}=-Q^{15}_{i}-P^{16}_{i}, \\ &\varSigma ^{21,21}_{i,j}=-Z^{1}_{i},\qquad \varSigma ^{21,22}_{i,j}=-{B^{1} _{i}}^{T}L^{3}_{j},\qquad \varSigma ^{21,25}_{i,j}={L^{5}_{i}}^{T},\qquad \varSigma ^{21,26}_{i,j}={L^{5}_{i}}^{T}, \\ &\varSigma ^{21,27}_{i,j}={L^{5}_{i}}^{T}C_{j},\qquad \varSigma ^{22,22}_{i,j}= {L^{3}_{i}}^{T}A^{2}_{j}+{A^{2}_{i}}^{T}L^{3}_{j}-Z^{2}_{i},\qquad \varSigma ^{22,25}_{i,j}={L^{3}_{i}}^{T}, \\ &\varSigma ^{22,26}_{i,j}={L^{3}_{i}}^{T},\qquad \varSigma ^{22,27}_{i,j}= {L^{3}_{i}}^{T}C_{j},\qquad \varSigma ^{23,23}_{i,j}=-P^{18}_{i},\qquad \varSigma ^{24,24}_{i,j}=-P^{18}_{i}, \\ &\varSigma ^{25,25}_{i,j}=-\epsilon _{1}I, \qquad\varSigma ^{26,26}_{i,j}=- \epsilon _{2}I, \qquad\varSigma ^{27,27}_{i,j}= -\frac{P^{19}_{i}}{w},\qquad G^{1}_{i}=P^{1}_{i}J, \\ & G^{2}_{i}=P^{1}_{i}W,\qquad \rho =h_{2}-h_{1}, \qquad\varphi (k)=h(k)-h _{1},\qquad \beta (k)=h_{2}-h(k), \\ &\psi (k)=\frac{1}{\varphi (k)} \Biggl[\sum_{i=k-h(k)}^{k-h_{1}-1} \xi (i) \Biggr],\qquad \phi (k)=\frac{1}{\beta (k)} \Biggl[\sum _{i=k-h _{2}}^{k-h(k)-1}\xi (i) \Biggr], \end{aligned}$$

and other terms are 0.

Theorem 3.1

System (1) is robustly stable if there exist positive definite symmetric matrices \(M_{i},Z^{1}_{i},Z^{2}_{i},P^{k}_{i},Q^{s}_{i}\ ( k=1,2, \dots,19, s=1,2,\dots,18, i=1,2,\dots,N)\), any matrices \(J, W, M^{r}_{i},N^{r}_{i}\), \(L_{i}^{l}\) (\(r=1,2,3,4\), \(l=1,2,3\), \(i=1,2,\dots,N\)) of appropriate dimensions, and positive real constants \(\epsilon _{1}\) and \(\epsilon _{2}\) satisfying the following LMIs:

$$\begin{aligned} &\prod _{i,i} < -I,\quad i=1,2,\dots,N, \end{aligned}$$
(7)
$$\begin{aligned} &\prod _{i,j}+\prod_{j,i} < \frac{2}{(N-1)}I,\quad i=1,\dots,N-1,j=i+1,\dots,N, \end{aligned}$$
(8)
$$\begin{aligned} & \begin{bmatrix} Q_{i}^{3t+1} & Q_{i}^{3t+2} \\ * & Q_{i}^{3t+3} \end{bmatrix} \geq 0,\quad t=0,1,2,\dots,5, i=1,2, \dots,N, \end{aligned}$$
(9)
$$\begin{aligned} & \begin{bmatrix} M_{i}^{l} & N_{i}^{l} \\ * & P_{i}^{9+l} \end{bmatrix} \geq 0,\quad l=1,2,\dots,4, i=1,2, \dots,N. \end{aligned}$$
(10)

Proof

Applying a model transformation method, system (1) can be replaced by the system

$$\begin{aligned} &{\xi }(k+1) = \xi (k)+\eta (k), \end{aligned}$$
(11)
$$\begin{aligned} &\eta (k) = \bigl[A(\varphi )-I \bigr]\xi (k)+B(\varphi )\xi \bigl(k-h(k) \bigr)+C( \varphi )\sum_{i=1}^{\infty }\delta (i) \xi (k-i) \\ &\phantom{\eta (k) =}{}+f_{1}\bigl(k,\xi (k)\bigr)+f_{2}\bigl(k,\xi \bigl(k-h(k) \bigr)\bigr). \end{aligned}$$
(12)

To improve the bound for \(h(k)\) in (1), we decompose constant matrices A and B as

$$\begin{aligned} &A(\varphi ) = A_{1}(\varphi ) + A_{2}( \varphi ), \end{aligned}$$
(13)
$$\begin{aligned} &B(\varphi ) = B_{1}(\varphi ) + B_{2}(\varphi ), \end{aligned}$$
(14)

where \(A_{1}(\varphi )=\sum_{i=1}^{N}\varphi _{i}A_{i}^{1}, A_{2}(\varphi )=\sum_{i=1}^{N}\varphi _{i}A_{i}^{2}, B_{1}(\varphi )=\sum_{i=1} ^{N}\varphi _{i}B_{i}^{1}, B_{2}(\varphi )=\sum_{i=1}^{N}\varphi _{i}B_{i}^{2} \), and \(A_{i}^{1}, A_{i}^{2}, B_{i}^{1}, B_{i}^{2} \in {R}^{n\times n}\) are constant matrices for all \(i=1,2,3,\dots,N\). By utilizing the following zero equations, we have

$$\begin{aligned} &\xi (k)-\xi \bigl(k-h(k)\bigr)-\sum_{i=k-h(k)}^{k-1} \eta (i) = 0, \end{aligned}$$
(15)
$$\begin{aligned} &\xi (k)-\xi \bigl(k-rh(k)\bigr)-\sum_{i=k-rh(k)}^{k-1} \eta (i) = 0. \end{aligned}$$
(16)

From (15) and (16), we obtain

$$\begin{aligned} & J\xi (k)-J\xi \bigl(k-h(k)\bigr)-J\sum _{i=k-h(k)}^{k-1}\eta (i) = 0, \end{aligned}$$
(17)
$$\begin{aligned} &W\xi (k)-W\xi \bigl(k-rh(k)\bigr)-W\sum_{i=k-rh(k)}^{k-1} \eta (i) = 0, \end{aligned}$$
(18)

where \(J,W\in R^{n\times n}\) are selected so that the stability criterion of the given system is guaranteed. Substituting (17)–(18) into the system (11)–(12), we obtain

$$\begin{aligned} &{\xi }(k+1) = \xi (k)+\eta (k)+J\xi (k)-J\xi \bigl(k-h(k)\bigr)-J \sum_{i=k-h(k)} ^{k-1}\eta (i) \\ &\phantom{{\xi }(k+1) =}{}+W\xi (k)-W\xi \bigl(k-rh(k)\bigr)-W\sum_{i=k-rh(k)}^{k-1} \eta (i), \end{aligned}$$
(19)
$$\begin{aligned} &\eta (k) = \bigl[A_{1}(\varphi ) + B_{1}(\varphi ) -I \bigr]\xi (k)+A _{2}(\varphi ) \xi \bigl(k-rh(k)\bigr) \\ &\phantom{\eta (k) =}{}+B_{2}(\varphi ) \xi \bigl(k-h(k)\bigr)-B_{1}(\varphi ) \sum_{i=k-h(k)}^{k-1} \eta (i)+A_{2}( \varphi ) \sum_{i=k-rh(k)}^{k-1}\eta (i) \\ &\phantom{\eta (k) =}{}+C(\varphi ) \sum_{i=1}^{\infty }\delta (i)\xi (k-i)+f_{1}\bigl(k,\xi (k)\bigr)+f _{2}\bigl(k,\xi \bigl(k-h(k)\bigr)\bigr). \end{aligned}$$
(20)

We define a parameter-dependent Lyapunov–Krasovskii functional for the system (19)–(20) as

$$\begin{aligned} V\bigl(k,\xi (k)\bigr) = \sum_{i=1}^{8}V_{i} \bigl(k,\xi (k)\bigr), \end{aligned}$$
(21)

where

$$\begin{aligned} V_{1}\bigl(k,\xi (k)\bigr) ={}&\xi ^{T}(k) P_{1}( \varphi )\xi (k), \\ V_{2}\bigl(k,\xi (k)\bigr) ={}&\sum_{i=k-h_{1}}^{k-1} \xi ^{T}(i) P_{2}(\varphi ) \xi (i)+ \sum _{i=k-rh_{1}}^{k-1}\xi ^{T}(i) P_{3}( \varphi )\xi (i) \\ &{}+ \sum_{i=k-h_{2}}^{k-1}\xi ^{T}(i) P_{4}(\varphi )\xi (i)+\sum_{i=k-rh _{2}}^{k-1} \xi ^{T}(i)P_{5}(\varphi )\xi (i) \\ &{}+\sum_{i=k-h_{2}}^{k-h_{1}-1}\xi ^{T}(i) P_{6}(\varphi )\xi (i)+ \sum_{i=k-rh_{2}}^{k-rh_{1}-1} \xi ^{T}(i) P_{7}(\varphi )\xi (i) \\ &{}+\sum_{i=k-h(k)}^{k-1}\xi ^{T}(i) P_{8}(\varphi )\xi (i)+\sum_{i=k-rh(k)} ^{k-1}\xi ^{T}(i) P_{9}(\varphi )\xi (i), \\ V_{3}\bigl(k,\xi (k)\bigr) ={}& h_{1}\sum _{j=-h_{1}}^{-1}\sum_{i=k+j}^{k-1} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{1}(\varphi ) & Q_{2}(\varphi ) \\ *& Q_{3}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} \\ &{}+rh_{1}\sum_{j=-rh_{1}}^{-1}\sum _{i=k+j}^{k-1} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{4}(\varphi ) & Q_{5}(\varphi ) \\ *& Q_{6}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} \\ &{}+h_{2}\sum_{j=-h_{2}}^{-1}\sum _{i=k+j}^{k-1} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{7}(\varphi ) & Q_{8}(\varphi ) \\ * & Q_{9}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} \\ &{}+rh_{2}\sum_{j=-rh_{2}}^{-1}\sum _{i=k+j}^{k-1} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{10}(\varphi ) & Q_{11}(\varphi ) \\ * & Q_{12}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} \\ &{}+\rho \sum_{j=-h_{2}}^{-h_{1}-1}\sum _{i=k+j}^{k-1} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{13}(\varphi ) & Q_{14}(\varphi ) \\ * & Q_{15}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} \\ &{}+r\rho \sum_{j=-rh_{2}}^{-rh_{1}-1}\sum _{i=k+j}^{k-1} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{16}(\varphi ) & Q_{17}(\varphi ) \\ * & Q_{18}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (i) \\ \eta (i) \end{bmatrix} , \\ V_{4}\bigl(k,\xi (k)\bigr) ={}& \sum_{j=-h_{1}+1}^{0} \sum_{i=k-1+j}^{k-1}\eta ^{T}(i) P_{10}(\varphi )\eta (i) \\ &{}+\sum_{j=-h_{2}+1}^{0}\sum _{i=k-1+j}^{k-1}\eta ^{T}(i) P_{11}( \varphi )\eta (i) \\ &{}+\sum_{j=-rh_{1}+1}^{0}\sum _{i=k-1+j}^{k-1}\eta ^{T}(i) P_{12}( \varphi )\eta (i) \\ &{}+\sum_{j=-rh_{2}+1}^{0}\sum _{i=k-1+j}^{k-1}\eta ^{T}(i) P_{13}( \varphi )\eta (i) \\ &{}+\sum_{i=1}^{\infty } \delta (i)\sum _{j=k-i}^{k-1} \xi _{j}^{T} M( \varphi ) \xi _{j}, \\ V_{5}\bigl(k,\xi (k)\bigr) ={}& h_{1}\sum _{j=-h_{1}}^{-1}\sum_{i=k+j}^{k-1} \eta ^{T}(i)P_{14}(\varphi )\eta (i) \\ &{}+h_{2}\sum_{j=-h_{2}}^{-1}\sum _{i=k+j}^{k-1}\eta ^{T}(i)P_{15}( \varphi )\eta (i) \\ &{}+\rho \sum_{j=-h_{2}}^{-h_{1}-1}\sum _{i=k+j}^{k-1}\eta ^{T}(i)P_{16}( \varphi )\eta (i), \\ V_{6}\bigl(k,\xi (k)\bigr) ={}& h_{2}\sum _{j=-h_{2}}^{-1}\sum_{i=k+j}^{k-1} \eta ^{T}(i)Z_{1}(\varphi )\eta (i) \\ &{}+rh_{2}\sum_{j=-rh_{2}}^{-1}\sum _{i=k+j}^{k-1}\eta ^{T}(i)Z_{2}( \varphi )\eta (i), \\ V_{7}\bigl(k,\xi (k)\bigr)= {}&\frac{h_{1}^{2}}{2}\sum _{i=-h_{1}}^{-1}\sum_{j=i} ^{0}\sum_{l=k+j}^{k-1}\eta ^{T}(l) P_{17}(\varphi )\eta (l) \\ &{}+\frac{1}{2}\sum_{i=k-h_{2}}^{k-h_{1}-1}\sum _{j=i}^{k-h_{1}-1}\sum _{l=j}^{k-h_{1}-1}\eta ^{T}(l) P_{18}( \varphi )\eta (l), \\ V_{8}\bigl(k,\xi (k)\bigr) ={}& \sum_{i=1}^{\infty } \delta (i)\sum_{j=k-i}^{k-1} \xi ^{T}(j) P_{19}(\varphi )\xi (j). \end{aligned}$$

From (21), the forward difference of \(V(k,\xi (k))\) is defined by

$$\begin{aligned} \Delta V\bigl(k,\xi (k)\bigr) = \sum _{i=1}^{8}\Delta V_{i}\bigl(k,\xi (k) \bigr), \end{aligned}$$
(22)

where

$$\begin{aligned} \Delta V_{i}\bigl(k,\xi (k)\bigr) = V_{i} \bigl(k+1,\xi (k+1)\bigr)-V_{i}\bigl(k,\xi (k)\bigr) \end{aligned}$$
(23)

for \(i=1,2,\dots,8\). The increments of \(V_{1}(k,\xi (k))\), \(V_{2}(k,\xi (k))\), and \(V_{3}(k,\xi (k))\) are obtained by taking the forward difference of \(V_{1}(k,\xi (k))\), \(V_{2}(k,\xi (k))\), and \(V_{3}(k,\xi (k))\), respectively. Then, we have

$$\begin{aligned} \Delta V_{1}\bigl(k,\xi (k)\bigr) ={}& \xi ^{T}(k+1)P_{1}(\varphi )\xi (k+1)-\xi ^{T}(k)P_{1}( \varphi )\xi (k) \\ ={}& \bigl[\xi ^{T}(k)+\eta ^{T}(k) \bigr]P_{1}( \varphi ) \Biggl[\xi (k)+\eta (k)+J \xi (k) \\ &{}-J\xi \bigl(k-h(k)\bigr)-J\sum_{i=k-h(k)}^{k-1} \eta (i)+W\xi (k) \\ &{}-W\xi \bigl(k-rh(k)\bigr)-W\sum_{i=k-rh(k)}^{k-1} \eta (i) \Biggr] \\ &{} + \Biggl[\xi ^{T}(k)J^{T}-\xi ^{T}\bigl(k-h(k) \bigr)J^{T} \\ &{}-\sum_{i=k-h(k)}^{k-1}\eta ^{T}(i)J^{T}+ \xi ^{T}(k)W^{T} -\xi ^{T}\bigl(k-rh(k)\bigr)W ^{T} \\ &{}-\sum_{i=k-rh(k)}^{k-1}\eta ^{T}(i)W^{T} \Biggr]P_{1}(\varphi ) \bigl[ \xi (k)+\eta (k) \bigr] \\ &{}+2\xi ^{T}(k)L_{1}^{T}(\varphi ) \Biggl[-\eta (k)+ \bigl[A_{1}(\varphi )+B _{1}(\varphi )-I \bigr]\xi (k) \\ &{}+A_{2}(\varphi )\xi \bigl(k-rh(k)\bigr)+B_{2}(\varphi )\xi \bigl(k-h(k)\bigr) \\ &{}-B_{1}(\varphi )\sum_{i=k-h(k)}^{k-1} \eta (i) +A_{2}(\varphi ) \sum_{i=k-rh(k)}^{k-1} \eta (i) \end{aligned}$$
(24)
$$\begin{aligned} &{}+C(\varphi ) \sum_{i=1}^{\infty }\delta (i)\xi (k-i)+f_{1}\bigl(k,\xi (k)\bigr)+f _{2}\bigl(k,\xi \bigl(k-h(k)\bigr)\bigr) \Biggr] \\ &{} +2\eta ^{T}(k)L_{2}^{T}(\varphi ) \Biggl[-\eta (k)+ \bigl[A_{1}(\varphi )+B_{1}(\varphi )-I \bigr]\xi (k) \\ &{}+A_{2}(\varphi )\xi \bigl(k-rh(k)\bigr)+B_{2}(\varphi )\xi \bigl(k-h(k)\bigr) \\ &{}-B_{1}(\varphi )\sum_{i=k-h(k)}^{k-1} \eta (i)+A_{2}(\varphi ) \sum_{i=k-rh(k)}^{k-1} \eta (i) \\ &{}+C(\varphi ) \sum_{i=1}^{\infty }\delta (i)\xi (k-i)+f_{1}\bigl(k,\xi (k)\bigr)+f _{2}\bigl(k,\xi \bigl(k-h(k)\bigr)\bigr) \Biggr] \\ &{}+2\sum_{i=k-rh(k)}^{k-1}\eta ^{T}(i)L_{3}^{T}(\varphi ) \Biggl[-\eta (k)+ \bigl[A_{1}(\varphi )+B_{1}(\varphi ) -I \bigr]\xi (k) \\ &{}+A_{2}(\varphi )\xi \bigl(k-rh(k) \bigr)+B_{2}(\varphi )\xi \bigl(k-h(k)\bigr) \\ &{}-B_{1}(\varphi )\sum_{i=k-h(k)}^{k-1} \eta (i)+A_{2}(\varphi ) \sum_{i=k-rh(k)}^{k-1} \eta (i) \\ &{}+C(\varphi ) \sum_{i=1}^{\infty }\delta (i)\xi (k-i)+f_{1}\bigl(k,\xi (k)\bigr)+f _{2}\bigl(k,\xi \bigl(k-h(k)\bigr)\bigr) \Biggr] \\ &{} -\xi ^{T}(k)P_{1}(\varphi )\xi (k), \end{aligned}$$
(25)
$$\begin{aligned} \Delta V_{2}\bigl(k,\xi (k)\bigr) ={}& \xi ^{T}(k)P_{2}(\varphi )\xi (k)-\xi ^{T}(k-h _{1})P_{2}(\varphi )\xi (k-h_{1}) \\ &{}+\xi ^{T}(k)P_{3}(\varphi )\xi (k)-\xi ^{T}(k-rh_{1})P_{3}(\varphi ) \xi (k-rh_{1}) \\ &{}+\xi ^{T}(k)P_{4}(\varphi )\xi (k)-\xi ^{T}(k-h_{2})P_{4}(\varphi ) \xi (k-h_{2}) \\ &{}+\xi ^{T}(k)P_{5}(\varphi )\xi (k)-\xi ^{T}(k-rh_{2})P_{5}(\varphi ) \xi (k-rh_{2}) \\ &{}+\xi ^{T}(k-h_{1})P_{6}(\varphi )\xi (k-h_{1})-\xi ^{T}(k-h_{2})P_{6}( \varphi )\xi (k-h_{2}) \\ &{}+\xi ^{T}(k-rh_{1})P_{7}(\varphi )\xi (k-rh_{1}) \\ &{}-\xi ^{T}(k-rh_{2})P_{7}(\varphi )\xi (k-rh_{2}) \\ &{}+\xi ^{T}(k)P_{8}(\varphi )\xi (k)-\xi ^{T} \bigl(k-h(k)\bigr)P_{8}(\varphi ) \xi \bigl(k-h(k)\bigr) \\ &{}+\xi ^{T}(k)P_{9}(\varphi )\xi (k)-\xi ^{T} \bigl(k-rh(k)\bigr)P_{9}(\varphi ) \xi \bigl(k-rh(k)\bigr), \end{aligned}$$
(26)
$$\begin{aligned} \Delta V_{3}\bigl(k,\xi (k)\bigr) \leq{}& \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} ^{T} \begin{bmatrix} h_{1}^{2}Q_{1}(\varphi ) & h_{1}^{2}Q_{2}(\varphi ) \\ * & h_{1}^{2}Q_{3}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} \\ &{}- \begin{bmatrix} \sum_{i=k-h_{1}}^{k-1}\xi (i) \\ \sum_{i=k-h_{1}}^{k-1}\eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{1}(\varphi ) & Q_{2}(\varphi ) \\ * & Q_{3}(\varphi ) \end{bmatrix} \begin{bmatrix} \sum_{i=k-h_{1}}^{k-1}\xi (i) \\ \sum_{i=k-h_{1}}^{k-1}\eta (i) \end{bmatrix} \\ &{}+ \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} ^{T} \begin{bmatrix} r^{2}h_{1}^{2}Q_{4}(\varphi ) & r^{2}h_{1}^{2}Q_{5}(\varphi ) \\ * & r^{2}h_{1}^{2}Q_{6}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} \\ &{}- \begin{bmatrix} \sum_{i=k-rh_{1}}^{k-1}\xi (i) \\ \sum_{i=k-rh_{1}}^{k-1}\eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{4}(\varphi ) & Q_{5}(\varphi ) \\ * & Q_{6}(\varphi ) \end{bmatrix} \begin{bmatrix} \sum_{i=k-rh_{1}}^{k-1}\xi (i) \\ \sum_{i=k-rh_{1}}^{k-1}\eta (i) \end{bmatrix} \\ &{}+ \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} ^{T} \begin{bmatrix} h_{2}^{2}Q_{7}(\varphi ) & h_{2}^{2}Q_{8}(\varphi ) \\ * & h_{2}^{2}Q_{9}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} \\ &{}- \begin{bmatrix} \sum_{i=k-h_{2}}^{k-1}\xi (i) \\ \sum_{i=k-h_{2}}^{k-1}\eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{7}(\varphi ) & Q_{8}(\varphi ) \\ * & Q_{9}(\varphi ) \end{bmatrix} \begin{bmatrix} \sum_{i=k-h_{2}}^{k-1}\xi (i) \\ \sum_{i=k-h_{2}}^{k-1}\eta (i) \end{bmatrix} \\ &{}+ \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} ^{T} \begin{bmatrix} r^{2}h_{2}^{2}Q_{10}(\varphi ) & r^{2}h_{2}^{2}Q_{11}(\varphi ) \\ * & r^{2}h_{2}^{2}Q_{12}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} \\ &{}- \begin{bmatrix} \sum_{i=k-rh_{2}}^{k-1}\xi (i) \\ \sum_{i=k-rh_{2}}^{k-1}\eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{10}(\varphi ) & Q_{11}(\varphi ) \\ * & Q_{12}(\varphi ) \end{bmatrix} \begin{bmatrix} \sum_{i=k-rh_{2}}^{k-1}\xi (i) \\ \sum_{i=k-rh_{2}}^{k-1}\eta (i) \end{bmatrix} \\ &{}+ \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} ^{T} \begin{bmatrix} \rho ^{2}Q_{13}(\varphi ) & \rho ^{2}Q_{14}(\varphi ) \\ * & \rho ^{2}Q_{15}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} \\ &{}- \begin{bmatrix} \sum_{i=k-h_{2}}^{k-h_{1}-1}\xi (i) \\ \sum_{i=k-h_{2}}^{k-h_{1}-1}\eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{13}(\varphi ) & Q_{14}(\varphi ) \\ * & Q_{15}(\varphi ) \end{bmatrix} \begin{bmatrix} \sum_{i=k-h_{2}}^{k-h_{1}-1}\xi (i) \\ \sum_{i=k-h_{2}}^{k-h_{1}-1}\eta (i) \end{bmatrix} \\ &{} + \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} ^{T} \begin{bmatrix} r^{2}\rho ^{2}Q_{16}(\varphi ) & r^{2}\rho ^{2}Q_{17}(\varphi ) \\ * & r^{2}\rho ^{2}Q_{18}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (k) \\ \eta (k) \end{bmatrix} \\ &{}- \begin{bmatrix} \sum_{i=k-rh_{2}}^{k-rh_{1}-1}\xi (i) \\ \sum_{i=k-rh_{2}}^{k-rh_{1}-1}\eta (i) \end{bmatrix} ^{T} \begin{bmatrix} Q_{16}(\varphi ) & Q_{17}(\varphi ) \\ * & Q_{18}(\varphi ) \end{bmatrix} \begin{bmatrix} \sum_{i=k-rh_{2}}^{k-rh_{1}-1}\xi (i) \\ \sum_{i=k-rh_{2}}^{k-rh_{1}-1}\eta (i) \end{bmatrix}. \end{aligned}$$
(27)

Taking the forward difference of \(V_{4}(k,\xi (k))\) yields

$$\begin{aligned} &\Delta V_{4}\bigl(k,\xi (k)\bigr) \\ &\quad= h_{1}\eta ^{T}(k)P_{10}(\varphi )\eta (k)- \sum _{i=-h_{1}+1}^{0}\eta ^{T}(k-1+i)P_{10}( \varphi )\eta (k-1+i) \\ &\qquad{}+h_{2}\eta ^{T}(k)P_{11}(\varphi )\eta (k)-\sum _{i=-h_{2}+1}^{0}\eta ^{T}(k-1+i)P_{11}( \varphi ) \\ &\qquad{}\times \eta (k-1+i)+rh_{1}\eta ^{T}(k)P_{12}( \varphi )\eta (k)- \sum_{i=-rh_{1}+1}^{0}\eta ^{T}(k-1+i) \\ &\qquad{}\times P_{12}(\varphi )\eta (k-1+i)+rh_{2}\eta ^{T}(k)P_{13}(\varphi )\eta (k) \\ &\qquad{}-\sum_{i=-rh_{2}+1}^{0}\eta ^{T}(k-1+i)P_{13}(\varphi )\eta (k-1+i). \end{aligned}$$

By (10), it is easy to see that

$$\begin{aligned} &2\xi ^{T}(k)N_{1}(\varphi )\sum _{i=-h_{1}+1}^{0}\eta (k-1+i)+ \sum _{i=-h_{1}+1}^{0}\eta ^{T}(k-1+i)P_{10}( \varphi )\eta (k-1+i) \\ &\qquad{}+h_{1}\xi ^{T}(k)M_{1}(\varphi )\xi (k) \\ &\quad=\sum_{i=-h_{1}+1}^{0} \begin{bmatrix} \xi (k) \\ \eta (k-1+i) \end{bmatrix} ^{T} \begin{bmatrix} M_{1}(\varphi ) & N_{1}(\varphi ) \\ * & P_{10}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (k) \\ \eta (k-1+i) \end{bmatrix} \geq 0, \end{aligned}$$
(28)
$$\begin{aligned} &2\xi ^{T}(k)N_{2}(\varphi )\sum _{i=-h_{2}+1}^{0}\eta (k-1+i)+ \sum _{i=-h_{2}+1}^{0}\eta ^{T}(k-1+i)P_{11}( \varphi )\eta (k-1+i) \\ &\qquad{} +h_{2}\xi ^{T}(k)M_{2}(\varphi )\xi (k) \\ &\quad =\sum_{i=-h_{2}+1}^{0} \begin{bmatrix} \xi (k) \\ \eta (k-1+i) \end{bmatrix} ^{T} \begin{bmatrix} M_{2}(\varphi ) & N_{2}(\varphi ) \\ * & P_{11}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (k) \\ \eta (k-1+i) \end{bmatrix} \geq 0, \end{aligned}$$
(29)
$$\begin{aligned} &2\xi ^{T}(k)N_{3}(\varphi )\sum _{i=-rh_{1}+1}^{0}\eta (k-1+i)+ \sum _{i=-rh_{1}+1}^{0}\eta ^{T}(k-1+i)P_{12}( \varphi ) \\ &\qquad{}\times \eta (k-1+i)+rh_{1}\xi ^{T}(k)M_{3}( \varphi )\xi (k) \\ &\quad=\sum_{i=-rh_{1}+1}^{0} \begin{bmatrix} \xi (k) \\ \eta (k-1+i) \end{bmatrix} ^{T} \begin{bmatrix} M_{3}(\varphi ) & N_{3}(\varphi ) \\ * & P_{12}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (k) \\ \eta (k-1+i) \end{bmatrix} \geq 0, \end{aligned}$$
(30)
$$\begin{aligned} &2\xi ^{T}(k)N_{4}(\varphi )\sum _{i=-rh_{2}+1}^{0}\eta (k-1+i)+ \sum _{i=-rh_{2}+1}^{0}\eta ^{T}(k-1+i)P_{13}( \varphi ) \\ &\qquad{}\times \eta (k-1+i)+rh_{2}\xi ^{T}(k)M_{4}( \varphi )\xi (k) \\ &\quad=\sum_{i=-rh_{2}+1}^{0} \begin{bmatrix} \xi (k) \\ \eta (k-1+i) \end{bmatrix} ^{T} \begin{bmatrix} M_{4}(\varphi ) & N_{4}(\varphi ) \\ * & P_{13}(\varphi ) \end{bmatrix} \begin{bmatrix} \xi (k) \\ \eta (k-1+i) \end{bmatrix} \geq 0. \end{aligned}$$
(31)

From (28), (29), (30), and (31), we obtain

$$\begin{aligned} &{}-\sum_{i=-h_{1}+1}^{0}\eta ^{T}(k-1+i)P_{10}(\varphi )\eta (k-1+i) \\ &\quad\leq h_{1}\xi ^{T}(k)M_{1}(\varphi )\xi (k)+2 \xi ^{T}(k)N_{1}(\varphi )\sum_{i=-h_{1}+1}^{0} \eta (k-1+i) \\ &\quad= h_{1}\xi ^{T}(k)M_{1}(\varphi )\xi (k)+2\xi ^{T}(k)N_{1}(\varphi ) \xi (k)-2\xi ^{T}(k)N_{1}( \varphi )\xi (k-h_{1}), \end{aligned}$$
(32)
$$\begin{aligned} &{-}\sum_{i=-h_{2}+1}^{0}\eta ^{T}(k-1+i)P_{11}(\varphi )\eta (k-1+i) \\ &\quad\leq h_{2}\xi ^{T}(k)M_{2}(\varphi )\xi (k)+2 \xi ^{T}(k)N_{2}(\varphi )\sum_{i=-h_{2}+1}^{0} \eta (k-1+i) \\ &\quad= h_{2}\xi ^{T}(k)M_{2}(\varphi )\xi (k)+2\xi ^{T}(k)N_{2}(\varphi ) \xi (k)-2\xi ^{T}(k)N_{2}( \varphi )\xi (k-h_{2}), \end{aligned}$$
(33)
$$\begin{aligned} &{-}\sum_{i=-rh_{1}+1}^{0}\eta ^{T}(k-1+i)P_{12}(\varphi )\eta (k-1+i) \\ &\quad\leq rh_{1}\xi ^{T}(k)M_{3}(\varphi )\xi (k)+2\xi ^{T}(k)N_{3}( \varphi )\sum _{i=-rh_{1}+1}^{0}\eta (k-1+i) \\ &\quad = rh_{1}\xi ^{T}(k)M_{3}(\varphi )\xi (k)+2\xi ^{T}(k)N_{3}(\varphi ) \xi (k) \\ &\qquad{}-2\xi ^{T}(k)N_{3}(\varphi )\xi (k-rh_{1}), \end{aligned}$$
(34)
$$\begin{aligned} &{-}\sum_{i=-rh_{2}+1}^{0}\eta ^{T}(k-1+i)P_{13}(\varphi )\eta (k-1+i) \\ &\quad\leq rh_{2}\xi ^{T}(k)M_{4}(\varphi )\xi (k)+2 \xi ^{T}(k)N_{4}( \varphi )\sum_{i=-rh_{2}+1}^{0} \eta (k-1+i) \\ &\quad= rh_{2}\xi ^{T}(k)M_{4}(\varphi )\xi (k)+2\xi ^{T}(k)N_{4}(\varphi ) \xi (k) \\ &\qquad{}-2\xi ^{T}(k)N_{4}(\varphi )\xi (k-rh_{2}). \end{aligned}$$
(35)

Therefore, we conclude that

$$\begin{aligned} &\Delta V_{4}\bigl(k,\xi (k)\bigr) \\ &\quad \leq h_{1}\eta ^{T}(k)P_{10}(\varphi ) \eta (k)+h_{1}\xi ^{T}(k)M_{1}(\varphi )\xi (k)+2\xi ^{T}(k)N_{1}( \varphi )\xi (k) \\ &\qquad{}-2\xi ^{T}(k)N_{1}(\varphi )\xi (k-h_{1})+h_{2} \eta ^{T}(k)P_{11}( \varphi )\eta (k) \\ &\qquad{}+h_{2}\xi ^{T}(k)M_{2}(\varphi )\xi (k)+2\xi ^{T}(k)N_{2}(\varphi ) \xi (k) \\ &\qquad{}-2\xi ^{T}(k)N_{2}(\varphi )\xi (k-h_{2})+rh_{1} \eta ^{T}(k)P_{12}( \varphi )\eta (k) \\ &\qquad{}+rh_{1}\xi ^{T}(k)M_{3}(\varphi )\xi (k)+2\xi ^{T}(k)N_{3}(\varphi ) \xi (k) \\ &\qquad{}-2\xi ^{T}(k)N_{3}(\varphi )\xi (k-rh_{1})+rh_{2} \eta ^{T}(k)P_{13}( \varphi )\eta (k) \\ &\qquad{}+rh_{2}\xi ^{T}(k)M_{4}(\varphi )\xi (k)+2\xi ^{T}(k)N_{4}(\varphi ) \xi (k) \\ &\qquad{}-2\xi ^{T}(k)N_{4}(\varphi )\xi (k-rh_{2}). \end{aligned}$$
(36)

By Lemma 2.2, the increments of \(V_{5}(k,\xi (k))\) and \(V_{6}(k,\xi (k))\) can be expressed mathematically as

$$\begin{aligned} &\Delta V_{5}\bigl(k,\xi (k)\bigr) \\ &\quad = h_{1}^{2} \eta ^{T}(k)P_{14}(\varphi ) \eta (k)-h_{1}\sum _{i=k-h_{1}}^{k-1}\eta ^{T}(i)P_{14}( \varphi )\eta (i) \\ &\qquad{}+h_{2}^{2}\eta ^{T}(k)P_{15}(\varphi ) \eta (k)-h_{2}\sum_{i=k-h_{2}} ^{k-1}\eta ^{T}(i)P_{15}(\varphi )\eta (i) \\ &\qquad{}+\rho ^{2}\eta ^{T}(k)P_{16}(\varphi )\eta (k)- \rho \sum_{i=k-h_{2}} ^{k-h_{1}-1}\eta ^{T}(i)P_{16}(\varphi )\eta (i) \\ &\quad\leq h_{1}^{2}\eta ^{T}(k)P_{14}( \varphi )\eta (k)- \Biggl(\sum_{i=k-h _{1}}^{k-1} \eta (i) \Biggr)^{T}P_{14}(\varphi ) \Biggl(\sum _{i=k-h_{1}}^{k-1} \eta (i) \Biggr) \\ &\qquad{}+h_{2}^{2}\eta ^{T}(k)P_{15}(\varphi ) \eta (k)- \Biggl(\sum_{i=k-h_{2}} ^{k-1}\eta (i) \Biggr)^{T}P_{15}(\varphi ) \Biggl(\sum _{i=k-h_{2}}^{k-1} \eta (i) \Biggr) \\ &\qquad{}+\rho ^{2}\eta ^{T}(k)P_{16}(\varphi )\eta (k)- \Biggl(\sum_{i=k-h_{2}} ^{k-h_{1}-1}\eta (i) \Biggr)^{T}P_{16}(\varphi ) \Biggl(\sum _{i=k-h_{2}} ^{k-h_{1}-1}\eta (i) \Biggr), \end{aligned}$$
(37)
$$\begin{aligned} &\Delta V_{6}\bigl(k,\xi (k)\bigr) \\ &\quad = h_{2}^{2}\eta ^{T}(k)Z_{1}(\varphi )\eta (k)-h _{2}\sum _{i=k-h_{2}}^{k-1}\eta ^{T}(i)Z_{1}( \varphi )\eta (i) \\ &\qquad{}+(rh_{2})^{2}\eta ^{T}(k)Z_{2}(\varphi )\eta (k)-rh_{2}\sum_{i=k-rh _{2}}^{k-1} \eta ^{T}(i)Z_{2}(\varphi )\eta (i) \\ &\quad \leq h_{2}^{2}\eta ^{T}(k)Z_{1}( \varphi )\eta (k)- \Biggl(\sum_{i=k-h(k)} ^{k-1} \eta (i) \Biggr)^{T}Z_{1}(\varphi ) \Biggl(\sum _{i=k-h(k)}^{k-1} \eta (i) \Biggr) \\ &\qquad{}+(rh_{2})^{2}\eta ^{T}(k)Z_{2}(\varphi )\eta (k)- \Biggl(\sum_{i=k-rh(k)} ^{k-1}\eta (i) \Biggr)^{T}Z_{2}(\varphi ) \\ &\qquad{}\times \Biggl(\sum_{i=k-rh(k)}^{k-1}\eta (i) \Biggr). \end{aligned}$$
(38)

The forward difference of \(V_{7}(k,\xi (k))\) can be obtained as

$$\begin{aligned} \Delta V_{7}\bigl(k,\xi (k)\bigr) = {}&\eta ^{T}(k) \biggl[ \frac{h_{1}^{4}}{4}P_{17}( \varphi )+\frac{(h_{2}-h_{1})^{2}}{4}P_{18}( \varphi ) \biggr]\eta (k) \\ &{}-\frac{h_{1}^{2}}{2}\sum_{i=-h_{1}}^{-1}\sum _{j=k+i}^{k-1}\eta ^{T}(j)P _{17}(\varphi )\eta (j) \\ &{}-\frac{1}{2}\sum_{i=k-h_{2}}^{k-h_{1}-1}\sum _{j=i}^{k-h_{1}-1}\eta ^{T}(j)P_{18}( \varphi )\eta (j). \end{aligned}$$
(39)

By Lemma 2.3, we obtain

$$\begin{aligned} &{-}\frac{h_{1}^{2}}{2}\sum_{i=-h_{1}}^{-1}\sum _{j=k+i}^{k-1}\eta ^{T}(j)P _{17}(\varphi )\eta (j) \\ &\quad\leq - \Biggl[h_{1}\xi (k)-\sum_{i=k-h_{1}}^{k-1} \xi (i) \Biggr]^{T}P _{17}(\varphi ) \Biggl[h_{1} \xi (k)-\sum_{i=k-h_{1}}^{k-1}\xi (i) \Biggr], \end{aligned}$$
(40)
$$\begin{aligned} &{-}\frac{1}{2}\sum_{i=k-h_{2}}^{k-h_{1}-1}\sum _{j=i}^{k-h_{1}-1}\eta ^{T}(j)P_{18}( \varphi )\eta (j) \\ &\quad= -\frac{1}{2}\sum_{i=k-h(k)}^{k-h_{1}-1}\sum _{j=i}^{k-h_{1}-1}\eta ^{T}(j)P_{18}( \varphi )\eta (j)-\frac{1}{2}\sum_{i=k-h_{2}}^{k-h(k)-1} \sum_{j=i}^{k-h(k)-1}\eta ^{T}(j)P_{18}( \varphi )\eta (j) \\ &\qquad{}-\frac{1}{2}\sum_{i=k-h_{2}}^{k-h(k)-1}\sum _{j=k-h(k)}^{k-h_{1}-1} \eta ^{T}(j)P_{18}( \varphi )\eta (j) \\ &\quad\leq - \Biggl[\sum_{i=k-h(k)}^{k-h_{1}-1}\sum _{j=i}^{k-h_{1}-1} \eta (j) \Biggr]^{T} \frac{1}{\varphi ^{2}(k)}P_{18}(\varphi ) \Biggl[\sum _{i=k-h(k)}^{k-h_{1}-1}\sum_{j=i}^{k-h_{1}-1} \eta (j) \Biggr] \\ &\qquad{}- \Biggl[\sum_{i=k-h_{2}}^{k-h(k)-1}\sum _{j=i}^{k-h(k)-1}\eta (j) \Biggr] ^{T} \frac{1}{\beta ^{2}(k)}P_{18}(\varphi ) \Biggl[\sum _{i=k-h_{2}}^{k-h(k)-1} \sum_{j=i}^{k-h(k)-1} \eta (j) \Biggr] \\ &\quad = - \bigl[\xi (k-h_{1})-\psi (k) \bigr]^{T}P_{18}( \varphi ) \bigl[\xi (k-h _{1})-\psi (k) \bigr] \\ &\qquad{}- \bigl[\xi \bigl(k-h(k)\bigr)-\phi (k) \bigr]^{T}P_{18}( \varphi ) \bigl[\xi \bigl(k-h(k)\bigr)- \phi (k) \bigr]. \end{aligned}$$
(41)

Substituting (40) and (41) into (39), we obtain

$$\begin{aligned} \Delta V_{7}\bigl(k,\xi (k)\bigr) \leq{}& \eta ^{T}(k) \biggl[\frac{h_{1}^{4}}{4}P _{17}(\varphi )+\frac{(h_{2}-h_{1})^{2}}{4}P_{18}( \varphi ) \biggr] \eta (k) \\ &{}- \Biggl[h_{1}\xi (k)-\sum_{i=k-h_{1}}^{k-1} \xi (i) \Biggr]^{T}P_{17}( \varphi ) \Biggl[h_{1} \xi (k)-\sum_{i=k-h_{1}}^{k-1}\xi (i) \Biggr] \\ &{}- \bigl[\xi (k-h_{1})-\psi (k) \bigr]^{T}P_{18}( \varphi ) \bigl[\xi (k-h _{1})-\psi (k) \bigr] \\ &{}- \bigl[\xi \bigl(k-h(k)\bigr)-\phi (k) \bigr]^{T}P_{18}( \varphi ) \bigl[\xi \bigl(k-h(k)\bigr)- \phi (k) \bigr]. \end{aligned}$$
(42)

Taking the forward difference of \(V_{8}(k,\xi (k))\), we obtain

$$\begin{aligned} \Delta V_{8}\bigl(k,\xi (k)\bigr) ={}& w\xi ^{T}(k)P_{19}( \varphi )\xi (k) \\ &{}-\frac{1}{w} \Biggl[\sum_{i=1}^{\infty } \delta (i)\xi (k-i) \Biggr]^{T}P _{19}(\varphi ) \Biggl[\sum _{i=1}^{\infty }\delta (i)\xi (k-i) \Biggr]. \end{aligned}$$
(43)

From (3) and (4), for any scalars \(\epsilon _{1}, \epsilon _{2}>0\), we obtain

$$\begin{aligned} &\epsilon _{1} \bigl(\varphi ^{2}\xi ^{T}(k)\xi (k)-f_{1}^{T}\bigl(k,\xi (k)\bigr)f _{1}\bigl(k, \xi (k)\bigr) \bigr)\geq 0, \end{aligned}$$
(44)
$$\begin{aligned} &\epsilon _{2} \bigl(\beta ^{2}\xi ^{T}\bigl(k-h(k) \bigr)\xi \bigl(k-h(k)\bigr)-f_{2}^{T}\bigl(k, \xi \bigl(k-h(k)\bigr)\bigr)f_{2}\bigl(k,\xi \bigl(k-h(k)\bigr)\bigr) \bigr) \geq 0. \end{aligned}$$
(45)

It follows from (22)–(45) that

$$\begin{aligned} \Delta V\bigl(k,\xi (k)\bigr) \leq \zeta ^{T}(k) \sum_{i=1}^{N}\sum _{j=1}^{N} \varphi _{i}\varphi _{j}\prod_{i,j} \zeta (k), \end{aligned}$$
(46)

where \(\zeta ^{T}(k)= [\xi ^{T}(k) \eta ^{T}(k) \xi ^{T}(k-h _{1}) \xi ^{T}(k-h_{2}) \xi ^{T}(k-h(k)) \xi ^{T}(k-rh _{1})\times \xi ^{T}(k-rh_{2}) \xi ^{T}(k-rh(k)) \sum_{i=k-h_{1}}^{k-1} \xi ^{T}(i) \sum_{i=k-rh_{1}}^{k-1}\xi ^{T}(i) \sum_{i=k-h _{2}}^{k-1}\xi ^{T}(i) \sum_{i=k-rh_{2}}^{k-1}\xi ^{T}(i) \times \sum_{i=k-h_{2}}^{k-h_{1}-1}\xi ^{T}(i) \sum_{i=k-rh_{2}}^{k-rh_{1}-1}\xi ^{T}(i) \sum_{i=k-h_{1}}^{k-1}\eta ^{T}(i) \sum_{i=k-rh_{1}}^{k-1}\eta ^{T}(i) \sum_{i=k-h_{2}}^{k-1}\eta ^{T}(i) \sum_{i=k-rh_{2}}^{k-1}\eta ^{T}(i) \times \sum_{i=k-h_{2}} ^{k-h_{1}-1}\eta ^{T}(i) \sum_{i=k-rh_{2}}^{k-rh_{1}-1}\eta ^{T}(i) \sum_{i=k-h(k)}^{k-1} \eta ^{T}(i) \sum_{i=k-rh(k)}^{k-1}\eta ^{T}(i) \psi (k) \phi (k) f_{1}(k,\xi (k)) f_{2}(k,\xi (k- h(k))) \sum_{i=1}^{\infty }\delta (i)\xi (k-i) ]\) and \(\prod_{i,j}\) is defined as in (6). The fact that \(\sum_{i=1}^{N}\varphi _{i}=1\) leads to the following identities:

$$\begin{aligned} & \sum_{i=1}^{N}\sum _{j=1}^{N}\varphi _{i}\varphi _{j}\prod_{i,j}= \sum _{i=1}^{N}\varphi _{i}^{2}{ \prod }_{i,i}+\sum_{i=1}^{N-1} \sum_{j=i+1} ^{N}\varphi _{i} \varphi _{j} \Bigl[\prod_{i,j}+\prod_{j,i} \Bigr], \end{aligned}$$
(47)
$$\begin{aligned} &(N-1)\sum_{i=1}^{N}\varphi _{i}^{2}-2\sum_{i=1}^{N-1} \sum_{j=i+1}^{N} \varphi _{i} \varphi _{j}= \sum_{i=1}^{N-1} \sum_{j=i+1}^{N}[\varphi _{i}- \varphi _{j}]^{2}\geq 0. \end{aligned}$$
(48)

By (46)–(48), if (7)–(10) are feasible, then

$$\begin{aligned} \Delta V\bigl(k,\xi (k)\bigr) < -\omega \Vert \xi \Vert ^{2}, \end{aligned}$$
(49)

where \(\omega >0\). Therefore, system (1)–(2) is robustly stable. This completes the proof. □

Remark 1

Theorem 3.1 reduces to LMIs stability condition for discrete-time system with time-varying delays and nonlinear uncertainties [21] if \(A(\varphi )=A\), \(B(\varphi )=B\), \(C(\varphi )=0\) where \(A, B \in R^{n\times n}\). We use the mixed model transformation in this paper, and utilize the various inequalities to reduce the possible conservatism of the obtained condition.

Remark 2

If \(A(\varphi )=A\), \(B(\varphi )=B\), \(C(\varphi )=0\) and \(f_{1}(k, \xi (k))=f_{2}(k,\xi (k-h(k)))=0\) where \(A, B \in R^{n\times n}\), model (1) is simplified to the discrete-time system with time-varying delays studied in [2, 4, 6, 7, 16, 18]. So the investigation in this paper expands and improves the studies in the literature.

Numerical examples

Example 4.1

Consider the system

$$\begin{aligned} {\xi }(k+1) = A\xi (k)+B\xi \bigl(k-h(k)\bigr)+f_{1} \bigl(k,\xi (k)\bigr)+f_{2}\bigl(k, \xi \bigl(k-h(k)\bigr)\bigr) \end{aligned}$$
(50)

with the parameters

$$\begin{aligned} A= \begin{bmatrix} 0.80 & 0 \\ 0.05 & 0.90 \end{bmatrix} ,\qquad B= \begin{bmatrix} -0.10 & 0 \\ -0.20 & -0.10 \end{bmatrix} , \quad\gamma \geq 0, \beta \geq 0, & \end{aligned}$$
(51)

which were considered in [21].

Decompose the matrices A and B as \(A=A_{1}+A_{2}\) and \(B=B_{1}+B _{2}\), where

$$\begin{aligned} &A_{1}= \begin{bmatrix} -0.5&-0.1 \\ -0.02&-0.3 \end{bmatrix} ,\qquad A_{2}= \begin{bmatrix} 1.3&0.1 \\ 0.07&1.2 \end{bmatrix} , \\ &B_{1}= \begin{bmatrix} -0.5&-0.1 \\ -0.3&-0.4 \end{bmatrix} ,\qquad B_{2}= \begin{bmatrix} 0.4&0.1 \\ 0.1&0.3 \end{bmatrix} . \end{aligned}$$

Applying Theorem 3.1 to system (50)–(51), the numerical results of our criterion are dramatically less conservative than those of [21], as shown in Table 1.

Table 1 The maximum upper bounds of \(h_{2}\) for Example 4.1

Example 4.2

Consider the system

$$\begin{aligned} {\xi }(k+1) = A\xi (k)+B\xi \bigl(k-h(k)\bigr) \end{aligned}$$
(52)

with the parameters

$$\begin{aligned} A= \begin{bmatrix} 0.80 & 0 \\ 0.05 & 0.90 \end{bmatrix} ,\qquad B= \begin{bmatrix} -0.10 & 0 \\ -0.20 & -0.10 \end{bmatrix} , \end{aligned}$$
(53)

which were considered in [2, 4, 6, 7, 16, 18].

Decompose the matrices A and B as \(A=A_{1}+A_{2}\) and \(B=B_{1}+B _{2}\), where

$$\begin{aligned} & A_{1}= \begin{bmatrix} -0.4&-0.1 \\ -0.02&-0.3 \end{bmatrix} ,\qquad A_{2}= \begin{bmatrix} 1.2&0.1 \\ 0.07&1.2 \end{bmatrix} , \\ & B_{1}= \begin{bmatrix} -0.5&-0.1 \\ -0.3&-0.4 \end{bmatrix} ,\qquad B_{2}= \begin{bmatrix} 0.4&0.1 \\ 0.1&0.3 \end{bmatrix} . \end{aligned}$$

Given various values of \(h_{1}\), the obtained upper bounds for \(h_{2}\) are shown in Table 2. The numerical results show the asymptotic stability of system (52) with (53). Moreover, by applying Theorem 3.1, less conservative results are achieved.

Table 2 The maximum upper bounds of \(h_{2}\) for Example 4.2

Conclusions

The problem of delay-interval-dependent robust stability for LPD discrete-time system with mixed time-varying delays and nonlinear uncertainties was studied. By utilizing various inequalities, a mixed model transformation, zero equations, and a new parameter-dependent Lyapunov–Krasovskii functional, the new delay-interval-dependent stability criterion is derived and formulated in terms of LMIs for the system. Some numerical examples are also illustrated to exhibit the effectiveness with less conservatism of the proposed stability criterion.

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Acknowledgements

The author would like to thank the editors and reviewers for careful reading of the original manuscript and for their valuable suggestions.

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This work was supported by the Faculty of Science at Sriracha, Kasetsart University Sriracha Campus, Thailand (Grant number P003/2018); the Thailand Research Fund (TRF); the Office of the Higher Education Commission (OHEC), Khon Kaen University (Grant number MRG6080042); Research and Academic Affairs Promotion Fund, Faculty of Science, Khon Kaen University, Fiscal Year 2019; and National Research Council of Thailand and Khon Kaen University, Thailand (Grant number 6200069).

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Keywords

  • Parameter dependence
  • Lyapunov–Krasovskii functional
  • Linear matrix inequality (LMI)
  • LPD discrete-time system
  • Interval time-varying delay