Novel robust stability condition for uncertain neutral systems with mixed time-varying delays

Abstract

In this article, the issue of robust stability analysis for a sort of uncertain neutral system with mixed time-varying delays is studied. A new Lyapunov–Krasovskii functional comprising quadruple-integral term is introduced so as to develop a less conservative stability condition. A novel discrete and neutral delay-dependent stability criterion based on linear matrix inequalities is given using delay-central point method as well as reciprocally convex combination approach, which is derived by integral inequality approach. Compared with the existing literature, this criterion can greatly reduce the complexity of theoretical derivation and computation. Finally, three numerical comparative examples are designated to verify the superiority of the proposed approach in reducing the conservation of conclusion.

Introduction

In the real world, a number of dynamic model systems, such as networked control systems, process control systems, and nuclear reactor control systems, contain significant time delays in the transmission of data and materials. Neutral delay is the leading example of many types of time delay, which not only exists in the system state but also has to do with the derivative of the system state. This neutral time-delay system is common in practical engineering application of population ecology, distribution network in lossless transmission line, heat exchanger, robot in the rigid environment, and so on. Therefore, the asymptotic stability analysis of neutral delay systems has become a hot research field recently.^1–25

For the stability analysis of neutral delay systems, the most common approach is directly to construct Lyapunov–Krasovskii functional (LKF) in the time domain combined with linear matrix inequality (LMI). In this framework, finding ways of reducing the conservatism of obtained results has become a core issue concerned by scholars home and abroad. There are many useful methods in terms of analysis, such as augmented functional method,^{1,2,22,26–36} free weighting matrix method,^12,14,22,35 and delayed decomposition method.^{5,6,8,26–28,32,33,35} The similarities of these methods are that they can make full use of the time-delay information and thus have positive effect on reducing the conservatism of the conclusion. However, with the introduction of too much matrix variables and the increasing number of segmentation, there has been unbearable burden on theoretical analysis and engineering calculation. In this case, a novel analysis method called independent influence analysis (IIA)^{1,2,7,11,13,14,26–30,32,33,36} which has the characteristics of simple form and less matrix variables is favored by everyone. Its characteristic is bound to promote the asymptotic stability analysis of time-delay systems. For instance, Gu et al. ³⁷ first introduces Jensen’s inequality to the asymptotic stability analysis of time-delay systems. Then, Ramakrishnan and Ray^26,27 and Zhang et al. ²⁸ apply Jensen’s inequality to do further promotion and get more effective conclusions. In this article, the novel stability criterion skillfully introduces new integral inequality to deal with cross terms. Therefore, it leads to the noticeable reduction in the complexity in calculation and conservativeness in conclusion.

In addition, convex combination approach^{29–31,38,39} also attracts considerable attention. A new stability criterion for time-delay systems is obtained by convex combination approach in the works by Zhu et al. ²⁹ and An et al. ³⁰ In the study by Zhu et al., ²⁹ by applying the reciprocally convex combination approach, a new lower conservative asymptotic stability criterion is obtained on the basis of LMI. Similarly, the new asymptotic stability criterion for nonlinear systems with time-varying delay is obtained as a result of Jensen’s inequality as well as the reciprocally convex combination approach in the work by An et al. ³⁰ In the study by Phat et al., ³¹ the convex combination approach is utilized to calculate the additional time-variable term, and a strictly constrained nonlinear time-varying coefficient is obtained which led to more advantageous results in stability analysis. In this article, reciprocally convex combination approach is adopted to limit the cross terms of LKF derivatives. Therefore, the lower conservative results were obtained.

In the study by Sun et al., ¹ triple-integral functional term is introduced in the newly constructed LKF, and it is concluded that the stability condition for time-delay system is significantly improved. Inspired by Sun et al.,^1,2,3 for linear systems with time-varying delays, a lower conservative stability criterion is obtained by constructing the LKF which consists triple-integral functional terms. In the study by Liu, ⁴ a new LKF including time-varying delay information is constructed. Based on IIA, the delay-dependent robust stability criteria for interval time-varying delay systems are obtained. In the study by Zhang et al., ³² by building a novel segmentation LKF and avoiding the convex combination approach and free weighting matrix method, which only applied tighter bounding inequality integral conditions, a new stability criterion is given for discrete systems with interval time-varying delay. In the study by Peng and Fei, ³³ the robust stability for a sort of uncertain Takagi–Sugeno (T-S) fuzzy systems with interval time-varying delay is researched. By constructing a proper LKF which contains delay segmentation characteristic, with the help of tighter bounding integral inequality to handle cross terms, the stability conditions are given to prove that the conclusion in terms of computational efficiency is less conservative.

In the works by Sun et al. ¹ and Fang et al., ⁶ it is not universal that discrete delay and neutral delay, which are known constants, are took into consideration, without considering time-varying delay. In previous studies,^6–8 the stability for neutral systems with time-varying delays is studied, but the results only were related to discrete delay and its derivative, without containing relevant neutral delay information. Therefore, the conclusion has certain conservatism. Recently, a novel stability criterion for a class of neutral systems with mixed time-varying delays is obtained in previous studies.^9–15 This criterion is related to both discrete delay and neutral delay. However, the conservatism of the research conclusion above needs further reduction. It is worthy of further study that how to select the appropriate LKF and apply tighter bounding approach, which is in favor of synthesis of controller and reduction in the conservation of conclusion, without increasing complexity in theory and computation. In this article, first of all, the delay range is divided into two equidistant subintervals, and in each subinterval, a new LKF comprising quadruple-integral term and quadratic forms of double-integral term is introduced. Second, double-integral functional term is added into new augmented vector in initial definition, such as ${\int }_{-h}^0{\int }_{t+\beta }^t\mathit{x}(s)\mathrm{ds}d\beta$ . Third, the state x is incorporated into the integrand of triple-integral functional term, and the lower bound of time-delay interval information is imported in LKF. Therefore, the stability of the results is significantly reduced.

For a sort of uncertain neutral system with mixed time-varying delays, a novel criterion is obtained, which is simple in form and less conservative in conclusion. In the case of the uncertainty for the norm bounded, the discrete and neutral delay-dependent stability condition in terms of LMIs is given by utilizing IIA, reciprocally convex combination approach combined with delay-central point (DCP) method, ²⁷ where the delay range is divided into two equidistant subintervals, and in each subinterval, a new LKF is introduced. Different from the previous methods, first, a new LKF comprising quadruple-integral term and augmented functional term is used. Second, applying tighter integral inequality bounding cross terms of functional derivatives is conducive to enlarge maximum allowable delay bound (MADB) and reduce conservatism in conclusion. Finally, the validity and superiority of proposed criterion are verified by the numerical comparative examples.

Problem description

Consider this uncertain neutral system with mixed time-varying delays as follows

$$\{\begin{array}{c}\stackrel{·}{x}(t)-C(t)\stackrel{·}{x}(t-\tau (t))=A(t)x(t)+B(t)x(t-h(t))\hfill \\ \multicolumn{1}{c}{x(t)=\varphi (t),\stackrel{·}{x}(t)=\phi (t),\forall t\in [-max(h_M,\tau ),0]}\end{array}$$

(1)

where $x(t)\in R^n$ is the system state variables; the initial functions $\varphi (t)$ and $\stackrel{·}{\phi }(t)$ are continuously differentiable belonging to $[-max(h_M,\tau ),0]$ ; $h(t)$ is the discrete time-varying delay satisfying $0\le h_m\le h(t)\le h_M$ , $\stackrel{·}{h}(t)\le \mu$ ; $\tau (t)$ is the neutral delay satisfying $0\le \tau (t)\le \tau$ , $\stackrel{·}{\tau }(t)\le \eta <1$ ; $A(t)$ , $B(t)$ , and $C(t)$ are unknown matrix functions with time-varying structural uncertainties satisfying the following

$$A(t)=A+\mathrm{\Delta }A(t),B(t)=B+\mathrm{\Delta }B(t),C(t)=C+\mathrm{\Delta }C(t)$$

(2)

where A, B, and C are known positive definite constant matrices with proper dimensions; $\mathrm{\Delta }A(t)$ , $\mathrm{\Delta }B(t)$ , and $\mathrm{\Delta }C(t)$ are unknown matrices with time-varying structural uncertainty; when they have norm-bounded uncertainty, they can be described as follows

$$\left[\begin{array}{ccc}\mathrm{\Delta }A(t)\hfill & \mathrm{\Delta }B(t)\hfill & \mathrm{\Delta }C(t)\hfill \end{array}\right]=\mathrm{DF}(t)\left[\begin{array}{ccc}{E}_a\hfill & E_b\hfill & E_c\hfill \end{array}\right]$$

(3)

where D, $E_a$ , $E_b$ , and $E_c$ are positive definite constant matrices with proper dimensions; $F(t)$ is uncertain matrix with measurable elements satisfying $F(t{)}^{T}F(t)\le I,\forall t$ , where I denotes identity matrix with proper dimensions. The system is transformed into nominal linear neutral system when $F(t)=0$ .

Lemma 1

For any positive definite constant matrix $W\in R^{n\times n}$ , $W=W^T>0$ , suppose that scalar $h=h(t)>0$ and vector function $\stackrel{·}{x}:[-h,0]\to {\mathbb{R}}^n$ , then the following inequalities hold ¹⁶

$$\begin{array}{c}-h\underset{t-h}{\overset{t}{\int }}{\stackrel{·}{x}}^T(s)W\stackrel{·}{x}(s)\mathrm{ds}\le {\varsigma }_1^T(t)\left[\begin{array}{cc}-W\hfill & W\hfill \\ \multicolumn{1}{c}{W}& -W\hfill \end{array}\right]{\varsigma }_1(t)\hfill \\ -\frac{h^2}{2}\underset{-h}{\overset{0}{\int }}\underset{t+\theta }{\overset{t}{\int }}{\stackrel{·}{x}}^T(t)W\stackrel{·}{x}(t)\mathrm{ds}\le {\varsigma }_2^T(t)\left[\begin{array}{cc}-W\hfill & W\hfill \\ \multicolumn{1}{c}{W}& -W\hfill \end{array}\right]{\varsigma }_2(t)\hfill \end{array}$$

where

$${\varsigma }_1^T(t)=\left[\begin{array}{cc}{x}^T(t)\hfill & x^T(t-h)\hfill \end{array}\right],{\varsigma }_2^T(t)=\left[\begin{array}{cc}h{x}^T(t)\hfill & \underset{t-h}{\overset{t}{\int }}{x}^T(s)\mathrm{ds}\hfill \end{array}\right]$$

Lemma 2

For any positive definite constant matrix $M=M^T>0$ , suppose that scalar $h>0$ and vector function $x(t):[0,h]\to {\mathbb{R}}^n$ , then the following inequalities hold ¹

$$\begin{array}{c}-h\underset{t-h}{\overset{t}{\int }}{x}^T(s)\mathrm{Mx}(s)\mathrm{ds}\le -\underset{t-h}{\overset{t}{\int }}{x}^T(s)\mathrm{dsM}\underset{t-h}{\overset{t}{\int }}x(s)\mathrm{ds}\hfill \\ -\frac{h^2}{2}\underset{-h}{\overset{0}{\int }}\underset{t+\beta }{\overset{t}{\int }}{x}^T(s)\mathrm{Mx}(s)\mathrm{dsd}\beta \le -\underset{-h}{\overset{0}{\int }}\underset{t+\beta }{\overset{t}{\int }}{x}^T(s)\mathrm{dsd}\beta M\hfill \\ \underset{-h}{\overset{0}{\int }}\underset{t+\beta }{\overset{t}{\int }}x(s)\mathrm{dsd}\beta \hfill \\ -\frac{h^3}{6}\underset{-h}{\overset{0}{\int }}\underset{\beta }{\overset{0}{\int }}\underset{t+\lambda }{\overset{t}{\int }}{x}^T(s)\mathrm{Mx}(s)\mathrm{dsd}\beta d\lambda \hfill \\ \le -\underset{-h}{\overset{0}{\int }}\underset{\beta }{\overset{0}{\int }}\underset{t+\lambda }{\overset{t}{\int }}{x}^T(s)\mathrm{dsd}\beta d\lambda M\underset{-h}{\overset{0}{\int }}\underset{\beta }{\overset{0}{\int }}\underset{t+\lambda }{\overset{t}{\int }}x(s)\mathrm{dsd}\beta d\lambda \hfill \end{array}$$

Lemma 3

For any positive definite constant matrix $M=M^T>0$ , suppose that scalars $0\le \alpha ,\epsilon \le 1$ , $h_m\le h(t)\le h_M$ and vector function $x(t):[0,h]\to {\mathbb{R}}^n$ , then the following inequalities hold ³⁸

$$\begin{array}{c}-(h_M-h_m)\underset{t-h_M}{\overset{t-h_m}{\int }}{\mathit{x}}^T(s)M\mathit{x}(s)\mathrm{ds}\le -{\mathbf{\zeta }}^T(t)(e_7{\mathrm{Me}}_7^T+e_6{\mathrm{Me}}_6^T)\mathbf{\zeta }(t)\hfill \\ -\alpha {\mathbf{\zeta }}^T(t)e_7{\mathrm{Me}}_7^T\zeta (t)-(1-\alpha ){\mathbf{\zeta }}^T(t)e_6{\mathrm{Me}}_6^T\mathbf{\zeta }(t)\hfill \\ -\frac{(h_M^2-h_m^2)}{2}\underset{-h_M}{\overset{-h_m}{\int }}\underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s)M\mathit{x}(s)\mathrm{dsd}\beta \hfill \\ \le -{\mathbf{\zeta }}^T(t)(e_{10}{\mathrm{Me}}_{10}^T+e_9{\mathrm{Me}}_9^T)\mathbf{\zeta }(t)-\epsilon {\mathbf{\zeta }}^T(t)e_{10}{\mathrm{Me}}_{10}^T\mathbf{\zeta }(t)\hfill \\ -(1-\epsilon ){\mathbf{\zeta }}^T(t)e_9{\mathrm{Me}}_9^T\mathbf{\zeta }(t)\hfill \end{array}$$

where

$$\begin{array}{l}\zeta (t)=[\begin{array}{cccc}x(t)& x(t-h(t))& x(t-h_{\mathrm{\Delta }})& x(t-h_M)\end{array}.\\ {\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}x(s)ds}{\displaystyle \underset{t-h(t)}{\overset{t-h_{\mathrm{\Delta }}}{\int }}x(s)ds}{\displaystyle \underset{t-h_M}{\overset{t-h(t)}{\int }}x(s)ds}\\ {\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}x(s)}ds}d\beta {\displaystyle \underset{-h(t)}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}x(s)}ds}d\beta \\ {\displaystyle \underset{-h_M}{\overset{-h(t)}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}x(s)}ds}d\beta ]\end{array}$$

Main results

Consider the nominal linear neutral condition of system (equation (1)) as follows

$$\{\begin{array}{c}\stackrel{·}{x}(t)-C\stackrel{·}{x}(t-\tau (t))=\mathrm{Ax}(t)+\mathrm{Bx}(t-h(t))\hfill \\ \multicolumn{1}{c}{x(t)=\varphi (t),\forall t\in [-max\{\tau ,h\},0]}\end{array}$$

(4)

Theorem 1

For scalars $h_m$ , $h_M$ , and ${\lambda }_1$ , ${\lambda }_2({\lambda }_1>{\lambda }_2)$ , it is asymptotically stable for the nominal system (equation (4)) if there exist positive definite symmetric matrices ${\mathit{P}}_i(i=1,2,3,4,5)$ , ${\mathit{Q}}_1$ , ${\mathit{Q}}_2$ , ${\mathit{U}}_1$ , ${\mathit{U}}_2$ , ${\mathit{X}}_j$ , ${\mathit{R}}_j(j=1,2,3,4)$ such that the following LMIs hold

$$\mathbf{\Phi }={\left({\mathit{\Phi }}_{i,j}\right)}_{10\times 10}<0$$

(5)

where

$$\begin{array}{c}\multicolumn{1}{c}{\mathbf{\Phi }={\left({\mathit{\Phi }}_{i,j}\right)}_{10\times 10}=\left[\begin{array}{cccccccccc}{\mathit{\Phi }}_{11}\hfill & {\mathit{\Phi }}_{12}\hfill & {\mathit{\Phi }}_{13}\hfill & {\mathit{\Phi }}_{14}\hfill & {\mathit{\Phi }}_{15}\hfill & {\mathit{\Phi }}_{16}\hfill & {\mathit{\Phi }}_{17}\hfill & {\mathit{\Phi }}_{18}\hfill & {\mathit{\Phi }}_{19}\hfill & {\mathit{\Phi }}_{110}\hfill \\ \multicolumn{1}{c}{*}& {\mathit{\Phi }}_{22}\hfill & {\mathit{\Phi }}_{23}\hfill & {\mathit{\Phi }}_{24}\hfill & {\mathit{\Phi }}_{25}\hfill & {\mathit{\Phi }}_{26}\hfill & {\mathit{\Phi }}_{27}\hfill & {\mathit{\Phi }}_{28}\hfill & {\mathit{\Phi }}_{29}\hfill & {\mathit{\Phi }}_{210}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\mathit{\Phi }}_{33}\hfill & {\mathit{\Phi }}_{34}\hfill & {\mathit{\Phi }}_{35}\hfill & {\mathit{\Phi }}_{36}\hfill & {\mathit{\Phi }}_{37}\hfill & {\mathit{\Phi }}_{38}\hfill & {\mathit{\Phi }}_{39}\hfill & {\mathit{\Phi }}_{310}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & {\mathit{\Phi }}_{44}\hfill & {\mathit{\Phi }}_{45}\hfill & {\mathit{\Phi }}_{46}\hfill & {\mathit{\Phi }}_{47}\hfill & {\mathit{\Phi }}_{48}\hfill & {\mathit{\Phi }}_{49}\hfill & {\mathit{\Phi }}_{410}\hfill \\ \multicolumn{1}{c}{\mathbf{*}}& \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & {\mathit{\Phi }}_{55}\hfill & {\mathit{\Phi }}_{56}\hfill & {\mathit{\Phi }}_{57}\hfill & {\mathit{\Phi }}_{58}\hfill & {\mathit{\Phi }}_{59}\hfill & {\mathit{\Phi }}_{510}\hfill \\ \multicolumn{1}{c}{\mathbf{*}}& \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & {\mathit{\Phi }}_{66}\hfill & {\mathit{\Phi }}_{67}\hfill & {\mathit{\Phi }}_{68}\hfill & {\mathit{\Phi }}_{69}\hfill & {\mathit{\Phi }}_{610}\hfill \\ \multicolumn{1}{c}{\mathbf{*}}& \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & {\mathit{\Phi }}_{77}\hfill & {\mathit{\Phi }}_{78}\hfill & {\mathit{\Phi }}_{79}\hfill & {\mathit{\Phi }}_{710}\hfill \\ \multicolumn{1}{c}{\mathbf{*}}& \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & {\mathit{\Phi }}_{88}\hfill & {\mathit{\Phi }}_{89}\hfill & {\mathit{\Phi }}_{810}\hfill \\ \multicolumn{1}{c}{\mathbf{*}}& \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & {\mathit{\Phi }}_{99}\hfill & {\mathit{\Phi }}_{910}\hfill \\ \multicolumn{1}{c}{\mathbf{*}}& \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & \mathbf{*}\hfill & {\mathit{\Phi }}_{1010}\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\mathit{\Phi }}_{11}={\mathit{P}}_1\mathit{A}+{\mathit{A}}^T{\mathit{P}}_1+{\mathit{Q}}_1+h_m^2{\mathit{X}}_1+h_m^2{\mathit{A}}^T{\mathit{X}}_2\mathit{A}-{\mathit{X}}_2}\\ +(h_M-h_m{)}^2{\mathit{X}}_3+(h_M-h_m{)}^2{\mathit{A}}^T{\mathit{X}}_4\mathit{A}+\frac{h_m^4}{4}{\mathit{R}}_1\hfill \\ +\frac{h_m^4}{4}{\mathit{A}}^T{\mathit{R}}_2\mathit{A}-h_m^2{\mathit{R}}_2+\frac{{(h_M^2-h_m^2)}^2}{4}{\mathit{R}}_3\hfill \\ +\frac{{(h_M^2-h_m^2)}^2}{4}{\mathit{A}}^T{\mathit{R}}_4\mathit{A}-3(h_M-h_m{)}^2{\mathit{R}}_4+\frac{h_m^6}{36}{\mathit{A}}^T{\mathit{U}}_1\mathit{A}\hfill \\ -\frac{h_m^4}{4}{\mathit{U}}_1+\frac{{(h_M^3-h_m^3)}^2}{36}{\mathit{A}}^T{\mathit{U}}_2\mathit{A}+\frac{{(h_M^2-h_m^2)}^2}{4}{\mathit{U}}_2\hfill \\ \multicolumn{1}{c}{{\mathit{\Phi }}_{12}=h_m^2{\mathit{A}}^T{\mathit{X}}_2\mathit{B}+(h_M-h_m{)}^2{\mathit{A}}^T{\mathit{X}}_4\mathit{B}+\frac{h_m^4}{4}{\mathit{A}}^T{\mathit{R}}_2\mathit{B}}\\ +\frac{{(h_M^2-h_m^2)}^2}{4}{\mathit{A}}^T{\mathit{R}}_4\mathit{B}+\frac{h_m^6}{36}{\mathit{A}}^T{\mathit{U}}_1\mathit{B}+\frac{{(h_M^3-h_m^3)}^2}{36}{\mathit{A}}^T{\mathit{U}}_2\mathit{B}\hfill \\ \multicolumn{1}{c}{{\mathit{\Phi }}_{13}={\mathit{X}}_2,{\mathit{\Phi }}_{15}=2{\mathit{P}}_2+h_m{\mathit{R}}_2,{\mathit{\Phi }}_{16}=(2-\epsilon )(h_M-h_m){\mathit{R}}_4,{\mathit{\Phi }}_{17}=(1+\epsilon )(h_M-h_m){\mathit{R}}_4}\\ \multicolumn{1}{c}{{\mathit{\Phi }}_{18}=2h_m{\mathit{P}}_4+\frac{h_m^2}{2}{\mathit{U}}_1,{\mathit{\Phi }}_{19}={\mathit{\Phi }}_{110}=2(h_M-h_m){\mathit{P}}_5+\frac{(h_M^2-h_m^2)}{2}{\mathit{U}}_2,{\mathit{\Phi }}_{14}=0}\\ \multicolumn{1}{c}{{\mathit{\Phi }}_{22}=h_m^2{\mathit{B}}^T{\mathit{X}}_2\mathit{B}+(h_M-h_m{)}^2{\mathit{B}}^T{\mathit{X}}_4\mathit{B}-{\mathit{X}}_4}\\ +\frac{h_m^4}{4}{\mathit{B}}^T{\mathit{R}}_2\mathit{B}+\frac{{(h_M^2-h_m^2)}^2}{4}{\mathit{B}}^T{\mathit{R}}_4\mathit{B}+\frac{h_m^6}{36}{\mathit{B}}^T{\mathit{U}}_1\mathit{B}\hfill \\ +\frac{{(h_M^3-h_m^3)}^2}{36}{\mathit{B}}^T{\mathit{U}}_2\mathit{B}\hfill \\ \multicolumn{1}{c}{{\mathit{\Phi }}_{23}=-(\alpha -2){\mathit{X}}_4,{\mathit{\Phi }}_{24}=(1+\alpha ){\mathit{X}}_4,{\mathit{\Phi }}_{25}={\mathit{\Phi }}_{26}={\mathit{\Phi }}_{27}={\mathit{\Phi }}_{28}={\mathit{\Phi }}_{29}={\mathit{\Phi }}_{210}=0}\\ \multicolumn{1}{c}{{\mathit{\Phi }}_{33}={\mathit{Q}}_2-{\mathit{Q}}_1-{\mathit{X}}_2+(\alpha -2){\mathit{X}}_4,{\mathit{\Phi }}_{35}=-2{\mathit{P}}_2,{\mathit{\Phi }}_{36}={\mathit{\Phi }}_{37}=2{\mathit{P}}_3,{\mathit{\Phi }}_{34}={\mathit{\Phi }}_{38}={\mathit{\Phi }}_{39}={\mathit{\Phi }}_{310}=0}\\ \multicolumn{1}{c}{{\mathit{\Phi }}_{44}=-{\mathit{Q}}_2-(1+\alpha ){\mathit{X}}_4,{\mathit{\Phi }}_{46}={\mathit{\Phi }}_{47}=-2{\mathit{P}}_3,{\mathit{\Phi }}_{45}={\mathit{\Phi }}_{48}={\mathit{\Phi }}_{49}={\mathit{\Phi }}_{410}=0,{\mathit{\Phi }}_{55}=-{\mathit{X}}_1-{\mathit{R}}_2}\\ \multicolumn{1}{c}{{\mathit{\Phi }}_{58}=-2{\mathit{P}}_4,{\mathit{\Phi }}_{56}={\mathit{\Phi }}_{57}={\mathit{\Phi }}_{59}={\mathit{\Phi }}_{510}=0,{\mathit{\Phi }}_{66}=(\alpha -2){\mathit{X}}_3-(2-\epsilon ){\mathit{R}}_4,{\mathit{\Phi }}_{69}={\mathit{\Phi }}_{610}=-2{\mathit{P}}_5}\\ \multicolumn{1}{c}{{\mathit{\Phi }}_{67}={\mathit{\Phi }}_{68}=0,{\mathit{\Phi }}_{77}=-(\alpha +1){\mathit{X}}_3-(1+\epsilon ){\mathit{R}}_4,{\mathit{\Phi }}_{79}={\mathit{\Phi }}_{710}=-2{\mathit{P}}_5,{\mathit{\Phi }}_{78}=0,{\mathit{\Phi }}_{88}=-{\mathit{R}}_1-{\mathit{U}}_1}\\ \multicolumn{1}{c}{{\mathit{\Phi }}_{89}={\mathit{\Phi }}_{810}=0,{\mathit{\Phi }}_{99}=-(2-\epsilon ){\mathit{R}}_3-{\mathit{U}}_2,{\mathit{\Phi }}_{910}=-{\mathit{U}}_2,{\mathit{\Phi }}_{1010}=(1+\epsilon ){\mathit{R}}_3-{\mathit{U}}_2}\\ \multicolumn{1}{c}{\alpha =\frac{h(t)-h_m}{h_M-h_m},\epsilon =\frac{h{(t)}^2-h_m^2}{h_M^2-h_m^2}}\end{array}$$

Proof

The delay range is divided into two equidistant subintervals at time-delay midpoint $h_{\mathrm{\Delta }}$ , namely, $[h_m,h_{\mathrm{\Delta }}]$ and $[h_{\mathrm{\Delta }},h_M]$ . Theorem 1 is true only if we manage to prove that Theorem 1 holds for the two subintervals.

• Case 1. When $h_{\mathrm{\Delta }}\le h(t)\le h_M$ , construct LKF as follows

$$V(x(t))=V_1(x(t))+V_2(x(t))+V_3(x(t))+V_4(x(t))+V_5(x(t))$$

(6)

where

$$\begin{array}{l}{V}_1(x(t))={\mathit{x}}^T(t){\mathit{P}}_1\mathit{x}(t)+{\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}{\mathit{x}}^T(s)ds}{P}_2{\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}\mathit{x}(s)ds}\\ +{\displaystyle \underset{t-h_M}{\overset{t-h_{\mathrm{\Delta }}}{\int }}{\mathit{x}}^T(s)}ds{\mathit{P}}_3{\displaystyle \underset{t-h_M}{\overset{t-h_{\mathrm{\Delta }}}{\int }}\mathit{x}(s)}ds\\ +{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{x}^T(s)dsd\beta }}{\mathit{P}}_4{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}\mathit{x}(s)dsd\beta }}\\ +{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s)dsd\beta }}{\mathit{P}}_5{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}\mathit{x}(s)dsd\beta }}\\ V_2(x(t))={\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}{\mathit{x}}^T(s)}{Q}_{1}x(s)ds+{\displaystyle \underset{t-h_M}{\overset{t-h_{\mathrm{\Delta }}}{\int }}{\mathit{x}}^T(s)}{\mathit{Q}}_{2}x(s)ds\\ V_3(\mathit{x}(t))=h_{\mathrm{\Delta }}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s)}}{\mathit{X}}_1\mathit{x}(s)dsd\beta \\ +h_{\mathrm{\Delta }}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)}}{\mathit{X}}_2\stackrel{.}{\mathit{x}}(s)dsd\beta \\ +(h_M-h_{\mathrm{\Delta }}){\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s)}}{\mathit{X}}_3\mathit{x}(s)dsd\beta \\ +(h_M-h_{\mathrm{\Delta }}){\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)}}{\mathit{X}}_4\stackrel{.}{\mathit{x}}(s)dsd\beta \\ V_4(\mathit{x}(t))=\frac{h_{\mathrm{\Delta }}^2}{2}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\mathit{x}}^T(s){\mathit{R}}_1\mathit{x}(s)dsd\lambda }}}d\beta \\ +\frac{h_{\mathrm{\Delta }}^2}{2}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s){\mathit{R}}_2\stackrel{.}{\mathit{x}}(s)dsd\lambda }}}d\beta \\ +\frac{(h_M^2-h_{\mathrm{\Delta }}^2)}{2}{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\mathit{x}}^T(s){\mathit{R}}_3\mathit{x}(s)dsd\lambda }}}d\beta \\ +\frac{(h_M^2-h_{\mathrm{\Delta }}^2)}{2}{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s){\mathit{R}}_4\stackrel{.}{\mathit{x}}(s)dsd\lambda }}}d\beta \\ V_5(\mathit{x}(t))=\frac{h_{\mathrm{\Delta }}^3}{6}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{\lambda }{\overset{0}{\int }}{\displaystyle \underset{t+\phi }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)U_1\stackrel{.}{\mathit{x}}(s)}dsd\phi }}}d\lambda d\beta \\ +\frac{(h_M^3-h_{\mathrm{\Delta }}^3)}{6}{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{\lambda }{\overset{0}{\int }}{\displaystyle \underset{t+\phi }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)U_2\stackrel{.}{\mathit{x}}(s)}}}}dsd\phi d\lambda d\beta \end{array}$$

The time derivatives of $V(x(t))$ along the trajectories of system (equation (4)) yields the following

$$\stackrel{·}{V}(x(t))={\stackrel{·}{V}}_1(t)+{\stackrel{·}{V}}_2(t)+{\stackrel{·}{V}}_3(t)+{\stackrel{·}{V}}_4(t)+{\stackrel{·}{V}}_5(t)$$

(7)

where

$$\begin{array}{c}{\stackrel{.}{V}}_1(t)=2{\mathit{x}}^T(t)A^T{\mathit{P}}_1\mathit{x}(t)+{\mathit{x}}^T(t-h(t))B^T{\mathit{P}}_{1}x(t)\\ +2{\mathit{x}}^T(t)P_2{\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}\mathit{x}(s)}ds-2x^T(t-h_{\mathrm{\Delta }}){\mathit{P}}_2{\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}x(s)}ds\\ +2x^T(t-h_{\mathrm{\Delta }})P_3{\displaystyle \underset{t-h_M}{\overset{t-h_{\mathrm{\Delta }}}{\int }}\mathit{x}(s)}ds-2{\mathit{x}}^T(t-h_M)P_3{\displaystyle \underset{t-h_M}{\overset{t-h_{\mathrm{\Delta }}}{\int }}\mathit{x}(s)}ds\\ +2h_{\mathrm{\Delta }}{\mathit{x}}^T(t){\mathit{P}}_4{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}x(s)ds}}d\beta \\ -2{\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}{x}^T(s)}ds{\mathit{P}}_4{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}x(s)}ds}d\beta \\ +2(h_M-h_{\mathrm{\Delta }}){\mathit{x}}^T(t){\mathit{P}}_5{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}\mathit{x}(s)}ds}d\beta \\ -2{\displaystyle \underset{t-h_M}{\overset{t-h_{\mathrm{\Delta }}}{\int }}{\mathit{x}}^T(s)}ds{\mathit{P}}_5{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}\mathit{x}(s)}ds}d\beta \\ {\stackrel{.}{V}}_2(t)=x^T(t){\mathit{Q}}_{1}x(t)-x^T(t-h_{\mathrm{\Delta }}){\mathit{Q}}_{1}x(t-h_{\mathrm{\Delta }})\\ +x^T(t-h_{\mathrm{\Delta }}){\mathit{Q}}_{2}x(t-h_{\mathrm{\Delta }})-x^T(t-h_M)Q_{2}x(t-h_M)\\ {\stackrel{.}{V}}_3(t)=h_{\mathrm{\Delta }}^2{\mathit{x}}^T(t){\mathit{X}}_{1}x(t)-h_{\mathrm{\Delta }}{\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}{\mathit{x}}^T(s)}{\mathit{X}}_{1}x(s)ds\\ +h_{\mathrm{\Delta }}^2{\stackrel{.}{\mathit{x}}}^T(t){\mathit{X}}_2\stackrel{.}{x}(t)-h_{\mathrm{\Delta }}{\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)}{\mathit{X}}_2\stackrel{.}{\mathit{x}}(s)ds\\ +{(h_M-h_{\mathrm{\Delta }})}^2{\mathit{x}}^T(t){\mathit{X}}_3\mathit{x}(t)-(h_M-h_{\mathrm{\Delta }}){\displaystyle \underset{t-h_M}{\overset{t-h_{\mathrm{\Delta }}}{\int }}{\mathit{x}}^T(s)}{\mathit{X}}_{3}x(s)ds\\ +{(h_M-h_{\mathrm{\Delta }})}^2{\stackrel{.}{\mathit{x}}}^T(t){\mathit{X}}_4\stackrel{.}{\mathit{x}}(t)-(h_M-h_{\mathrm{\Delta }}){\displaystyle \underset{t-h_M}{\overset{t-h_{\mathrm{\Delta }}}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)}{\mathit{X}}_4\stackrel{.}{\mathit{x}}(s)ds\\ {\stackrel{.}{V}}_4(t)=\frac{h_{\mathrm{\Delta }}^4}{4}{\mathit{x}}^T(t){\mathit{R}}_1\mathit{x}(t)-\frac{h_{\mathrm{\Delta }}^2}{2}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s){\mathit{R}}_{1}x(s)}}dsd\beta \\ +\frac{h_{\mathrm{\Delta }}^4}{4}{\stackrel{.}{\mathit{x}}}^T(t)R_2\stackrel{.}{\mathit{x}}(t)-\frac{h_{\mathrm{\Delta }}^2}{2}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\stackrel{.}{x}}^T(s){\mathit{R}}_2\stackrel{.}{x}(s)}}dsd\beta \\ +\frac{{(h_M^2-h_{\mathrm{\Delta }}^2)}^2}{4}{\mathit{x}}^T(t){\mathit{R}}_3\mathit{x}(t)+\frac{{(h_M^2-h_{\mathrm{\Delta }}^2)}^2}{4}{\stackrel{.}{\mathit{x}}}^T(t){\mathit{R}}_4\stackrel{.}{x}(t)\\ -\frac{(h_M^2-h_{\mathrm{\Delta }}^2)}{2}{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s){\mathit{R}}_3\mathit{x}(s)}}dsd\beta \\ -\frac{(h_M^2-h_{\mathrm{\Delta }}^2)}{2}{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s){\mathit{R}}_4\stackrel{.}{\mathit{x}}(s)}}dsd\beta \\ {\stackrel{.}{V}}_5(t)=\frac{h_{\mathrm{\Delta }}^6}{36}{\stackrel{.}{\mathit{x}}}^T(t){\mathit{U}}_1\stackrel{.}{\mathit{x}}(t)-\frac{h_{\mathrm{\Delta }}^3}{6}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s){\mathit{U}}_1\stackrel{.}{\mathit{x}}(s)ds}d\lambda }}d\beta \\ +\frac{{(h_M^3-h_{\mathrm{\Delta }}^3)}^2}{36}{\stackrel{.}{\mathit{x}}}^T(t){\mathit{U}}_2\stackrel{.}{x}(t)\\ -\frac{(h_M^3-h_{\mathrm{\Delta }}^3)}{6}{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)}}}{\mathit{U}}_2\stackrel{.}{x}(s)dsd\lambda d\beta \end{array}$$

As can be seen from Lemma 1 and Lemma 2, respectively, we could have the inequalities as follows

$$-h_{\mathrm{\Delta }}\underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}{\mathit{x}}^T(s)X_1\mathit{x}(s)\mathrm{ds}\le -{\mathbf{\zeta }}^{\mathbf{T}}(t)e_5{X}_1{e}_5^T\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}$$

(8)

$$-h_{\mathrm{\Delta }}\underset{t-h_{\mathrm{\Delta }}}{\overset{t}{\int }}{\stackrel{·}{\mathit{x}}}^T(s)X_2\stackrel{·}{\mathit{x}}(s)\mathrm{ds}\le -{\mathbf{\zeta }}^{\mathbf{T}}(t)(e_1-e_3)X_2(e_1^T-e_3^T)\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}$$

(9)

where $\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}$ is defined in Lemma 3.

From Lemma 3, we have the following inequality

$$\begin{array}{c}-(h_M-h_{\mathrm{\Delta }})\underset{t-h_M}{\overset{t-h_{\mathrm{\Delta }}}{\int }}{\mathit{x}}^T(s)X_3\mathit{x}(s)\mathrm{ds}\le -{\mathbf{\zeta }}^T(t)(e_7{X}_3{e}_7^T+e_6{X}_3{e}_6^T)\mathbf{\zeta }(t)\hfill \\ -\alpha {\mathbf{\zeta }}^T(t)e_7{\mathbf{X}}_3{e}_7^T\mathbf{\zeta }(t)-(1-\alpha ){\mathbf{\zeta }}^T(t)e_6{\mathbf{X}}_3{e}_6^T\mathbf{\zeta }(t)\hfill \end{array}$$

(10)

Similarly, the following inequalities can also be obtained

$$\begin{array}{c}-(h_M-h_{\mathrm{\Delta }})\underset{t-h_M}{\overset{t-h_{\mathrm{\Delta }}}{\int }}{\stackrel{·}{\mathit{x}}}^T(s)X_4\stackrel{·}{\mathit{x}}(s)\mathrm{ds}\le -{\mathbf{\zeta }}^T(t)(e_2-e_4)X_4(e_2^T-e_4^T)\mathbf{\zeta }(t)\hfill \\ -{\mathbf{\zeta }}^T(t)(e_3-e_2){\mathbf{X}}_4(e_3^T-e_2^T)\mathbf{\zeta }(t)-\alpha {\mathbf{\zeta }}^T(t)(e_2-e_4)X_4(e_2^T-e_4^T)\zeta (t)\hfill \\ -(1-\alpha ){\mathbf{\zeta }}^T(t)(e_3-e_2){\mathbf{X}}_4(e_3^T-e_2^T)\mathbf{\zeta }(t)\hfill \end{array}$$

(11)

$$-\frac{h_{\mathrm{\Delta }}^2}{2}\underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}\underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s)R_1\mathit{x}(s)\mathrm{dsd}\beta \le -{\mathbf{\zeta }}^T(t)e_8{R}_1{e}_8^T\mathbf{\zeta }(t)$$

(12)

$$\begin{array}{l}-\frac{h_{\mathrm{\Delta }}^2}{2}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\stackrel{.}{x}}^T(s)R_2\stackrel{.}{x}(s)}}dsd\beta \\ \le -{\zeta }^T(t)(h_{\mathrm{\Delta }}{e}_1-e_5)R_3(h_{\mathrm{\Delta }}{e}_1^T-e_5^T)\zeta (t)\end{array}$$

(13)

$$\begin{array}{c}-\frac{(h_M^2-h_{\mathrm{\Delta }}^2)}{2}\underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}\underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s)R_3\mathit{x}(s)\mathrm{dsd}\beta \hfill \\ \le -{\mathbf{\zeta }}^{\mathbf{T}}(t)(e_{10}{R}_3{e}_{10}^T+e_9{R}_3{e}_9^T)\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}\hfill \\ -\mathbf{\epsilon }{\mathbf{\zeta }}^{\mathbf{T}}\mathbf{(}\mathbf{t}\mathbf{)}{\mathbf{e}}_{10}{\mathbf{R}}_3{\mathbf{e}}_{10}^{\mathbf{T}}\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}-\mathbf{(}1-\mathbf{\epsilon }\mathbf{)}{\mathbf{\zeta }}^{\mathbf{T}}\mathbf{(}\mathbf{t}\mathbf{)}{\mathbf{e}}_9{\mathbf{R}}_3{\mathbf{e}}_9^{\mathbf{T}}\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}\hfill \end{array}$$

(14)

$$\begin{array}{c}-\frac{(h_M^2-h_{\mathrm{\Delta }}^2)}{2}\underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}\underset{t+\beta }{\overset{t}{\int }}{\stackrel{·}{\mathit{x}}}^T(s)R_4\stackrel{·}{\mathit{x}}(s)\mathrm{dsd}\beta \hfill \\ \le -{\mathbf{\zeta }}^{\mathbf{T}}(t)((h_M-h_{\mathrm{\Delta }})e_1-e_7)R_4((h_M-h_{\mathrm{\Delta }})e_1^T-e_7^T)\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}\hfill \\ -{\mathbf{\zeta }}^{\mathbf{T}}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{(}\mathbf{(}{\mathbf{h}}_{\mathbf{M}}-{\mathbf{h}}_{\mathbf{\Delta }}\mathbf{)}{\mathbf{e}}_1-{\mathbf{e}}_6\mathbf{)}{\mathbf{R}}_4\mathbf{(}\mathbf{(}{\mathbf{h}}_{\mathbf{M}}-{\mathbf{h}}_{\mathbf{\Delta }}\mathbf{)}{\mathbf{e}}_1^{\mathbf{T}}-{\mathbf{e}}_6^{\mathbf{T}}\mathbf{)}\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}\hfill \\ -\mathbf{\epsilon }{\mathbf{\zeta }}^{\mathbf{T}}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{(}\mathbf{(}{\mathbf{h}}_{\mathbf{M}}-{\mathbf{h}}_{\mathbf{\Delta }}\mathbf{)}{\mathbf{e}}_1-{\mathbf{e}}_7\mathbf{)}{\mathbf{R}}_4\mathbf{(}\mathbf{(}{\mathbf{h}}_{\mathbf{M}}-{\mathbf{h}}_{\mathbf{\Delta }}\mathbf{)}{\mathbf{e}}_1^{\mathbf{T}}-{\mathbf{e}}_7^{\mathbf{T}}\mathbf{)}\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}\hfill \\ -\mathbf{(}1-\mathbf{\epsilon }\mathbf{)}{\mathbf{\zeta }}^{\mathbf{T}}\mathbf{(}\mathbf{t}\mathbf{)}\mathbf{(}\mathbf{(}{\mathbf{h}}_{\mathbf{M}}-{\mathbf{h}}_{\mathbf{\Delta }}\mathbf{)}{\mathbf{e}}_1-{\mathbf{e}}_6\mathbf{)}{\mathbf{R}}_4\mathbf{(}\mathbf{(}{\mathbf{h}}_{\mathbf{M}}-{\mathbf{h}}_{\mathbf{\Delta }}\mathbf{)}{\mathbf{e}}_1^{\mathbf{T}}-{\mathbf{e}}_6^{\mathbf{T}}\mathbf{)}\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}\hfill \end{array}$$

(15)

$$\begin{array}{l}-\frac{h_{\mathrm{\Delta }}^3}{6}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{0}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\stackrel{.}{x}}^T(s)U_1\stackrel{.}{x}(s)ds}d\lambda }}d\beta \\ \le -{\zeta }^T(t)\left(\frac{h_{\mathrm{\Delta }}^2}{2}{e}_1-e_8\right)U_1\left(\frac{h_{\mathrm{\Delta }}^2}{2}{e}_1^T-e_8^T\right)\zeta (t)\end{array}$$

(16)

$$\begin{array}{l}-\frac{(h_M^3-h_{\mathrm{\Delta }}^3)}{6}{\displaystyle \underset{-h_M}{\overset{-h_{\mathrm{\Delta }}}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\stackrel{.}{x}}^T(s)U_2\stackrel{.}{x}(s)dsd\lambda }}}d\beta \\ \le -{\zeta }^T(t)\left(\frac{(h_M^2-h_{\mathrm{\Delta }}^2)}{2}{e}_1-e_9-e_{10}\right)U_2\left(\frac{(h_M^2-h_{\mathrm{\Delta }}^2)}{2}{e}_1^T-e_9^T-e_{10}^T\right)\zeta (t)\end{array}$$

(17)

Substituting equations (8)–(17) in equation (7), $\stackrel{·}{V}(x(t))$ can be expressed as follows

$$\stackrel{·}{V}(x(t))\le {\mathbf{\zeta }}^{\mathbf{T}}(t)[\alpha {\mathrm{\Gamma }}_1+(1-\alpha ){\mathrm{\Gamma }}_2+\epsilon {\mathrm{\Gamma }}_3+(1-\epsilon ){\mathrm{\Gamma }}_4]\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}$$

(18)

where

$$\begin{array}{c}{\mathrm{\Gamma }}_1=-e_7{X}_3{e}_7^T-(e_2-e_4)X_4(e_2^T-e_4^T)\hfill \\ {\mathrm{\Gamma }}_2=-e_6{X}_3{e}_6^T-(e_3-e_2)X_4(e_3^T-e_2^T)\hfill \\ {\mathrm{\Gamma }}_3=-e_{10}{R}_3{e}_{10}^T-((h_M-h_{\mathrm{\Delta }})e_1-e_7){\mathrm{R}}_4((h_M-h_{\mathrm{\Delta }})e_1^T-e_7^T)\hfill \\ {\mathrm{\Gamma }}_4=-e_9{R}_3{e}_9^T-((h_M-h_{\mathrm{\Delta }})e_1-e_6){\mathrm{R}}_4((h_M-h_{\mathrm{\Delta }})e_1^T-e_6^T)\hfill \end{array}$$

For $0\le \alpha ,\epsilon \le 1$ , based on reciprocally convex combination approach, the following inequalities hold

$$\alpha ({\mathrm{\Gamma }}_1+{\lambda }_{1}I)+(1-\alpha )({\mathrm{\Gamma }}_2+{\lambda }_{1}I)<0$$

(19)

$$\epsilon ({\mathrm{\Gamma }}_3-{\lambda }_{2}I)+(1-\epsilon )({\mathrm{\Gamma }}_4-{\lambda }_{2}I)<0$$

(20)

namely

$$\alpha {\mathrm{\Gamma }}_1+(1-\alpha ){\mathrm{\Gamma }}_2<-{\lambda }_{1}I$$

(21)

$$\epsilon {\mathrm{\Gamma }}_3+(1-\epsilon ){\mathrm{\Gamma }}_4<{\lambda }_{2}I$$

(22)

Since ${\lambda }_1>{\lambda }_2$ , by adding equation (21) to equation (22), we can obtain the following

$$\alpha {\mathrm{\Gamma }}_1+(1-\alpha ){\mathrm{\Gamma }}_2+\epsilon {\mathrm{\Gamma }}_3+(1-\epsilon ){\mathrm{\Gamma }}_4<({\lambda }_2-{\lambda }_1)I<0$$

(23)

If $\alpha {\mathrm{\Gamma }}_1+(1-\alpha ){\mathrm{\Gamma }}_2+\epsilon {\mathrm{\Gamma }}_3+(1-\epsilon ){\mathrm{\Gamma }}_4<0$ , according to Lyapunov stability, there exists a given fully small positive $\delta$ , then inequality $\stackrel{·}{V}(x(t))<-\delta \Vert x(t){\Vert }^2$ holds. Therefore, the nominal system (equation (4)) is asymptotically stable.

• Case 2. When $h_m\le h(t)\le h_{\mathrm{\Delta }}$ , construct LKF as follows

$$\begin{array}{l}{V}_0(x(t))=V_{01}(x(t))+V_{02}(x(t))+V_{03}(x(t))\\ +V_{04}(x(t))+V_{05}(x(t))\end{array}$$

(24)

where

$$\begin{array}{c}{V}_{01}(x(t))={\mathit{x}}^T(t){\mathit{P}}_1\mathit{x}(t)+{\displaystyle \underset{t-h_m}{\overset{t}{\int }}{x}^T(s)ds}{\mathit{P}}_2{\displaystyle \underset{t-h_m}{\overset{t}{\int }}\mathit{x}(s)ds}\\ +{\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t-h_m}{\int }}{x}^T(s)}ds{\mathit{P}}_3{\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t-h_m}{\int }}\mathit{x}(s)}ds\\ +{\displaystyle \underset{-h_m}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{x}^T(s)dsd\beta }}{\mathit{P}}_4{\displaystyle \underset{-h_m}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}\mathit{x}(s)dsd\beta }}\\ +{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{-h_m}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s)dsd\beta }}{\mathit{P}}_5{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{-h_m}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}\mathit{x}(s)dsd\beta }}\\ V_{02}(\mathit{x}(t))={\displaystyle \underset{t-h_m}{\overset{t}{\int }}{x}^T(s)}{Q}_{1}x(s)ds+{\displaystyle \underset{t-h_{\mathrm{\Delta }}}{\overset{t-h_m}{\int }}{\mathit{x}}^T(s)}{Q}_2\mathit{x}(s)ds\\ V_{03}(\mathit{x}(t))=h_m{\displaystyle \underset{-h_m}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s)}}{\mathit{X}}_1\mathit{x}(s)dsd\beta \\ +h_m{\displaystyle \underset{-h_m}{\overset{0}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)}}{\mathit{X}}_2\stackrel{.}{\mathit{x}}(s)dsd\beta \\ +(h_{\mathrm{\Delta }}-h_m){\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{-h_m}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\mathit{x}}^T(s)}}{\mathit{X}}_{3}x(s)dsd\beta \\ +(h_{\mathrm{\Delta }}-h_m){\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{-h_m}{\int }}{\displaystyle \underset{t+\beta }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)}}{\mathit{X}}_4\stackrel{.}{\mathit{x}}(s)dsd\beta \\ V_{04}(\mathit{x}(t))=\frac{h_m^2}{2}{\displaystyle \underset{-h_m}{\overset{0}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\mathit{x}}^T(s)}}}{\mathit{R}}_1\mathit{x}(s)dsd\lambda d\beta \\ +\frac{h_m^2}{2}{\displaystyle \underset{-h_m}{\overset{0}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)}{\mathit{R}}_2\stackrel{.}{\mathit{x}}(s)dsd\lambda }}d\beta \\ +\frac{(h_{\mathrm{\Delta }}^2-h_m^2)}{2}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{-h_m}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\mathit{x}}^T(s)}{\mathit{R}}_3\mathit{x}(s)dsd\lambda }}d\beta \\ +\frac{(h_{\mathrm{\Delta }}^2-h_m^2)}{2}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{-h_m}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{t+\lambda }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s)}{\mathit{R}}_4\stackrel{.}{\mathit{x}}(s)dsd\lambda }}d\beta \\ V_{05}(\mathit{x}(t))=\frac{h_m^3}{6}{\displaystyle \underset{-h_m}{\overset{0}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{\lambda }{\overset{0}{\int }}{\displaystyle \underset{t+\phi }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s){\mathit{U}}_1\stackrel{.}{x}(s)}dsd\phi }}}d\lambda d\beta \\ +\frac{(h_{\mathrm{\Delta }}^3-h_m^3)}{6}{\displaystyle \underset{-h_{\mathrm{\Delta }}}{\overset{-h_m}{\int }}{\displaystyle \underset{\beta }{\overset{0}{\int }}{\displaystyle \underset{\lambda }{\overset{0}{\int }}{\displaystyle \underset{t+\phi }{\overset{t}{\int }}{\stackrel{.}{\mathit{x}}}^T(s){\mathit{U}}_2\stackrel{.}{x}(s)}}}}dsd\phi d\lambda d\beta \end{array}$$

where

$$\begin{array}{c}\multicolumn{1}{c}{\overline{\mathit{\zeta }}(t)=[\begin{array}{cccc}\mathit{x}(t)\hfill & \mathit{x}(t-h(t))\hfill & \mathit{x}(t-h_m)\hfill & \mathit{x}(t-h_{\mathrm{\Delta }})\hfill \end{array}}\\ \begin{array}{ccc}\underset{t-h_m}{\overset{t}{\int }}\mathit{x}(s)\mathrm{ds}\hfill & \underset{t-h(t)}{\overset{t-h_m}{\int }}\mathit{x}(s)\mathrm{ds}\hfill & \underset{t-h_{\mathrm{\Delta }}}{\overset{t-h(t)}{\int }}\mathit{x}(s)\mathrm{ds}\hfill \end{array}\hfill \\ \begin{array}{cc}\underset{-h_m}{\overset{0}{\int }}\underset{t+\beta }{\overset{t}{\int }}\mathit{x}(s)\mathrm{ds}d\beta \hfill & \underset{-h(t)}{\overset{-h_m}{\int }}\underset{t+\beta }{\overset{t}{\int }}\mathit{x}(s)\mathrm{ds}d\beta \hfill \end{array}\hfill \\ \underset{-h_{\mathrm{\Delta }}}{\overset{-h(t)}{\int }}\underset{t+\beta }{\overset{t}{\int }}\mathit{x}(s)\mathrm{ds}d\beta ]\hfill \end{array}$$

${\mathit{P}}_i(i=1,2,3,4,5)$ , ${\mathit{Q}}_1$ , ${\mathit{Q}}_2$ , ${\mathit{U}}_1$ , ${\mathit{U}}_2$ , ${\mathit{X}}_j$ , ${\mathit{R}}_j(j=1,2,3,4)$ . These matrices are defined in equation (6). In the same way, we can obtain as follows

$${\stackrel{·}{V}}_0(x(t))\le {\mathbf{\zeta }}^{\mathbf{T}}(t)[\alpha {\mathrm{\Gamma }}_1+(1-\alpha ){\mathrm{\Gamma }}_2+\epsilon {\mathrm{\Gamma }}_3+(1-\epsilon ){\mathrm{\Gamma }}_4]\mathbf{\zeta }\mathbf{(}\mathbf{t}\mathbf{)}$$

(25)

where

$$\begin{array}{c}{\mathrm{\Gamma }}_1=-e_7{X}_3{e}_7^T-(e_2-e_4)X_4(e_2^T-e_4^T)\hfill \\ {\mathrm{\Gamma }}_2=-e_6{X}_3{e}_6^T-(e_3-e_2)X_4(e_3^T-e_2^T)\hfill \\ {\mathrm{\Gamma }}_3=-e_{10}{R}_3{e}_{10}^T-((h_{\mathrm{\Delta }}-h_m)e_1-e_7){\mathrm{R}}_4((h_{\mathrm{\Delta }}-h_m)e_1^T-e_7^T)\hfill \\ {\mathrm{\Gamma }}_4=-e_9{R}_3{e}_9^T-((h_{\mathrm{\Delta }}-h_m)e_1-e_6){\mathrm{R}}_4((h_{\mathrm{\Delta }}-h_m)e_1^T-e_6^T)\hfill \end{array}$$

If $\alpha {\mathrm{\Gamma }}_1+(1-\alpha ){\mathrm{\Gamma }}_2+\epsilon {\mathrm{\Gamma }}_3+(1-\epsilon ){\mathrm{\Gamma }}_4<0$ , according to Lyapunov stability, there exists a given fully small positive $\delta$ , then inequality ${\stackrel{·}{V}}_0(x(t))<-\delta \Vert x(t){\Vert }^2$ holds. Therefore, the nominal system (equation (4)) is asymptotically stable.

Since $h_M-h_{\mathrm{\Delta }}=h_{\mathrm{\Delta }}-h_m$ , applying Lemma 3 to equation (18) or (25), we have conclusion which is equivalent to the LMIs (equation (5)) in Theorem 1. This fulfills the proof.

Remark 1

In this article, the uniqueness of new LKF is reflected in the two following sides. First, the delay range is divided into two equidistant subintervals, and in each subinterval, a new LKF comprising quadruple-integral term and quadratic forms of double-integral term is introduced. Different from the previous methods, a new LKF is used. Second, unlike in the study by Ramakrishnan and Ray, ²⁷ double-integral functional term is added into new augmented vector in initial definition, such as ${\int }_{-h}^0{\int }_{t+\beta }^{t}x(s)\mathrm{ds}d\beta$ . Third, the state is incorporated into the integrand of triple-integral functional term, and the lower bound of time-delay interval information is imported in LKF. The stability of the results is significantly reduced owing to the presence of quadruple-integral functional term and quadratic term such as $\int \int x^T(s)\mathrm{dsd}\beta$ .

Remark 2

In formula (5), the novel stability criterion does not involve redundant-free weighting matrices but only skillfully introduces new integral inequality to deal with cross terms. This leads to the noticeable reduction in the complexity in calculation and conservativeness in conclusion.

Remark 3

In formulas (10), (11), and (14), as an alternative method, reciprocally convex combination approach is adopted to limit the cross terms of LKF derivatives. Therefore, the lower conservative results were obtained.

Remark 4

For a given scalar $\mu$ and delay variation rate $\stackrel{·}{h}(t)$ satisfying $0<\stackrel{·}{h}(t)\le \mu$ , substituting ${\int }_{t-h(t)}^t{x}^T(s){\mathit{Q}}_{3}x(s)\mathrm{ds}$ in constructing LKF, the stability criteria including delay variation rate can be obtained in Theorem 2.

Theorem 2

For the scalars $h_m$ , $h_M$ , and $\mu$ , ${\lambda }_1$ , ${\lambda }_2({\lambda }_1>{\lambda }_2)$ , it is asymptotically stable for the nominal system (equation (4)), if there exist positive definite symmetric matrices $P_i(i=1,2,3,4,5)$ , $Q_1$ , $Q_2$ , ${\mathit{Q}}_3$ , $U_1$ , $U_2$ , $X_j$ , $R_j(j=1,2,3,4)$ such that the following LMIs hold

$$\tilde{\mathbf{\Phi }}={\left({\tilde{\mathit{\Phi }}}_{i,j}\right)}_{10\times 10}<0$$

(26)

where ${\tilde{\mathit{\Phi }}}_{11}={\mathit{\Phi }}_{11}+{\mathit{Q}}_3$ , ${\tilde{\mathit{\Phi }}}_{22}={\mathit{\Phi }}_{22}-\mu {\mathit{Q}}_3$ . Other items in $\tilde{\mathit{\Phi }}$ are defined as same as in $\mathit{\Phi }$ Theorem 1.

Next, we investigate robust stability issue for uncertain neutral system with mixed time-varying delays in the following section.

Theorem 3

For the scalars $0<h_m<h_M$ and $\mu$ , ${\lambda }_1$ , ${\lambda }_2({\lambda }_1>{\lambda }_2)$ , it is asymptotically stable for the system (equation (1)), if there exist positive definite symmetric matrices $P_i(i=1,2,3,4,5)$ , $Q_1$ , $Q_2$ , ${\mathit{Q}}_3$ , $U_1$ , $U_2$ , $X_j$ , $R_j(j=1,2,3,4)$ , for scalar $\gamma >0$ and free matrices with appropriate dimensions ${\mathit{T}}_1$ , ${\mathit{T}}_2$ , such that the following LMIs hold

$$\left[\begin{array}{ccc}\tilde{\mathit{\Phi }}\hfill & {\mathbf{\Gamma }}_1^{\mathrm{T}}D\hfill & \gamma {\mathbf{\Gamma }}_2^{\mathrm{T}}\hfill \\ \multicolumn{1}{c}{*}& -\gamma \mathit{I}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -\gamma \mathit{I}\hfill \end{array}\right]<0$$

(27)

where ${\mathbf{\Gamma }}_1=\left[\begin{array}{ccccccccc}{\mathit{T}}_1^T\hfill & 0\hfill & 0\hfill & 0\hfill & {\mathit{T}}_2^T\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]$ , ${\mathbf{\Gamma }}_2=\left[\begin{array}{ccccccccc}{\mathit{E}}_a\hfill & 0\hfill & {\mathit{E}}_b\hfill & 0\hfill & 0\hfill & {\mathit{E}}_c\hfill & 0\hfill & 0\hfill & 0\hfill \end{array}\right]$ .

Proof

Replacing $\mathit{A}$ , $\mathit{B}$ , and $\mathit{C}$ in the nominal system (equation (4)) with $\mathit{A}+\mathrm{\Delta }\mathit{A}$ , $\mathit{B}+\mathrm{\Delta }\mathit{B}$ , and $\mathit{C}+\mathrm{\Delta }\mathit{C}$ , respectively, in the uncertain system (equation (1)) and following the way in Theorem 1, the LMIs (equation (27)) stated in Theorem 3 are obtained. The proof is completed.

Numerical examples

In this section, two typical numerical examples are used to demonstrate superiority and less conservatism of this method. It is easy to obtain the feasible solution using the LMI toolbox of MATLAB. MADB is the most common measurement to evaluate the stability of time-delay system.

Example 1

Consider the uncertain neutral system with mixed time-varying delays as follows

$$\stackrel{.}{x}(t)-\left[\begin{array}{cc}c& 0\\ 0& c\end{array}\right]\stackrel{.}{x}(t-\tau (t))=\left[\begin{array}{cc}-2+{\delta }_1& 0\\ 0& -1+{\delta }_2\end{array}\right]x(t)+\left[\begin{array}{cc}-1+{\delta }_3& 0\\ -1& -1+{\delta }_4\end{array}\right]x(t-h(t))$$

where $0\le |c|<1$ ; ${\delta }_1$ , ${\delta }_2$ , ${\delta }_3$ , and ${\delta }_4$ are unknown parameters satisfying $|{\delta }_1|\le 1.6$ , $|{\delta }_2|\le 0.05$ , $|{\delta }_3|\le 0.1$ , and $|{\delta }_4|\le 0.3$ .

The stability of the system is discussed in two cases:

• 1. When $\mu =0.1$ , $\eta =0$ , for a given c, the MADB value can be obtained by Theorem 3, as shown in Table 1.

Table 1.

MADB for a given $c(\mu =0.1,\eta =0)$ .

Method	c
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7
Karimi et al. 10	0.97	0.78	0.60	0.45	0.31	0.19	0.10	0.02
Rakkiyappan et al. 13	1.1072	0.9208	0.7516	0.5991	0.4625	0.3402	0.2310	0.1277
Lakshmanan et al. 14	1.3209	1.0777	0.8701	0.6827	0.5123	0.3596	0.2419	0.1450
Yu and Lien 15	1.3519	1.1127	0.8972	0.7055	0.5366	0.3882	0.2568	0.1355
Yue et al. 18	1.3913	1.1439	0.9158	0.7097	0.5366	0.3734	0.2445	0.1346
Theorem 3	1.4326	1.1781	0.9386	0.7232	0.5398	0.3665	0.2329	0.1242

MADB: maximum allowable delay bound.

Bold value lies in the superiority of the proposed approach in reducing the conservation of conclusion.

As can be seen from Table 1, with the increase in c, MADB will gradually reduce. Compared with the existing literatures,^9,12–14,17 the proposed method (Theorem 3) is less conservative, especially when c is small.

• 2. When $c=0.1$ , $\eta =0.1$ , for a given $\mu$ , the MADB value can be obtained by Theorem 3, as shown in Table 2.

Table 2.

MADB for a given $\mu (c=0.1,\eta =0.1)$ .

Method	$\mu$
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	$\ge 1$
Karimi et al. 10	0.81	0.77	0.73	0.68	0.62	0.57	0.50	0.42	−
Rakkiyappan et al. 13	0.9404	0.9109	0.8825	0.8537	0.8253	0.7980	0.7711	0.7472	0.7216
Lakshmanan et al. 14	1.1454	1.0657	0.9974	0.9426	0.9007	0.8732	0.8592	0.8558	0.9376
Yu and Lien 15	1.1485	1.1004	1.0576	1.0266	1.0032	0.9858	0.9754	0.9709	0.9708
Yue et al. 18	1.1859	1.1310	1.0779	1.0275	0.9866	0.9766	0.9748	0.9748	0.9748
Theorem 3	1.2062	1.1463	1.0902	1.0314	0.9725	0.9702	0.9636	0.9636	0.9636

MADB: maximum allowable delay bound.

Bold value lies in the superiority of the proposed approach in reducing the conservation of conclusion.

As can be seen from Table 2, the approach of this article extends the MADB value. Compared with the existing literatures,^9,12–14,17 the results are less conservative.

Example 2

Considering another uncertain neutral system, the parameter matrices are as follows

$$\begin{array}{c}A=\left[\begin{array}{cc}-2\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -1\hfill \end{array}\right],B=\left[\begin{array}{cc}-1\hfill & 0\hfill \\ \multicolumn{1}{c}{-1}& -1\hfill \end{array}\right],\hfill \\ C=\left[\begin{array}{cc}c\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& c\hfill \end{array}\right],D=\left[\begin{array}{cc}0.01\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0.04\hfill \end{array}\right]\hfill \\ E_a=\left[\begin{array}{cc}160\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1.25\hfill \end{array}\right],E_b=\left[\begin{array}{cc}10\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 7.5\hfill \end{array}\right],\hfill \\ E_c=\left[\begin{array}{cc}0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]\hfill \end{array}$$

1. When $\mu =0.1$ , $\eta =0$ , for a given c, the MADB value can be shown in Table 3.

2. When $c=0.1$ , $\eta =0.1$ , for a given $\mu$ , the MADB value can be shown in Table 4.

Table 3.

MADB for a given $c(\mu =0.1,\eta =0)$ .

Method	c
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7
Rakkiyappan et al. 13	1.107	0.920	0.751	0.599	0.462	0.340	0.231	0.127
Zhang and Han 16	1.166	0.962	0.778	0.616	0.472	0.346	0.235	0.130
Fang et al. 12	1.177	0.971	0.786	0.622	0.478	0.349	0.235	0.130
Yue et al. 18	1.3933	1.1353	0.9024	0.6977	0.5215	0.3718	0.2453	0.1353
Theorem 3	1.4887	1.2071	0.9529	0.7306	0.5413	0.3824	0.2501	0.1382

MADB: maximum allowable delay bound.

Bold value lies in the superiority of the proposed approach in reducing the conservation of conclusion.

Table 4.

MADB for a given $\mu (c=0.1,\eta =0.1)$ .

Method	$\mu$
	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	$\ge 1$
Rakkiyappan et al. 13	0.9404	0.9109	0.8825	0.8537	0.8253	0.7980	0.7711	0.7472	0.7216
Fang et al. 12	0.9770	0.9608	0.9516	0.9503	0.9502	0.9501	0.9499	0.9499	0.9499
Yue et al. 18	1.1642	1.1220	1.0819	1.0444	1.0179	1.0148	1.0142	1.0142	1.0142
Theorem 3	1.2761	1.2276	1.1816	1.1398	1.1142	1.1132	1.1129	1.1129	1.1129

MADB: maximum allowable delay bound.

Bold value lies in the superiority of the proposed approach in reducing the conservation of conclusion.

As can be seen from Tables 3 and 4, the approach of this article extends the MADB value. Compared with the existing literatures,^11,12,15,17 the results are less conservative.

Conclusion

In this article, we discuss the issue of delay-dependent robust stability for a sort of uncertain neutral system with mixed time-varying delays. Based on the new LKF comprising quadruple-integral augmented term, a new delay-dependent stability criterion in terms of LMIs is proposed. Both the discrete delay and the neutral delay are related to the criterion. In order to improve the computational efficiency and simplify the conclusion, the criterion is gained by tighter integral inequality bounding method and reciprocally convex combination approach instead of model transformation method and free weighting matrix method. Therefore, it is concluded that the criterion can obtain a less conservative conclusion due to take full advantage of the delay lower bound information. Finally, the numerical simulation results demonstrate that the proposed method is more superior and competitive than the existing methods in the literature.