Stability Analysis for Time-Delay Systems via a New Negativity Condition on Quadratic Functions

Abstract

This article studies the stability problem of linear systems with time-varying delays. First, a new negative condition is established for a class of quadratic functions whose variable is within a closed set. Then, based on this new condition, a couple of stability criteria for the system under study are derived by constructing an appropriate Lyapunov–Krasovskii functional. Finally, it is demonstrated through two numerical examples that the proposed stability criteria are efficient and outperform some existing methods.

1. Introduction

It is well known that time-delays may not be avoided in practical systems such as production processes, mechanical transmission, and networked control systems [1,2,3,4,5,6,7]. Therefore, a great number of results on delay-dependent stability analysis of time-delay systems have been obtained in the past decades [8,9,10,11,12,13,14].

The Lyapunov–Krasovskii functional (LKF) method is widely used to analyze the stability of time-delay systems. The conservativeness of the LKF method can be reduced in two ways: constructing an appropriate LKF, and obtaining a strict lower bound of LKF derivative. To derive less conservative stability conditions, several LKFs were proposed such as augmented LKF [15], multiple-integral based LKF [16], matrix-refined-function-based LKF [17], and delay-product-type LKF [18]. On the other hand, the free-weighting-matrix approach [19], the model transformation method [20], and several integral inequality methods are proposed to deal with the integral term in an LKF derivative. With further research, a few novel integral inequalities are found such as Jensen’s inequality [21], Wirtinger-based inequality [22], free-matrix-based integral inequality [23], Bessel–Legendre inequality [24], auxiliary function-based integral inequalities [25], and double free-matrix-based integral inequality [26]. Using Bessel–Legendre inequality, reciprocally convex terms are obtained. These reciprocally convex terms are usually treated with reciprocally convex combination lemma [27] and reciprocally convex inequalities [28,29]. Therefore, affine Bessel–Legendre integral inequality (ABLII) [30] and generalized free-matrix-based integral inequality (GFMBII) [31] were proposed to overcome the reciprocally convex terms, and the less conservative stability criterion was obtained. With further development, high-order polynomials with respect to the time-varying delay appear in the LFK derivative. It is difficult to obtain negativity conditions for such polynomials. For quadratic polynomial functions, it can be expressed as: $\ell (d(t))=a_2{d}^2(t)+a_{1}d(t)+a_0,$ where $a_i\in ℝ(i=0,1,2)$ are real symmetric matrices and independent of $d(t)$, $d(t)\in [0,\tilde{h}]$ is time delay, and $\tilde{h}$ is a constant. If $\ell (d(t))<0$ for $d(t)\in [0,\tilde{h}]$, then the stability of the time-delay system is guaranteed. Therefore, it is significant to derive the negative condition of the quadratic polynomial function to obtain a less conservative criterion. Recently, some negativity conditions were reported in [32,33]. Some improved negative conditions of quadratic polynomial functions were proposed in [34,35] by a new quadratic-partitioning method. In addition, there is still room for improvement for the quadratic-partitioning method. This inspired the research of this paper.

The current work focuses on the stability analysis of linear systems with time-varying delay. The main contribution can be summarized as follows. (1) Using the fine division of intervals, a new negative condition of the quadratic function with multiple variable parameters is obtained. Based on this condition, less conservative results of the time-varying time-delay system is derived. (2) An LKF of the Lyapunov matrix parameterized by the delay is constructed, which is helpful to obtain less conservative stability criteria.

The rest of this article is organized as follows. The system statement and two lemmas are given in Section 2. The main results are presented in Section 3. Section 4 provides two numerical examples to demonstrate the effectiveness of the proposed approach. Finally, the research is summarized in Section 5.

Throughout this paper, ${ℝ}^n$ denotes the n-dimensional Euclidean space, ${ℝ}^{n\times n}$ represents the set of $n\times n$ real matrices, ${\mathbb{S}}^n$ denotes $n\times n$ real symmetric matrices, the superscripts $T$ and $-1$ stand for the transpose and the inverse of a matrix, $R>0$ means that $R$ is a real symmetric and positive-definite matrix, and $\mathrm{Sym}\left\{X\right\}=X+X^T$; $\mathrm{col}\{\cdots \}$ and $\mathrm{diag}\{\cdots \}$ stand for a block-column vector and a block-diagonal, respectively; $I$ and $0$ denote the identity matrix and the zero-matrix with appropriated dimensions. The other notations used in this paper are standard.

2. System Statement and Preliminaries

Consider the following system with time-varying delay $d(t)$

$$\{\begin{array}{l}\dot{x}(t)=Ax(t)+A_{1}x(t-d(t))\hfill \\ x(t)=\tilde{\phi }(t),t\in [-\tilde{h},0]\hfill \end{array}$$

(1)

where $A,A_1\in {ℝ}^{n\times n}$ are system matrices; $x(t)\in {ℝ}^n$ is the state vector; $d(t)$ is a continuous function of time that satisfies (2), and $\tilde{\phi }(t)$ is the initial condition;

$$0\le d(t)\le \tilde{h},{\mu }_1\le \dot{d}(t)\le {\mu }_2$$

(2)

Before presenting a new stability criterion of the system (1), the following definitions and lemmas are necessary.

Lemma 1

([32]). Consider a quadratic function: $\ell (x)=a_2{x}^2+a_{1}x+a_0$, where $x\in [0,\tilde{h}]$, $a_i\in ℝ(i=0,1,2)$, if the following inequalities hold:

(i) 

$\ell (0)<0$

(ii) 

$\ell (\tilde{h})<0$

(iii) 

$-{\tilde{h}}^2{a}_2+\ell (0)<0$

Then$\ell (x)<0$.

A new negative condition with variable parameters for the quadratic polynomial function is presented as follows.

Lemma 2.

For a quadratic function:$\ell (x)=a_2{x}^2+a_{1}x+a_0$, where$x\in [0,\tilde{h}]$$a_i\in ℝ(i=0,1,2)$, if the following inequalities hold:

(i) 

$\ell (0)<0$

(ii) 

$\ell (\tilde{h})<0$

(iii) 

$-a_2{\left[\frac{(m_j-j+1)\tilde{h}}{2^N}\right]}^2+\ell (\frac{(j-1)\tilde{h}}{2^N})<0,m_j\in [j-1,j],j=1,2\cdots ,2^N$

(iv) 

$-a_2{\left[\frac{(j-m_j)\tilde{h}}{2^N}\right]}^2+\ell (\frac{j\tilde{h}}{2^N})<0,m_j\in [j-1,j],j=1,2\cdots ,2^N$

Then$\ell (x)<0$.

Proof.

In case of $a_2>0$, if satisfying $\ell (0)<0$, $\ell (\tilde{h})<0$, then $\ell (x)<0$ is guaranteed for all $x\in [0,\tilde{h}]$. In case of $a_2<0$, a tangent function $g(x)=\dot{\ell }(m)(x-m)+$$\ell (m)$ is given, where $m\in [0,\tilde{h}]$. If the coordinates of the point where the tangent function $g(x)$ intersects both ends of the interval are negatively determined, then $\ell (x)<0$. Second, the interval $[0,\tilde{h}]$ is evenly divided into $2^N$ subintervals $[{\tilde{h}}_{j-1},{\tilde{h}}_j]$, where ${\tilde{h}}_j({\tilde{h}}_0=0)=$$\frac{j}{2^N}\tilde{h},j=1,2\cdots ,2^N$. In these subintervals, let $m=m_j\frac{\tilde{h}}{2^N}$ with $m_j\in [j-1,j]$. Finally, if $g(0)<0$, $g(\frac{m_j\tilde{h}}{2^N})<0$ holds, then $g(x)<0$ is guaranteed for $x\in [0,\tilde{h}]$. Thus, condition $(iii)$ and $(iv)$ lead to $\ell (x)<0$. □

Remark 1.

It is worth pointing out that Lemma 2 in [32] and Lemma 1 in [33] are special cases of Lemma 2 with$N=0(m_1=1.0)$and$N=1(m_1=0.0,m_2=2.0)$, respectively. Therefore, Lemma 2 provides a more general negative condition for the quadratic function.

3. Main Results

To simplify the matrix terminology, define

${\zeta }_0(t)=col[x(t),x(t-d(t)),x(t-\tilde{h})]$

${\varpi }_1(t)=\frac{1}{d(t)}{\displaystyle {\int }_{t-d(t)}^{t}x}(s)ds$

${\varpi }_2(t)=\frac{1}{\tilde{h}-d(t)}{\displaystyle {\int }_{t-\tilde{h}}^{t-d(t)}x}(s)ds$

${\varpi }_3(t)=\frac{1}{d^2(t)}{\displaystyle {\int }_{t-d(t)}^t{\displaystyle {\int }_{t-d(t)}^{\theta }x}}(s)dsd\theta$

${\varpi }_4(t)=\frac{1}{{(\tilde{h}-d(t))}^2}{\displaystyle {\int }_{t-\tilde{h}}^{t-d(t)}{\displaystyle {\int }_{t-\tilde{h}}^{\theta }x}}(s)dsd\theta$

${\zeta }_1(t)=col[{\zeta }_0(t),{\varpi }_1(t),{\varpi }_3(t)]$

${\zeta }_2(t)=col[{\zeta }_0(t),{\varpi }_2(t),{\varpi }_4(t)]$

${\zeta }_3(t,s)=col[{\zeta }_0(t),\dot{x}(s),x(s),{\displaystyle {\int }_s^{t}x}(\theta )d\theta ,{\displaystyle {\int }_{t-d(t)}^{s}x}(\theta )d\theta ]$

${\zeta }_4(t,s)=col[{\zeta }_0(t),\dot{x}(s),x(s),{\displaystyle {\int }_s^{t-d(t)}x}(\theta )d\theta ,{\displaystyle {\int }_{t-\tilde{h}}^{s}x}(\theta )d\theta ]$

$\xi (t)=[{\zeta }_0(t),{\xi }_0(t),{\xi }_1(t)],{\xi }_1(t)=col[{\varpi }_3(t),{\varpi }_4(t)]$

${\xi }_0(t)=col[\dot{x}(t-d(t)),\dot{x}(t-\tilde{h}),{\varpi }_1(t),{\varpi }_2(t)]$

$e_i=[0_{n\times (i-1)n}{I}_n{0}_{n\times (9-i)n}],i=1,2,\cdots ,9$

A new stability criterion can be obtained as follows.

Theorem 1.

For given scalars$\tilde{h}>0$and${\mu }_1<{\mu }_2<1$, system (1) is asymptotically stable if there exist matrices${\widehat{Q}}_1(\in {ℝ}^{7n\times 7n})>0$,${\widehat{Q}}_2(\in {ℝ}^{7n\times 7n})>0$,$R_1(\in {ℝ}^{n\times n})>0$,$R_2(\in {ℝ}^{n\times n})>0$,$P_{10}$,$P_{11}$,$P_{20}$, $P_{21}$and any matrices$Y_1$and$Y_2\in {ℝ}^{9n\times 3n}$such that the following inequalities (3)–(5) are satisfied.

$$\Omega (d(t),\dot{d}(t))={\eta }_2(\dot{d}(t))d{(t)}^2+{\eta }_1(\dot{d}(t))d(t)+{\eta }_0(\dot{d}(t))<0$$

(3)

$$P_{10}+d(t)P_{11}>0$$

(4)

$$P_{20}+(\tilde{h}-d(t))P_{21}>0$$

(5)

where

$${\eta }_0(\dot{d}(t))={\sigma }_0(\dot{d}(t))+\tilde{h}{Y}_2{\overline{R}}_1^{-1}{Y}_2^T,{\eta }_1(\dot{d}(t))={\sigma }_1(\dot{d}(t))+Y_1{\overline{R}}_2^{-1}{Y}_1^T-Y_2{\overline{R}}_1^{-1}{Y}_2^T$$

$$\begin{array}{cc}\hfill {\eta }_2(\dot{d}(t))=& \mathrm{Sym}\{C_{11}^T{P}_{11}{\lambda }_1+C_{21}^T{P}_{21}{\lambda }_1+C_{53}^T{\widehat{Q}}_1{\lambda }_4+C_{83}^T{\widehat{Q}}_2{\lambda }_5\}\hfill \\ & -(1-\dot{d}(t))C_{42}^T{\widehat{Q}}_1{C}_{42}+C_{32}^T{\widehat{Q}}_1{C}_{32}-C_{72}^T{\widehat{Q}}_2{C}_{72}+(1-\dot{d}(t))C_{62}^T{\widehat{Q}}_2{C}_{62}\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\sigma }_1(\dot{d}(t))=& \mathrm{Sym}\{C_{11}^T{P}_{10}{\lambda }_1+C_{11}^T{P}_{11}{\lambda }_2-C_{21}^T(P_{20}+2\tilde{h}{P}_{21}){\lambda }_1-C_{21}^T{P}_{21}{\lambda }_3+C_{31}^T{\widehat{Q}}_1{C}_{32}\hfill \\ & -(1-\dot{d}(t))C_{41}^T{\widehat{Q}}_1{C}_{42}+C_{52}^T{\widehat{Q}}_1{\lambda }_4-C_{82}^T{\widehat{Q}}_2{\lambda }_5+(1-\dot{d}(t))C_{61}^T{\widehat{Q}}_2{C}_{62}\hfill \\ & -C_{71}^T{\widehat{Q}}_2{C}_{72}\}+2\dot{d}(t)C_{11}^T{P}_{11}{C}_{11}+2\dot{d}(t)C_{21}^T{P}_{21}{C}_{21}-(1-\dot{d}(t))e_4^T(R_1-R_2)e_4\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\sigma }_0(\dot{d}(t))=& \mathrm{Sym}\{C_{11}^T{P}_{10}{\lambda }_2+C_{21}^T(\tilde{h}{P}_{20}+{\tilde{h}}^2{P}_{21}){\lambda }_1+C_{51}^T{\widehat{Q}}_1{\lambda }_4+C_{21}^T(P_{20}+\tilde{h}{P}_{21}){\lambda }_3\hfill \\ & +C_{81}^T{\widehat{Q}}_2{\lambda }_5+Y_1{\Delta }_1+Y_2{\Delta }_2\}+\dot{d}(t)C_{11}^T{P}_{10}{C}_{11}-\dot{d}(t)C_{21}^T(P_{20}+2\tilde{h}{P}_{21})C_{21}\hfill \\ & +C_{31}^T{\widehat{Q}}_1{C}_{31}-(1-\dot{d}(t))C_{41}^T{\widehat{Q}}_1{C}_{41}+(1-\dot{d}(t))C_{61}^T{\widehat{Q}}_2{C}_{61}-C_{71}^T{\widehat{Q}}_2{C}_{71}\hfill \\ & +\tilde{h}(1-\dot{d}(t))e_4^T(R_1-R_2)e_4+\tilde{h}{e}_0^T{R}_2{e}_0\hfill \end{array}$$

with

$C_{11}=col[e_1,e_2,e_3,e_6,e_6-e_8],C_{21}=col[e_1,e_2,e_3,e_7,e_7-e_9]$

$C_{31}=col[e_0,e_1,e_1,e_2,e_3,0,0],C_{32}=col[0,0,0,0,0,0,e_6]$

$C_{41}=col[e_4,e_2,e_1,e_2,e_3,0,0],C_{42}=col[0,0,0,0,0,e_6,0]$

$C_{51}=col[e_1-e_2,0,0,0,0,0,0],C_{52}=col[0,e_6,e_1,e_2,e_3,0,0]$

$C_{53}=col[0,0,0,0,0,e_8,e_6-e_8],C_{61}=col[e_4,e_2,e_1,e_2,e_3,0,\tilde{h}{e}_7]$

$C_{62}=col[0,0,0,0,0,0,-e_7]$

$C_{71}=col[e_5,e_3,e_1,e_2,e_3,\tilde{h}{e}_7,0],C_{72}=col[0,0,0,0,0,-e_7,0]$

$C_{81}=col[e_2-e_3,\tilde{h}{e}_7,\tilde{h}{e}_1,\tilde{h}{e}_2,\tilde{h}{e}_3,{\tilde{h}}^2{e}_9,{\tilde{h}}^2(e_7-e_9)]$

$C_{82}=[0,e_7,e_1,e_2,e_3,2\tilde{h}{e}_9,2\tilde{h}(e_7-e_9)],C_{83}=col[0,0,0,0,0,e_9,e_7-e_9]$

${\lambda }_1=col[e_0,(1-\dot{d}(t))e_4,e_5,0,0],{\lambda }_2=col[{\tau }_1,{\tau }_2],{\lambda }_3=col[{\tau }_3,{\tau }_4]$

${\tau }_1=col[0,0,0,e_1-(1-\dot{d}(t))e_2-\dot{d}(t)e_6]$

${\tau }_2=col[e_6-(1-\dot{d}(t))e_2-2\dot{d}(t)(e_6-e_8)]$

${\tau }_3=col[0,0,0,(1-\dot{d}(t))e_2-e_3+\dot{d}(t)e_7]$

${\tau }_4=col[(1-\dot{d}(t))e_7-e_3+2\dot{d}(t)(e_7-e_9)]$

${\lambda }_4=col[0,0,e_0,(1-\dot{d}(t))e_4,e_5,e_1,-(1-\dot{d}(t))e_2]$

${\lambda }_5=col[0,0,e_0,(1-\dot{d}(t))e_4,e_5,(1-\dot{d}(t))e_2,-e_3]$

${\overline{R}}_i=\mathrm{diag}\{R_i,3R_i,5R_i\},i=1,2$

$e_0=A{e}_1+A_1{e}_2$

${\Delta }_1=\left[\begin{array}{c}{e}_1-e_2\\ e_1+e_2-2e_6\\ e_1-e_2+6e_6-12e_8\end{array}\right]$, ${\Delta }_2=\left[\begin{array}{c}{e}_2-e_3\\ e_2+e_3-2e_7\\ e_2-e_3+6e_7-12e_9\end{array}\right]$

Proof.

A suitable LKF is constructed as

$$V_{\kappa }(t)={\displaystyle \sum _{i=1}^3{V}_{\kappa i}(t)}$$

(6)

where

$$\begin{array}{c}{V}_{\kappa 1}(t)=d(t){\zeta }_1^T(t)P_1{\zeta }_1(t)+(\tilde{h}-d(t)){\zeta }_2^T(t)P_2{\zeta }_2(t)\hfill \\ V_{\kappa 2}(t)={\displaystyle {\int }_{t-d(t)}^t{\zeta }_3^T}(t,s){\widehat{Q}}_1{\zeta }_3(t,s)ds+{\displaystyle {\int }_{t-\tilde{h}}^{t-d(t)}{\zeta }_4^T}(t,s){\widehat{Q}}_2{\zeta }_4(t,s)ds\hfill \\ V_{\kappa 3}(t)={\displaystyle {\int }_{t-d(t)}^t(\tilde{h}-t+s)}{\dot{x}}^T(s)R_2\dot{x}(s)ds+{\displaystyle {\int }_{t-\tilde{h}}^{t-d(t)}(\tilde{h}-t+s)}{\dot{x}}^T(s)R_1\dot{x}(s)ds\hfill \end{array}$$

with

$$P_1=P_{10}+d(t)P_{11}$$

$$P_2=P_{20}+(\tilde{h}-d(t))P_{21}$$

The derivative of $V_{\kappa }(t)$ can be obtained as:

$$\begin{array}{cc}\hfill {\dot{V}}_{\kappa 1}(t)=& 2d(t){\zeta }_1^T(t)P_1{\dot{\zeta }}_1(t)+d(t){\zeta }_1^T(t){\dot{P}}_1{\zeta }_1(t)+\dot{d}(t){\zeta }_1^T(t)P_1{\zeta }_1(t)\hfill \\ & +2(\tilde{h}-d(t)){\zeta }_2^T(t)P_2{\dot{\zeta }}_2(t)+(\tilde{h}-d(t)){\zeta }_2^T(t){\dot{P}}_2{\zeta }_2(t)-\dot{d}(t){\zeta }_2^T(t)P_2{\zeta }_2(t)\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\dot{V}}_{\kappa 2}(t)=& {\zeta }_3^T(t,t){\widehat{Q}}_1{\zeta }_3(t,t)-{\zeta }_4^T(t,t-\tilde{h}){\widehat{Q}}_2{\zeta }_4(t,t-\tilde{h})\hfill \\ & -(1-\dot{d}(t)){\zeta }_3^T(t,t-d(t)){\widehat{Q}}_1{\zeta }_3(t,t-d(t))\hfill \\ & +(1-\dot{d}(t)){\zeta }_4^T(t,t-d(t)){\widehat{Q}}_2{\zeta }_4(t,t-d(t))\\ & +2{\displaystyle {\int }_{t-d(t)}^t{\zeta }_3^T}(t,s){\widehat{Q}}_1\frac{\partial {\zeta }_3(t,s)}{\partial t}ds+2{\displaystyle {\int }_{t-\tilde{h}}^{t-d(t)}{\zeta }_4^T}(t,s){\widehat{Q}}_2\frac{\partial {\zeta }_4(t,s)}{\partial t}ds\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\dot{V}}_{\kappa 3}(t)=& (1-\dot{d}(t))(\tilde{h}-d(t)){\dot{x}}^T(t-d(t))(R_1-R_2)\dot{x}(t-d(t))\hfill \\ & +\tilde{h}{\dot{x}}^T(t)R_2\dot{x}(t)-{\displaystyle {\int }_{t-\tilde{h}}^{t-d(t)}{\dot{x}}^T}(s)R_1\dot{x}(s)ds-{\displaystyle {\int }_{t-d(t)}^t{\dot{x}}^T}(s)R_2\dot{x}(s)ds\hfill \end{array}$$

It follows from Lemma 3 in [31] that

$$-{\displaystyle {\int }_{t-d(t)}^t{\dot{x}}^T}(s)R_2\dot{x}(s)ds-{\displaystyle {\int }_{t-\tilde{h}}^{t-d(t)}{\dot{x}}^T}(s)R_1\dot{x}(s)ds\le {\xi }^T(t)\Lambda (d(t))\xi (t)$$

(7)

where

$$\Lambda (d(t))=\mathrm{Sym}\{Y_1{\Delta }_1+Y_2{\Delta }_2\}+d(t)Y_1{\overline{R}}_2^{-1}{Y}_1^T+(\tilde{h}-d(t))Y_2{\overline{R}}_1^{-1}{Y}_2^T$$

According to (7), yields

$${\dot{V}}_{\kappa }(t)\le {\xi }^T(t)(\Omega (d(t),\dot{d}(t)))\xi (t)$$

where $\Omega (d(t),\dot{d}(t))$ is defined in Theorem 1. Noting (2), if the condition (3) is satisfied, then ${\dot{V}}_{\kappa }(t)<0$. Thus, we conclude that the system (1) is asymptotically stable based on the Lyapunov stability theory. □

Remark 2.

Lyapunov matrices$P_1=P_{10}+d(t)P_{11}$and$P_{20}+(\tilde{h}-d(t))P_{21}$parameterized by the delay are included in$V_{\kappa 1}(t)$of the LKF (6). They are different from the LKF in [29,31] and are helpful to reduce the conservativeness of the obtained condition.

Remark 3.

Note that$\Omega (d(t),\dot{d}(t))$in Theorem 1 is a quadratic polynomial function about$d(t)$, which is nonlinear on$d(t)$. Thus, Theorem 1 is not workable when checking the stability of the system (1). Fortunately, Lemmas 1 and 2 provide two ways to derive a stability criterion from Theorem 1. In the following, in order to show the merit of Lemma 2 proposed in this paper, we present two stability criteria using Lemmas 1 and 2, respectively, without detailed proofs.

The following result is derived from Theorem 1 using Lemma 2.

Theorem 2.

For given scalars$\tilde{h}>0$and${\mu }_1<{\mu }_2<1$, system (1) is asymptotically stable if there exist matrices${\widehat{Q}}_1(\in {ℝ}^{7n\times 7n})>0$,${\widehat{Q}}_2(\in {ℝ}^{7n\times 7n})>0$,$R_1(\in {ℝ}^{n\times n})>0$,$R_2(\in {ℝ}^{n\times n})>0$,$P_{10}$,$P_{11}$,$P_{20}$,$P_{21}$and any matrices$Y_1$and$Y_2\in {ℝ}^{9n\times 3n}$such that the following inequalities (8)–(11) are satisfied for$k=1,2$.

$$\left[\begin{array}{cc}{\sigma }_0({\mu }_k)& \sqrt{\tilde{h}}{Y}_2\\ *& -{\overline{R}}_1\end{array}\right]<0$$

(8)

$$\left[\begin{array}{cc}{\tilde{h}}^2{\eta }_2({\mu }_k)+\tilde{h}{\sigma }_1({\mu }_k)+{\sigma }_0({\mu }_k)& \sqrt{\tilde{h}}{Y}_1\\ *& -{\overline{R}}_2\end{array}\right]<0$$

(9)

$$\left[\begin{array}{ccc}({(\frac{(j-1)\tilde{h}}{2^N})}^2-& & \\ {(\frac{(m_j-j+1)\tilde{h}}{2^N})}^2){\eta }_2({\mu }_k)& \sqrt{\frac{(2^N-j+1)\tilde{h}}{2^N}}{Y}_2& \sqrt{\frac{(j-1)\tilde{h}}{2^N}}{Y}_1\\ +\frac{(j-1)\tilde{h}}{2^N}{\sigma }_1({\mu }_k)+{\sigma }_0({\mu }_k)& & \\ *& -{\overline{R}}_1& 0\\ *& *& -{\overline{R}}_2\end{array}\right]<0$$

(10)

$$\left[\begin{array}{ccc}({(\frac{j\tilde{h}}{2^N})}^2-{(\frac{(j-m_j)\tilde{h}}{2^N})}^2){\eta }_2({\mu }_k)& \sqrt{\frac{(2^N-j)\tilde{h}}{2^N}}{Y}_2& \sqrt{\frac{j\tilde{h}}{2^N}}{Y}_1\\ +\frac{j\tilde{h}}{2^N}{\sigma }_1({\mu }_k)+{\sigma }_0({\mu }_k)& & \\ *& -{\overline{R}}_1& 0\\ *& *& -{\overline{R}}_2\end{array}\right]<0$$

(11)

where the notations are given in Theorem 1.

The following result is derived from Theorem 1 using Lemma 1.

Theorem 3.

For given scalars$\tilde{h}>0$and${\mu }_1<{\mu }_2<1$, system (1) is asymptotically stable if there exist matrices${\widehat{Q}}_1(\in {ℝ}^{7n\times 7n})>0$,${\widehat{Q}}_2(\in {ℝ}^{7n\times 7n})>0$,$R_1(\in {ℝ}^{n\times n})>0$,$R_2(\in {ℝ}^{n\times n})>0$,$P_{10}$,$P_{11}$,$P_{20}$,$P_{21}$and any matrices$Y_1$and$Y_2\in {ℝ}^{9n\times 3n}$such that the following inequalities (12)–(14) are satisfied for$k=1,2$.

$$\left[\begin{array}{cc}{\sigma }_0({\mu }_k)& \sqrt{\tilde{h}}{Y}_2\\ *& -{\overline{R}}_1\end{array}\right]<0$$

(12)

$$\left[\begin{array}{cc}{\tilde{h}}^2{\eta }_2({\mu }_k)+\tilde{h}{\sigma }_1({\mu }_k)+{\sigma }_0({\mu }_k)& \sqrt{\tilde{h}}{Y}_1\\ *& -{\overline{R}}_2\end{array}\right]<0$$

(13)

$$\left[\begin{array}{cc}-{\tilde{h}}^2{\eta }_2({\mu }_k)+{\sigma }_0({\mu }_k)& \sqrt{\tilde{h}}{Y}_2\\ *& -{\overline{R}}_1\end{array}\right]<0$$

(14)

where the notations are given in Theorem 1.

4. Numerical Examples

In this section, in order to validate the proposed method, two numerical examples are used in comparison with the other existing methods.

Example 1.

Consider System (1) with

$$\begin{array}{c}A=\left[\begin{array}{cc}-2& 0\\ 0& -0.9\end{array}\right],A_1=\left[\begin{array}{cc}-1& 0\\ -1& -1\end{array}\right]\end{array}$$

As shown in

Table 1. Maximum allowable upper bounds of $\tilde{h}$ for different $\mu$.

Methods	$\mathit{\mu }=0.1$	$\mathit{\mu }=0.5$	$\mathit{\mu }=0.8$
[32]	4.753	2.429	2.183
[29]	4.910	3.233	2.789
[31]	4.921	3.221	2.792
[34]	4.939	3.298	2.869
[36]	4.942	3.309	2.882
[37]	4.966	3.395	2.983
Theorem 2 (N = 1)	4.978	3.414	2.969
Theorem 2 (N = 2)	4.997	3.424	3.029
Theorem 2 (N = 3)	5.015	3.452	3.030
Theorem 3	4.946	3.337	2.918

. For given$m_j\in [j-1,j]$, where$j=1,2\cdots ,2^N$, integer$N\ge 0$. The maximum value of$\tilde{h}$calculated using Theorem 2, Theorem 3, and the other existing methods for different$\mu ={\mu }_1={\mu }_2$, where$N=1(m_1=1.00,m_2=1.50)$,$N=2(m_1=1.00,m_2=1.7,$$m_3=2.55,m_4=3.45)$,$N=3(m_1=1.00,m_2=1.8,m_3=2.55,m_4=3.40,m_5=4.55,m_6=5.45,$$m_7=6.55,m_8=7.45)$it is found that the method in this paper provides less conservative results than [29,31,32,34,36,37].

Example 2.

Consider System (1) with

$$\begin{array}{c}A=\left[\begin{array}{cc}0& 1\\ -1& -2\end{array}\right],A_1=\left[\begin{array}{cc}0& 0\\ -1& 1\end{array}\right]\end{array}$$

In Example 2, for given$m_j\in [j-1,j]$, where$j=1,2\cdots ,2^N$, integer$N\ge 0$. The maximum value of$\tilde{h}$calculated by using Theorems 2 and 3, where$N=1(m_1=1.00,m_2=1.70)$,$N=2(m_1=1.00,m_2=1.8,m_3=2.55,m_4=3.45)$, and the methods in [29,31,34,36,37] are listed in

Table 2. Maximum allowable upper bounds of $\tilde{h}$ for different $\mu$.

Methods	$\mathit{\mu }=0.1$	$\mathit{\mu }=0.2$	$\mathit{\mu }=0.5$	$\mathit{\mu }=0.8$
[29]	7.230	4.556	2.496	1.922
[31]	7.308	4.670	2.509	1.940
[34]	7.401	4.765	2.709	2.091
[36]	7.400	4.795	2.717	2.089
[37]	7.572	4.947	2.801	2.137
Theorem 2 (N = 1)	7.586	4.987	2.800	2.137
Theorem 2 (N = 2)	7.656	4.992	2.868	2.172
Theorem 3	7.428	4.825	2.749	2.114

. It is demonstrated in this numerical example that the proposed approach is efficient.

5. Conclusions

This paper examined the stability problem of time-varying delay systems. A delay-type LFK was constructed and a novel negativity condition was presented for a class of quadratic polynomial functions. Based on them, a couple of stability criteria have been derived with less conservatism. The effectiveness and superiority of the proposed approach have been demonstrated by two numerical examples.