A New Slack Lyapunov Functional for Dynamical System with Time Delay

Abstract

The traditional method of constructing a Lyapunov functional for dynamical systems with time delay is usually dependent on positive definite matrices in the quadratic form. In this paper, a new Lyapunov functional is proposed and the corresponding proof is given. It do not require that all matrices in the quadratic form of Lyapunov functionals are positive definite, while the quadratic form is still positive definite, which makes the estimate more relaxed due to special construction of matrices. It is a general form and can be used in the performance analysis of a variety of dynamical systems. Moreover, a lemma concerning the quadratic function is applied to deal with the quadratic term of time-varying delay. Lastly, in the case of classical dynamical systems with time delay, the verification results are given to illustrate the usefulness of the new slack Lyapunov functional.

1. Introduction

Consider a system with a time-varying delay:

$$\left\{\begin{array}{c}\dot{x}\left(t\right)=Ax\left(t\right)+A_{d}x(t-\phi \left(t\right)),\hfill \\ x\left(s\right)=\psi \left(s\right),s\in [-\phi ,0],\hfill \end{array}\right.$$

(1)

where $x\left(t\right)={[x_1\left(t\right),x_2\left(t\right),\dots ,x_n\left(t\right)]}^T\in {\mathbb{R}}^n$ stands for the state vector, the equation $x\left(s\right)=\psi \left(s\right),s\in [-\phi ,0]$ denotes the initial condition, A∈${\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$ and $A_d$∈${\mathbb{R}}^{\mathrm{n}\times \mathrm{n}}$ are the given known matrices, the continuous function $\phi \left(t\right)$ denotes the time-varying delayed signal and satisfies

$$0\le \phi \left(t\right)\le \phi ,{\nu }_1\le \dot{\phi }\left(t\right)\le {\nu }_2.$$

In the recent few years, the performance analysis of time delay dynamical systems has become a hot research topic [1,2,3,4,5,6,7]. Integral inequalities (Jensen inequality [8], Wirtinger inequality [9] and their morphing forms [10,11]), free-weight-matrix based inequalities and their morphing forms [12,13], and relevant upgraded inequalities [14,15] are implemented to reduce the conservatism of performance conditions for dynamical systems with time-varying delay. In order to reduce the conservatism of system conditions accurately, it is necessary to redesign the LKF. The contributions in this paper are presented as follows:

First, the new construction of LKF does not strictly limit the positive nature of the matrix in LKF, which can fundamentally reduce the conservatism of the stability conditions. It is not a simple construction of LKF, but from the LKF itself, to reduce the limitations of the positive definite matrices in the quadratic form while it still guarantees the positive properties of the quadratic form. Second, it is crucial that the multiplicity of the heavy integral in LKF is n dimensional, so that the LKF becomes a generalized form that could be widely used. Furthermore, in the process of obtaining the performance conditions, the delay-related terms ${\phi }^2\left(t\right)$, $\phi \left(t\right)$, $\dot{\phi }\left(t\right)$ need to be handled well, especially the quadratic term ${\phi }^2\left(t\right)$. By Lemma 1, they are handled to obtain better performance conditions.

Throughout the text, the notations that are important to describe the mathematical properties are presented in Table 1.

Table 1. The notations and their explanations throughout the text.

Notation	Description and Explanation of the Notation
${\mathbb{R}}^n$	n-dimensional Euclidean space
${\mathbb{R}}^{n\times n}$	the set of all $n\times n$ real matrices
${\mathbb{S}}^n$	the set of all $n\times n$ real symmetric matrices
$U>V$	$U-V$ is positive definite
$U\ge V$	$U-V$ is positive semi-definite
$Sym\left\{U\right\}$	$U+U^T$
0	zero or zero matrix
$(\genfrac{}{}{0pt}{}{k}{l})$	the value $\frac{k!}{(k-l)!l!}$
$l!$	$1\times 2\times 3\times \cdots \times (l-1)\times l$
*	the lower-left block of a symmetric matrix

2. Preliminaries

Lemma 1

([16,17]). $f\left(\phi \right)=c_2{\phi }^2+c_1\phi +c_0$ is the quadratic function, where $c_2,c_1,c_0\in \mathbb{R}$, if the following inequalities hold:

$$\left(1\right)f\left(0\right)<0,\left(2\right)f\left(\overline{\phi }\right)<0,\left(3\right)f\left(0\right)-{\overline{\phi }}^2{c}_2<0,$$

then $f\left(\phi \right)<0,\forall \phi \in \left[0,\overline{\phi }\right]$.

For Lemma 1, it can be deduced that as constant terms $c_0$, $c_1$, and $c_2$ are extended to n-dimensional matrices $C_0$, $C_1$, and $C_2$, and the corresponding result can still be obtained.

3. Main Results

Lemma 2.

For any constant $\alpha ,\beta$ satisfying $\alpha <\beta$, $x\left(t\right)$ is a differentiable function in $[\alpha ,\beta ]\to {\mathbb{R}}^n$, if matrices $P_{11},P_{22},P_{m-1,m-1}\in {\mathbb{S}}^{mn},P_{m,m}\in {\mathbb{S}}^n,P_{1,m},P_{2,m},P_{m-1,m}\in {\mathbb{R}}^{mn\times n}$, other matrices $\in {\mathbb{R}}^{mn\times mn}$ satisfy $\mathcal{P}>0$, where

$$\mathcal{P}=\left[\begin{array}{cccc}{P}_{11}& P_{12}& \cdots & P_{1,m}\\ *& P_{22}& \cdots & P_{2,m}\\ ⋮& ⋮& \ddots & ⋮\\ *& *& *& P_{m,m}\end{array}\right],$$

then

$$V_g\left(t\right)={\eta }^T\left(t\right)\mathcal{P}\eta \left(t\right)+{\int }_{\alpha }^{\beta }\stackrel{.}{x}{}^T\left(s\right)P_{m,m}\stackrel{.}{x}\left(s\right)ds\ge 0,$$

where

$$\begin{array}{cc}\hfill & \mathcal{P}=({\mathcal{P}}_1+(\beta -\alpha ){\mathcal{P}}_2+{(\beta -\alpha )}^2{\mathcal{P}}_3+\cdots +{(\beta -\alpha )}^{m-2}{\mathcal{P}}_{m-1}),\hfill \end{array}$$

$${\mathcal{P}}_1=[-P_{1,m},P_{1,m},P_{2,m}{J}_{1,1},P_{3,m}{J}_{2,2},\dots ,P_{m-1,m}{J}_{m-2,m-2},\underset{lelements}{\underbrace{0,\dots ,0,}}],$$

$${\mathcal{P}}_{l+1}={\mathrm{\Lambda }}_1^{l+1}+{\mathrm{\Lambda }}_2^{l+1}+{\mathrm{\Lambda }}_3^{l+1},(l=1,2,\cdots ,m-3),$$

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_1^{l+1}=Sym\left\{\left[0,0,P_{l+2,m}{J}_{l+1,1},P_{l+3,m}{J}_{l+2,2},\dots ,P_{m-1,m}{J}_{m-2,m-2-l},\underset{lelements}{\underbrace{0,\dots ,0,}}\right]\right\},\end{array}$$

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_2^{l+1}=Sym\left\{\left[\left(-\sum _{i=1}^l\frac{J_{l,i}}{i!}{P}_{l+1,m}+{(-1)}^{l+1}{P}_{l+1,m}\right),-{(-1)}^{l+1}{P}_{l+1,m},\underset{m-2elements}{\underbrace{0,\dots ,0,}}\right]\right\},\end{array}$$

$${\mathrm{\Lambda }}_3^{l+1}=\left\{\begin{array}{cc}\frac{1}{l}{P}_{\left(l+1\right)/2,\left(l+1\right)/2}\hfill & lisoddand1\le l\le m-3,\hfill \\ \\ -{\displaystyle \sum _{i=m-2,iisodd}^{2(m-2)+1}}\frac{{(\beta -\alpha )}^{i-(m-2)}}{i}{P}_{\left(i+1\right)/2,\left(i+1\right)/2}\hfill & lisoddandl=m-2,\hfill \\ \\ 0_{mn,mn}\hfill & liseven.\hfill \end{array}\right.,$$

$$\begin{array}{c}\hfill J_{k,l}={(-1)}^{k+l}\left(\genfrac{}{}{0pt}{}{k}{l}\right)\left(\genfrac{}{}{0pt}{}{k+l}{l}\right)l!.\end{array}$$

Proof of Lemma 2.

Let $f_0=-1,f_1(s,\alpha ,\beta )=\alpha +\beta -2s,f_2(s,\alpha ,\beta )=-{(\beta -\alpha )}^2+6(s-\alpha )(\beta -\alpha )-6{(s-\alpha )}^2$ $,\cdots ,f_k(s,\alpha ,\beta )={\displaystyle \sum _{l=0}^k}{(-1)}^{k+l+1}(\genfrac{}{}{0pt}{}{k}{l})(\genfrac{}{}{0pt}{}{k+l}{l})\frac{{(s-\alpha )}^l}{{(\beta -\alpha )}^{l-k}},$ then we get

$${\int }_{\alpha }^{\beta }{f}_0\left(s\right)\stackrel{.}{x}\left(s\right)ds=x\left(\alpha \right)-x\left(\beta \right),$$

$${\int }_{\alpha }^{\beta }{f}_1\left(s\right)\stackrel{.}{x}\left(s\right)ds=2{\int }_{\alpha }^{\beta }x\left(s\right)ds-(\beta -\alpha )(x\left(\alpha \right)+x\left(\beta \right)),$$

$${\int }_{\alpha }^{\beta }{f}_2\left(s\right)\stackrel{.}{x}\left(s\right)ds=-{(\beta -\alpha )}^2(x\left(\beta \right)-x\left(\alpha \right))-6(\beta -\alpha ){\int }_{\alpha }^{\beta }x\left(s\right)ds+12{\int }_{\alpha }^{\beta }{\int }_{t_1}^{\beta }x\left(s\right)dsd{t}_1,$$

$$⋮$$

$$\begin{array}{cc}\hfill {\int }_{\alpha }^{\beta }{f}_k\left(s\right)\stackrel{.}{x}\left(s\right)ds& =-\sum _{l=0}^k{\int }_{\alpha }^{\beta }{(\beta -\alpha )}^{k-l}{(-1)}^{k+l}\left(\genfrac{}{}{0pt}{}{k}{l}\right)\left(\genfrac{}{}{0pt}{}{k+l}{l}\right){\left(s-\alpha \right)}^l\stackrel{.}{x}\left(s\right)ds\hfill \\ \hfill & =-\sum _{l=0}^k{(-1)}^{k+l}{(\beta -\alpha )}^{k-l}\left(\genfrac{}{}{0pt}{}{k}{l}\right)\left(\genfrac{}{}{0pt}{}{k+l}{l}\right)l!{\int }_{\alpha }^{\beta }\left(\frac{{(s-\alpha )}^l}{l!}\right)\stackrel{.}{x}\left(s\right)ds\hfill \\ \hfill & =-\sum _{l=1}^k{(-1)}^{k+l}{(\beta -\alpha )}^{k-l}\left(\genfrac{}{}{0pt}{}{k}{l}\right)\left(\genfrac{}{}{0pt}{}{k+l}{l}\right)l!{\int }_{\alpha }^{\beta }{\int }_{t_1}^{\beta }\dots {\int }_{t_l}^{\beta }\stackrel{.}{x}\left(s\right)dsd{t}_l\dots d{t}_1\hfill \\ \hfill & +{(\beta -\alpha )}^k{(-1)}^{k+1}(x\left(\beta \right)-x\left(\alpha \right))\hfill \\ \hfill & =-\sum _{l=1}^k{(\beta -\alpha )}^{k-l}(J_{k,l}(I_l+\frac{{(\beta -\alpha )}^l}{l!}x\left(\beta \right)))\hfill \\ \hfill & +{(\beta -\alpha )}^k{(-1)}^{k+1}(x\left(\beta \right)-x\left(\alpha \right)),\hfill \\ & (k\ge l\ge 1),\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\int }_{\alpha }^{\beta }{\int }_{t_1}^{\beta }\dots {\int }_{t_l}^{\beta }\stackrel{.}{x}\left(s\right)dsd{t}_l\dots d{t}_1=\frac{{(\beta -\alpha )}^l}{l!}x\left(\beta \right)-{\int }_{\alpha }^{\beta }{\int }_{s_1}^{\beta }\dots {\int }_{t_{l-1}}^{\beta }x\left(s\right)dsd{t}_{l-1}\dots d{t}_1,(l\ge 1),\end{array}$$

$$I_l\triangleq -{\int }_{\alpha }^{\beta }{\int }_{s_1}^{\beta }\dots {\int }_{t_{l-1}}^{\beta }x\left(s\right)dsd{t}_{l-1}\dots d{t}_1.$$

Consider the following equations,

$${\int }_{\alpha }^{\beta }{f}_1^2\left(s\right)ds=\frac{{(\beta -\alpha )}^3}{3},{\int }_{\alpha }^{\beta }{f}_2^2\left(s\right)ds=\frac{{(\beta -\alpha )}^5}{5},\cdots ,{\int }_{\alpha }^{\beta }{f}_k^2\left(s\right)ds=\frac{{(\beta -\alpha )}^{2k+1}}{2k+1},$$

$${\int }_{\alpha }^{\beta }{f}_i\left(s\right)f_j\left(s\right)ds=0,(i\ne j).$$

Let $\xi \left(s\right)={\left[f_0\eta {}^T\left(t\right),f_1\left(s\right)\eta {}^T\left(t\right),f_2\left(s\right)\eta {}^T\left(t\right),\cdots ,f_{m-2}\left(s\right)\eta {}^T\left(t\right),\stackrel{.}{x}{}^T\left(s\right)\right]}^T$ and $\eta \left(t\right)=$$[x^T\left(\beta \right),$$x^T\left(\alpha \right),{\int }_{\alpha }^{\beta }{x}^T\left(s\right)ds,{\int }_{\alpha }^{\beta }{\int }_{t_1}^{\beta }{x}^T\left(s\right)dsd{t}_1,\cdots ,{\int }_{\alpha }^{\beta }{\int }_{t_1}^{\beta }\dots {\int }_{t_{m-3}}^{\beta }{x}^T\left(s\right)dsd{t}_{m-3}\dots d{t}_1{]}^T$, one has

$${\int }_{\alpha }^{\beta }{\xi }^T\left(s\right)\mathcal{P}\xi \left(s\right)ds\le {\eta }^T\left(t\right)\mathcal{P}\eta \left(t\right)+{\int }_{\alpha }^{\beta }\stackrel{.}{x}{}^T\left(s\right)P_{m,m}\stackrel{.}{x}\left(s\right)ds.$$

It is obvious that $V_g\left(t\right)\ge 0$ and $V_g\left(t\right)>0$ for $\xi \left(s\right)\ne 0$. The proof is completed. □

Remark 1.

In the previous LKF’s construction, the positive definite quadratic form with definite matrices is usually used. However, in Lemma 2, ${\mathcal{P}}_1$, ${\mathcal{P}}_2$, ⋯, ${\mathcal{P}}_{m-1}$ might not be positive definite, which relaxes the restriction of LMIs. $\eta \left(t\right)$ contains multiple integrals, which adds certain cross terms and helps to decrease the conservatism of conditions for the system’s stability. It is the upgraded and generalized form in reference [15,18]. If it is deformed or degraded slightly, the following Lemma 3 and Lemma 4 can be obtained.

Remark 2.

It is noted that in Lemma 2, the matrix $\mathcal{P}$ is complex, and $\eta \left(t\right)$ contains many terms, which increases the difficulty of applying it to some extent. In fact, if using Lemma 2 directly, it is almost impossible to use in Lyapunov analysis. However, Lemma 2 is a unified form, and m can take small values. Reducing the complexity of Lemma 2 makes it easy to use. In particular, using setting $m=3$ and 4, one can get Lemma 3 and Lemma 4. At this time, $\mathcal{P}$ contains much less terms. $\eta \left(t\right)$ also contains significantly fewer terms. In return, they can be easily used in stability analysis. The matrix variables can also become easily solvable.

Lemma 3.

For any constant $\alpha ,\beta$ satisfying $\alpha <\beta$, $x\left(t\right)$ is a differentiable function in $[\alpha ,\beta ]\to {\mathbb{R}}^n$, if block matrices in $\mathcal{P}$: $P_{11},P_{22}\in {\mathbb{S}}^{3n},P_{33}\in {\mathbb{S}}^n,P_{12}\in {\mathbb{R}}^{3n\times 3n},P_{13},P_{23},\in {\mathbb{R}}^{3n\times n}$ satisfy $\mathcal{P}>0$, where

$$\mathcal{P}=\left[\begin{array}{ccc}{P}_{11}& P_{12}& P_{13}\\ *& P_{22}& P_{23}\\ *& *& P_{33}\end{array}\right],$$

(2)

then

$$V_g\left(t\right)={\eta }^T\left(t\right)\mathcal{P}\eta \left(t\right)+{\int }_{\alpha }^{\beta }\stackrel{.}{x}{}^T\left(t\right)P_{33}\stackrel{.}{x}\left(t\right)dt,$$

(3)

where

$$\mathcal{P}=({\mathcal{P}}_1+(\beta -\alpha ){\mathcal{P}}_2+{(\beta -\alpha )}^3{\mathcal{P}}_3),$$

(4)

$$\eta \left(t\right)={\left[x^T\left(\beta \right),x^T\left(\alpha \right),{\int }_{\alpha }^{\beta }{x}^T\left(t\right)dt\right]}^T,$$

(5)

$${\mathcal{P}}_1=Sym\left\{\left[-P_{13},P_{13},2P_{23}\right]\right\},$$

(6)

$${\mathcal{P}}_2=Sym\left\{\left[-P_{23},-P_{23},0\right]\right\}+P_{11},$$

(7)

$${\mathcal{P}}_3=\frac{1}{3}{P}_{22}.$$

(8)

Proof of Lemma 3.

Based on the proof in Lemma 2, it is not difficult to prove $V_g\left(t\right)\ge 0$. □

Lemma 4.

For any constant $\alpha ,\beta$ satisfying $\alpha <\beta$, $x\left(t\right)$ is a differentiable function in $[\alpha ,\beta ]\to {\mathbb{R}}^n$, if block matrices in $\mathcal{P}$: $P_{11},P_{22},P_{33}\in {\mathbb{S}}^{4n},P_{44}\in {\mathbb{S}}^n,P_{12},P_{13},P_{23}\in {\mathbb{R}}^{4n\times 4n},P_{14},P_{24},P_{34}\in {\mathbb{R}}^{4n\times n}$ satisfy $\mathcal{P}>0$, where

$$\mathcal{P}=\left[\begin{array}{cccc}{P}_{11}& P_{12}& P_{13}& P_{14}\\ *& P_{22}& P_{23}& P_{24}\\ *& *& P_{33}& P_{34}\\ *& *& *& P_{44}\end{array}\right],$$

(9)

then

$$V_g\left(t\right)={\eta }^T\left(t\right)\mathcal{P}\eta \left(t\right)+{\int }_{\alpha }^{\beta }\stackrel{.}{x}{}^T\left(t\right)P_{44}\stackrel{.}{x}\left(t\right)dt\ge 0,$$

(10)

where

$$\mathcal{P}=({\mathcal{P}}_1+(\beta -\alpha ){\mathcal{P}}_2+{(\beta -\alpha )}^3{\mathcal{P}}_3),$$

(11)

$$\eta \left(t\right)={\left[x^T\left(\beta \right),x^T\left(\alpha \right),{\int }_{\alpha }^{\beta }{x}^T\left(t\right)dt,\frac{1}{\beta -\alpha }{\int }_{\alpha }^{\beta }{\int }_{t_1}^{\beta }{x}^T\left(s\right)dsd{t}_1\right]}^T,$$

(12)

$${\mathcal{P}}_1=Sym\left\{\left[-P_{14},P_{14},2P_{24}-6P_{34},12P_{34}\right]\right\},$$

(13)

$${\mathcal{P}}_2=Sym\left\{\left[-P_{24}+P_{34},-P_{24}-P_{34},0,0\right]\right\}+P_{11},$$

(14)

$${\mathcal{P}}_3=\frac{1}{3}{P}_{22}+\frac{1}{5}{P}_{33}.$$

(15)

Proof of Lemma 4.

Let $f_2(s,\alpha ,\beta )=\frac{-{(\beta -\alpha )}^2+6(s-\alpha )(\beta -\alpha )-6{(s-\alpha )}^2}{\beta -\alpha },$ the other conditions are the same as Lemma 1. It is not difficult to prove $V_g\left(t\right)\ge 0$. □

In this section, based on Lemma 1 and the new slack LKF that was constructed in Lemmas 2–4, the lower conservative conditions for system (1) is obtained.

In order to simplify the statement, the following abbreviated symbols are given:

$$\begin{array}{cc}& {\mathrm{\Theta }}_1\left(\phi \left(t\right)\right)={\left[\begin{array}{ccc}{\theta }_1^T\hfill & {\theta }_2^T\hfill & \phi \left(t\right){\theta }_6^T\phi \left(t\right){\theta }_8^T\hfill \end{array}\right]}^T={\mathrm{\Theta }}_{11}+\phi \left(t\right){\mathrm{\Theta }}_{12},\hfill \\ & {\mathrm{\Theta }}_{11}={\left[\begin{array}{cccc}{\theta }_1^T\hfill & {\theta }_2^T\hfill & 0\hfill & 0\hfill \end{array}\right]}^T,{\mathrm{\Theta }}_{12}={\left[\begin{array}{cccc}0\hfill & 0\hfill & {\theta }_6^T\hfill & {\theta }_8^T\hfill \end{array}\right]}^T,\hfill \\ & {\mathrm{\Theta }}_2\left(\phi \left(t\right)\right)={\left[\begin{array}{cccc}{\theta }_2^T\hfill & {\theta }_3^T\hfill & (\phi -\phi \left(t\right)){\theta }_7^T\hfill & (\phi -\phi \left(t\right)){\theta }_9^T\hfill \end{array}\right]}^T={\mathrm{\Theta }}_{21}+\phi \left(t\right){\mathrm{\Theta }}_{22},\hfill \\ & {\mathrm{\Theta }}_{21}={\left[\begin{array}{cccc}{\theta }_2^T\hfill & {\theta }_3^T\hfill & \phi {\theta }_7^T\hfill & \phi {\theta }_9^T\hfill \end{array}\right]}^T,{\mathrm{\Theta }}_{22}={\left[\begin{array}{cccc}0\hfill & 0\hfill & -{\theta }_7^T\hfill & -{\theta }_9^T\hfill \end{array}\right]}^T,\hfill \\ & {\mathrm{\Theta }}_3\left(\dot{\phi }\left(t\right)\right)={\left[{\theta }_0^T(1-\dot{\phi }\left(t\right)){\theta }_4^T{\theta }_1^T-(1-\dot{\phi }\left(t\right)){\theta }_2^T{\theta }_1^T-(1-\dot{\phi }\left(t\right)){\theta }_6^T-\dot{\phi }\left(t\right){\theta }_8^T\right]}^T,\hfill \\ & {\mathrm{\Theta }}_4\left(\dot{\phi }\left(t\right)\right)=\left[(1-\dot{\phi }\left(t\right)){\theta }_4^T{\theta }_5^T(1-\dot{\phi }\left(t\right)){\theta }_2^T-{\theta }_3^T{\left.(1-\dot{\phi }\left(t\right)){\theta }_2^T-{\theta }_7^T+\dot{\phi }\left(t\right){\theta }_9^T\right]}^T\right.,\hfill \end{array}$$

$$\begin{array}{cc}& {\mathrm{\Pi }}_1\left(\phi \left(t\right)\right)={\left[\begin{array}{ccc}{\theta }_1^T\hfill & {\theta }_2^T\hfill & \phi \left(t\right){\theta }_6^T\hfill \end{array}\right]}^T={\mathrm{\Pi }}_{11}+\phi \left(t\right){\mathrm{\Pi }}_{12},\hfill \\ & {\mathrm{\Pi }}_{11}={\left[\begin{array}{ccc}{\theta }_1^T\hfill & {\theta }_2^T\hfill & 0\hfill \end{array}\right]}^T,{\mathrm{\Pi }}_{12}={\left[\begin{array}{ccc}0\hfill & 0\hfill & {\theta }_6^T\hfill \end{array}\right]}^T,\hfill \\ & {\mathrm{\Pi }}_2\left(\phi \left(t\right)\right)={\left[\begin{array}{ccc}{\theta }_2^T\hfill & {\theta }_3^T\hfill & (\phi -\phi \left(t\right)){\theta }_7^T\hfill \end{array}\right]}^T={\mathrm{\Pi }}_{21}+\phi \left(t\right){\mathrm{\Pi }}_{22},\hfill \\ & {\mathrm{\Pi }}_{21}={\left[\begin{array}{ccc}{\theta }_2^T\hfill & {\theta }_3^T\hfill & \phi {\theta }_7^T\hfill \end{array}\right]}^T,{\mathrm{\Pi }}_{22}={\left[\begin{array}{ccc}0\hfill & 0\hfill & -{\theta }_7^T\hfill \end{array}\right]}^T,\hfill \\ & {\mathrm{\Pi }}_3\left(\dot{\phi }\left(t\right)\right)={\left[{\theta }_0^T(1-\dot{\phi }\left(t\right)){\theta }_4^T{\theta }_1^T-(1-\dot{\phi }\left(t\right)){\theta }_2^T\right]}^T,\hfill \\ & {\mathrm{\Pi }}_4\left(\dot{\phi }\left(t\right)\right)={\left[(1-\dot{\phi }\left(t\right)){\theta }_4^T{\theta }_5^T(1-\dot{\phi }\left(t\right)){\theta }_2^T-{\theta }_3^T\right]}^T,\hfill \end{array}$$

$$\begin{array}{cc}& \xi \left(t\right)=\left[x^T\left(t\right),x^T(t-\phi \left(t\right)),x^T(t-\phi ),{\dot{x}}^T(t-\phi \left(t\right)),{\dot{x}}^T(t-\phi ),\frac{1}{\phi \left(t\right)}{\int }_{t-\phi \left(t\right)}^t{x}^T\left(s\right)ds,\right.\hfill \\ & \frac{1}{\phi -\phi \left(t\right)}{\int }_{t-\phi }^{t-\phi \left(t\right)}{x}^T\left(s\right)ds,\frac{1}{\phi {\left(t\right)}^2}{\int }_{t-\phi \left(t\right)}^t{\int }_v^t{x}^T\left(s\right)dsdv,\hfill \\ & {\left.\frac{1}{{(\phi -\phi \left(t\right))}^2}{\int }_{t-\phi }^{t-\phi \left(t\right)}{\int }_v^{t-\phi \left(t\right)}{x}^T\left(s\right)dsdv\right]}^T,\hfill \end{array}$$

$$\begin{array}{c}\hfill {\theta }_k=\left[0_{n\times (k-1)n},I_{n\times n},0_{n\times (9-k)n}\right],k=1,2,3,\cdots ,9,\end{array}$$

$$\begin{array}{c}\hfill {\theta }_0=A{\theta }_1+A_d{\theta }_2.\end{array}$$

Remark 3.

In the following, Theorem 1 and Theorem 2, which ensure the system (1) is stable, can be obtained, respectively, based on the two simplified forms of Lemma 2: Lemma 4 and Lemma 3. The condition in Lemma 4 and Lemma 3 that does not require positive definite $\mathcal{P}$ can relax the linear matrix inequalities (LMIs) condition, which will be verified in the simulation examples. Lemma 1 helps to deal with the quadratic term of the delay: ${\phi }^2\left(t\right)$.

Theorem 1.

For the given constants $\phi >0$,${\nu }_1<{\nu }_2<1$, system (1) can be guaranteed to be asymptotically stable, if there exist positive definite matrices $\mathcal{S}\in {\mathbb{S}}^{13n}$, $\mathcal{T}\in {\mathbb{S}}^{13n}$, $P\in {\mathbb{S}}^{7n}$, $Q_1,Q_2\in {\mathbb{S}}^{6n}$, $Z\in {\mathbb{S}}^n$, and arbitrary matrices $N_1\in {\mathbb{R}}^{9n\times 3n},N_2\in {\mathbb{R}}^{9n\times 3n},$ such that for $i=1,2$, LMIs (16)–(18) hold,

$$\left[\begin{array}{cc}\mathrm{\Psi }\left(0,{\nu }_i\right)& \sqrt{\phi }{N}_2\\ \sqrt{\phi }{N}_2^T& -\widehat{Z}\end{array}\right]<0,$$

(16)

$$\left[\begin{array}{cc}\mathrm{\Psi }\left(\phi ,{\nu }_i\right)& \sqrt{\phi }{N}_1\\ \sqrt{\phi }{N}_1^T& -\widehat{Z}\end{array}\right]<0,$$

(17)

$$-{\phi }^2{\mathrm{\Omega }}_a\left({\nu }_i\right)+\left[\begin{array}{cc}\mathrm{\Psi }\left(0,{\nu }_i\right)& \sqrt{\phi }{N}_2\\ \sqrt{\phi }{N}_2^T& -\widehat{Z}\end{array}\right]<0,$$

(18)

$$\mathcal{S}=\left[\begin{array}{cccc}{S}_{11}& S_{12}& S_{13}& S_{14}\\ *& S_{22}& S_{23}& S_{24}\\ *& *& S_{33}& S_{34}\\ *& *& *& S_{44}\end{array}\right]>0,$$

(19)

$$\mathcal{T}=\left[\begin{array}{cccc}{T}_{11}& T_{12}& T_{13}& T_{14}\\ *& T_{22}& T_{23}& T_{24}\\ *& *& T_{33}& T_{34}\\ *& *& *& T_{44}\end{array}\right]>0,$$

(20)

where $\mathrm{\Psi }(\phi \left(t\right),\dot{\phi }\left(t\right))=\mathrm{\Phi }(\phi \left(t\right),\dot{\phi }\left(t\right))+{\phi }^2\left(t\right){\mathrm{\Omega }}_a\left(\dot{\phi }\left(t\right)\right)+{\mathrm{\Omega }}_b(\phi \left(t\right),\dot{\phi }\left(t\right))$, and $\mathrm{\Phi }(\phi \left(t\right),\dot{\phi }\left(t\right))$ denotes $\mathrm{\Phi }(\tau \left(t\right),\dot{\tau }\left(t\right))$ in [19].

Proof of Theorem 1.

Choose the following Lyapunov functional

$$V\left(t\right)=V_w\left(t\right)+V_S\left(t\right)+V_T\left(t\right),$$

(21)

where $V_w\left(t\right)$ is equal to the Lyapunov functional $V\left(t\right)$ in Theorem 1 of Ref. [19]. In order to demonstrate the usefulness of the new proposed slack Lyapunov functional clearer and make the structure of the article more explicit, all contents concerning $V_w\left(t\right)$ are not listed here. Please refer to the work in Ref. [19].

$$V_S\left(t\right)={\rho }_1^T\left(t\right)\mathcal{S}{\rho }_1\left(t\right)+{\int }_{t-\phi \left(t\right)}^t{\dot{x}}^T\left(s\right)S_{44}\dot{x}\left(s\right)ds,$$

(22)

$$V_T\left(t\right)={\rho }_2^T\left(t\right)\mathcal{T}{\rho }_2\left(t\right)+{\int }_{t-\phi }^{t-\phi \left(t\right)}{\dot{x}}^T\left(s\right)T_{44}\dot{x}\left(s\right)ds,$$

(23)

with

$$\begin{array}{cc}\hfill & \mathcal{S}={\mathcal{S}}_1+\phi \left(t\right){\mathcal{S}}_2+{\phi }^2·\phi \left(t\right){\mathcal{S}}_3\hfill \\ \hfill & \mathcal{T}={\mathcal{T}}_1+(\phi -\phi \left(t\right)){\mathcal{T}}_2+{\phi }^2(\phi -\phi \left(t\right)){\mathcal{T}}_3,\hfill \end{array}$$

$${\mathcal{S}}_1=Sym\left\{\left[-S_{14},S_{14},2S_{24}-6S_{34},12S_{34}\right]\right\},$$

$${\mathcal{S}}_2=Sym\left\{\left[-S_{24}+S_{34},-S_{24}-S_{34},0,0\right]\right\}+S_{11},$$

$${\mathcal{S}}_3=\frac{1}{3}{S}_{22}+\frac{1}{5}{S}_{33},$$

$${\mathcal{T}}_1=Sym\left\{\left[-T_{14},T_{14},2T_{24}-6T_{34},12T_{34}\right]\right\},$$

$${\mathcal{T}}_2=Sym\left\{\left[-T_{24}+T_{34},-T_{24}-T_{34},0,0\right]\right\}+T_{11},$$

$${\mathcal{T}}_3=\frac{1}{3}{T}_{22}+\frac{1}{5}{T}_{33},$$

$${\rho }_1\left(\iota \right)={\left[x^T\left(\iota \right),x^T(\iota -\phi \left(\iota \right)),{\int }_{\iota -\phi \left(\iota \right)}^{\iota }{x}^T\left(s\right)ds,\frac{1}{\phi \left(\iota \right)}{\int }_{\iota -\phi \left(\iota \right)}^{\iota }{\int }_v^{\iota }{x}^T\left(s\right)dsdv\right]}^T,$$

$${\rho }_2\left(\iota \right)={\left[x^T(\iota -\phi \left(\iota \right)),x^T(\iota -\phi ),{\int }_{\iota -\phi }^{\iota -\phi \left(\iota \right)}{x}^T\left(s\right)ds,\frac{1}{\phi -\phi \left(\iota \right)}{\int }_{\iota -\phi }^{\iota -\phi \left(\iota \right)}{\int }_v^{\iota -\phi \left(\iota \right)}{x}^T\left(s\right)dsdv\right]}^T.$$

Calculating the derivative of (22) and (23) gives

$$\begin{array}{cc}\hfill {\dot{V}}_S\left(t\right)& =2{\rho }_1^T\left(t\right)\mathcal{S}{\dot{\rho }}_1\left(t\right)\hfill \\ \hfill & ={\xi }^T\left(t\right)\left[Sym\left\{{\mathrm{\Theta }}_1^T\left(\phi \left(t\right)\right)\mathcal{S}{\mathrm{\Theta }}_3\left(\dot{\phi }\left(t\right)\right)\right\}+{\theta }_0^T{S}_{44}{\theta }_0-(1-\dot{\phi }\left(t\right)){\theta }_4^T{S}_{44}{\theta }_4\right.\hfill \\ \hfill & \left.+{\mathrm{\Theta }}_1^T\left(\phi \left(t\right)\right)\left(\dot{\phi }\left(t\right)S_2+{\phi }^2\dot{\phi }\left(t\right)S_3\right){\mathrm{\Theta }}_1\left(\phi \left(t\right)\right)\right]\xi \left(t\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\dot{V}}_T\left(t\right)& =2{\rho }_2^T\left(t\right)\mathcal{T}{\dot{\rho }}_2\left(t\right)\hfill \\ \hfill & ={\xi }^T\left(t\right)[Sym\left\{{\mathrm{\Theta }}_2^T\left(\phi \left(t\right)\right)\mathcal{T}{\mathrm{\Theta }}_4\left(\dot{\phi }\left(t\right)\right)\right\}-{\theta }_5^T{T}_{44}{\theta }_5+(1-\dot{\phi }\left(t\right)){\theta }_4^T{T}_{44}{\theta }_4\hfill \\ \hfill & +{\mathrm{\Theta }}_2^T\left(\phi \left(t\right)\right)\left(-\dot{\phi }\left(t\right){\mathcal{T}}_2-{\phi }^2\dot{\phi }\left(t\right){\mathcal{T}}_3\right){\mathrm{\Theta }}_2\left(\phi \left(t\right)\right)]\xi \left(t\right).\hfill \end{array}$$

(25)

By combining (24) and (25), Theorem 1 of Ref. [19], one has

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)<& {\xi }^T\left(t\right)({\phi }^2\left(t\right){\mathrm{\Omega }}_a\left(\dot{\phi }\left(t\right)\right)+{\mathrm{\Omega }}_b(\phi \left(t\right),\dot{\phi }\left(t\right))+{\mathrm{\Omega }}_0(\phi \left(t\right),\dot{\phi }\left(t\right))\hfill \\ \hfill & -{\mathrm{\Omega }}_1\left(\phi \left(t\right)\right))\xi \left(t\right),\hfill \end{array}$$

(26)

where

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_a\left(\dot{\phi }\left(t\right)\right)& ={\mathrm{\Theta }}_{12}^T\left(\dot{\phi }\left(t\right){\mathcal{S}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{S}}_3\right){\mathrm{\Theta }}_{12}+Sym\left\{{\mathrm{\Theta }}_{12}^T\left({\mathcal{S}}_2+{\phi }^2{\mathcal{S}}_3\right){\mathrm{\Theta }}_3\left(\dot{\phi }\left(t\right)\right)\right\}\hfill \\ \hfill & -{\mathrm{\Theta }}_{22}^T\left(\dot{\phi }\left(t\right){\mathcal{T}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{T}}_3\right){\mathrm{\Theta }}_{22}-Sym\left\{{\mathrm{\Theta }}_{22}^T\left({\mathcal{T}}_2+{\phi }^2{\mathcal{T}}_3\right){\mathrm{\Theta }}_4\left(\dot{\phi }\left(t\right)\right)\right\},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_b(\phi \left(t\right),\dot{\phi }\left(t\right))=& Sym\left\{{\mathrm{\Theta }}_{11}^T\left({\mathcal{S}}_1+\phi \left(t\right){\mathcal{S}}_2+{\phi }^2\phi \left(t\right){\mathcal{S}}_3\right){\mathrm{\Theta }}_3\left(\dot{\phi }\left(t\right)\right)\right\}\hfill \\ \hfill & +Sym\left\{{\mathrm{\Theta }}_1\left(\phi \left(t\right)\right){\mathcal{S}}_1{\mathrm{\Theta }}_3\left(\dot{\phi }\left(t\right)\right)\right\}\hfill \\ \hfill & -Sym\left\{{\mathrm{\Theta }}_{21}^T\left(T_1+\phi \left(t\right){\mathcal{T}}_2+{\phi }^2\phi \left(t\right){\mathcal{T}}_3\right){\mathrm{\Theta }}_4\left(\dot{\phi }\left(t\right)\right)\right\}\hfill \\ \hfill & +Sym\left\{{\mathrm{\Theta }}_2\left(\phi \left(t\right)\right)\left({\mathcal{T}}_1+\phi {\mathcal{T}}_2+{\phi }^3{\mathcal{T}}_3\right){\mathrm{\Theta }}_4\left(\dot{\phi }\left(t\right)\right)\right\}\hfill \\ \hfill & +{\theta }_0^T{S}_{44}{\theta }_0+(1-\dot{\phi }\left(t\right)){\theta }_4^T\left(T_{44}-S_{44}\right){\theta }_4\hfill \\ \hfill & -{\theta }_5^T{T}_{44}{\theta }_5+{\mathrm{\Theta }}_{11}^T\left(\dot{\phi }\left(t\right){\mathcal{S}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{S}}_3\right){\mathrm{\Theta }}_{11}\hfill \\ \hfill & +\phi \left(t\right)Sym\left\{{\mathrm{\Theta }}_{11}^T\left(\dot{\phi }\left(t\right){\mathcal{S}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{S}}_3\right){\mathrm{\Theta }}_{12}\right\}\hfill \\ \hfill & -\phi \left(t\right)Sym\left\{{\mathrm{\Theta }}_{21}^T\left(\dot{\phi }\left(t\right){\mathcal{T}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{T}}_3\right){\mathrm{\Theta }}_{22}\right\}\hfill \\ & -{\mathrm{\Theta }}_{21}^T\left(\dot{\phi }\left(t\right){\mathcal{T}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{T}}_3\right){\mathrm{\Theta }}_{21}.\hfill \end{array}$$

(28)

${\mathrm{\Omega }}_0(\phi \left(t\right),\dot{\phi }\left(t\right))$, ${\mathrm{\Omega }}_1\left(\phi \left(t\right)\right)$ and the variables in them are defined in Ref. [19]. After computing the derivative of $V\left(t\right)$, it can get a matrix of this form: ${\phi }^2\left(t\right){\mathrm{\Omega }}_a\left(\dot{\phi }\left(t\right)\right)+{\mathrm{\Omega }}_b(\phi \left(t\right),\dot{\phi }\left(t\right))+{\mathrm{\Omega }}_0(\phi \left(t\right),\dot{\phi }\left(t\right))-{\mathrm{\Omega }}_1\left(\phi \left(t\right)\right)$. Based on Schur complement and Lemma 1, one has ${\phi }^2\left(t\right){\mathrm{\Omega }}_a$$\left(\dot{\phi }\left(t\right)\right)+{\mathrm{\Omega }}_b(\phi \left(t\right),\dot{\phi }\left(t\right))+{\mathrm{\Omega }}_0(\phi \left(t\right),\dot{\phi }\left(t\right))-{\mathrm{\Omega }}_1\left(\phi \left(t\right)\right)<0$ if LMIs (16)–(18) hold for all $0<\phi \left(t\right)<$$\phi ,{\nu }_1<\dot{\phi }\left(t\right)<{\nu }_2,$ we have $\dot{V}\left(t\right)<0.$ Following the Lyapunov theorem, system (1) can be deduced to be asymptotically stable. □

Theorem 2.

For given constants $\phi >0$,${\nu }_1<{\nu }_2<1$, system (1) can be guaranteed to be asymptotically stable, if there exist positive definite matrices $\mathcal{S}\in {\mathbb{S}}^{10n}$, $\mathcal{T}\in {\mathbb{S}}^{10n}$, $P\in {\mathbb{S}}^{7n}$, $Q_1,Q_2\in {\mathbb{S}}^{6n}$, $Z\in {\mathbb{S}}^n$, and arbitrary matrices $N_1\in {\mathbb{R}}^{9n\times 3n},N_2\in {\mathbb{R}}^{9n\times 3n},$ such that for $i=1,2$, LMIs (29)–(31) hold,

$$\left[\begin{array}{cc}\mathrm{\Psi }\left(0,{\nu }_i\right)& \sqrt{\phi }{N}_2\\ \sqrt{\phi }{N}_2^T& -\widehat{Z}\end{array}\right]<0,$$

(29)

$$\left[\begin{array}{cc}\mathrm{\Psi }\left(\phi ,{\nu }_i\right)& \sqrt{\phi }{N}_1\\ \sqrt{\phi }{N}_1^T& -\widehat{Z}\end{array}\right]<0,$$

(30)

$$-{\phi }^2{\mathrm{\Omega }}_a\left({\nu }_i\right)+\left[\begin{array}{cc}\mathrm{\Psi }\left(0,{\nu }_i\right)& \sqrt{\phi }{N}_2\\ \sqrt{\phi }{N}_2^T& -\widehat{Z}\end{array}\right]<0,$$

(31)

$$\mathcal{S}=\left[\begin{array}{ccc}{S}_{11}& S_{12}& S_{13}\\ *& S_{22}& S_{23}\\ *& *& S_{33}\end{array}\right]>0$$

(32)

$$\mathcal{T}=\left[\begin{array}{ccc}{T}_{11}& T_{12}& T_{13}\\ *& T_{22}& T_{23}\\ *& *& T_{33}\\ *& *& *\end{array}\right]>0$$

(33)

where $\mathrm{\Psi }(\phi \left(t\right),\dot{\phi }\left(t\right))=\mathrm{\Phi }(\phi \left(t\right),\dot{\phi }\left(t\right))+{\phi }^2\left(t\right){\mathrm{\Omega }}_a\left(\dot{\phi }\left(t\right)\right)+{\mathrm{\Omega }}_b(\phi \left(t\right),\dot{\phi }\left(t\right))$, and $\mathrm{\Phi }(\phi \left(t\right),\dot{\phi }\left(t\right))$ denote $\mathrm{\Phi }(\tau \left(t\right),\dot{\tau }\left(t\right))$ in Ref. [19].

Proof of Theorem 2.

Choose the following Lyapunov functional

$$V\left(t\right)=V_w\left(t\right)+V_S\left(t\right)+V_T\left(t\right),$$

(34)

where

$$V_S\left(t\right)={\rho }_1^T\left(t\right)\mathcal{S}{\rho }_1\left(t\right)+{\int }_{t-\phi \left(t\right)}^t{\dot{x}}^T\left(s\right)S_{33}\dot{x}\left(s\right)ds,$$

(35)

$$V_T\left(t\right)={\rho }_2^T\left(t\right)\mathcal{T}{\rho }_2\left(t\right)+{\int }_{t-\phi }^{t-\phi \left(t\right)}{\dot{x}}^T\left(s\right)T_{33}\dot{x}\left(s\right)ds,$$

(36)

with

$$\begin{array}{cc}\hfill & \mathcal{S}={\mathcal{S}}_1+\phi \left(t\right){\mathcal{S}}_2+{\phi }^2·\phi \left(t\right){\mathcal{S}}_3,\hfill \\ \hfill & \mathcal{T}={\mathcal{T}}_1+(\phi -\phi \left(t\right)){\mathcal{T}}_2+{\phi }^2(\phi -\phi \left(t\right)){\mathcal{T}}_3,\hfill \end{array}$$

$${\mathcal{S}}_1=Sym\left\{\left[-S_{13},S_{13},2S_{23}\right]\right\},$$

$${\mathcal{S}}_2=Sym\left\{\left[-S_{23},-S_{23},0,\right]\right\}+S_{11},$$

$${\mathcal{S}}_3=\frac{1}{3}{S}_{22},$$

$${\mathcal{T}}_1=Sym\left\{\left[-T_{13},T_{13},2T_{23}\right]\right\},$$

$${\mathcal{T}}_2=Sym\left\{\left[-T_{23},-T_{23},0,\right]\right\}+T_{11},$$

$${\mathcal{T}}_3=\frac{1}{3}{T}_{22},$$

$${\rho }_1\left(\iota \right)={\left[x^T\left(\iota \right),x^T(\iota -\phi \left(\iota \right)),{\int }_{\iota -\phi \left(\iota \right)}^{\iota }{x}^T\left(s\right)ds\right]}^T,$$

$${\rho }_2\left(\iota \right)={\left[x^T(\iota -\phi \left(\iota \right)),x^T(\iota -\phi ),{\int }_{\iota -\phi }^{\iota -\phi \left(\iota \right)}{x}^T\left(s\right)ds\right]}^T.$$

In this theorem, ${\mathrm{\Omega }}_a\left(\dot{\phi }\left(t\right)\right)$ and ${\mathrm{\Omega }}_b(\phi \left(t\right),\dot{\phi }\left(t\right))$ are

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_a\left(\dot{\phi }\left(t\right)\right)& ={\mathrm{\Pi }}_{12}^T\left(\dot{\phi }\left(t\right){\mathcal{S}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{S}}_3\right){\mathrm{\Pi }}_{12}+Sym\left\{{\mathrm{\Pi }}_{12}^T\left({\mathcal{S}}_2+{\phi }^2{\mathcal{S}}_3\right){\mathrm{\Pi }}_3\left(\dot{\phi }\left(t\right)\right)\right\}\hfill \\ \hfill & -{\mathrm{\Pi }}_{22}^T\left(\dot{\phi }\left(t\right){\mathcal{T}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{T}}_3\right){\mathrm{\Pi }}_{22}-Sym\left\{{\mathrm{\Pi }}_{22}^T\left({\mathcal{T}}_2+{\phi }^2{\mathcal{T}}_3\right){\mathrm{\Pi }}_4\left(\dot{\phi }\left(t\right)\right)\right\}.\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_b(\phi \left(t\right),\dot{\phi }\left(t\right))=& Sym\left\{{\mathrm{\Pi }}_{11}^T\left({\mathcal{S}}_1+\phi \left(t\right){\mathcal{S}}_2+{\phi }^2\phi \left(t\right){\mathcal{S}}_3\right){\mathrm{\Pi }}_3\left(\dot{\phi }\left(t\right)\right)\right\}\hfill \\ \hfill & +Sym\left\{{\mathrm{\Pi }}_1\left(\phi \left(t\right)\right){\mathcal{S}}_1{\mathrm{\Pi }}_3\left(\dot{\phi }\left(t\right)\right)\right\}\hfill \\ \hfill & -Sym\left\{{\mathrm{\Pi }}_{21}^T\left(T_1+\phi \left(t\right){\mathcal{T}}_2+{\phi }^2\phi \left(t\right){\mathcal{T}}_3\right){\mathrm{\Pi }}_4\left(\dot{\phi }\left(t\right)\right)\right\}\hfill \\ \hfill & +Sym\left\{{\mathrm{\Pi }}_2\left(\phi \left(t\right)\right)\left({\mathcal{T}}_1+\phi {\mathcal{T}}_2+{\phi }^3{\mathcal{T}}_3\right){\mathrm{\Pi }}_4\left(\dot{\phi }\left(t\right)\right)\right\}\hfill \\ \hfill & +{\theta }_0^T{S}_{33}{\theta }_0+(1-\dot{\phi }\left(t\right)){\theta }_4^T\left(T_{33}-S_{33}\right){\theta }_4\hfill \\ \hfill & -{\theta }_5^T{T}_{33}{\theta }_5+{\mathrm{\Pi }}_{11}^T\left(\dot{\phi }\left(t\right){\mathcal{S}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{S}}_3\right){\mathrm{\Pi }}_{11}\hfill \\ \hfill & +\phi \left(t\right)Sym\left\{{\mathrm{\Pi }}_{11}^T\left(\dot{\phi }\left(t\right){\mathcal{S}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{S}}_3\right){\mathrm{\Pi }}_{12}\right\}\hfill \\ \hfill & -\phi \left(t\right)Sym\left\{{\mathrm{\Pi }}_{21}^T\left(\dot{\phi }\left(t\right){\mathcal{T}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{T}}_3\right){\mathrm{\Pi }}_{22}\right\}\hfill \\ & -{\mathrm{\Pi }}_{21}^T\left(\dot{\phi }\left(t\right){\mathcal{T}}_2+{\phi }^2\dot{\phi }\left(t\right){\mathcal{T}}_3\right){\mathrm{\Pi }}_{21}.\hfill \end{array}$$

(38)

Do the same as Theorem 1, and it is then not hard to prove. □

4. Results

Here, it is noted that all numerical results have been performed by LMI toolbox of MATLAB. Two numerical examples are presented to illustrate the validity of the theorems.

Example 1.

Consider dynamical system (1) containing the following given matrices:

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

Let ${\nu }_1=-\nu$ and ${\nu }_2=\nu$, for different known scalar $\nu$, the maximal allowed upper bounds (MAUB) of $\phi$, which make the system stable in Refs. [2,9,13,19,20,21,22,23] are presented in Table 2. We validate our theorems, which are also presented in Table 2. By comparison, the simulations based on Theorems 1 and 2 in this paper have better performance. The validity of Lemmas 2–4 is verified.

Table 2. As for different $\nu$, MUAB of $\phi$ obtained by simulation in Example 1.

Different $\mathit{\nu }$	[9]	[13]	[2]	[23] (N = 1)	[23] (N = 2)	[22]	[21]	[20]	[19] (Corollary 1)	[19] (Theorem 1)	Theorem 2	Theorem 1
$\nu =0.1$	4.703	4.788	4.83	4.8	4.93	4.91	4.908	4.918	4.841	4.921	4.942	4.949
$\nu =0.2$	3.834	4.06	4.14	3.99	4.22	4.216	4.199	4.209	4.154	4.218	4.284	4.291
$\nu =0.5$	2.42	3.055	3.14	2.79	3.09	3.233	3.166	3.196	3.159	3.221	3.314	3.322
$\nu =0.8$	2.137	2.615	2.71	2.42	2.66	2.789	2.735	2.759	2.729	2.792	2.926	2.932

Example 2.

Consider dynamical system (1) containing the following given matrices:

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

Let ${\nu }_1=-\nu$ and ${\nu }_2=\nu$, for different known scalar $\nu$, MAUB of $\phi$ that make the system stable in Refs. [2,9,13,18,19,22] are presented in Table 3. We validate our theorems, which are also presented in Table 3. By comparison, the simulations based on Theorems 1 and 2 in this paper have better performance. The validity of Lemmas 2, 3, and 4 is verified.

Table 3. As for different known $\nu$, MUAB of $\phi$ obtained by simulation in Example 2.

Different $\mathit{\nu }$	[9]	[13]	[2]	[18]	[22]	[19] (Corollary 1)	[19] (Theorem 1)	Theorem 2	Theorem 1
$\nu =0.1$	6.59	7.148	7.167	7.176	7.23	7.189	7.308	7.362	7.368
$\nu =0.2$	3.672	4.466	4.517	4.543	4.556	4.549	4.67	4.741	4.749
$\nu =0.5$	1.411	2.352	2.415	2.496	2.509	2.503	2.664	2.698	2.703
$\nu =0.8$	1.275	1.768	1.838	1.922	1.940	1.950	2.072	2.087	2.093

5. Conclusions

Based on a specially constructed matrix and integral calculation, a new slack Lyapunov functional was derived, in which the quadratic matrices might not be positive definite, therefore relaxing the criterion of stability of dynamical systems. It can be applied to the performance analysis of the dynamic system, and can even be extended to be used in the exponential stability analysis, while it is further upgraded into the exponential form of LKF. In combination with Lemma 1 for the matrix form of quadratic function type, we dealt with the the second power term of delay: ${\phi }^2\left(t\right)$ and $\dot{\phi }\left(t\right)$. In addition, some cross-terms after calculating the derivative of the new slack Lyapunov functional were established, making LMIs more slack.