New Lyapunov–Krasovskii Functional for Mixed-Delay-Dependent Stability of Uncertain Linear Neutral Systems

Abstract

The robust stability in a class of uncertain linear neutral systems with time-varying delays is studied. Through choosing multiple integral Lyapunov terms and using some novel integral inequalities, a much tighter estimation on derivative of Lyapunov–Krasovskii (L–K) functional is presented and two stability criteria are expressed in terms of linear matrix inequalities, in which those previously ignored information can be considered. In particular, the proposed Lyapunov technique can effectively consider the interconnection between neutral delay and state one. Finally, two numerical examples with comparing results can show the application area and benefits of the proposed conditions.

Appendix

In what follows, some lemmas will be presented for the proof procedure of Theorems 1–2.

Lemma 1

[50] For $d(t)\in [0,d]$, a symmetric matrix $R>0$ and any matrix $S_1$ satisfying $\left[ \begin{array}{cc} R_1 &{}\quad S_1 \\ *&{}\quad R_1 \end{array}\right] \ge 0$ with $R_1=\mathrm {diag}\{R,3R,5R\}$, the following inequality can be true

$$\begin{aligned}&-\int _{t-d(t)}^{t} \dot{x}(s)\hbox {d}s -\int _{t-d}^{t-d(t)}\dot{x}(s)\hbox {d}s \\&\le -\frac{1}{d}\zeta ^\mathrm{T}(t)\left[ \begin{array}{cc} E_1 \\ E_2\end{array}\right] ^\mathrm{T}\left( \left[ \begin{array}{cc} R_1 &{}\quad S_1 \\ *&{} R_1 \end{array}\right] +\left[ \begin{array}{cc} \frac{d-d(t)}{d}T &{}\quad 0 \\ *&{} \frac{d(t)}{d}T \end{array}\right] \right) \left[ \begin{array}{cc} E_1 \\ E_2 \end{array}\right] \zeta (t), \end{aligned}$$

where

$$\begin{aligned} \zeta ^\mathrm{T}(t)= & {} \Big [x^\mathrm{T}(t)~~x^\mathrm{T}(t-d(t))~~x^\mathrm{T}(t-d) ~~\varphi ^\mathrm{T}(t)~~\varrho ^\mathrm{T}(t)~~\nu ^\mathrm{T}(t)~~\omega ^\mathrm{T}(t)\Big ];\\ E_1= & {} \left[ \begin{array}{cc} e_1-e_2 \\ e_1+e_2-2e_4 \\ e_1-e_2+6e_4-12e_6 \end{array}\right] ,~~E_2=\left[ \begin{array}{cc} e_2-e_3 \\ e_2+e_3-2e_5 \\ e_2-e_3+6e_5-12e_7 \end{array}\right] ;\\ e_i= & {} \Big [0_{i-1}~~~I_n~~~0_{7-i}\Big ]~(1\le i\le 7),~~~ T=R_1-S_1^\mathrm{T}R_1^{-1}S_1; \\ \varphi (t)= & {} \frac{1}{d(t)}\int _{t-d(t)}^{t}x(s)\hbox {d}s,~~\nu (t)=\frac{2}{d^2(t)}\int _{t-d(t)}^{t}\int _{t-d(t)}^{s} x(u)\hbox {d}u\hbox {d}s; \\ \varrho (t)= & {} \frac{1}{d-{d}(t)}\int _{t-d}^{t-d(t)}x(s)\hbox {d}s,~~ \omega (t)=\frac{2}{(d-{d}(t))^2}\int _{t-d}^{t-d(t)}\int _{t-d}^{s} x(u)\hbox {d}u\hbox {d}s.~~ \end{aligned}$$

Lemma 2

[28, 50] For an any constant matrix $M>0$, the following inequalities hold for all continuously differentiable function $\varphi$ in $[a,b]\rightarrow \mathbf{R}^n$:

$$\begin{aligned}&-(b-a)\int _{a}^{b} \varphi ^\mathrm{T}(s)M\varphi (s)\hbox {d}s \le -\left( \int _{a}^{b} \varphi (s)\hbox {d}s\right) ^\mathrm{T} M\left( \int _{a}^{b} \varphi (s)\hbox {d}s\right) -3\Theta ^\mathrm{T}M\Theta , \\&-\frac{b^2-a^2}{2}\int _{a}^{b}\int _{t+\theta }^{t} \varphi ^\mathrm{T}(s)M\varphi (s)\hbox {d}s\hbox {d}\theta \\&\le -\left( \int _{a}^{b}\int _{t+\theta }^{t} \varphi (s)\hbox {d}s\hbox {d}\theta \right) ^\mathrm{T} M\left( \int _{a}^{b}\int _{t+\theta }^{t} \varphi (s)\hbox {d}s\hbox {d}\theta \right) ,\\&-\frac{b^3-a^3}{6}\int _{-b}^{-a}\int _{\varrho }^{0}\int _{t+\theta }^t \varphi ^\mathrm{T}(s)M\varphi (s)\hbox {d}s\hbox {d}\theta \hbox {d}\varrho \\&\le -\left( \int _{-b}^{-a}\int _{\varrho }^{0}\int _{t+\theta }^t \varphi (s)\hbox {d}s\hbox {d}\theta \hbox {d}\varrho \right) ^\mathrm{T} M\left( \int _{-b}^{-a}\int _{\varrho }^{0}\int _{t+\theta }^t \varphi (s)\hbox {d}s\hbox {d}\theta \hbox {d}\varrho \right) , \end{aligned}$$

where $\Theta =\int _{a}^{b} {\varphi }(s)\hbox {d}s-\frac{2}{b-a}\int _{a}^{b}\int _{a}^{s} {\varphi }(u)\hbox {d}u\hbox {d}s$.

Lemma 3

[28] For an any constant matrix $M>0$, the following inequality holds for all continuously differentiable function $\varphi$ in $[a,b]\rightarrow \mathbf{R}^n$:

$$\begin{aligned}&-\frac{(b-a)^2}{2}\int _{a}^{b}\int _{a}^{s}{\varphi }^\mathrm{T}(u)M{\varphi }(u)\hbox {d}u\hbox {d}s \\&\le -\left( \int _{a}^{b}\int _{a}^{s} {\varphi }(u)\hbox {d}u\hbox {d}s\right) ^\mathrm{T} M\left( \int _{a}^{b}\int _{a}^{s} {\varphi }(u)\hbox {d}u\hbox {d}s\right) -2\Theta ^\mathrm{T}M\Theta , \end{aligned}$$

where $\Theta =\int _{a}^{b}\int _{a}^{s} {\varphi }(u)\hbox {d}u\hbox {d}s-\frac{3}{b-a}\int _{a}^{b}\int _{a}^{s}\int _{a}^{u} {\varphi }(v)\hbox {d}v\hbox {d}u\hbox {d}s$.

Lemma 4

[29] For vector $\omega$, real scalars $a\le b$, symmetric matrix $R>0$ such that the integration is well defined, then the following inequality holds,

$$\begin{aligned} (b-a)\int _{a}^{b}\dot{\omega }^\mathrm{T}(s) R\dot{\omega }(s)\hbox {d}s\ge \chi ^\mathrm{T}_1R\chi _1+3\chi ^\mathrm{T}_2R\chi _2+5\chi ^\mathrm{T}_3R\chi _3, \end{aligned}$$

where

$$\begin{aligned} \chi _1= & {} \omega (b)-\omega (a),~~\chi _2=\omega (b)+\omega (a)-\frac{2}{b-a}\int _{a}^{b}\omega (s)\hbox {d}s,\\ \chi _3= & {} \omega (b)-\omega (a)+\frac{6}{b-a}\int _{a}^{b}\omega (s)\hbox {d}s-\frac{12}{(b-a)^2}\int _{a}^{b}\int _{s}^{b}\omega (\theta )\hbox {d}\theta \hbox {d}s. \end{aligned}$$

As an extended case of Lemma 2 in [25], we can derive the following Lemma easily.

Lemma 5

[25] Suppose that $\Omega ,\Xi _{ij},\Xi _{mn}~(i,m=1,2,3,4;j,n=1,2)$ are the constant matrices of appropriate dimensions, $\alpha \in [0,1]$, $\beta \in [0,1]$, $\gamma \in [0,1]$, and $\delta \in [0,1]$, then

$$\begin{aligned}&\Omega +\big [\alpha \Xi _{11}+(1-\alpha )\Xi _{12}\big ]+\big [\beta \Xi _{21}+(1-\beta )\Xi _{22}\big ]\\&+\big [\gamma \Xi _{31}+(1-\gamma )\Xi _{32}\big ]+\big [\delta \Xi _{41}+(1-\delta )\Xi _{42}\big ]<0 \nonumber \end{aligned}$$

holds, if and only if the following inequalities hold simultaneously,

$$\begin{aligned}&\Omega +\Xi _{ij}+\Xi _{mn}<0~~(i,m=1,2,3,4;j,n=1,2).\nonumber \end{aligned}$$

Lemma 6

[26, 38] Let $I-G^\mathrm{T}G>0$ define the set $\Upsilon =\big \{\Delta (t)=\Sigma (t)[I-G\Sigma (t)]^{-1}, \Sigma ^\mathrm{T}(t)\Sigma (t)\le I \big \}$, for given matrices H, J, and R of appropriate dimensions and symmetric one H, then $H+J\Delta (t)R+R^\mathrm{T}\Delta ^\mathrm{T}(t)J^\mathrm{T}<0$, iff there exists $\rho >0$ such that

$$\begin{aligned}&H+\left[ \begin{array}{cc} \rho ^{-1}R \\ \rho J^\mathrm{T} \end{array}\right] ^\mathrm{T}\left[ \begin{array}{cc} I &{}\quad -G\\ -G^\mathrm{T} &{}\quad I \end{array}\right] ^{-1}\left[ \begin{array}{cc} \rho ^{-1}R \\ \rho J^\mathrm{T} \end{array}\right] <0. \end{aligned}$$