Stability of neutral-type descriptor systems with multiple time-varying delays

Abstract

This article deals with the problem of stability of descriptor neutral systems with multiple delays. Using Lyapunov functional and free-weighting matrix method, a delay-dependent stability criterion is obtained and formulated in the form of linear matrix inequalities, which can easily be checked by utilizing Matlab linear matrix inequality toolbox. Finally, a numerical example is presented to illustrate the effectiveness of the method.

1. Introduction

Since the time delay is frequently viewed as a source of instability and encountered in various engineering systems such as chemical processes, long transmission lines in pneumatic systems, networked control systems, etc., the study of delay systems has received much attention and various topics have been discussed over the past years. Commonly, the existing results can be classified into two types: delay-independent conditions and delay-dependent conditions. In general, the delay-dependent case is more conservative than delay-independent case.

A neutral system with time-delays which contains delays both in its state and in its derivatives of state is encountered in many dynamic systems and their presences must be taken into account in real dynamic process such as circuit systems, population dynamics, automatic control, and heat exchangers, etc. Due to its profound and practical background, much attention has been focused on the problems of stability analysis for neutral time-delay system from mathematics and control communities [1–7]. Using Lyapunov method, Park [1] presented new sufficient conditions for the stability of the systems in terms of linear matrix inequality (LMI) which can be easily solved by various convex optimization algorithms. Some delay-independent stability criteria were given in terms of the characteristic equation of system, involving the measures, eigenvalues, spectral radius, and spectral norms of the corresponding matrices [3]. Although the conditions are easy to check, they require the matrix measure to be Hurwitz matrix. The problem of delay-dependent stability criteria for a class of constant time-delay neutral systems with time-varying structured uncertainties was investigated [4]. Han [5] obtained delay-dependent stability conditions for uncertain neutral time-varying system by model transformation method, due to cross terms of model transformation, results are less conservative. Zhao [6] dealt with the problem of delay-dependent robust stability for delay neutral type control system with time-varying structured uncertainties and time-varying delay. Some new delay and its derivative-dependent criteria were derived. He [7] concerned the problem of the delay-dependent robust stability of neutral systems with mixed delays and time-varying structured uncertainties. A new method based on linear matrix inequalities was presented that makes it easy to calculate both the upper stability bounds on the delays and the free weighting matrices. Since the criteria take the sizes of the neutral- and discrete-delays into account, it is less conservative than previous methods.

Recently, Li [8] studied the stability of the neutral-type descriptor system with mixed delays, and derived some stability criteria, but the criteria are all delay independent which do not include the information on delay, therefore have a some conservative in. However, the descriptor delay neutral system stability and control have not yet fully investigated, and their stability conditions are not given a strict linear matrix inequalities, it is difficult to achieve through the LMI toolbox in Matlab. Particularly delay-dependent sufficient conditions are few even non-existing in the published works.

In this article, the problem of stability of neutral type descriptor systems with time-varying delays is researched. Using free-weight matrix method in combination with Lyapunov-Krasovskii functional method is used to obtain the LMI-based delay-dependent sufficient conditions for stability. And we consider parameter uncertainties both in its state and in the derivatives of its state. Examples are given to illustrate the effectiveness of the condition.

Notations: The notation in this article is quite standard. Rⁿand R^{n × m}denote, respectively the n- dimensional Euclidean space and the set of all n × m real matrices. The superscript X^T and X^-1 denote, respectively, the transpose and the inverse of any square matrix X. I is the identity matrix of appropriate dimension. || · || will refer to the Euclidean vector norm. The symbol * always denotes the symmetric block in one symmetric matrix.

2. System description

Consider the following uncertain neutral type descriptor time-delay systems:

$$\left\{\begin{array}{c}Eẋ\left(t\right)-\sum _{i=1}^m\left(D_i+\Delta D_i\right)ẋ\left(t-h_i\left(t\right)\right)=\left(A_0+\Delta A_0\right)x\left(t\right)+\sum _{i=1}^m\left(A_i+\Delta A_i\right)x\left(t-d_i\left(t\right)\right)\\ x\left(t\right)=\phi \left(t\right),t\in \left[-\text{max}\left\{h,d\right\},0\right]\end{array}\right.$$

(1)

where x(t) ∈ Rⁿis the state, φ(t) is a continuous vector-valued initial function, $0<h_1\left(t\right)<h_2\left(t\right)<\cdots <h_m\left(t\right)\le h,0=d_0\left(t\right)<d_1\left(t\right)<\cdots <d_m\left(t\right)\le d,0<{ḣ}_t\left(t\right)<{\tau }_i\left(t\right)<{\tau }_i\le \tau \le 1,0<{ḋ}_i\left(t\right)<{\mu }_i\le \mu \le 1$, A₀, A _i , D _i ∈ Rⁿ, A _i , D _i are known constant matrices with appropriate dimensions. Where ΔA _i , ΔD _i are the constant matrices which denote time-varying parameter uncertainties and are assumed to belong to certain bounded compact sets. The parameter uncertainties are assumed to be of the following form:

$$\left[\Delta A_0\left(t\right)\Delta A_i\left(t\right)\Delta D_i\left(t\right)\right]=HF\left(t\right)\left[E_0{E}_{i1}{E}_{i2}\right]$$

(2)

where H, E _ik (k = 1, 2) are known real constant matrices with appropriate dimensions, and F(t) is the uncertain matrix satisfying F^T (t)F(t) ≤ I, ∀t, I is unit matrix with appropriate dimensions.

Remark 1. When E = I, the system (1) reduces to the traditional uncertain neutral system with time-varying delays.

Remark 2. Li [8] considered the stability of neutral type descriptor systems with constant time-delay, and the system did not include parameter uncertainty in the derivative of its state. So the system (1) is more widely in our article.

Lemma 1 (Schur-complement) For any matrix $S=\left[\begin{array}{cc}{S}_{11}\hfill & \hfill S_{12}\hfill \\ S_{12}^{\mathsf{\text{T}}}\hfill & \hfill S_{22}\hfill \end{array}\right]$, with $S_{11}=S_{11}^{\mathsf{\text{T}}},S_{12}=S_{12}^{\mathsf{\text{T}}}$, then the following conditions are equivalent:

$$\left(1\right)S<0,\left(2\right)S_{11}<0,S_{22}-S_{12}^{\mathsf{\text{T}}}{S}_{11}^{-1}{S}_{12}<0,\left(3\right)S_{22}<0,S_{11}-S_{12}{S}_{22}^{-1}{S}_{12}^{\mathsf{\text{T}}}<0$$

Lemma 2 [9] If there is symmetric matrix X, $\left[\begin{array}{cc}{P}_1+X\hfill & \hfill Q_1\hfill \\ Q_1^{\mathsf{\text{T}}}\hfill & \hfill R_1\hfill \end{array}\right]>0,\left[\begin{array}{cc}{P}_2+X\hfill & \hfill Q_2\hfill \\ Q_2^{\mathsf{\text{T}}}\hfill & \hfill R_2\hfill \end{array}\right]>0$ if and only if

$$\left[\begin{array}{ccc}{P}_1+P_2\hfill & \hfill Q_1\hfill & \hfill Q_2\hfill \\ Q_1^{\mathsf{\text{T}}}\hfill & \hfill R_1\hfill & \hfill 0\hfill \\ Q_2^{\mathsf{\text{T}}}\hfill & \hfill 0\hfill & \hfill R_2\hfill \end{array}\right]>0$$

Lemma 3 [10] Given matrices Q = Q^T, H, E with appropriate dimensions, we have

$$Q+HFE+E^{\mathsf{\text{T}}}{F}^{\mathsf{\text{T}}}{H}^{\mathsf{\text{T}}}<0,$$

for all F(t) satisfying F^TF ≤ I if and only if there exists a constant ε > 0, such that

$$Q+\epsilon H{H}^{\mathsf{\text{T}}}+{\epsilon }^{-1}{E}^{\mathsf{\text{T}}}E<0.$$

3. Main results

Theorem 1 The nominal system of the system (1) is asymptotically stable, if there exist nonsingular symmetric matrix P, and positive-definite symmetric matrices Q _i , S _i , R _i and any appropriate dimensional matrices N_{i 0}, N _ij , M _ij (i, j = 1, 2,. .., m), such that the following LMI holds:

$$E^{\mathsf{\text{T}}}P=P^{\mathsf{\text{T}}}E\ge 0$$

(3a)

$$\left[\begin{array}{cccc}\Omega \hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{S}\hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{R}\hfill & \hfill -\bar{N}\hfill \\ *\hfill & \hfill -S\hfill & \hfill 0\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill -R\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill -R\hfill \end{array}\right]<0$$

(3b)

where

$$\begin{array}{c}\Omega =\left[\begin{array}{cccccc}{\Omega }_{00}\hfill & \hfill \cdots \hfill & \hfill {\Omega }_{0m}\hfill & \hfill {\Omega }_{0m+1}\hfill & \hfill \cdots \hfill & \hfill {\Omega }_{02m+1}\hfill \\ ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \cdot \hfill & \hfill \cdots \hfill & \hfill {\Omega }_{mm}\hfill & \hfill {\Omega }_{mm+1}\hfill & \hfill \cdots \hfill & \hfill {\Omega }_{m2m+1}\hfill \\ \cdot \hfill & \hfill \cdots \hfill & \hfill *\hfill & \hfill {\Omega }_{m+1m+1}\hfill & \hfill \cdots \hfill & \hfill {\Omega }_{m+12m+1}\hfill \\ ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ *\hfill & \hfill \cdots \hfill & \hfill *\hfill & \hfill *\hfill & \hfill \cdots \hfill & \hfill {\Omega }_{2m+12m+1}\hfill \end{array}\right],{\bar{N}}_i=\left[\begin{array}{c}{N}_{i0}\hfill \\ ⋮\hfill \\ N_{im}\hfill \\ M_{i1}\hfill \\ ⋮\hfill \\ M_{im}\hfill \end{array}\right]\\ \bar{S}=\left[S_1{S}_2\cdots S_m\right],\bar{R}=\left[d_1{R}_1{d}_2{R}_2\dots d_m{R}_m\right],\Gamma =\left[A_0{A}_1\dots A_m{D}_1\dots D_m\right],\\ \bar{N}=\left[d_1{\bar{N}}_1{d}_2{\bar{N}}_2\dots d_m{\bar{N}}_m\right],S=\mathsf{\text{diag}}\left\{S_1,S_2,\dots ,S_m\right\},R=\mathsf{\text{diag}}\left\{d_1{R}_1,d_2{R}_2,\dots ,d_m{R}_m\right\},\\ {\Omega }_{00}=P^{\mathsf{\text{T}}}{A}_0+A_0^{\mathsf{\text{T}}}P+\sum _{i=0}^m\left[N_{i0}E+E^{\mathsf{\text{T}}}{N}_{i0}^{\mathsf{\text{T}}}\right]+\sum _{i=1}^m{Q}_i,{\Omega }_{0k}=P{A}_k-N_{k0}E+\sum _{i=0}^m{E}^{\mathsf{\text{T}}}{N}_{ik}^{\mathsf{\text{T}}},\\ {\Omega }_{0m+k}=\mathsf{\text{P}}{\mathsf{\text{D}}}_k+\sum _{i=0}^m{E}^{\mathsf{\text{T}}}{M}_{ik}^{\mathsf{\text{T}}},{\Omega }_{kk}=-\left(1-{\mu }_i\right)Q_k-\left(N_{kk}E+E^{\mathsf{\text{T}}}{N}_{kk}^{\mathsf{\text{T}}}\right),k=1,2,\dots ,m,\\ {\Omega }_{lm+k}=-E^{\mathsf{\text{T}}}{M}_{lk}^{\mathsf{\text{T}}},l,k=1,2,\dots ,m,{\Omega }_{lk}=-N_{kl}E-E^{\mathsf{\text{T}}}{N}_{lk}^{\mathsf{\text{T}}},l=1,2,\dots ,m,l<k\le m,\\ {\Omega }_{m+km+k}=-\left(1-{\tau }_i\right)E^{\mathsf{\text{T}}}{S}_{k}E,k=1,2,\dots ,m,\end{array}$$

Proof: Constructing a Lyapunov-Krasovskii functional as follows:

$$V\left(x,t\right)=V_1+V_2+V_3+V_4$$

(4)

in which

$$\begin{array}{c}{V}_1=x^{\mathsf{\text{T}}}\left(t\right)E^{\mathsf{\text{T}}}Px\left(t\right),V_2=\sum _{i=0}^m\underset{t-d_i\left(t\right)}{\overset{t}{\int }}{x}^{\mathsf{\text{T}}}\left(s\right)Q_{i}x\left(s\right)ds,\\ V_3=\sum _{i=0}^m\underset{t-h_i\left(t\right)}{\overset{t}{\int }}{ẋ}^{\mathsf{\text{T}}}\left(s\right)E^{\mathsf{\text{T}}}{S}_{i}Eẋ\left(s\right)ds,V_4=\sum _{i=0}^m\underset{-d_i\left(t\right)}{\overset{0}{\int }}\underset{t+\theta }{\overset{t}{\int }}{ẋ}^{\mathsf{\text{T}}}\left(s\right)E^{\mathsf{\text{T}}}{R}_{i}Eẋ\left(s\right)dsd\theta ,\end{array}$$

The time derivative of V(x, t) along the trajectory of system (1) is given by

$$\begin{array}{c}\stackrel{·}{V}={\stackrel{·}{V}}_1+{\stackrel{·}{V}}_2+{\stackrel{·}{V}}_3+{\stackrel{·}{V}}_4\\ {\stackrel{·}{V}}_1=2x^{\mathsf{\text{T}}}\left(t\right)\mathsf{\text{PE}}ẋ\left(t\right)=2x^{\mathsf{\text{T}}}\left(t\right)P\left(A_{0}x\left(t\right)+\sum _{i=0}^m{A}_{i}x\left(t-d_i\left(t\right)\right)+\sum _{i=0}^m{D}_iẋ\left(t-h_i\left(t\right)\right)\right)\\ {\stackrel{·}{V}}_2=\sum _{i=1}^m{x}^{\mathsf{\text{T}}}\left(t\right)Q_{i}x\left(t\right)-\sum _{i=1}^m\left(1-{\mu }_i\right)x^{\mathsf{\text{T}}}\left(t-d_i\left(t\right)\right)Q_{i}x\left(t-d_i\left(t\right)\right),\\ {\stackrel{·}{V}}_3=\sum _{i=1}^m{ẋ}^{\mathsf{\text{T}}}\left(t\right)E^{\mathsf{\text{T}}}{S}_{i}Eẋ\left(t\right)-\sum _{i=1}^m\left(1-{\tau }_i\right){ẋ}^{\mathsf{\text{T}}}\left(t-h_i\left(t\right)\right)E^{\mathsf{\text{T}}}{S}_{i}Eẋ\left(t-h_i\left(t\right)\right)\\ {\stackrel{·}{V}}_4=\sum _{i=1}^m{d}_i{ẋ}^{\mathsf{\text{T}}}\left(t\right)E^{\mathsf{\text{T}}}{R}_{i}Eẋ\left(t\right)-\sum _{i=1}^m\underset{t-d_i\left(t\right)}{\overset{t}{\int }}{ẋ}^{\mathsf{\text{T}}}\left(s\right)E^{\mathsf{\text{T}}}{R}_{i}Eẋ\left(s\right)ds\end{array}$$

(5)

According to Newton-Leibniz formula, apparently for any appropriate dimensional matrices N_{i 0}, N _ij , M _ij (i, j = 1, 2,..., m), then

$$2\sum _{i=0}^m\left[x^{\mathsf{\text{T}}}\left(t\right)N_{i0}+\sum _{j=1}^m{x}^{\mathsf{\text{T}}}\left(t-d_i\left(t\right)\right)N_{ij}+\sum _{j=1}^m{ẋ}^{\mathsf{\text{T}}}\left(t-h_j\left(t\right)\right)M_{ij}\right]\left[Ex\left(t\right)-Ex\left(t-d_i\left(t\right)\right)-\underset{t-d_i\left(t\right)}{\overset{t}{\int }}Eẋ\left(s\right)ds\right]=0,$$

(6)

As for any appropriate dimensional matrix X _i ≥ 0, (i = 1, 2,..., m), then

$$\sum _{i=0}^m\left[{\xi }_1^{\mathsf{\text{T}}}\left(t\right)X_i{\xi }_1\left(t\right)-\underset{t-d_i\left(t\right)}{\overset{t}{\int }}{\xi }_1^{\mathsf{\text{T}}}\left(t\right)X_i{\xi }_1\left(t\right)ds\right]=0,$$

(7)

where

$$\begin{array}{c}{X}_i=\left[\begin{array}{cccccc}{X}_{i00}\hfill & \hfill \dots \hfill & \hfill X_{i0m}\hfill & \hfill X_{i0m+1}\hfill & \hfill \cdots \hfill & \hfill X_{i02m+1}\hfill \\ ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \cdot \hfill & \hfill \cdots \hfill & \hfill X_{imm}\hfill & \hfill X_{mm+1}\hfill & \hfill \cdots \hfill & \hfill X_{im2m+1}\hfill \\ \cdot \hfill & \hfill \cdots \hfill & \hfill *\hfill & \hfill X_{im+1m+1}\hfill & \hfill \cdots \hfill & \hfill X_{im+12m+1}\hfill \\ ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ *\hfill & \hfill \cdots \hfill & \hfill *\hfill & \hfill *\hfill & \hfill \cdots \hfill & \hfill X_{i2m+12m+1}\hfill \end{array}\right],\\ {\xi }_1^{\mathsf{\text{T}}}\left(t\right)=\left[x^{\mathsf{\text{T}}}\left(t\right)x^{\mathsf{\text{T}}}\left(t-d_1\left(t\right)\right)\dots x^{\mathsf{\text{T}}}\left(t-d_m\left(t\right)\right){ẋ}^{\mathsf{\text{T}}}\left(t-h_m\left(t\right)\right)\dots {ẋ}^{\mathsf{\text{T}}}\left(t-h_m\left(t\right)\right)\right]\end{array}$$

from (6),(7) and (5), lead to

$$\stackrel{·}{V}={\xi }_1^{\mathsf{\text{T}}}\left(t\right)\left(\Omega +\sum _{i=0}^m{d}_i{X}_i+\sum _{i=0}^m{\Gamma }^{\mathsf{\text{T}}}\left(S_i+d_i{R}_i\right)\Gamma \right){\xi }_1\left(t\right)-\underset{t-d_i\left(t\right)}{\overset{t}{\int }}{\xi }_2^{\mathsf{\text{T}}}\left(t,s\right){\Psi }_i{\xi }_2\left(t,s\right)ds$$

(8)

in which

$${\xi }_2^{\mathsf{\text{T}}}\left(t,s\right)=\left[{\xi }_1^{\mathsf{\text{T}}}\left(t\right){\left(Eẋ\left(s\right)\right)}^{\mathsf{\text{T}}}\right],{\Psi }_i=\left[\begin{array}{cc}{d}_i{X}_i\hfill & \hfill d_i{N}_i\hfill \\ *\hfill & \hfill d_i{R}_i\hfill \end{array}\right]$$

If

$$\left(\Omega +\sum _{i=0}^m{d}_i{X}_i+\sum _{i=0}^m{\Gamma }^{\mathsf{\text{T}}}\left(S_i+d_i{R}_i\right)\Gamma \right)<0,{\Psi }_i\ge 0$$

(9)

According to Lyapunov-Krasovskii stability theorem, the system (1) is asymptotically stable. According to Schur-complement,

$$\left(\Omega +\sum _{i=0}^m{d}_i{X}_i+\sum _{i=0}^m{\Gamma }^{\mathsf{\text{T}}}\left(S_i+h_i{R}_i\right)\Gamma \right)<0$$

$$\begin{array}{c}\iff \left[\begin{array}{ccc}\Omega +\sum _{i=0}^m{d}_i{X}_i\hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{S}\hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{R}\hfill \\ *\hfill & \hfill -S\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill -R\hfill \end{array}\right]<0,\\ \iff \left[\begin{array}{ccc}-\Omega -\sum _{i=0}^m{d}_i{X}_i\hfill & \hfill -{\Gamma }^{\mathsf{\text{T}}}\bar{S}\hfill & \hfill -{\Gamma }^{\mathsf{\text{T}}}\bar{R}\hfill \\ *\hfill & \hfill S\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill R\hfill \end{array}\right]>0\end{array}$$

(10)

$${\Psi }_i\ge 0\iff \left[\begin{array}{cc}{d}_i{X}_i\hfill & \hfill d_i{N}_i\hfill \\ *\hfill & \hfill d_i{R}_i\hfill \end{array}\right]>0$$

(11)

According to Lemma 2, from (10) and (11), if and only if

$$\left[\begin{array}{cccc}-\Omega -\sum _{i=0}^m{d}_i{X}_i\hfill & \hfill -{\Gamma }^{\mathsf{\text{T}}}\bar{S}\hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{R}\hfill & \hfill -\bar{N}\hfill \\ *\hfill & \hfill S\hfill & \hfill 0\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill R\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill R\hfill \end{array}\right]>0\iff \left[\begin{array}{cccc}\Omega \hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{S}\hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{R}\hfill & \hfill -\bar{N}\hfill \\ *\hfill & \hfill -S\hfill & \hfill 0\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill -R\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill -R\hfill \end{array}\right]<0$$

Then, we can get the theorem easily.

According to Theorem 1 and Lemma 3, it can be generalized to its structure uncertain neutral generalized time-delay systems, we have the following theorem:

Theorem 2 The system (1) is robustly asymptotically stable, if there exists constant ε₁ > 0, nonsingular symmetric matrix P, positive-definite symmetric matrices Q _i , S _i , R _i and any appropriate dimensional matrices N_{i 0}, N _ij , M _ij (i, j = 1, 2,. .., m), such that the following LMI holds:

$$E^{\mathsf{\text{T}}}P=P^{\mathsf{\text{T}}}E\ge 0$$

(12a)

$$\left[\begin{array}{cccccc}\Omega \hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{S}\hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{R}\hfill & \hfill \epsilon {\Theta }_1\hfill & \hfill {\Theta }_2^{\mathsf{\text{T}}}\hfill & \hfill -\bar{N}\hfill \\ *\hfill & \hfill -S\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill -R\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill -\epsilon I\hfill & \hfill 0\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill -\epsilon I\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill -R\hfill \end{array}\right]<0$$

(12b)

in which

$${\Theta }_1^{\mathsf{\text{T}}}=\left[H^{\mathsf{\text{T}}}P\dots \dots H^{\mathsf{\text{T}}}\bar{S}{H}^{\mathsf{\text{T}}}\bar{R}\right],{\Theta }_2=\left[E_0{E}_{11}\dots E_{m1}{E}_{12}\dots E_{m2}\right]$$

Proof: Replacing A _i , D _i in (3b) with A _i + ΔA _i , D _i + ΔD _i respectively, we find that (2) for (1) is equivalent to the following condition:

$$\left[\begin{array}{cccc}\Omega \hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{S}\hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{R}\hfill & \hfill -\bar{N}\hfill \\ *\hfill & \hfill -S\hfill & \hfill 0\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill -R\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill -R\hfill \end{array}\right]+{\Theta }_{1}F\left(t\right){\Theta }_2+{\Theta }_2^{\mathsf{\text{T}}}{F}^{\mathsf{\text{T}}}\left(t\right){\Theta }_1^{\mathsf{\text{T}}}<0$$

(13)

By Lemma 3, a sufficient condition guaranteeing (13) for (1) is that there exists a positive number ε > 0 such that

$$\left[\begin{array}{cccc}\Omega \hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{S}\hfill & \hfill {\Gamma }^{\mathsf{\text{T}}}\bar{R}\hfill & \hfill -\bar{N}\hfill \\ *\hfill & \hfill -S\hfill & \hfill 0\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill -R\hfill & \hfill 0\hfill \\ *\hfill & \hfill *\hfill & \hfill *\hfill & \hfill -R\hfill \end{array}\right]+\epsilon {\Theta }_1{\Theta }_1^{\mathsf{\text{T}}}+{\epsilon }^{-1}{\Theta }_2^{\mathsf{\text{T}}}{\Theta }_2<0$$

(14)

Applying the Schur complement shows that (14) is equivalent to (12b).The proof is completed.

Remark 3. In the proof of the theorem, it is worth noting that the method taking the relationship between Ex(t) and $Ex\left(t-d_i\left(t\right)\right)-{\int }_{t-d_i\left(t\right)}^{t}Eẋ\left(s\right)ds$ into account is suitable or deriving LMI conditions of the stability.

4. Numerical examples

Consider the system (1) described by

$$\begin{array}{c}E=\left[\begin{array}{cc}1\hfill & \hfill 0\hfill \\ 0\hfill & \hfill 0\hfill \end{array}\right],A_0=\left[\begin{array}{cc}1\hfill & \hfill 0.3\hfill \\ 1\hfill & \hfill -2\hfill \end{array}\right],A_1=\left[\begin{array}{cc}2\hfill & \hfill 1\hfill \\ -0.3\hfill & \hfill 0.5\hfill \end{array}\right],A_2=\left[\begin{array}{cc}0.5\hfill & \hfill 0\hfill \\ 0\hfill & \hfill -0.2\hfill \end{array}\right],D_1=\left[\begin{array}{cc}0.1\hfill & \hfill 0.3\hfill \\ 0\hfill & \hfill 0.2\hfill \end{array}\right],D_2=\left[\begin{array}{cc}0.2\hfill & \hfill 0\hfill \\ 0.3\hfill & \hfill 0.1\hfill \end{array}\right],\\ H=\left[\begin{array}{cc}0.1\hfill & \hfill 0.5\hfill \\ 0.2\hfill & \hfill -0.1\hfill \end{array}\right],E_0=\left[\begin{array}{cc}-0.2\hfill & \hfill 0.1\hfill \\ 0.1\hfill & \hfill 0.3\hfill \end{array}\right],E_1=\left[\begin{array}{cc}0.1\hfill & \hfill 0\hfill \\ 0\hfill & \hfill 0.3\hfill \end{array}\right],E_2=\left[\begin{array}{cc}0.1\hfill & \hfill 0.3\hfill \\ -0.15\hfill & \hfill 0.2\hfill \end{array}\right],E_3=\left[\begin{array}{cc}0.2\hfill & \hfill 0.3\hfill \\ -0.1\hfill & \hfill 0.2\hfill \end{array}\right],\\ E_4=\left[\begin{array}{cc}0.2\hfill & \hfill 0\hfill \\ -0.15\hfill & \hfill 0.1\hfill \end{array}\right],d_1=1.2,d_2=1.5,{\tau }_1=0.3,{\tau }_2=0.4,{\mu }_1=0.6,{\mu }_2=0.8,\epsilon =0.01\end{array}$$

According to the theorem, form (12a), (12b) by LMI toolbox in Matlab, lead to

$$\begin{array}{c}p=\left[\begin{array}{cc}-0.3494\hfill & \hfill 3.7287\hfill \\ 3.7287\hfill & \hfill -4.6081\hfill \end{array}\right],Q_1=10^8\left[\begin{array}{cc}3.0945\hfill & \hfill -0.0510\hfill \\ -0.0510\hfill & \hfill -3.1127\hfill \end{array}\right],Q_2=10^8\left[\begin{array}{cc}4.7370\hfill & \hfill 0.0538\hfill \\ 0.0538\hfill & \hfill 4.7511\hfill \end{array}\right],\\ S_1=10^{-3}\left[\begin{array}{cc}0.2771\hfill & \hfill -0.3533\hfill \\ -0.3533\hfill & \hfill 0.6516\hfill \end{array}\right],S_2=\left[\begin{array}{cc}0.0010\hfill & \hfill -0.0016\hfill \\ -0.0016\hfill & \hfill -0.0031\hfill \end{array}\right],R_1=10^{-3}\left[\begin{array}{cc}0.0558\hfill & \hfill -0.1204\hfill \\ -0.1204\hfill & \hfill 0.3384\hfill \end{array}\right],\\ R_2=10^{-3}\left[\begin{array}{cc}0.0446\hfill & \hfill -0.0963\hfill \\ -0.0963\hfill & \hfill 0.2707\hfill \end{array}\right],N_{10}=10^{-3}\left[\begin{array}{cc}-0.0575\hfill & \hfill 0.2862\hfill \\ -0.0725\hfill & \hfill -0.0215\hfill \end{array}\right],N_{11}=\left[\begin{array}{cc}0.0018\hfill & \hfill 0.0010\hfill \\ 0.0006\hfill & \hfill 0.0006\hfill \end{array}\right],\\ N_{12}=\left[\begin{array}{cc}-0.0001\hfill & \hfill 0.0001\hfill \\ -0.0013\hfill & \hfill -0.0007\hfill \end{array}\right],M_{11}=10^{-8}\left[\begin{array}{cc}-0.1552\hfill & \hfill -0.1066\hfill \\ 0.2627\hfill & \hfill 0.1663\hfill \end{array}\right],M_{12}=10^{-8}\left[\begin{array}{cc}0.2665\hfill & \hfill 0.2039\hfill \\ -0.2238\hfill & \hfill -0.1461\hfill \end{array}\right],\\ N_{20}=10^{-3}\left[\begin{array}{cc}-0.3006\hfill & \hfill -0.0287\hfill \\ 0.0327\hfill & \hfill -0.0163\hfill \end{array}\right],N_{21}=\left[\begin{array}{cc}-0.0011\hfill & \hfill -0.0004\hfill \\ -0.0005\hfill & \hfill -0.0002\hfill \end{array}\right],N_{22}=10^{-3}\left[\begin{array}{cc}0.2400\hfill & \hfill -0.0615\hfill \\ 0.5240\hfill & \hfill 0.2985\hfill \end{array}\right],\\ M_{21}=10^{-9}\left[\begin{array}{cc}0.7342\hfill & \hfill 0.4191\hfill \\ -0.9933\hfill & \hfill -0.6986\hfill \end{array}\right],M_{22}=10^{-8}\left[\begin{array}{cc}-0.2313\hfill & \hfill -0.1270\hfill \\ 0.0793\hfill & \hfill 0.0460\hfill \end{array}\right]\end{array}$$

5. Conclusion

The stability of neutral type descriptor systems with time-varying delays has been solved in terms of LMI approach. Using Lyapunov-Krasovskii functional method, and free-weight matrix method, a criterion for stability of systems is given. In the criterion, the relationship between Ex(t) and $Ex\left(t-d_i\left(t\right)\right)-{\int }_{t-d_i\left(t\right)}^{t}Eẋ\left(s\right)ds$ is taken into account. The criterion is presented in terms o linear matrix inequalities, which can be easily solved by Matlab Toolbox. Finally, a numerical example is presented to illustrate the effectiveness of the method.