Improving Delay-Range-Dependent Stability Condition for Systems with Interval Time-Varying Delay

Abstract

This paper discusses the delay-range-dependent stability for systems with interval time-varying delay. Through defining the new Lyapunov-Krasovskii functional and estimating the derivative of the LKF by introducing new vectors, using free matrices and reciprocally convex approach, the new delay-range-dependent stability conditions are obtained. Two well-known examples are given to illustrate the less conservatism of the proposed theoretical results.

1. Introduction

It is well known that time-varying delays are frequently encountered in many practical control systems, and they are usually regarded as a source of instability and poor performance. So the stability issue of time delay systems has received considerable attention [1–16]. In the last years, in order to further reduce conservatism of the stability results, some methods were developed, such as the delay-fraction approach in [1, 2], free weighting matrices method in [3, 4], the convex analysis method in [5, 6], the reciprocally convex approach in [7], LKF constructing method with matrices that depend on the time delays [8], LKF constructing method with triple-integral terms in [9], LKF constructing method with quadruple-integral terms in [10], and simple LKF having quadratic terms multiplied by a higher degree scalar function [11, 12]. These methods reduced the conservatism of the stability results. But when delay is interval time-varying, the information of the delay derivative is not full used, which causes the conservatism of the stability results.

Motivated by recent methods, in this paper, we further discuss the stability of linear systems with interval time-varying delay. Firstly, a novel LKF is introduced. Then, by introducing new vectors, using free matrices and reciprocally convex approach, the derivative of LKF is estimated less conservatively, and as a result, the stability criterion is obtained in terms of LMI. Finally, two examples are given to illustrate the effectiveness of the proposed method.

Throughout the note, the used notations are standard. 𝐑^𝑛 denotes the 𝑛-dimensional Euclidean space, 𝐑^𝑛×𝑚 is a set of 𝑛×𝑚 real matrix, 𝐴^𝑇 is the transpose of 𝐴, 𝑃>0(𝑃<0) means symmetric positive (negative) definite matrix, and ∗ in the matrix denotes the symmetric element; 𝐼 is the identity matrix of appropriate dimensions, 𝑥_𝑡=𝑥(𝑡+𝜃),𝜃∈[−ℎ,0]. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulations

Consider the following time-delay system: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴₁𝑥((𝑡,𝑥(𝑡)(𝑡)[],𝑡−ℎ))=𝜙,𝑡∈−ℎ,0 (1) where 𝑥(𝑡)∈𝐑^𝑛 is the state vector, the initial condition 𝜙(𝑡) is a continuously differentiable vector-valued function, 𝐴,𝐴₁∈𝐑^𝑛×𝑛 are known real constant matrices, and ℎ(𝑡) is the time-varying delay satisfying ℎ₁≤ℎ(𝑡)≤ℎ₂,ℎ̇(𝑡)≤𝜇≤∞, (2) where ℎ₁, ℎ₂, and 𝜇 are constants.

To obtain the main results, the following lemmas are needed.

Lemma 1 (see [9]). For any symmetric matrix 𝑍=𝑍^𝑇≥0, scalars ℎ₂>ℎ₁>0 such that the integration is well defined; then −ℎ₂−ℎ₁^−ℎ1_−ℎ2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑍𝑥(𝑠)𝑑𝑠𝑑𝜃≤−^−ℎ1_−ℎ2^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑑𝑠𝑑𝜃𝑍^−ℎ1_−ℎ2^𝑡_𝑡+𝜃𝑥(𝑠)𝑑𝑠𝑑𝜃. (3)

Lemma 2 (see [7]). Let 𝑓₁,𝑓₂,…,𝑓_𝑁∶𝐑^𝑚→𝐑 have positive values in an open subset 𝐃 of  𝐑^𝑚. Then, the reciprocally convex combination of 𝑓_𝑖 over 𝐃 satisfies min_{𝛼𝑖∣𝛼𝑖>0,∑𝛼𝑖=1𝑖}1𝛼_𝑖𝑓_𝑖(𝑡)=_𝑖𝑓_𝑖(𝑡)+max_{𝑔𝑖𝑗(𝑡)𝑖≠𝑗}𝑔_𝑖𝑗(𝑡)𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑡𝑜𝑔_𝑖,𝑗∶𝐑^𝑚⟶𝐑,𝑔_𝑖𝑗(𝑡)=𝑔_𝑗𝑖(𝑡),𝑓_𝑖(𝑡)𝑔_𝑖𝑗(𝑡)𝑔_𝑖𝑗(𝑡)𝑓_𝑗(𝑡)≥0. (4)

Lemma 3 (see [16]). The symmetric appropriately dimensional matrices Θ>0,Ξ,Υ, if  Ξ−ΥΘΥ^𝑇<0 hold; then there exists a matrix of appropriate dimension Π such that the following LMI holds: Ξ+ΥΠ^𝑇+ΠΥ^𝑇Π∗−Θ<0. (5)

3. Main Results

In this section, the stability of system (1) is investigated. Through constructing a novel LKF and estimating the derivative of it, new stability condition is provided.

Firstly, define the following vector as 𝜁^𝑇(𝑡)=𝑥^𝑇(𝑡)^𝑡_𝑡−ℎ1𝑥^𝑇(𝑠),𝜉𝑑𝑠_𝑇(𝑡)=𝑥̇^𝑇(𝑡)𝑥^𝑇(𝑡), (6) and then construct the L-K functional as follows: 𝑉𝑥_𝑡=⁴_𝑖=1𝑉_𝑖𝑥_𝑡, (7) where 𝑉₁𝑥_𝑡=𝜁^𝑇(𝑡)(𝑡),𝑉𝑃𝜁₂𝑥_𝑡=^𝑡_𝑡−ℎ1𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)𝑑𝑠+^𝑡−ℎ1_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)+𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ2}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)𝑉𝑑𝑠,₃𝑥_𝑡=ℎ₁⁰_−ℎ1^𝑡_𝑡+𝜃𝜉^𝑇(𝑠)𝑅₁0∗𝑅₂𝜉(𝑠)+ℎ𝑑𝑠𝑑𝜃₂−ℎ₁^−ℎ1_−ℎ2^𝑡_𝑡+𝜃𝜉^𝑇(𝑠)𝑋₁0∗𝑋₂𝜉(𝑠)𝑉𝑑𝑠𝑑𝜃,₄𝑥_𝑡=ℎ²₁2⁰_−ℎ1⁰_𝜃^𝑡_𝑡+𝜆𝑥̇^𝑇(𝑠)𝑈₁𝑥̇(𝑠)+ℎ𝑑𝑠𝑑𝜆𝑑𝜃₂₂−ℎ²₁2^−ℎ2_−ℎ1⁰_𝜃^𝑡_𝑡+𝜆𝑥̇^𝑇(𝑠)𝑈₂𝑥̇(𝑠)+1𝑑𝑠𝑑𝜆𝑑𝜃2^𝑡−ℎ1_𝑡−ℎ2^𝑡−ℎ1_𝜃^𝑡_𝜆𝑥̇^𝑇(𝑠)𝑈₃𝑥̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃, (8) and matrices 𝑄₁,𝑄₂,𝑄₃,𝑅₁,𝑅₂,𝑋₁,𝑋₂,𝑈₁,𝑈₂,𝑈₃∈𝐑^𝑛×𝑛>0, 𝑃=[𝑃_𝑖𝑗](1≤𝑖≤𝑗≤2)∈𝐑^2𝑛×2𝑛>0.

In order to make the proof process clear, the following notations are used:𝛼(𝑡)=ℎ(𝑡)−ℎ₁,𝛽(𝑡)=ℎ₂(𝑡)(𝑡)=1−ℎ,𝜑𝛼(𝑡)^𝑡−ℎ1_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)(𝑡)=1𝑑𝑠,𝜙𝛽(𝑡)^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥^𝑇(𝑠)𝜒𝑑𝑠,_𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑥^𝑇𝑡−ℎ₂^𝑡_𝑡−ℎ1𝑥^𝑇(𝑠)𝜑^𝑇(𝑡)𝜙^𝑇(𝑡)𝑥̇^𝑇(𝑡) (9)and block entry matrices 𝑒_𝑖(𝑖=1,2,…,5) (e.g., 𝑒^𝑇₂=[0𝐼000000]).

Then taking the time derivatives of 𝑉_𝑖(𝑥_𝑡) along the trajectory of system (1) yields𝑉̇₁𝑥_𝑡=2𝜁^𝑇(𝑡)𝑃𝜁̇(𝑡)=2𝜒^𝑇(𝑡)𝑒₁𝑒₅𝑃𝑒₈𝑒₁−𝑒₃^𝑇𝜒(𝑡),𝑉̇₂𝑥_𝑡=𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)−𝑥^𝑇𝑡−ℎ₁𝑄₁−𝑄₂𝑥𝑡−ℎ₁−1−ℎ̇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑄₂−𝑄₃𝑥(𝑡−ℎ(𝑡))−𝑥^𝑇𝑡−ℎ₂𝑄₃𝑥𝑡−ℎ₂≤𝜒^𝑇(𝑡)𝑒₁𝑄₁𝑒^𝑇₁−𝑒₃𝑄₁−𝑄₂𝑒^𝑇₃−1−𝜇𝑒₂𝑄₂−𝑄₃𝑒^𝑇₂−𝑒₄𝑄₃𝑒^𝑇₄𝜒(𝑡), (10) where 𝑄₂−𝑄₃>0.

To reduce the conservatism of the main result, by using the Jesen inequality and Lemma 2 with introducing appropriate matrices 𝑌₁,𝑌₂∈𝐑^𝑛×𝑛, we can calculate 𝑉̇₃(𝑥_𝑡) as 𝑉̇₃𝑥_𝑡=ℎ²₁𝜉^𝑇(𝑡)𝑅₁0∗𝑅₂𝜉(𝑡)+ℎ₂−ℎ₁²𝜉^𝑇(𝑡)𝑋₁0∗𝑋₂𝜉(𝑡)−ℎ₁^𝑡_𝑡−ℎ1𝜉^𝑇(𝑠)𝑅₁0∗𝑅₂𝜉(𝑠)−ℎ𝑑𝑠₂−ℎ₁^𝑡−ℎ1_𝑡−ℎ2𝜉^𝑇(𝑠)𝑋₁0∗𝑋₂𝜉(𝑠)𝑑𝑠≤𝜒^𝑇(𝑡)𝑒₈ℎ²₁𝑅₁+ℎ₂−ℎ₁²𝑋₁𝑒^𝑇₈+𝑒₁ℎ²₁𝑅₂+ℎ₂−ℎ₁²𝑋₂𝑒^𝑇₁−𝑒₁−𝑒₃𝑅₁𝑒₁−𝑒₃^𝑇−𝑒₅𝑅₂𝑒^𝑇₅𝜒(𝑡)−𝜒^𝑇(𝑡)𝑒₂−𝑒₃𝑒₄−𝑒₂×𝑋₁𝑌₁∗𝑋₁𝑒₂−𝑒₃𝑒₄−𝑒₂^𝑇𝜒(𝑡)−Γ₁, (11) where by using Lemma 2 with matrix 𝑌₁, one can get −ℎ₂−ℎ₁^𝑡−ℎ1_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑋₁𝑥̇(𝑠)ℎ𝑑𝑠=−₂−ℎ₁^𝑡−ℎ1_{𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑋₁𝑥̇(𝑠)−ℎ𝑑𝑠₂−ℎ₁^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑋₁𝑥̇(𝑠)𝑑𝑠≤−𝑥(𝑡−ℎ(𝑡))−𝑥(𝑡−ℎ₁)𝑥(𝑡−ℎ₂)−𝑥(𝑡−ℎ(𝑡))^𝑇×𝑋₁𝑌₁∗𝑋₁𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₁𝑥𝑡−ℎ₂−𝑥(𝑡−ℎ(𝑡))=−𝜒^𝑇(𝑡)𝑒₂−𝑒₃𝑒₄−𝑒₂𝑋₁𝑌₁∗𝑋₁×𝑒₂−𝑒₃𝑒₄−𝑒₂^𝑇𝜒(𝑡), (12) and by using Lemma 2 with matrix 𝑌₂, one can get −ℎ₂−ℎ₁^𝑡−ℎ1_𝑡−ℎ2𝑥^𝑇(𝑠)𝑋₂𝑥(𝑠)ℎ𝑑𝑠=−₂−ℎ₁^𝑡−ℎ1_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝑋₂𝑥(𝑠)−ℎ𝑑𝑠₂−ℎ₁^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥^𝑇(𝑠)𝑋₂𝑥(𝑠)∫𝑑𝑠≤−^{𝑡−ℎ1𝑡−ℎ(𝑡)}∫𝑥(𝑠)𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ2}𝑥(𝑠)𝑑𝑠^𝑇×𝑋₂𝑌₂∗𝑋₂∫^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥(𝑠)∫𝑑𝑠_{𝑡−ℎ(𝑡)𝑡−ℎ2}𝑥(𝑠)𝑑𝑠=Γ₁. (13)

To the time-derivative of 𝑉₄(𝑥_𝑡), it can be calculated as 𝑉̇₄𝑥_𝑡=ℎ⁴₁4𝑥̇^𝑇(𝑡)𝑈₁𝑥̇(𝑡)+ℎ²₂−ℎ²₁²4𝑥̇^𝑇(𝑡)𝑈₂𝑥̇(𝑡)+ℎ₂−ℎ₁²4𝑥̇^𝑇(𝑡)𝑈₃𝑥̇(𝑡)−ℎ²₁2⁰_−ℎ1^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑈₁𝑥̇(𝑠)−ℎ𝑑𝑠𝑑𝜃₂₂−ℎ²₁2^−ℎ1_−ℎ2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑈₂𝑥̇(𝑠)−1𝑑𝑠𝑑𝜃2^𝑡−ℎ1_𝑡−ℎ2^𝑡−ℎ1_𝜃𝑥̇^𝑇(𝑠)𝑈₃𝑥̇(𝑠)𝑑𝑠𝑑𝜃=𝜒^𝑇(𝑡)𝑒₈ℎ²₁2𝑈₁+ℎ²₂−ℎ²₁²4𝑈₂+ℎ₂−ℎ₁²4𝑈₃𝑒^𝑇₈𝜒(𝑡)−𝜒^𝑇(𝑡)ℎ₁𝑒₁−𝑒₅𝑈₁ℎ₁𝑒₁−𝑒₅^𝑇𝜒(𝑡)−𝜒^𝑇(𝑡)𝑒₃−𝑒₆^𝑇𝑈₃𝑒₃−𝑒₆−𝑒₂−𝑒₇^𝑇𝑈₃𝑒₂−𝑒₇𝜒(𝑡)−Γ₂, (14) where by using Lemma 1, we can get −ℎ²₁2⁰_−ℎ1^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑈₁𝑥̇(𝑠)ℎ𝑑𝑠𝑑𝜃≤−₁𝑥(𝑡)−^𝑡_𝑡−ℎ1𝑥(𝑠)𝑑𝑠^𝑇×𝑈₁ℎ₁𝑥(𝑡)−^𝑡_𝑡−ℎ1𝑥(𝑠)𝑑𝑠=−𝜒^𝑇(𝑡)ℎ₁𝑒₁−𝑒₅𝑈₁ℎ₁𝑒₁−𝑒₅^𝑇𝜒(𝑡),−12^𝑡−ℎ1_𝑡−ℎ2^𝑡−ℎ1_𝜃𝑥̇^𝑇(𝑠)𝑈₃𝑥̇(𝑠)1𝑑𝑠𝑑𝜃≤−2^𝑡−ℎ1_{(𝑡)𝑡−ℎ}^𝑡−ℎ1_𝜃𝑥̇^𝑇(𝑠)𝑈₃𝑥̇(𝑠)−1𝑑𝑠𝑑𝜃2^{𝑡−ℎ(𝑡)}_{𝑡−ℎ2𝜃}^{𝑡−ℎ(𝑡)}𝑥̇^𝑇(𝑠)𝑈₃𝑥̇(𝑠)ℎ(𝑡)𝑑𝑠𝑑𝜃≤−−ℎ₁𝑥𝑡−ℎ₁−^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥(𝑠)𝑑𝑠^𝑇×𝑈₃𝛼²(𝑡)ℎ(𝑡)−ℎ₁𝑥𝑡−ℎ₁−^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥(𝑠)−ℎ𝑑𝑠₂−ℎ(𝑡)𝑥(𝑡−ℎ(𝑡))−^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥(𝑠)𝑑𝑠^𝑇×𝑈₃𝛽²(𝑡)ℎ₂−ℎ(𝑡)𝑥(𝑡−ℎ(𝑡))−^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥(𝑠)𝑑𝑠=−𝜒^𝑇(𝑡)𝑒₃−𝑒₆^𝑇𝑈₃𝑒₃−𝑒₆−𝑒₂−𝑒₇^𝑇𝑈₃𝑒₂−𝑒₇𝜒(𝑡). (15)

Similar to the method in (12)-(13), by introducing matrix 𝑌₃∈𝐑^𝑛×𝑛 and using Lemma 2, we can estimate the following: −ℎ²₂−ℎ²₁2^−ℎ1_−ℎ2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑈₂𝑥̇(𝑠)ℎ𝑑𝑠𝑑𝜃=−²²−ℎ²₁2^−ℎ1_−ℎ(𝑡)^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑈₂𝑥̇(𝑠)−ℎ𝑑𝑠𝑑𝜃₂₂−ℎ²₁2^−ℎ(𝑡)_−ℎ2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑈₂𝑥̇(𝑠)𝑑𝑠𝑑𝜃≤−(ℎ(𝑡)−ℎ₁)𝑥(𝑡)−∫^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥(𝑠)𝑑𝑠(ℎ₂−ℎ(𝑡))𝑥(𝑡)−∫^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥(𝑠)𝑑𝑠^𝑇×𝑈₂𝑌₃∗𝑈₂ℎ(𝑡)−ℎ₁𝑥(𝑡)−∫^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥(𝑠)ℎ𝑑𝑠₂−ℎ(𝑡)𝑥(𝑡)−∫^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥(𝑠)𝑑𝑠=Γ₂. (16)

In the last, in order to obtain the stability result base on LMI, we can deal with Γ₁,Γ₂ as follows: Γ₁+Γ₂=−𝜒^𝑇(𝑡)00000𝛼(𝑡)00000000𝛽(𝑡)0^𝑇×𝑋₂𝑌₂∗𝑋₂(𝑡)(𝑡)0𝜒(𝑡)00000𝛼00000000𝛽−𝜒^𝑇(𝑡)𝛼(𝑡)0000−𝛼(𝑡)00𝛽(𝑡)00000−𝛽(𝑡)0^𝑇×𝑈₂𝑌₃∗𝑈₂×𝛼(𝑡)(𝑡)𝛽(𝑡)(𝑡)0𝜒(𝑡)0000−𝛼0000000−𝛽=−𝜒^𝑇(𝑡)Υ(𝑡)ΘΥ^𝑇(𝑡)𝜒(𝑡), (17) where Υ^𝑇(𝑡)=𝛼(𝑡)(𝑡)𝛽(𝑡)(𝑡)0(𝑡)(𝑡)0,𝑈0000−𝛼0000000−𝛽00000𝛼00000000𝛽Θ=²𝑌₃00∗𝑈₂0000𝑋₂𝑌₂00∗𝑋₂. (18) Also, for appropriate matrices 𝑁₁,𝑁₂, one can have 2𝑥^𝑇(𝑡)𝑁₁+𝑥̇^𝑇(𝑡)𝑁₂×𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡−ℎ(𝑡))−𝑥̇(𝑡)=0. (19)

Therefore, combining (10)–(19), we can obtain 𝑉̇𝑥_𝑡=⁴_𝑖=1𝑉̇_𝑖𝑥_𝑡≤𝜒^𝑇(𝑡)Ξ−Υ(𝑡)ΘΥ^𝑇(𝑡)𝜒(𝑡), (20) where Ξ=𝑒₁𝑒₅𝑃𝑒₈𝑒₁−𝑒₃^𝑇+𝑒₈𝑒₁−𝑒₃𝑃𝑒₁𝑒₅^𝑇−𝑒₃𝑄₁−𝑄₂𝑒^𝑇₃−1−𝜇𝑒₂𝑄₂−𝑄₃𝑒^𝑇₂−𝑒₄𝑄₃𝑒^𝑇₄+𝑒₈ℎ²₁𝑅₁+ℎ₂−ℎ₁²𝑋₁+ℎ²₁2𝑈₁+ℎ²₂−ℎ²₁²4𝑈₂+ℎ₂−ℎ₁²4𝑈₃𝑒^𝑇₈+𝑒₁𝑄₁+ℎ²₁𝑅₂+ℎ₂−ℎ₁²𝑋₂𝑒^𝑇₁−𝑒₁−𝑒₃𝑅₁𝑒₁−𝑒₃^𝑇−𝑒₅𝑅₂𝑒^𝑇₅−𝑒₂−𝑒₃𝑒₄−𝑒₂𝑋₁𝑌₁∗𝑋₁𝑒₂−𝑒₃𝑒₄−𝑒₂^𝑇−ℎ₁𝑒₁−𝑒₅𝑈₁ℎ₁𝑒₁−𝑒₅^𝑇−𝑒₃−𝑒₆^𝑇𝑈₃𝑒₃−𝑒₆−𝑒₂−𝑒₇^𝑇𝑈₃𝑒₂−𝑒₇+𝑒₁𝑁₁+𝑒₈𝑁₂𝑒^𝑇₁𝐴+𝑒^𝑇₂𝐴₁−𝑒₈^𝑇+𝑒^𝑇₁𝐴+𝑒^𝑇₂𝐴₁−𝑒₈𝑒₁𝑁₁+𝑒₈𝑁₂^𝑇. (21) As we know, if Ξ−Υ(𝑡)ΘΥ^𝑇(𝑡)<0 holds, then 𝑉̇(𝑥_𝑡)<0, which means that system (1) is asymptotically stable. So, by Lemma 3, there exists a matrix of appropriate dimension Π such that the following LMI holds: Ξ+ΥΠ^𝑇+ΠΥ^𝑇Π∗−Θ<0. (22)

So, we give the main theorem of this paper as follows.

Theorem 4. The time-delayed system (1) having contrains (2) is asymptotically stable, if there exist matrices 𝑄₁,𝑄₂,𝑄₃,𝑅₁,𝑅₂,𝑋₁,𝑋₂,𝑈₁,𝑈₂,𝑈₃∈𝐑^𝑛×𝑛>0, matrix 𝑃∈𝐑^2𝑛×2𝑛>0, and appropriate matrices 𝑌₁,𝑌₂,𝑌₃,𝑁₁,𝑁₂,Π such that the following LMIs hold: 𝑄₂−𝑄₃>0,Ξ+Υ₁Π^𝑇+ΠΥ^𝑇₁Π∗−Θ<0,Ξ+Υ₂Π^𝑇+ΠΥ^𝑇₂Π∗−Θ<0, (23) where Ξ, Υ, Π, and Θ are defined in (18) and (21).

Proof. From above, one can see that Ξ+ΥΠ^𝑇+ΠΥ^𝑇 is a convex combination of matrix Π as Ξ+ΥΠ^𝑇+ΠΥ^𝑇=Ξ+ℎ(𝑡)−ℎ₁Υ₁Π^𝑇+ΠΥ^𝑇₁+ℎ₂−ℎ(𝑡)Υ₂Π^𝑇+ΠΥ^𝑇₂<0. (24) So (22) can be handled nonconservatively by the two corresponding boundary LMIs: Ξ+Υ₁Π^𝑇+ΠΥ^𝑇₁Π∗−Θ<0,Ξ+Υ₂Π^𝑇+ΠΥ^𝑇₂Π∗−Θ<0, (25) where Υ₁=Υ(ℎ(𝑡)−ℎ₁=0), Υ₂=Υ(ℎ₂−ℎ(𝑡)=0). This completes the proof.

Remark 5. From (17) and (22), it can be seen that, in order to obtain the less conservative stability condition in terms of LMI, 𝛼(𝑡), 𝛽(𝑡) are used, but free matrices Π=[Π_𝑖𝑗](1≤𝑖≤𝑗≤8) are also introduced, which increases the computational burden; this is a disadvantage of the proposed method.

Remark 6. When ℎ₁≤ℎ(𝑡)≤ℎ₂, 𝜇₁≤ℎ̇(𝑡)≤𝜇₂, let Ξ₁=Ξ(𝜇=𝜇₁), Ξ₂=Ξ(𝜇=𝜇₂), and using the convex combination again with ^{Ξ1+Υ1Π𝑇+ΠΥ𝑇1Π}_∗−Θ<0, ^{Ξ1+Υ2Π𝑇+ΠΥ𝑇2Π}_∗−Θ<0, ^{Ξ2+Υ1Π𝑇+ΠΥ𝑇1Π}_∗−Θ<0, ^{Ξ2+Υ2Π𝑇+ΠΥ𝑇2Π}_∗−Θ<0, the corresponding stability criterion can also be obtained.

4. Numerical Examples

In this section, the effectiveness of the obtained results in this paper is shown by the two well-known numerical examples.

Example 1. Consider the linear system (1) with 𝐴=−200−0.9,𝐴₁=−10−1−1. (26)
The purpose is to compare the admissible upper bounds ℎ₂ given lower bound ℎ₁ and 𝜇.
For ℎ₁=0, 𝜇=0.5,0.9, in [14], the result is ℎ₂=2.04,1.37, in [17], the result is ℎ₂=2.33,1.87, and our results are ℎ₂=2.33,1.87.
For ℎ₁=1, 𝜇=0.5,0.9, in [14], the result is ℎ₂=2.07,1.74, in [17], the result is ℎ₂=2.33,2.07, and our results are ℎ₂=2.43,2.08.
For ℎ₁=2, 𝜇=0.5,0.9, in [14], the result is ℎ₂=2.43,2.43, in [17], the result is ℎ₂=2.61,2.61, and our results are ℎ₂=2.64,2.64.
For various ℎ₁,𝜇, the maximum upper bounds on delay ℎ₂ by different methods are also listed in Table 1. It can be seen that the result obtained in this paper is less conservative.

Table 1 

Admissible upper bound ℎ₂ for various ℎ₁ and 𝜇 in Example 1.

Example 2. Consider the linear system (1) with 𝐴=01−1−2,𝐴₁=00−11. (27)
The purpose is to compare the admissible upper bounds ℎ₂ which guarantee the asymptotic stability of the above system for given lower bound ℎ₁ and 𝜇. This example is used in many recent papers, such as [6, 9], For 𝜇=0.3, Table 2 lists the comparison of our results with some recent ones, and it is easy to see that the results obtained in this paper are less conservative.

Table 2 

Admissible upper bound ℎ₂ for various ℎ₁ and 𝜇=0.3 in Example 1.

5. Conclusions

In this note, the stability of interval time-varying delay systems has been discussed. Through constructing a novel LKF and using some new analysis methods, the delay-range-dependent stability criteria were derived. Compared with some previous stability conditions, the obtained main results in this paper have less conservatism. In the end, numerical examples were given to show the superiority of the obtained criteria and their improvements over the existing results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the National Nature Science Foundation under Grants 61104119, the Science and Technology Innovation Talents Project of Henan University under Grant 13HASTIT044, the Doctoral Foundation from Henan Polytechnic University under Grants B2010-50, and the Young Core Instructor Foundation from the Department of Education of Henan Province under Grant 2011GGJS-054.