New Delay-Dependent Stability Conditions for Time-Varying Delay Systems

Abstract

This paper addresses the delay-dependent stability for systems with time-varying delay. First, by taking multi-integral terms into consideration, new Lyapunov-Krasovskii functional is defined. Second, in order to reduce the computational complexity of the main results, reciprocally convex approach and some special transformations are introduced, and new delay-dependent stability criteria are proposed, which are less conservative and have less decision variables than some previous results. Finally, two well-known examples are given to illustrate the correctness and advantage of our theoretical results.

1. Introduction

Time-varying delays are often source of the degradation of performance and the instability of practical control systems. In the last decades, the stability of time-delay systems was one of the hot issues in control theory, and many approaches are proposed to study this topic, such as the properly chosen LKFs [1–3], the delay-fraction approach [4–6], descriptor model transformation method [7], integral inequality lemma [8–10], free weighting matrices [11], and Jensen inequality [12, 13].

Recently, in order to further reduce conservatism of the stability results, some new methods were developed. In [14], by constructing novel LKF with matrices that depended on the time delays, stability conditions that depend on both the upper and lower bounds on delay derivatives were obtained. In [15], the convex analysis method was proposed, and in [16], reciprocally convex approach was used to consider the stability of time-varying delay systems. In [17], new LKF including quadratic terms multiplied by a higher degree scalar function was proposed and this method was further improved in [18], where the quadratic convex combination technique was used. In [19], some triple-integral terms were introduced to LKF, and in [20], some quadruple-integral terms were used in LKF to discuss the stability of time-varying delay systems. These methods reduced the conservatism of the stability results, but more decision variables were introduced; that is, the advantage in conservatism is obtained at the cost of high computational complexity. The above analysis naturally leads us to find possible solutions to this problem.

In this paper, we further discuss the stability of linear systems with time-varying delay. The main contribution of this paper lies in two aspects. First, a novel LKF which contains some multi-integral terms is introduced; second, the derivative of LKF is estimated by using reciprocally convex approach and some special transformations, which reduces the conservatism as well as the computational burden of the main results.

Throughout the paper, the used notations are standard. 𝐑^𝑛 denotes the 𝑛-dimensional Euclidean space, 𝐑^𝑛×𝑚 is a set of 𝑛×𝑚 real matrices, 𝐴^𝑇 is the transpose of 𝐴, 𝑃>0(𝑃<0) means symmetric positive (negative) definite matrix, ∗ in the matrix denotes the symmetric element, 𝐼 is the identity matrix of appropriate dimensions, and 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 (𝑡)ℎ̇(𝑡)0≤ℎ≤ℎ,≤𝑑≤∞, (2)(3) where 0<ℎ and 𝑑≥0 are constants.

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

Lemma 1 (see [16]). 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) (4) subject to 𝑔_𝑖,𝑗∶𝐑^𝑚⟶𝐑,𝑔_𝑖𝑗(𝑡)=𝑔_𝑗𝑖(𝑡),𝑓_𝑖(𝑡)𝑔_𝑖𝑗(𝑡)𝑔_𝑖𝑗(𝑡)𝑓_𝑗(𝑡)≥0. (5)

Lemma 2 (see [18]). For any symmetric matrix 𝑋₀,𝑋₁,𝑋₂ and a vector 𝜉, let 𝑓(𝛼)=𝜉^𝑇𝑋₀𝜉+𝛼𝜉^𝑇𝑋₁𝜉+𝛼²𝜉^𝑇𝑋₂𝜉 (6) with 𝑋₂>0. Then for all 𝛼∈[𝛼₁,𝛼₂], one has 𝑓(𝛼₁)<0 and 𝑓(𝛼₂)<0⇒𝑓(𝛼)<0.

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, construct the following LKF candidate: 𝑉𝑥_𝑡=⁵_𝑖=1𝑉_𝑖𝑥_𝑡, (7) where 𝑉₁𝑥_𝑡=𝑥^𝑇(𝑡)(𝑡),𝑉𝑃𝑥₂𝑥_𝑡=^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑡)𝑥^𝑇(𝑠)𝑄𝑥^𝑇(𝑡)𝑥^𝑇(𝑠)^𝑇𝑉𝑑𝑠,₃𝑥_𝑡=^𝑡_𝑡−ℎ𝑥^𝑇(𝑡)𝑥^𝑇(𝑠)𝑅𝑥^𝑇(𝑡)𝑥^𝑇(𝑠)^𝑇𝑉𝑑𝑠,₄𝑥_𝑡=ℎ⁰_−ℎ^𝑡_𝑡+𝜃𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)𝑀𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)^𝑇𝑉𝑑𝑠𝑑𝜃,₅𝑥_𝑡=2⁰_−ℎ⁰_𝜃^𝑡_𝑡+𝜆𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃, (8) and matrices 𝑃,𝑈∈𝐑^𝑛×𝑛,𝑄={𝑄_𝑖𝑗(1≤𝑖≤𝑗≤2)},𝑅={𝑅_𝑖𝑗(1≤𝑖≤𝑗≤2)},𝑀={𝑀_𝑖𝑗𝐑(1≤𝑖≤𝑗≤2)}∈_2𝑛×2𝑛>0.

Next, let us define a vector as𝜒^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))𝑥^𝑇(𝑡−ℎ)∫^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠∫^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝑑𝑠 (9)and 𝐴_𝑐=𝐴𝐴₁000, block entry matrices 𝑒_𝑖(𝑖=1,2,…,5) (e.g., 𝑒^𝑇₂=0𝐼000).

Taking the time derivatives of 𝑉_𝑖(𝑥_𝑡) along the trajectory of system (1) yields 𝑉̇₁𝑥_𝑡=𝜒^𝑇(𝑡)𝑒₁𝑃𝐴_𝑐+𝐴^𝑇_𝑐𝑃𝑒^𝑇₁𝜒(𝑡),𝑉̇₂𝑥_𝑡=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)𝑄𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)^𝑇−1−ℎ̇(𝑡)𝑥^𝑇(𝑡)𝑥^𝑇((𝑡𝑥𝑡−ℎ))×𝑄^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ(𝑡))^𝑇+2𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡−ℎ(𝑡))^𝑇×𝑄₁₁ℎ(𝑡)𝑥(𝑡)+𝑄₁₂^𝑡_{(𝑡)𝑡−ℎ}𝑥(𝑠)𝑑𝑠≤𝜒^𝑇(𝑡)𝑒₁𝑒₁𝑄𝑒₁𝑒₁^𝑇−()×𝑒1−𝑑₁𝑒₂𝑄𝑒₁𝑒₂^𝑇𝜒(𝑡)+2𝜒^𝑇(𝑡)𝐴^𝑇_𝑐𝑒₀𝑄ℎ(𝑡)𝑒₁𝑒₅^𝑇𝜒(𝑡),𝑉̇₃𝑥_𝑡=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)𝑅𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)^𝑇−𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ)𝑅𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ)^𝑇+2𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡−ℎ(𝑡))^𝑇×𝑅₁₁ℎ𝑥(𝑡)+𝑅₁₂^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠=𝜒^𝑇(𝑡)𝑒₁𝑒₁𝑅𝑒₁𝑒₁^𝑇−𝑒₁𝑒₃𝑅𝑒₁𝑒₃^𝑇𝜒(𝑡)+2𝜒^𝑇(𝑡)𝐴^𝑇_𝑐𝑒₀𝑅ℎ𝑒₁𝑒₄+𝑒₅^𝑇𝜒(𝑡). (10)(11)

To the time derivative of 𝑉₄(𝑥_𝑡), it can be calculated as 𝑉̇₄𝑥_𝑡=ℎ²𝑥^𝑇(𝑡)𝑥̇^𝑇(𝑡)𝑀𝑥^𝑇(𝑡)𝑥̇^𝑇(𝑡)^𝑇−ℎ^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)𝑀𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)^𝑇𝑑𝑠=ℎ²𝑥^𝑇(𝑡)𝑥̇^𝑇(𝑡)𝑀𝑥^𝑇(𝑡)𝑥̇^𝑇(𝑡)^𝑇−ℎ^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)𝑀𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)^𝑇𝑑𝑠−ℎ^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)𝑀𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)^𝑇𝑑𝑠. (12)

To reduce the conservatism of the main results, the last two terms of 𝑉̇₄(𝑥_𝑡) were estimated by different methods. In [11], the free matrices were introduced. Reference [13] used convex combination by introducing an approximation on the difference between delay bounds. Reference [16] adopted reciprocally convex approach, which achieved performance behavior identical to approaches based on the integral inequality lemma but with much less decision variables. Therefore, in the following, by introducing appropriate matrix 𝑁∈𝐑^2𝑛×2𝑛 and using Lemma 1, we can estimate 𝑉̇₄(𝑥_𝑡) as 𝑉̇₄𝑥_𝑡≤ℎ²𝑥^𝑇(𝑡)𝑥̇^𝑇(𝑡)𝑀𝑥^𝑇(𝑡)𝑥̇^𝑇(𝑡)^𝑇−ℎℎ(𝑡)^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)𝑑𝑠^𝑇×𝑀^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)−ℎ𝑑𝑠ℎ−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)𝑑𝑠^𝑇×𝑀^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥^𝑇(𝑠)𝑥̇^𝑇(𝑠)𝑑𝑠≤𝜒^𝑇(𝑡)ℎ²𝑒₁𝐴^𝑇_𝑐𝑀𝑒₁𝐴^𝑇_𝑐^𝑇−𝑒₅𝑒₁−𝑒₂𝑀𝑒₅𝑒₁−𝑒₂^𝑇+𝑒₄𝑒₂−𝑒₃𝑀𝑒₄𝑒₂−𝑒₃^𝑇+2𝑒₅𝑒₁−𝑒₂𝑁𝑒₄𝑒₂−𝑒₃^𝑇𝜒(𝑡), (13) where ^𝑀𝑁_∗𝑀≥0.

To the time derivative of 𝑉₅(𝑥_𝑡), on the first step, the following transformation is used: 𝑉̇₅𝑥_𝑡=ℎ²𝑥̇^𝑇(𝑡)𝑈𝑥̇(𝑡)−2⁰_−ℎ^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠𝑑𝜃=ℎ²𝜒^𝑇(𝑡)𝐴^𝑇_𝑐𝑈𝐴_𝑐𝜒(𝑡)−2^𝑡_{𝑡−ℎ(𝑡)}(𝑠−𝑡+ℎ)𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠−2^{𝑡−ℎ(𝑡)}_𝑡−ℎ(𝑠−𝑡+ℎ)𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠, (14) and then, in order to reduce the conservatism of the main result, the appropriate matrices 𝐹₁,𝐹₂∈𝐑^𝑛×5𝑛 are introduced to establish the relations between different vectors of 𝜒(𝑡), so the last two terms in (14) can be estimated as −2^𝑡_{𝑡−ℎ(𝑡)}(𝑠−𝑡+ℎ)𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠≤−2^𝑡_{𝑡−ℎ(𝑡)}(𝑠−𝑡+ℎ(𝑡))𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠≤2^𝑡_{𝑡−ℎ(𝑡)}(𝑠−𝑡+ℎ(𝑡))𝜒^𝑇(𝑡)𝐹^𝑇₁𝑈⁻¹𝐹₁𝜒(𝑡)+4𝜒^𝑇(𝑡)𝐹^𝑇₁^𝑡_{𝑡−ℎ(𝑡)}(𝑠−𝑡+ℎ(𝑡))𝑥̇(𝑠)𝑑𝑠=ℎ²(𝑡)𝜒^𝑇(𝑡)𝐹^𝑇₁𝑈⁻¹𝐹₁𝜒(𝑡)+4𝜒^𝑇(𝑡)𝐹^𝑇₁ℎ(𝑡)𝑥(𝑡)−^𝑡_{(𝑡)𝑡−ℎ}𝑥(𝑠)𝑑𝑠=ℎ²(𝑡)𝜒^𝑇(𝑡)𝐹^𝑇₁𝑈⁻¹𝐹₁𝜒(𝑡)+4𝜒^𝑇(𝑡)𝐹^𝑇₁ℎ(𝑡)𝑒₁−𝑒₅^𝑇𝜒(𝑡),−2^{𝑡−ℎ(𝑡)}_𝑡−ℎ(𝑠−𝑡+ℎ)𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠≤2^{𝑡−ℎ(𝑡)}_𝑡−ℎ(𝑠−𝑡+ℎ)𝜒^𝑇(𝑡)𝐹^𝑇₂𝑈⁻¹𝐹₂𝜒(𝑡)+4𝜒^𝑇(𝑡)𝐹^𝑇₂^{𝑡−ℎ(𝑡)}_𝑡−ℎ()𝑥̇(𝑠)=((𝑡𝑠−𝑡+ℎ𝑑𝑠ℎ−ℎ))²𝜒^𝑇(𝑡)𝐹^𝑇₂𝑈⁻¹𝐹₂𝜒(𝑡)+4𝜒^𝑇(𝑡)𝐹^𝑇₂(𝑡𝑥((𝑡−(ℎ−ℎ))𝑡−ℎ))_{𝑡−ℎ(𝑡)𝑡−ℎ}𝑥(𝑠)=((𝑡𝑑𝑠ℎ−ℎ))²𝜒^𝑇(𝑡)𝐹^𝑇₂𝑈⁻¹𝐹₂𝜒(𝑡)+4𝜒^𝑇(𝑡)𝐹^𝑇₂(ℎ−ℎ(𝑡))𝑒₂−𝑒₄^𝑇. (15)

Therefore, combining (11)–(15), we can obtain 𝑉̇𝑥_𝑡≤𝜒^𝑇(𝑡)Ξ₀+ℎ(𝑡)Ξ₁+Ξ₂𝜒(𝑡), (16) where Ξ₀=𝑒₁𝑃𝐴_𝑐+𝐴^𝑇_𝑐𝑃𝑒^𝑇₁+𝑒₁𝑒₁𝑅𝑒₁𝑒₁^𝑇−𝑒₁𝑒₃𝑅𝑒₁𝑒₃^𝑇+𝑒₁𝑒₁𝑄𝑒₁𝑒₁^𝑇−(1−𝑑)𝑒₁𝑒₂𝑄𝑒₁𝑒₂^𝑇+𝐴^𝑇_𝑐𝑒₀𝑅ℎ𝑒₁𝑒₄+𝑒₅^𝑇+ℎ𝑒₁𝑒₄+𝑒₅𝑅𝐴^𝑇_𝑐𝑒₀^𝑇+𝐴^𝑇_𝑐𝑒₀𝑄𝑒₀𝑒₅^𝑇+𝑒₀𝑒₅𝑄𝐴^𝑇_𝑐𝑒₀^𝑇+ℎ²𝑒₁𝐴^𝑇_𝑐𝑀𝑒₁𝐴^𝑇_𝑐^𝑇−𝑒₅𝑒₁−𝑒₂𝑀𝑒₅𝑒₁−𝑒₂^𝑇+𝑒₄𝑒₂−𝑒₃𝑀𝑒₄𝑒₂−𝑒₃^𝑇+𝑒₅𝑒₁−𝑒₂𝑁𝑒₄𝑒₂−𝑒₃^𝑇+𝑒₄𝑒₂−𝑒₃𝑁𝑒₅𝑒₁−𝑒₂^𝑇+ℎ²𝐴^𝑇_𝑐𝑈𝐴_𝑐−4𝐹^𝑇₁𝑒^𝑇₅+4𝐹^𝑇₂ℎ𝑒₂−𝑒₄^𝑇,Ξ₁=𝐴^𝑇_𝑐𝑒₀𝑄𝑒₁𝑒₀^𝑇+𝑒₁𝑒₀𝑄𝐴^𝑇_𝑐𝑒₀^𝑇+4𝐹^𝑇₁𝑒^𝑇₁−4𝐹^𝑇₂𝑒^𝑇₂,Ξ₂=ℎ²(𝑡)𝐹^𝑇₁𝑈⁻¹𝐹₁+(ℎ−ℎ(𝑡))²𝐹^𝑇₂𝑈⁻¹𝐹₂. (17)

It can be found that the function 𝑉̇(𝑥_𝑡)=𝜒^𝑇(𝑡)[Ξ₀+ℎ(𝑡)Ξ₁+Ξ₂]𝜒(𝑡) is a quadratic function on the scalar ℎ(𝑡) and the coefficient of second order is 𝜒^𝑇(𝑡)[𝐹^𝑇₁𝑈⁻¹𝐹₁+𝐹^𝑇₂𝑈⁻¹𝐹₂]𝜒(𝑡)≥0 since 𝑈>0, which means the 𝑉̇(𝑥_𝑡) is a convex quadratic function for ℎ(𝑡). Therefore, apply Lemma 2, if the following holds: Ξ₀+ℎ(𝑡)Ξ₁+Ξ_2ℎ(𝑡)=0Ξ<0,₀+ℎ(𝑡)Ξ₁+Ξ_{2ℎ(𝑡)=ℎ}<0; (18) then we have Ξ₀+ℎ(𝑡)Ξ₁+Ξ₂<0∀ℎ(𝑡)∈[0,ℎ], (19) which implies 𝑉̇(𝑥_𝑡)<0, which means that system (1) is asymptotically stable. So, we give the main theorem of this paper as follows.

Theorem 3. Given scalars 0<ℎ and 𝜇≥0, system (1) with a time-varying delay satisfying (2) and (3) is asymptotically stable, if there exist 𝑛×𝑛 matrices 𝑃>0,𝑈>0; 2𝑛×2𝑛 matrices 𝑄>0,𝑅>0,𝑀>0; 2𝑛×2𝑛 matrix 𝑁; and 𝑛×5𝑛 matrices 𝐹₁,𝐹₂ such that the following LMIs hold: Ξ𝑀𝑁∗𝑀≥0,₀ℎ𝐹^𝑇₂Ξ∗−𝑈<0,₀+ℎΞ₁ℎ𝐹^𝑇₁∗−𝑈<0. (20)

Remark 4. In (14), the term −2∫⁰_−ℎ∫^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠𝑑𝜃 also can be estimated by Jesen inequality as −2⁰_−ℎ^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)4𝑑𝑠𝑑𝜃≤−ℎ²ℎ𝑥(𝑡)−^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠^𝑇×𝑈ℎ𝑥(𝑡)−^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠, (21) but in this case, only relations between vectors 𝑥(𝑡),∫^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑑𝑠,∫^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠 were established; the corresponding result would be conservative. Therefore, we converted the term to a quadratic term multiplied by a scalar function and used free matrices 𝐹₁,𝐹₂ to establish the relations between 𝑥(𝑡),𝑥(𝑡−ℎ(𝑡)),𝑥(𝑡−ℎ),∫^{𝑡−ℎ(𝑡)}_𝑡−ℎ𝑥(𝑠)𝑑𝑠,and∫^𝑡_{𝑡−ℎ(𝑡)}𝑥(𝑠)𝑑𝑠, which reduced the conservatism of the stability result.

Remark 5. In this paper, it can be seen that LKFs in [5, 11, 13, 16] are just reduced forms of LKF (7), so Theorem 3 is less conservative than those stability conditions. Moreover, the reciprocally convex approach and the LKF transformation in (14) are used to estimate the derivative of the LKF in this paper. Those are the main reasons why Theorem 3 is less conservative and has less decision variables.
When the information of the time derivative of delay is unknown, just let 𝑄=0 in LKF (7); we can get the following result from Theorem 3.

Corollary 6. Given a scalar 0<ℎ, if there exist 𝑛×𝑛 matrices 𝑃>0,𝑈>0; 2𝑛×2𝑛 matrices 𝑅>0,𝑀>0; 2𝑛×2𝑛 matrix 𝑁; and 𝑛×5𝑛 matrices 𝐹₁,𝐹₂ such that LMIs (20) with 𝑄=0 are feasible, then system (1) with a time-varying delay satisfying (2) is asymptotically stable.

4. Numerical Examples

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

Example 7. Consider the linear system (1) with 𝐴=−200−0.9,𝐴₁=−10−1−1. (22)
The purpose of this example is to compare the admissible upper bounds ℎ which guarantee the asymptotic stability of the above system by different methods. Using the method in [12], when 𝑑=0.5,𝑑=0.9, and 𝑑>1, the upper bounds for the time delay are ℎ=2.08,ℎ=1.66, and ℎ=1.66; ours are ℎ=2.33,ℎ=1.87, and ℎ=1.86. Table 1 also lists the comparison of our results with some recent ones in [2, 7, 11, 15, 16, 18]. From Table 1, it can be seen that our results are less conservative.

Table 1 

Admissible upper bound ℎ for various 𝜇 in Example 7.

Moreover, the maximum upper bounds ℎ obtained by Theorem 3 are almost equal to those in [18], but the number of variables is 49𝑛²+5𝑛 in [18], and ours is just 21𝑛²+4𝑛. That is, the variables in [18] are around 2.2 times more than those in Theorem 3. Therefore, from both mathematical and practical points of view, our condition is more desirable than that in [18] even for some cases when the two methods give almost the same upper bound on the delay.

Example 8. Consider the linear system (1) with 𝐴=01−1−1,𝐴₁=000−1. (23)
This example is used in many recent papers, such as [7, 15, 18]. For various 𝜇, the maximum upper bounds on delay ℎ by different methods are listed in Table 2. It can be seen that the result obtained in this paper is less conservative.

Table 2 

Admissible upper bound ℎ for various 𝜇 in Example 8.

5. Conclusions

In this paper, the stability of time-varying delay systems has been discussed. Through constructing a novel LKF which contains multi-integral terms and using some new analysis methods, some delay-dependent stability criteria were derived. Compared with some previous stability conditions, the obtained main results in this paper have less conservatism and less decision variables. 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 Grant 61104119, the Science and Technology Innovation Talents Project of Henan University under Grant 13HASTIT044, the Doctoral Foundation from Henan Polytechnic University under Grant B2010-50, and the Young Core Instructor Foundation from Department of Education of Henan Province under Grant 2011GGJS-054.