Improved Results on Robust Stability for Systems with Interval Time-Varying Delays and Nonlinear Perturbations

Abstract

This paper investigated delay-dependent robust stability criteria for systems with interval time-varying delays and nonlinear perturbations. A delay-partitioning approach is used in this paper, the delay-interval is partitioned into multiple equidistant subintervals, a new Lyapunov-Krasovskii (L-K) functional contains some triple-integral terms, and augment terms are introduced on these intervals. Then, by using integral inequalities method together with free-weighting matrix approach, a new less conservative delay-dependent stability criterion is formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by optimization algorithms. Numerical examples are given to show the effectiveness and the benefits of the proposed method.

1. Introduction

Time-delay phenomena are ubiquitous in many practical systems such as communication systems, nuclear reactors, aircraft stabilization, and process control systems, which are often major sources of instability and poor performance. Hence, stability analysis and stabilization of systems with time-delays have received considerable attention in the past few decades [1–14]. Recently, many researchers pay attention to the interval time-varying delay, wherein the delay varies in a range for which the lower bound is not restricted to be zero. A typical example with interval time delay is the network control systems [2].

The basic framework for stability analysis and synthesis of stabilizing controllers is L-K functional and linear matrix inequality (LMI). Under this framework, an important issue was to enlarge the feasible region of stability criteria. To derive the delay-dependent stability conditions, many methods have been reported in the literature. For example, the free-weighting matrix method was used in [3–6]; Jensen’s integral inequality method was adopted in [7–13]. Recently, inspired by the discretized Lyapunov method, delay-partitioning approach was proposed in [14, 15], wherein the delay-interval was uniformly divided into multiple segments, choosing proper functional with different weighted matrices corresponding to different segments. A new technique called delay-central point method was proposed in [16]. Based on the delay-central point method and decomposition technique, [17] proposed a less conservative stability criterion for computing the maximum allowable bound of the delay range. As an extension of delay-central point method, a new delay-partitioning approach was proposed in [18] for the uncertain stochastic systems with interval time-varying delay. Referring to the nonlinearities, as time delays, also can cause instability and poor performance of practical systems. Therefore, the stability problem of time-delay systems with nonlinear perturbations has received increasing attention [19–24]. A descriptor model transformation was employed in [19]. The free-weighting matrices approach was adopt in [20, 21]. Recently, a less conservative delay-dependent stability criterion was provided in [22] by partitioning the delay-interval into two segments of equal length and evaluating the time-derivative of a candidate L-K functional in each segment of the delay-interval. The main advantage of the method [22] is that more information on the variation interval of the delay is employed, but we can employ information of time-delay much more if we partition the delay-interval into more segments.

Inspired by the idea of [18, 22], in this paper, we divide the variation interval of the delay into 𝑁 parts with equal length and construct a new L-K functional with triple-integral terms and augment terms for this delay-interval. Based on integral inequalities method together with free-weighting matrix approach, a new delay-dependent stability criterion for the system is formulated in terms of linear matrix inequalities, which can be easily calculated by using MATLAB LMI control toolbox. Numerical examples are given to illustrate the effectiveness and less conservatism of the proposed method.

2. Problem Description and Preliminaries

Consider the following neutral system with mixed time-varying delays and nonlinear perturbations: 𝑥̇(𝑡)=𝐀𝑥(𝑡)+𝐁𝑥(𝑡−ℎ(𝑡))+𝑓₁(𝑥(𝑡),𝑡)+𝑓₂(𝑥((𝑡),𝑥(𝜃)(𝜃),𝑥̇(𝜃)(𝜃),ℎ𝑡−ℎ)),𝑡=𝜙=𝜑∀𝜃∈−max_𝑀,𝜏,0, (1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector, 𝚨 and 𝚩 are constant matrices with appropriate dimensions, and ℎ(𝑡) is a time-varying delay satisfying 0≤ℎ_𝑚≤ℎ(𝑡)≤ℎ_𝑀,ℎ̇(𝑡)≤𝜇,∀𝑡≥0, (2) where ℎ_𝑚 and ℎ_𝑀 represent the lower and upper bounds of the time-varying delay ℎ(𝑡), respectively, 𝜇 is the bound on the delay-derivative, and initial condition 𝜑(𝑡) is a continuous vector-valued function. The functions 𝑓₁(𝑥(𝑡),𝑡) and 𝑓₂(𝑥(𝑡−ℎ(𝑡)),𝑡) are unknown nonlinear perturbations with respect to the current state 𝑥(𝑡) and in the delay state 𝑥(𝑡−ℎ(𝑡)), respectively. They satisfy 𝑓₁(0,𝑡)=0, 𝑓₂(0,𝑡)=0, and 𝑓^𝑇₁(𝑥(𝑡),𝑡)𝑓(𝑥(𝑡),𝑡)≤𝛼²𝑥^𝑇(𝑡)𝑥(𝑡),𝑓^𝑇₂(𝑥(𝑡−ℎ(𝑡)),𝑡)𝑓₂(𝑥(𝑡−ℎ(𝑡)),𝑡)≤𝛽²𝑥^𝑇(𝑡−ℎ(𝑡))𝑥(𝑡−ℎ(𝑡)), (3) where 𝛼≥0, 𝛽≥0 are given constants; for simplicity we denote 𝑓₁=𝑓₁(𝑥(𝑡),𝑡), 𝑓₂=𝑓₂(𝑥(𝑡−ℎ(𝑡)),𝑡).

Before moving on, we need the following instrumental lemma.

Lemma 1 (see [10]). For any scalar ℎ(𝑡)≥0 and constant matrix 𝐐∈ℝ^𝑛×𝑛, 𝐐=𝐐^𝑇>0, the following inequality holds: −^𝑡_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐐𝑥̇(𝑠)𝑑𝑠≤ℎ(𝑡)𝜁^𝑇(𝑡)𝐕𝐐⁻¹𝐕^𝑇𝜁(𝑡)+2𝜁^𝑇(𝑡)𝐕[𝑥(𝑡)−𝑥(𝑡−ℎ(𝑡))], (4) where 𝜁^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇𝑡−ℎ_𝑖𝑥^𝑇(𝑡−ℎ(𝑡))𝑥^𝑇𝑡−ℎ_𝑖+1𝑥̇^𝑇(𝑡)^𝑡_{𝑡−ℎ𝑖}𝑥^𝑇(𝑠)𝑑𝑠^{𝑡−ℎ𝑖}_{𝑡−ℎ𝑖+1}𝑥^𝑇(𝑠)𝑑𝑠𝑓^𝑇₁𝑓^𝑇₂. (5)𝐕 is a free-weighting matrix with appropriate dimensions.

Lemma 2 (see [9]). For any constant matrix 𝐖∈ℝ^𝑛×𝑛, 𝐖=𝐖^𝑇>0, a scalar function ℎ∶=ℎ(𝑡)>0 and a vector-valued function 𝑥̇∶[−ℎ,0]→𝑅^𝑛, such that the following integrations are well defined; then −ℎ^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝐖𝑥̇(𝑠)𝑑𝑠≤𝜁^𝑇₁(𝑡)𝜁−𝐖𝐖𝐖−𝐖¹(𝑡),−ℎ²2⁰_−ℎ^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑡)𝐖𝑥̇(𝑡)𝑑𝑠≤𝜁^𝑇₂(𝑡)𝜁−𝐖𝐖𝐖−𝐖²(𝑡) (6) with 𝜁^𝑇₁(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−ℎ)] and 𝜁^𝑇₂(𝑡)=[ℎ𝑥^𝑇(𝑡)∫^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠].

Lemma 3 (see [25]). Suppose 𝛾₁≤𝛾(𝑡)≤𝛾₂, where 𝛾(⋅)∶𝑅₊→𝑅₊. Then, for any constant matrices Ξ₁, Ξ₂, and Ω with proper dimensions, the following matrix inequality Ω+𝛾(𝑡)−𝛾₁Ξ₁+𝛾₂−𝛾(𝑡)Ξ₂<0 (7) holds, if and only if Ω+𝛾₂−𝛾₁Ξ₁𝛾<0,Ω+₂−𝛾₁Ξ₂<0. (8)

3. Main Result

In this section, we study the delay-dependent robust stability of system (1) based on the delay-partitioning approach.

Theorem 4. For given positive scalars ℎ_𝑚, ℎ_𝑀, 𝜇, 𝛼, and 𝛽 system (1) with uncertainty (3) is asymptotically stable if there exist symmetric matrices 𝐏𝐏=¹¹𝐏₁₂𝐏₁₃∗𝐏₂₂𝐏₂₃∗∗𝐏₃₃ (9) and 𝐐₁, 𝐐₂, 𝐐₃, 𝐙₂, 𝐙₃, 𝐑₂, 𝐑₃, any matrices 𝐊_𝑎, 𝐕_𝑎, 𝐖_𝑎(𝑎=1,2), and scalars 𝜀_𝑖≥0(𝑖=1,2) such that the following LMIs hold: 𝚽ℎ_𝛿𝐖∗−𝐙₃<0,𝑖=1,2,…,𝑁,𝚽ℎ_𝛿𝐕∗−𝐙₃<0,𝑖=1,2,…,𝑁, (10)(11) where 𝚽=(𝚽_𝑖,𝑗)_9×9 with 𝚽₁₁=𝐏₁₂+𝐏^𝑇₁₂+𝐐₁+𝐐₂+𝐐₃−𝐙₂−ℎ²_𝑖𝐑₂−ℎ²_𝛿𝐑₃+𝐊₁𝐀+𝐀^𝑇𝐊^𝑇₁+𝜀₁𝑎²,𝚽₁₂=𝐙₂−𝐏₁₂+𝐏₁₃,𝚽₁₃=𝐊₁𝚽𝐁,₁₄=−𝐏₁₃,𝚽₁₅=𝐏₁₁−𝐊₁+𝐀^𝑇𝐊^𝑇₂,𝚽₁₆=𝐏^𝑇₂₂+ℎ_𝑖𝐑₂,𝚽₁₇=𝐏^𝑇₃₂+ℎ_𝛿𝐑₃,𝚽₁₈=𝚽₁₉=𝐊₁,𝚽₂₂=−𝐐₂−𝐙₂+𝐕₁+𝐕^𝑇₁,𝚽₂₃=−𝐕₁+𝐕^𝑇₂,𝚽₂₄=𝚽₂₅𝚽=0,₂₆=−𝐏^𝑇₂₂+𝐏^𝑇₂₃,𝚽₂₇=−𝐏^𝑇₃₂+𝐏^𝑇₃₃,𝚽₂₈=𝚽₂₉𝚽=0,₃₃=−1−𝜇𝐐₁−𝐕₂−𝐕^𝑇₂+𝐖₁+𝐖^𝑇₁+𝜀₂𝛽²,𝚽₃₄=−𝐖₁+𝐖^𝑇₂,𝚽₃₅=𝐁^𝑇𝐊^𝑇₂,𝚽₃₆=⋯=𝚽₃₉=0,𝚽₄₄=−𝐐₃−𝐖₂−𝐖^𝑇₂,𝚽₄₅=0,𝚽₄₆=−𝐏^𝑇₂₃,𝚽₄₇=−𝐏^𝑇₃₃,𝚽₄₈=𝚽₄₉𝚽=0,₅₅=ℎ²_𝑖𝐙₂+ℎ_𝛿𝐙₃+14ℎ⁴_𝑖𝐑₂+14ℎ²_𝑖+1−ℎ²_𝑖²𝐑₃−𝐊₂−𝐊^𝑇₂,𝚽₅₆=𝐏^𝑇₂₁,𝚽₅₇=𝐏^𝑇₃₁,𝚽₅₈=𝚽_5,9=𝐊₂,𝚽₆₆=−𝐑₂,𝚽₆₇=⋯=𝚽₆₉𝚽=0,₇₇=−𝐑₃,𝚽₇₈=𝚽₇₉𝚽=0,₈₈=−𝜀₁𝐈,𝚽₈₉=0,𝚽₉₉=−𝜀₂ℎ𝐈,_𝛿=ℎ_𝑖+1−ℎ_𝑖=ℎ_𝑀−ℎ_𝑚𝑁,ℎ_𝑖=ℎ₁+(𝑖−1)ℎ_𝑀−ℎ_𝑚𝑁,𝐊=𝐊^𝑇₁000𝐊^𝑇₂0000^𝑇,𝐕=0𝐕^𝑇₁𝐕^𝑇₂000000^𝑇,𝐖=00𝐖^𝑇₁𝐖^𝑇₂00000^𝑇. (12)

Proof. First, we decompose the delay-interval [ℎ_𝑚,ℎ_𝑀] into 𝑁 equidistant subinterval, where 𝑁 is a given integer; that is, ℎ_𝑚,ℎ_𝑀=ℎ₁,ℎ₂∪ℎ₂,ℎ₃∪⋯∪ℎ_𝑖,ℎ_𝑖+1∪⋯∪ℎ_𝑁,ℎ_𝑁+1. (13)
Then, the L-K functional corresponding to the time-variation ℎ(𝑡)∈[ℎ₂,ℎ₃] is chosen as 𝑉₂(𝑡)=𝜉^𝑇(𝑡)𝐏𝜉(𝑡)+^𝑡_{(𝑡)𝑡−ℎ}𝑥^𝑇(𝑠)𝐐₁𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ2}𝑥^𝑇(𝑠)𝐐₂𝑥(𝑠)𝑑𝑠+^𝑡_𝑡−ℎ3𝑥^𝑇(𝑠)𝐐_𝟑𝑥(𝑠)𝑑𝑠+ℎ₂⁰_−ℎ2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝐙₂𝑥̇(𝑠)𝑑𝑠𝑑𝜃+^−ℎ2_−ℎ3^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝐙₃𝑥̇(𝑠)+ℎ𝑑𝑠𝑑𝜃₂₂2⁰_−ℎ2⁰_𝜃^𝑡_𝑡+𝜆𝑥̇^𝑇(𝑠)𝐑₂𝑥̇(𝑠)+ℎ𝑑𝑠𝑑𝜆𝑑𝜃₂₃−ℎ²₂2^−ℎ2_−ℎ3⁰_𝜃^𝑡_𝑡+𝜆𝑥̇^𝑇(𝑠)𝐑₃𝑥̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃 (14) with 𝜉^𝑇(𝑡)=[𝑥^𝑇(𝑡)∫^𝑡_𝑡−ℎ2𝑥^𝑇(𝑡)𝑑𝑠∫^𝑡−ℎ2_𝑡−ℎ3𝑥^𝑇(𝑡)𝑑𝑠].
The time-derivative of the L-K functional along the trajectory of (1) is given by 𝑉̇₂(𝑡)=2𝜉^𝑇(𝑡)𝐏𝜉̇(𝑡)+𝑥^𝑇(𝑡)𝐐₁+𝐐₂+𝐐₃𝑥(𝑡)−1−𝜇𝑥^𝑇(𝑡−ℎ(𝑡))𝐐₁𝑥(𝑡−ℎ(𝑡))−𝑥^𝑇𝑡−ℎ₂𝐐₂𝑥𝑡−ℎ₂−𝑥^𝑇𝑡−ℎ₃𝐐₃𝑥𝑡−ℎ₃+𝑥̇^𝑇(𝑡)ℎ²₂𝐙₂+ℎ₃−ℎ₂𝐙₃𝑥̇(𝑡)−ℎ₂^𝑡_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝐙₂𝑥̇(𝑠)𝑑𝑠−^𝑡−ℎ2_𝑡−ℎ3𝑥̇^𝑇(𝑠)𝐙₃𝑥̇(𝑠)+1𝑑𝑠4ℎ⁴₂𝑥̇^𝑇(𝑡)𝐑₂𝑥̇(𝑡)+14ℎ²₃−ℎ²₂²𝑥̇^𝑇(𝑡)𝐑₃𝑥̇(𝑡)−ℎ²₂2⁰_−ℎ2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝐑₂𝑥̇(𝑡)−ℎ𝑑𝑠𝑑𝜃₂₃−ℎ²₂2^−ℎ2_−ℎ3^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝐑₃𝑥̇(𝑡)𝑑𝑠𝑑𝜃. (15)
Note that −^𝑡−ℎ2_𝑡−ℎ3𝑥̇^𝑇(𝑠)𝐙₃𝑥̇(𝑠)𝑑𝑠=−^{𝑡−ℎ(𝑡)}_𝑡−ℎ3𝑥̇^𝑇(𝑠)𝐙₃𝑥̇(𝑠)−𝑑𝑠_{𝑡−ℎ2(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐙₃𝑥̇(𝑠)𝑑𝑠. (16)
From Lemmas 1 and 2, we have −^𝑡−ℎ2_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝐙₃𝑥̇(𝑠)≤ℎ(𝑡)𝑑𝑠−ℎ₂𝜁^𝑇(𝑡)𝐕𝐙₃⁻¹𝐕^𝑇𝜁(𝑡)+2𝜁^𝑇(𝑡)𝐕𝑥𝑡−ℎ₂−𝑥(𝑡−ℎ(𝑡)),−^{𝑡−ℎ(𝑡)}_𝑡−ℎ3𝑥̇^𝑇(𝑠)𝐙₃𝑥̇(𝑠)≤ℎ𝑑𝑠₃−ℎ(𝑡)𝜁^𝑇(𝑡)𝐖𝐙₃⁻¹𝐖^𝑇𝜁(𝑡)+2𝜁^𝑇(𝑡)𝐖𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₃,−ℎ₂^𝑡_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝐙₂𝑥̇(𝑠)≤𝑥(𝑡)𝑥𝑑𝑠𝑡−ℎ₂^𝑇−𝐙₂𝐙₂∗−𝐙₂𝑥(𝑡)𝑥𝑡−ℎ₂,−ℎ²₂2⁰_−ℎ2^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝐑₂𝑥̇(𝑡)≤ℎ𝑑𝑠𝑑𝜃₂𝑥(𝑡)_{𝑡𝑡−ℎ2}𝑥(𝑠)𝑑𝑠^𝑇−𝐑₂𝐑₂𝐑₂−𝐑₂ℎ₂𝑥(𝑡)_{𝑡𝑡−ℎ2}𝑥(𝑠),−ℎ𝑑𝑠₂₃−ℎ²₂2^−ℎ2_−ℎ3^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝐑₃𝑥̇(𝑡)≤ℎ𝑑𝑠𝑑𝜃₃−ℎ₂𝑥(𝑡)_{𝑡−ℎ2𝑡−ℎ3}𝑥(𝑠)𝑑𝑠−𝐑₃𝐑₃𝐑₃−𝐑₃ℎ₃−ℎ₂𝑥(𝑡)_{𝑡−ℎ2𝑡−ℎ3}𝑥(𝑠).𝑑𝑠 (17)(18)
From (3), we can obtain, for any scalars 𝜀₁>0, 𝜀₂>0, 𝜀₁𝛼²𝑥^𝑇(𝑡)𝑥(𝑡)−𝑓^𝑇₁(𝑥(𝑡),𝑡)𝑓₁(𝑥(𝑡)𝜀,𝑡)≥0,₂𝛽²𝑥^𝑇(𝑡−ℎ(𝑡))𝑥(𝑡−ℎ(𝑡))−𝑓^𝑇₂(𝑥(𝑡−ℎ(𝑡)),𝑡)𝑓₂(𝑥(𝑡−ℎ(𝑡)),𝑡)≥0. (19)
From the system (1), we have the following equation: 2𝑥^𝑇(𝑡)𝐊₁+𝑥̇(𝑡)𝐊₂×𝐀𝑥(𝑡)+𝐁𝑥(𝑡−ℎ(𝑡))+𝑓₁(𝑥(𝑡),𝑡)+𝑓₂(𝑥(𝑡−ℎ(𝑡)),𝑡)−𝑥̇(𝑡)=0. (20)
By substituting (17)~(20) in (15) and defining an augmented state vector, 𝜁^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇𝑡−ℎ_𝑖𝑥^𝑇(𝑡−ℎ(𝑡))𝑥^𝑇𝑡−ℎ_𝑖+1𝑥̇^𝑇(𝑡)^𝑡_{𝑡−ℎ𝑖+1}𝑥^𝑇(𝑠)𝑑𝑠^{𝑡−ℎ𝑖}_{𝑡−ℎ𝑖+1}𝑥^𝑇(𝑠)𝑑𝑠𝑓^𝑇₁𝑓^𝑇₂. (21)
Then the time-derivative 𝑉̇₂(𝑥(𝑡),𝑡) can be expressed as follows: 𝑉̇₂(𝑥(𝑡),𝑡)≤𝜁^𝑇(𝑡)𝚽+ℎ₃−ℎ(𝑡)𝐖𝐙₃⁻¹𝐖^𝑇+ℎ(𝑡)−ℎ₂𝐕𝐙₃⁻¹𝐕^𝑇𝜁(𝑡). (22)
One can see that if ℎ(𝑡)∈[ℎ₂,ℎ₃], 𝚽+ℎ₃−ℎ(𝑡)𝐖𝐙₃⁻¹𝐖^𝑇+ℎ(𝑡)−ℎ₂𝐕𝐙₃⁻¹𝐕^𝑇<0. (23)
Then 𝑉̇(𝑡)<−𝜀‖𝑥(𝑡)‖² for some scalar 𝜀>0, from which we conclude that system (1) is asymptotically stable according to L-K stability theory [1].
Applying Lemma 3 to (23) yields the following inequality: 𝚽+ℎ₃−ℎ(𝑡)𝐖𝐙₃⁻¹𝐖^𝑇ℎ(𝑡)<0,𝚽+−ℎ₂𝐕𝐙₃⁻¹𝐕^𝑇<0. (24)
By Schur complement on (24), we can obtain (10) and (11) with 𝑖=2.
Without loss of generality, when ℎ(𝑡)∈[ℎ_𝑖,ℎ_𝑖+1](𝑖=1,3,…,𝑁), construct the following L-K functional: 𝑉_𝑖(𝑡)=𝜉^𝑇(𝑡)𝐏𝜉(𝑡)+^𝑡_{𝑡−ℎ(𝑡)}𝑥^𝑇(𝑠)𝐐₁𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ𝑖}𝑥^𝑇(𝑠)𝐐₂𝑥(𝑠)𝑑𝑠+^𝑡_{𝑡−ℎ𝑖+1}𝑥^𝑇(𝑠)𝐐₃𝑥(𝑠)𝑑𝑠+ℎ_𝑖⁰_−ℎ𝑖^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)𝐙₂𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{−ℎ𝑖−ℎ𝑖+1𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝐙₃𝑥̇(𝑠)+ℎ𝑑𝑠𝑑𝜃_2𝑖2⁰_−ℎ𝑖⁰_𝜃^𝑡_𝑡+𝜆𝑥̇^𝑇(𝑠)𝐑₂𝑥̇(𝑠)+ℎ𝑑𝑠𝑑𝜆𝑑𝜃_2𝑖+1−ℎ²_𝑖2^−ℎ𝑖_−ℎ𝑖+1⁰_𝜃^𝑡_𝑡+𝜆𝑥̇^𝑇(𝑠)𝐑₃𝑥̇(𝑠)𝑑𝑠𝑑𝜆𝑑𝜃 (25) with 𝜉^𝑇(𝑡)=[𝑥^𝑇(𝑡)∫^𝑡_{𝑡−ℎ𝑖}𝑥^𝑇(𝑡)𝑑𝑠∫^{𝑡−ℎ𝑖}_{𝑡−ℎ𝑖+1}𝑥^𝑇(𝑡)𝑑𝑠], and 𝐏, 𝐐₁, 𝐐₂, 𝐐₃, 𝐙₂, 𝐙₃, 𝐑₂, 𝐑₃ are the same matrix variables used in 𝑉₂(𝑡). Now, define an augmented state vector: 𝜁^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇𝑡−ℎ_𝑖𝑥^𝑇(𝑡−ℎ(𝑡))𝑥^𝑇𝑡−ℎ_𝑖+1𝑥̇^𝑇(𝑡)^𝑡_{𝑡−ℎ𝑖+1}𝑥^𝑇(𝑠)𝑑𝑠^{𝑡−ℎ𝑖}_{𝑡−ℎ𝑖+1}𝑥^𝑇(𝑠)𝑑𝑠𝑓^𝑇₁𝑓^𝑇₂. (26)
Using the same method, we have the conditions (10) and (11) as that for ℎ(𝑡)∈[ℎ_𝑖,ℎ_𝑖+1](𝑖=1,3,…,𝑁). The proof is completed.

Remark 5. As an extension of the method used in [18, 22], we divide the delay-interval into 𝑁 subintervals, constructing a new L-K functional that contains some triple-integral terms and augment term for each delay-interval. This treatment makes us employ more information on the time delay and yields less conservative delay-range bounds.

Remark 6. If there is no perturbation, that is, 𝑓=0, 𝑔=0, then the stability problem of system (1) is reduced to analyze the stability of the system: 𝑥̇(𝑡)(𝑡)((𝑡,𝑥(𝜃)(𝜃)=𝐀𝑥+𝐁𝑥𝑡−ℎ))=𝜙,∀𝜃∈−ℎ_𝑀,0. (27)
According to Theorem 4, we can obtain the following corollary for the delay-dependent stability of system (27).

Corollary 7. For given positive scalars, ℎ_𝑚, ℎ_𝑀, 𝜇, system (27) with uncertainty (10) is asymptotically stable if there exist real symmetric positive definitive matrices 𝐏𝐏=¹¹𝐏₁₂𝐏₁₃∗𝐏₂₂𝐏₂₃∗∗𝐏₃₃ (28) and 𝐐₁, 𝐐₂, 𝐐₃, 𝐙₂, 𝐙₃, 𝐑₂, 𝐑₃, any matrices 𝐊_𝑎, 𝐕_𝑎, 𝐖_𝑎(𝑎=1,2), and scalars 𝜀_𝑖≥0(𝑖=1,2) such that the following LMIs hold: 𝚽ℎ_𝛿𝐖∗−𝐙₃<0,𝑖=1,2,…,𝑁,𝚽ℎ_𝛿𝐕∗−𝐙₃<0,𝑖=1,2,…,𝑁, (29) where 𝚽=(𝚽_𝑖,𝑗)_7×7 with 𝚽₁₁=𝐏₁₂+𝐏^𝑇₁₂+𝐐₁+𝐐₂+𝐐₃−𝐙₂−ℎ²_𝑖𝐑₂−ℎ²_𝛿𝐑₃+𝐊₁𝐀+𝐀^𝑇𝐊^𝑇₁,𝚽₁₂=𝚽₁₂,𝚽₁₃=𝚽₁₃,𝚽₁₄=𝚽₁₄,𝚽₁₅=𝚽₁₅,𝚽₁₆=𝚽₁₆,𝚽₁₇=𝚽₁₇,𝚽₂₂=𝚽₂₂,𝚽₂₃=𝚽₂₃,𝚽₂₄=𝚽₂₅=0,𝚽₂₆=𝚽₂₆,𝚽₂₇=𝚽₂₇,𝚽₃₃=−1−𝜇𝐐₁−𝐕₂−𝐕^𝑇₂+𝐖₁+𝐖^𝑇₁,𝚽₃₄=𝚽₃₄,𝚽₃₅=𝚽₃₅,𝚽₃₆=𝚽₃₇=0,𝚽₄₄=𝚽₄₄,𝚽₄₅=0,𝚽₄₆=𝚽₄₆,𝚽₄₇=𝚽₄₇,𝚽₅₅=𝚽₅₅,𝚽₅₆=𝚽₅₆,𝚽₅₇=𝚽₅₇,𝚽₆₆=𝚽₆₆,𝚽₆₇𝚽=0,₇₇=−𝐑₃,ℎ_𝛿=ℎ_𝑖+1−ℎ_𝑖=ℎ_𝑀−ℎ_𝑚𝑁,ℎ_𝑖=ℎ₁+(𝑖−1)ℎ_𝑀−ℎ_𝑚𝑁,𝐊=𝐊^𝑇₁000𝐊^𝑇₂00^𝑇,𝐕=0𝐕^𝑇₁𝐕^𝑇₂0000^𝑇,𝐖=00𝐖^𝑇₁𝐖^𝑇₂000^𝑇. (30)

4. Numerical Examples

Example 1. Consider the following neutral time-delay system with 𝐀=.−1.20.1−0.1−1,𝐁=−0.60.7−1−0.8 (31)
For given values of 𝛼, 𝛽, and 𝜇, we apply Theorem 4 to calculate the maximal allowable value ℎ_𝑀 that guarantees that the asymptotical stability of the system is listed in Table 1. From Table 1, it is easy to see that our proposed stability criterion gives much less conservative results than those in [21, 22].

Table 1 

Admissible upper bounds ℎ₂ for various 𝜇 and ℎ_𝑚=0,0.5,1.

Example 2. Consider the system (27) with the following matrices: 𝐀=.01−1−2,𝐁=00−11 (32)
For given ℎ_𝑚 and 𝜇, we calculate the allowable upper bound of ℎ_𝑀 that guarantees the asymptotical stability of system (27). Using different methods, computational results are obtained and these are listed in Table 2. From Table 2, it can be seen that our results are less conservative than the existing criteria.

Table 2 

Maximum bounds ℎ_𝑀 for a given ℎ_𝑚.

5. Conclusion

This paper studies the problem of robust delay-dependent stability for a class of linear systems with interval time-varying delay and nonlinear perturbations. Based on the delay-partitioning approach, the less conservative delay-dependent stability conditions are derived. The reduction in the conservatism of the proposed stability criteria is mainly attributed to the new L-K functional which contains some triple-integral terms and augment terms for each divided segment. Numerical examples have illustrated the effectiveness of the proposed method.

Conflict of Interests

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

Acknowledgment

The research is supported by the National Natural Science Foundation of China, Grant nos. 61304001 and 61304103.