Robust Stability of a Class of Uncertain Lur'e Systems of Neutral Type

Abstract

This paper deals with the problem of stability for a class of Lur’e systems with interval time-varying delay and sector-bounded nonlinearity. The interval time-varying delay function is not assumed to be differentiable. We analyze the global exponential stability for uncertain neutral and Lur’e dynamical systems with some sector conditions. By constructing a set of improved Lyapunov-Krasovskii functional combined with Leibniz-Newton’s formula, we establish some stability criteria in terms of linear matrix inequalities. Numerical examples are given to illustrate the effectiveness of the results.

1. Introduction

In many practical systems, models of system are described by neutral differential equations, in which the models depend on the delays of state and state derivatives. Heat exchanges, distributed networks containing lossless transmission lines, and population ecology are examples of neutral systems. Because of its wider application, therefore, several researchers have studied neutral systems and provided sufficient conditions to guarantee the asymptotic stability of neutral time delay systems, see [1–6] and references cited therein.

It is well known that nonlinearities may cause instability and poor performance of practical systems, which have driven many researchers to study [3–9]. Many nonlinear control systems can be modeled as a feedback connection of a linear neutral system and a nonlinear element. One of the important classes of nonlinear systems is the Lur’e system whose nonlinear element satisfies certain sector constraints. Lur’e systems with sector bound have been widely interested in the control system such as Kalman-YaKubovih-Popov lemma, Popov criterion, and Circle criterion [10–12]. On the other hand, it is well known that the existence of time delay in a system may cause instability and oscillations. Example of time delay systems are chemical engineering systems, biological modeling, electrical networks, physical networks, and many others, [3–5].

The stability criteria for system with time delays can be classified into two categories: delay-independent and delay-dependent. Delay-independent criteria does not employ any information on the size of the delay; while delay-dependent criteria makes use of such information at different levels. Delay-dependent stability conditions are generally less conservative than delay-independent ones especially when the delay is small. In most of the existing results, the range of time-varying delay considered in paper is form 0 to an upper bound. In practice, the range of delay may vary in a range for which the lower bound is not restricted to be 0, that is, interval time-varying delay. A typical example with interval time delay is the networked control system, which has been widely studied in the recent literature (see, e.g., [3, 13, 14]). However there are fewer results for removing restriction to the derivative of interval time-varying delays. Therefore their methods have a conservatism which can be improved upon.

It is known that exponential stability is more favorite property than asymptotic stability since it gives a faster convergence rate to the equilibrium point, the decay rates (i.e., convergent rates) are important indices of practical systems, and the exponential stability analysis of time-delay systems has been a popular topic in the past decades; see, for example, [3, 13, 15] and their references. In [13], delay-dependent exponential stability for uncertain linear systems with interval time-varying delays, in [15], global exponential periodicity and global exponential stability of a class of recurrent neural networks with time-varying delays.

Recently, there are many research study on the asymptotic stability of a class of neutral and Lur’e dynamical systems with time delay, see, for example, [8, 9, 16, 17]. The problems have been dealt with stability analysis for neutral systems with mixed delays and sector-bounded nonlinearity [8], robust absolute stability criteria for uncertain Lur’e systems of neutral type [16], and robust stability criteria for a class of Lur’e systems with interval time-varying delay [9]. However, it is worth pointing out that, even though these results above were elegant, there still exist some points waiting for the improvement. Firstly, most of the work above the time-varying delays are required to be differentiable. In fact, the constraint on the derivative of the time-varying delay is not required which allows the time-delay to be a fast time-varying function. Secondly, in most studies on the asymptotic stability of Lur’e dynamical systems still need to be improved to the exponential stability.

Based on the above discussions, we consider the problem of robust stability for a class of uncertain neutral and Lur’e dynamical systems with sector-bounded nonlinearity. The time delay is a continuous function belonging to a given interval, which means that the lower and upper bounds for the time varying delay are available, but the delay function is not necessary to be differentiable. To the best of the authors knowledge, there were no global stability results for uncertain neutral and Lur’e dynamical systems with some sector conditions [8, 9, 16, 17]. Based on the construction of improved Lyapunov-Krasovskii functionals combined with Liebniz-Newton’s formula and the integral terms, new delay-dependent sufficient conditions for the uncertain neutral and Lur’e dynamical of system are established of LMIs. The new stability condition is much less conservative and more general than some existing results. Numerical examples are given to illustrate the effectiveness of our theoretical results.

The rest of this paper is organized as follows. In Section 2, we give notations, definition, propositions, and lemma to be used in the proof of the main results. Delay-dependent sufficient conditions for uncertain neutral and Lur’e dynamical systems with sector-bounded nonlinearity are presented in Section 3. Numerical examples illustrated the obtained results and are given in Section 4. The paper ends with conclusions in Section 5 and cited references.

2. Problem Formulation and Preliminaries

The following notation will be used in this paper: ℝ⁺ denotes the set of all real nonnegative numbers; ℝ^𝑛 denotes the 𝑛-dimensional space and the vector norm ‖⋅‖; 𝑀^𝑛×𝑟 denotes the space of all matrices of (𝑛×𝑟)-dimensions. 𝐴^𝑇 denotes the transpose of matrix 𝐴; 𝐴 is symmetric if 𝐴=𝐴^𝑇; 𝐼 denotes the identity matrix; 𝜆(𝐴) denotes the set of all eigenvalues of 𝐴; 𝜆_max(𝐴)=max{Re𝜆;𝜆∈𝜆(𝐴)}. 𝑥_𝑡∶={𝑥(𝑡+𝑠)∶𝑠∈[−ℎ,0]}, ‖𝑥_𝑡‖=sup_{𝑠∈[−ℎ,0]}‖𝑥(𝑡+𝑠)‖; 𝐶([0,𝑡],ℝ^𝑛) denotes the set of all ℝ^𝑛-valued continuous functions on [0,𝑡]; Matrix 𝐴 is called semipositive definite (𝐴≥0) if 𝑥^𝑇𝐴𝑥≥0, for all 𝑥∈ℝ^𝑛; 𝐴 is positive definite (𝐴>0) if 𝑥^𝑇𝐴𝑥>0 for all 𝑥≠0;𝐴>𝐵 means 𝐴−𝐵>0; diag(𝑐₁,𝑐₂,…,𝑐_𝑚) denotes block diagonal matrix with diagonal elements 𝑐_𝑖, 𝑖=1,2,…,𝑚. The symmetric term in a matrix is denoted by *.

Consider the following uncertain Lur’e system of neutral type with interval time-varying delays and sector-bounded nonlinearity: 𝑥̇(𝑡)−𝐶𝑥̇𝑡−𝜂(𝑡)=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+𝐴₁+Δ𝐴₁(𝑡)𝑥((𝑡+((𝑡𝑓(𝜎(𝑡,𝜎(𝑡)𝑡−ℎ))𝐵+Δ𝐵))))=𝐻^𝑇𝑥(𝑡)=ℎ₁ℎ₂⋯ℎ_𝑚^𝑇𝑥(𝑡)𝑥()(),𝑥̇()()[]ℎ,∀𝑡≥0,𝑡+𝑠=𝜙𝑡+𝑠𝑡+𝑠=𝜑𝑡+𝑠,𝑠∈−𝑚,0,𝑚=max₂,𝜂, (2.1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector; 𝜎(𝑡)∈ℝ^𝑛 is the output vector; 𝐴∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑚, 𝐶∈ℝ^𝑛×𝑛, 𝐴₁∈ℝ^𝑛×𝑛, 𝐻∈ℝ^𝑛×𝑚 are constant known matrices; 𝑓(𝜎(𝑡))∈ℝ^𝑚 is the nonlinear function in the feedback path, which is denoted as 𝑓 for simplicity in the sequel. Its form is formulated as 𝑓(𝜎(𝑡))=𝑓₁𝜎₁(𝑡)𝑓₂𝜎₂(𝑡)⋯𝑓_𝑚𝜎_𝑚(𝑡)^𝑇,𝜎(𝑡)=𝜎₁(𝑡)𝜎₂(𝑡)⋯𝜎_𝑚(𝑡)^𝑇=ℎ^𝑇₁𝑥(𝑡)ℎ^𝑇₂𝑥(𝑡)⋯ℎ^𝑇_𝑚𝑥(𝑡), (2.2) wherein, each term 𝑓_𝑖(𝜎_𝑖(𝑡)),𝑖=1,2,…,𝑚 satisfies any one of the following sector conditions: 𝑓_𝑖𝜎_𝑖(𝑡)∈𝐾_[0,𝑘𝑖]=𝑓_𝑖𝜎_𝑖(𝑡)∣𝑓_𝑖(0)=0,0<𝜎_𝑖(𝑡)𝑓_𝑖𝜎_𝑖(𝑡)≤𝑘_𝑖𝜎_𝑖(𝑡)²,𝜎_𝑖(𝑡)≠0 (2.3) or 𝑓_𝑖𝜎_𝑖(𝑡)∈𝐾_[0,∞]=𝑓_𝑖𝜎_𝑖(𝑡)∣𝑓_𝑖(0)=0,𝜎_𝑖(𝑡)𝑓_𝑖𝜎_𝑖(𝑡)>0,𝜎_𝑖(𝑡)≠0. (2.4)Δ𝐴(𝑡), Δ𝐵(𝑡), and  Δ𝐴₁(𝑡) are time-varying uncertainties of appropriate dimensions, which are assumed to be of the following form: Δ𝐴(𝑡)Δ𝐵(𝑡)Δ𝐴₁(𝑡)=𝐷𝐹(𝑡)𝐸₁𝐸₂𝐸₃, (2.5) where 𝐷, 𝐸₁, 𝐸₂, and 𝐸₃ are known matrices of appropriate dimensions, and the time-varying matrix 𝐹(𝑡) satisfies 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼,∀𝑡≥0. (2.6) The delays ℎ(𝑡) and 𝜏(𝑡) are time-varying continuous functions that satisfy 0≤ℎ₁≤ℎ(𝑡)≤ℎ₂,0≤𝜂(𝑡)≤𝜂,𝜂̇(𝑡)≤𝜂_𝑑<1. (2.7) We introduce the following technical well-known propositions and definition, which will be used in the proof of our results.

Definition 2.1. If there exist 𝛾>0 and 𝜓(𝛾)>0 such that ‖𝑥(𝑡)‖≤𝜓𝛾𝑒^−𝛾𝑡,∀𝑡>0, (2.8) system (2.1) is said to be exponentially stable at the equilibrium point, where 𝛾 is called the degree of exponential stability.

Proposition 2.2 (Cauchy inequality). For any symmetric positive definite matrix 𝑁∈𝑀^𝑛×𝑛 and 𝑥,𝑦∈ℝ^𝑛 we have ±2𝑥^𝑇𝑦≤𝑥^𝑇𝑁𝑥+𝑦^𝑇𝑁⁻¹𝑦. (2.9)

Proposition 2.3 (see [2]). For any symmetric positive definite matrix 𝑀>0, scalar 𝛾>0, and vector function 𝜔∶[0,𝛾]→ℝ^𝑛 such that the integrations concerned are well defined, the following inequality holds ^𝛾₀𝜔(𝑠)𝑑𝑠^𝑇𝑀^𝛾₀𝜔(𝑠)𝑑𝑠≤𝛾^𝛾₀𝜔^𝑇(𝑠)𝑀𝜔(𝑠)𝑑𝑠. (2.10)

Proposition 2.4 (Schur complement lemma, [2]). Given constant symmetric matrices 𝑋, 𝑌, 𝑍 with appropriate dimensions satisfying 𝑋=𝑋^𝑇, 𝑌=𝑌^𝑇>0. Then 𝑋+𝑍^𝑇𝑌⁻¹𝑍<0 if and only if 𝑋𝑍^𝑇𝑍−𝑌𝑍<0𝑜𝑟−𝑌𝑍_𝑇𝑋<0. (2.11)

3. Main Results

Now we present a new delay-dependent condition for the uncertain system (2.1) satisfying the sector conditions (2.3).

Assumption 3.1. All the eigenvalues of matrix C are inside the unit circle.

Theorem 3.2. Under Assumption 3.1, given 𝛼>0. The system (2.1) satisfying the sector condition (2.3) is 𝛼-exponentially stabilizable if there exist symmetric positive definite matrices 𝑃, 𝑄, 𝑅, 𝑈, 𝐹,𝐿; symmetric positive semidefinite matrices 𝑍=diag(𝑧₁,𝑧₂,…,𝑧_𝑚) and 𝐽=diag(𝑗₁,𝑗₂,…,𝑗_𝑚); scalars 𝜖₁>0 and 𝜖₂>0; matrices 𝑁_𝑖; 𝑖=1,2,3 of appropriate dimension such that the following LMI holds: ℳ₁=ℳ−0𝐼−𝐼000000^𝑇×𝑒^−2𝛼ℎ2𝑈ℳ0−𝐼𝐼000000<0,₂=ℳ−0−𝐼0𝐼00000^𝑇×𝑒^−2𝛼ℎ2𝑈𝜙0𝐼0−𝐼00000<0,ℳ=¹¹𝜙₁₂𝜙₁₃𝜙₁₄𝜙₁₅𝜙₁₆𝜙₁₇𝜙₁₈𝜙₁₉∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅𝜙₂₆𝜙₂₇𝜙₂₈𝜙₂₉∗∗𝜙₃₃000000∗∗∗𝜙₄₄00000∗∗∗∗𝜙₅₅𝜙₅₆𝜙₅₇𝜙₅₈𝜙₅₉∗∗∗∗∗𝜙₆₆𝜙₆₇𝜙₆₈0∗∗∗∗∗∗𝜙₇₇00∗∗∗∗∗∗∗𝜙₈₈0∗∗∗∗∗∗∗∗𝜙₉₉, (3.1)(3.2) where 𝜙₁₁=𝑃𝐴+𝐴^𝑇𝑃+𝑄+𝑄^𝑇−𝑒^−2𝛼ℎ1𝑅−𝑒^−2𝛼ℎ2𝑅+2𝛼𝑥^𝑇(𝑡)𝑃𝑥(𝑡),𝜙₁₂=𝑃𝐴₁+𝐴^𝑇𝑁^𝑇₁,𝜙₁₃=𝑒^−2𝛼ℎ1𝑅,𝜙₁₄=𝑒^−2𝛼ℎ2𝑅,𝜙₁₇=𝑃𝐶,𝜙₁₈=𝑃𝐷,𝜙₁₉=𝜖_𝑖𝐸^𝑇₁,𝜙₁₆=𝐴^𝑇𝐿+𝐴^𝑇𝑁^𝑇₃,𝜙₁₅=𝐴^𝑇𝐻𝑍^𝑇+𝑃𝐵+𝐻𝐾𝐽+𝐴^𝑇𝑁^𝑇₂𝜙+2𝛼𝐻𝑍,₂₂=−2𝑒^−2𝛼ℎ2𝑈+𝑁₁𝐴₁+𝐴^𝑇₁𝑁^𝑇₁,𝜙₂₃=𝑒^−2𝛼ℎ2𝑈,𝜙₂₄=𝑒^−2𝛼ℎ2𝜙𝑈,₂₅=𝐴^𝑇₁𝐻𝑍^𝑇+𝑁₁𝐵+𝐴^𝑇₁𝑁^𝑇₂,𝜙₂₆=𝐴^𝑇₁𝐿−𝑁₁+𝐴^𝑇₁𝑁^𝑇₃,𝜙₂₇=𝑁₁𝐶,𝜙₂₈=𝑁₁𝜙𝐷,₂₉=𝜖_𝑖𝐸^𝑇₃,𝜙₃₃=−𝑒^−2𝛼ℎ1𝑄−𝑒^−2𝛼ℎ1𝑅−𝑒^−2𝛼ℎ2𝑈,𝜙₄₄=−𝑒^−2𝛼ℎ2𝑄−𝑒^−2𝛼ℎ2𝑅−𝑒^−2𝛼ℎ2𝜙𝑈,₅₇=𝑍𝐻^𝑇𝐶+𝑁₂𝐶,𝜙₅₅=𝑍𝐻^𝑇𝐵+𝐵^𝑇𝐻𝑍^𝑇−𝐽−𝐽^𝑇+𝑁₂𝐵+𝐵^𝑇𝑁^𝑇₂,𝜙₅₆=𝐵^𝑇𝐿−𝑁₂+𝐵^𝑇𝑁^𝑇₃,𝜙₅₉=𝜖_𝑖𝐸^𝑇₂,𝜙₅₈=𝑍𝐻^𝑇𝐷+𝑁₂𝐷,𝜙₈₈=−𝜖_𝑖𝜙𝐼,₉₉=−𝜖_𝑖𝐼,𝜙₆₆=ℎ²₁𝑅+ℎ²₂𝑅+𝐹+ℎ₂−ℎ₁²𝑈−𝐿−𝐿^𝑇−𝑁₃−𝑁^𝑇₃,𝜙₆₇=𝐿𝐶+𝑁₃𝐶,𝜙₆₈=𝐿𝐷+𝑁₃𝐷,𝜙₇₇=−𝑒^−2𝛼𝜂1−𝜂_𝐷𝐹. (3.3)

The solution 𝑥(𝑡) of the system satisfies, ‖𝑥(𝑡)‖≤𝑎𝜙²+𝑏𝑀₁²+𝑐𝑀₂²𝜆_𝑚(𝑃), (3.4) where 𝑎=𝜆_𝑀(𝑃)+2ℎ₂𝜆_𝑀(𝑅)+ℎ₂𝜆_𝑀(𝑈)+2𝜆_𝑀(𝐻𝑍𝐾𝐻^𝑇), 𝑏=2𝜆_𝑀(𝑄)(1−𝑒^−2𝛼ℎ2)/2𝛼+2ℎ₂𝜆_𝑀(𝑅)((1−𝑒^−2𝛼ℎ2)/2𝛼)+ℎ₂𝜆_𝑀(𝑈)((1−𝑒^−2𝛼ℎ2)/2𝛼), and 𝑐=𝜆_𝑀(𝐹)((1−𝑒^−2𝛼𝜂)/2𝛼), ‖𝑀₁‖=sup_{−𝑚≤𝑠≤0}‖𝑥(𝑠)‖, ‖𝑀₂‖=sup_{−𝑚≤𝑠≤0}‖𝑥̇(𝑠)‖.

Proof. Using (2.5), the uncertain system (2.1) can be represented as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴₁𝑥((𝑡(𝜎(𝑡𝑥̇(𝑡)(𝑡),𝑝(𝑡)(𝑡)𝐸𝑡−ℎ))+𝐵𝑓))+𝐶𝑡−𝜂+𝐷𝑝=𝐹₁𝑥(𝑡)+𝐸₂𝑓(𝜎(𝑡))+𝐸₃𝑥((𝑡,𝜎(𝑡)𝑡−ℎ))=𝐻^𝑇𝑥(𝑡)=ℎ₁ℎ₂⋯ℎ_𝑚^𝑇𝑥(𝑡)𝑥(𝑠)(𝑠)ℎ,∀𝑡≥0,=𝜙,𝑠∈−max₂,𝜂₂,0. (3.5) We consider the following Lyapunov-Krasovskii functional 𝑉(𝑥(𝑡))=⁸_𝑖=1𝑉_𝑖, (3.6) where 𝑉₁(𝑥(𝑡))=𝑒^2𝛼𝑡𝑥^𝑇(𝑡)(𝑡),𝑉𝑃𝑥₂(𝑥(𝑡))=^𝑡_𝑡−ℎ1𝑒^2𝛼𝑠𝑥^𝑇(𝑠)(𝑠)𝑉𝑄𝑥𝑑𝑠,₃(𝑥(𝑡))=^𝑡_𝑡−ℎ2𝑒^2𝛼𝑠𝑥^𝑇(𝑠)(𝑠)𝑉𝑄𝑥𝑑𝑠,₄(𝑥(𝑡))=ℎ₁⁰_−ℎ1^𝑡_𝑡+𝑠𝑒^2𝛼𝑠𝑥̇^𝑇(𝜏)𝑅𝑥̇(𝜏)𝑉𝑑𝜏𝑑𝑠,₅(𝑥(𝑡))=ℎ₂⁰_−ℎ2^𝑡_𝑡+𝑠𝑒^2𝛼𝑠𝑥̇^𝑇(𝜏)𝑅𝑥̇(𝜏)𝑉𝑑𝜏𝑑𝑠,₆(𝑥(𝑡))=ℎ₂−ℎ₁^−ℎ1_−ℎ2^𝑡_𝑡+𝑠𝑒^2𝛼𝑠𝑥̇^𝑇(𝜏)𝑈𝑥̇(𝜏)𝑉𝑑𝜏𝑑𝑠,₇(𝑥(𝑡))=^𝑡_{(𝑡)𝑡−𝜂}𝑒^2𝛼𝑠𝑥̇^𝑇(𝑠)𝐹𝑥̇(𝑠)𝑉𝑑𝑠,₈(𝑥(𝑡))=2^𝑚_𝑖=1𝜆_𝑖𝑒^{2𝛼𝑡𝜎𝑖}₀𝑓_𝑖𝜎_𝑖𝑑𝜎_𝑖. (3.7) Taking the derivative of 𝑉(𝑥_𝑡) along the solution of system (3.5), we have 𝑉̇₁(𝑥(𝑡))=2𝛼𝑒^2𝛼𝑡𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+2𝑒^2𝛼𝑡𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡),=𝑒^2𝛼𝑡2𝛼𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐴𝑥(𝑡)+𝐴₁𝑥((𝑡(𝜎(𝑡𝑥̇(𝑡)(𝑡),𝑉̇𝑡−ℎ))+𝐵𝑓))+𝐶𝑡−𝜂𝐷𝑝₂(𝑥(𝑡))=𝑒^2𝛼𝑡𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑒^−2𝛼ℎ1𝑥^𝑇𝑡−ℎ₁𝑄𝑥𝑡−ℎ₁,𝑉̇₃(𝑥(𝑡))=𝑒^2𝛼𝑡𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑒^−2𝛼ℎ2𝑥^𝑇𝑡−ℎ₂𝑄𝑥𝑡−ℎ₂,𝑉̇₄(𝑥(𝑡))≤𝑒^2𝛼𝑡ℎ²₁𝑥̇^𝑇(𝑡)𝑅𝑥̇(𝑡)−ℎ₁𝑒^{−2𝛼ℎ1𝑡}_𝑡−ℎ1𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠,𝑉̇₅(𝑥(𝑡))≤𝑒^2𝛼𝑡ℎ²₂𝑥̇^𝑇(𝑡)𝑅𝑥̇(𝑡)−ℎ₂𝑒^{−2𝛼ℎ2𝑡}_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠,𝑉̇₆(𝑥(𝑡))≤𝑒^2𝛼𝑡ℎ₂−ℎ₁²𝑥̇^𝑇(𝑡)𝑈𝑥̇(𝑡)−ℎ₂−ℎ₁𝑒^{−2𝛼ℎ2𝑡−ℎ1}_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠,𝑉̇₇(𝑥(𝑡))≤𝑒^2𝛼𝑡𝑥̇^𝑇(𝑡)𝐹𝑥̇(𝑡)−𝑒^−2𝛼𝜂1−𝜂_𝑑𝑥̇^𝑇𝑡−𝜂(𝑡)𝐹𝑥̇𝑡−𝜂(𝑡),𝑉̇₈(𝑥(𝑡))=2^𝑚_𝑖=1𝜆_𝑖𝑒^2𝛼𝑡2𝛼^𝜎𝑖₀^(𝑡)𝑓_𝑖𝜎_𝑖𝑑𝜎_𝑖+𝑓_𝑖𝜎_𝑖(𝑡)𝜎̇_𝑖(𝑡)≤𝑒^2𝛼𝑡4𝛼𝑓^𝑇(𝜎(𝑡))𝑍𝜎(𝑡)+2𝑓^𝑇(𝜎(𝑡))𝑍𝜎̇(𝑡)≤𝑒^2𝛼𝑡4𝛼𝑓^𝑇(𝜎(𝑡))𝑍𝐻𝑥(𝑡)+2𝑓^𝑇(𝜎(𝑡))𝑍𝐻^𝑇𝑥̇(𝑡)=𝑒^2𝛼𝑡4𝛼𝑓^𝑇(𝜎(𝑡))𝑍𝐻𝑥(𝑡)+2𝑓^𝑇(𝜎(𝑡))𝑍𝐻^𝑇×𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡−ℎ(𝑡))+𝐵𝑓(𝜎(𝑡))+𝐶𝑥̇𝑡−𝜂(𝑡)+𝐷𝑝(𝑡). (3.8) Applying Proposition 2.3 and the Leibniz-Newton formula, we have −ℎ₁^𝑡_𝑡−ℎ1𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠≤−^𝑡_𝑡−ℎ1𝑥̇(𝑠)^𝑇𝑅^𝑡_𝑡−ℎ1𝑥̇(𝑠𝑥(𝑡))≤−−𝑥𝑡−ℎ₁^𝑇𝑅𝑥(𝑡)−𝑥𝑡−ℎ₁=−𝑥^𝑇(𝑡)𝑅𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑅𝑥𝑡−ℎ₁−𝑥^𝑇𝑡−ℎ₁𝑅𝑥𝑡−ℎ₁,−ℎ₂^𝑡_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)𝑑𝑠≤−^𝑡_𝑡−ℎ2𝑥̇(𝑠)^𝑇𝑅^𝑡_𝑡−ℎ2𝑥̇(𝑠𝑥(𝑡))≤−−𝑥𝑡−ℎ₂^𝑇𝑅𝑥(𝑡)−𝑥𝑡−ℎ₂=−𝑥^𝑇(𝑡)𝑅𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑅𝑥𝑡−ℎ₂−𝑥^𝑇𝑡−ℎ₂𝑅𝑥𝑡−ℎ₂. (3.9) Note that −ℎ₂−ℎ₁^𝑡−ℎ1_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠=−ℎ₂−ℎ₁^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)−ℎ𝑑𝑠₂−ℎ₁^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)ℎ𝑑𝑠=−₂−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)−ℎ(𝑡)𝑑𝑠−ℎ₁^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)−ℎ(𝑡)𝑑𝑠−ℎ₁^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)−ℎ𝑑𝑠₂−ℎ(𝑡)^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠. (3.10) Using Proposition 2.3 gives −ℎ₂−ℎ(𝑡)^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠≤−^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥̇(𝑠)𝑑𝑠^𝑇𝑈^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥̇(𝑠)𝑥((𝑡𝑑𝑠≤−𝑡−ℎ))−𝑥𝑡−ℎ₂^𝑇𝑈𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₂,−ℎ(𝑡)−ℎ₁^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠≤−^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑑𝑠^𝑇𝑈^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥̇(𝑠)𝑥𝑑𝑠≤−𝑡−ℎ₁−𝑥(𝑡−ℎ(𝑡))^𝑇𝑈𝑥𝑡−ℎ₁−𝑥(𝑡−ℎ(𝑡)). (3.11) Let 𝛽=(ℎ₂−ℎ(𝑡))/(ℎ₂−ℎ₁)≤1, so we have −ℎ₂−ℎ(𝑡)^𝑡−ℎ1_{(𝑡)𝑡−ℎ}𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠=−𝛽^𝑡−ℎ1_{(𝑡)𝑡−ℎ}ℎ₂−ℎ₁𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠≤−𝛽^𝑡−ℎ1_{(𝑡)𝑡−ℎ}ℎ(𝑡)−ℎ₁𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑥𝑑𝑠≤−𝛽𝑡−ℎ₁−𝑥(𝑡−ℎ(𝑡))^𝑇𝑈𝑥𝑡−ℎ₁((𝑡,−ℎ(𝑡)−𝑥𝑡−ℎ))−ℎ₁^{𝑡−ℎ(𝑡)}_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠=−1−𝛽^{𝑡−ℎ(𝑡)}_𝑡−ℎ2ℎ₂−ℎ₁𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠≤−1−𝛽^{𝑡−ℎ(𝑡)}_𝑡−ℎ2ℎ₂−ℎ(𝑡)𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑥((𝑡𝑑𝑠≤−1−𝛽𝑡−ℎ))−𝑥𝑡−ℎ₂^𝑇×𝑈𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₂. (3.12) Therefore from (3.10)–(3.12), we obtain −ℎ₂−ℎ₁^𝑡−ℎ1_𝑡−ℎ2𝑥̇^𝑇(𝑠)𝑈𝑥̇(𝑠)𝑑𝑠≤−𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₂^𝑇𝑈𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₂−𝑥𝑡−ℎ₁−𝑥(𝑡−ℎ(𝑡))^𝑇𝑈𝑥𝑡−ℎ₁((𝑡𝑥−𝑥𝑡−ℎ))−𝛽𝑡−ℎ₁−𝑥(𝑡−ℎ(𝑡))^𝑇𝑈𝑥𝑡−ℎ₁((𝑡−𝑥((𝑡−𝑥𝑡−ℎ))1−𝛽𝑡−ℎ))−𝑥𝑡−ℎ₂^𝑇𝑈𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₂. (3.13) We add the following zero equation: 2𝜉^𝑇(𝑡)𝑁𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡−ℎ(𝑡))+𝐵𝑓(𝜎(𝑡))+𝐶𝑥̇𝑡−𝜂(𝑡)+𝐷𝑝(𝑡)−𝑥̇(𝑡)=0, (3.14) where 𝑁=[𝑁^𝑇₁𝑁^𝑇₂𝑁^𝑇₃]^𝑇, 𝜉(𝑡)=[𝑥^𝑇(𝑡−ℎ(𝑡))𝑓^𝑇(𝜎(𝑡))𝑥̇^𝑇(𝑡)]^𝑇 and by using the identity relation −𝑥̇(𝑡)+𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡−ℎ(𝑡))+𝐵𝑓(𝜎(𝑡))+𝐶𝑥̇𝑡−𝜂(𝑡)+𝐷𝑝(𝑡)=0, (3.15) we have −2𝑥̇^𝑇(𝑡)𝐿𝑥̇(𝑡)+2𝑥̇^𝑇(𝑡)𝐿𝐴𝑥(𝑡)+2𝑥̇^𝑇(𝑡)𝐿𝐴₁𝑥(𝑡−ℎ(𝑡))+2𝑥̇^𝑇(𝑡)(𝜎(𝑡𝑥̇𝐿𝐵𝑓))+2^𝑇(𝑡)𝐿𝐶𝑥̇𝑡−𝜂(𝑡)+2𝑥̇^𝑇(𝑡)𝐿𝐷𝑝(𝑡)=0. (3.16) For system (2.1) with nonlinearity located in the sectors [0,𝑘_𝑗](𝑗=1,2,…,𝑚), if there exists 𝐽=diag(𝑗₁,𝑗₂,…,𝑗_𝑚), then we have 𝑗_𝑖𝑓_𝑖𝜎_𝑖𝑘_𝑖ℎ^𝑇_𝑖𝑥(𝑡)−𝑓_𝑖𝜎_𝑖≥0,𝑖=1,2,…,𝑚, (3.17) which is equivalent to 𝑥^𝑇(𝑡)𝐻𝐾𝐽𝑓(𝜎(𝑡))−𝑓^𝑇(𝜎(𝑡))𝐽𝑓(𝜎(𝑡))≥0. (3.18) Similarly, for any 𝜖>0, from (3.4), we have −𝜖𝑝^𝑇(𝑡)𝑝(𝑡)+𝜖𝐸₁𝑥(𝑡)+𝐸₂𝑓(𝜎(𝑡))+𝐸₃𝑥(𝑡−ℎ(𝑡))^𝑇×𝐸₁𝑥(𝑡)+𝐸₂𝑓(𝜎(𝑡))+𝐸₃𝑥(𝑡−ℎ(𝑡))≥0. (3.19) Hence, according to (3.9), (3.13), and by adding the zero term (3.16) and (3.18)-(3.19), we get 𝑉̇(𝑥(𝑡))≤𝑒^2𝛼𝑡2𝛼𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐴𝑥(𝑡)+𝐴₁𝑥((𝑡(𝜎(𝑡𝑥̇(𝑡)(𝑡)𝑡−ℎ))+𝐵𝑓))+𝐶𝑡−𝜂+𝐷𝑝+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑒^−2𝛼ℎ1𝑥^𝑇𝑡−ℎ₁𝑄𝑥𝑡−ℎ₁−𝑒^−2𝛼ℎ2𝑥^𝑇𝑡−ℎ₂𝑄𝑥𝑡−ℎ₂+ℎ²₁𝑥̇^𝑇(𝑡)𝑅𝑥̇(𝑡)+ℎ²₂𝑥̇^𝑇(𝑡)𝑅𝑥̇(𝑡)−𝑒^−2𝛼ℎ1𝑥^𝑇(𝑡)𝑅𝑥(𝑡)+2𝑒^−2𝛼ℎ1𝑥^𝑇(𝑡)𝑅𝑥𝑡−ℎ₁+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑒^−2𝛼ℎ1𝑥^𝑇𝑡−ℎ₁𝑅𝑥𝑡−ℎ₁−𝑒^−2𝛼ℎ2𝑥^𝑇𝑡−ℎ₂𝑅𝑥𝑡−ℎ₂−𝑒^−2𝛼ℎ2𝑥^𝑇(𝑡)𝑅𝑥(𝑡)+2𝑒^−2𝛼ℎ2𝑥^𝑇(𝑡)𝑅𝑥𝑡−ℎ₂×𝑥̇^𝑇(𝑡)𝐹𝑥̇(𝑡)−𝑒^−2𝛼𝜂1−𝜂_𝐷𝑥̇^𝑇𝑡−𝜂(𝑡)𝐹𝑥̇𝑡−𝜂(𝑡)2𝛼𝑓^𝑇(𝜎(𝑡))𝑍𝐻𝑥(𝑡)+2𝑓^𝑇(𝜎(𝑡))𝑍𝐻^𝑇𝐴𝑥(𝑡)+𝐴₁𝑥(𝑡−ℎ(𝑡))+𝐵𝑓(𝜎(𝑡))+𝐶𝑥̇𝑡−𝜂(𝑡)+𝐷𝑝(𝑡)−𝑒^−2𝛼ℎ2𝑥^𝑇(𝑡)𝑅𝑥(𝑡)+2𝑒^−2𝛼ℎ2𝑥^𝑇(𝑡)𝑅𝑥𝑡−ℎ₂ℎ₂−ℎ₁²𝑥̇^𝑇(𝑡)𝑈𝑥̇(𝑡)+2𝑥^𝑇(𝑡)𝐻𝐾𝐽𝑓(𝜎(𝑡))−2𝑓^𝑇(𝜎(𝑡))𝐽𝑓(𝜎(𝑡))−2𝑥̇^𝑇(𝑡)𝐿𝑥̇(𝑡)+2𝑥̇^𝑇(𝑡)(𝑡)𝑥̇𝐿𝐴𝑥+2^𝑇(𝑡)𝐿𝐴₁𝑥(𝑡−ℎ(𝑡))+2𝑥̇^𝑇(𝑡)𝐿𝐵𝑓(𝜎(𝑡))+2𝑥̇^𝑇(𝑡)𝑥̇(𝑡)𝑥̇𝐿𝐶𝑡−𝜂+2^𝑇(𝑡)𝐿𝐷𝑝(𝑡)−𝜖𝑝^𝑇(𝑡)𝑝(𝑡)+𝜖𝐸₁𝑥(𝑡)+𝐸₂𝑓(𝜎(𝑡))+𝐸₃𝑥(𝑡−ℎ(𝑡))^𝑇×𝐸₁𝑥(𝑡)+𝐸₂𝑓(𝜎(𝑡))+𝐸₃𝑥(𝑡−ℎ(𝑡))−𝑒^−2𝛼ℎ2𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₂^𝑇×𝑈𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₂−𝑒^−2𝛼ℎ2𝑥𝑡−ℎ₁−𝑥(𝑡−ℎ(𝑡))^𝑇𝑈𝑥𝑡−ℎ₁((𝑡𝑥−𝑥𝑡−ℎ))−𝛽𝑡−ℎ₁−𝑥(𝑡−ℎ(𝑡))^𝑇𝑈𝑥𝑡−ℎ₁((𝑡−𝑥((𝑡−𝑥𝑡−ℎ))1−𝛽𝑡−ℎ))−𝑥𝑡−ℎ₂^𝑇𝑈𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₂=𝑒^2𝛼𝑡𝜁^𝑇(𝑡)ℳ𝜁(𝑡)−𝛽𝑥𝑡−ℎ₁−𝑥(𝑡−ℎ(𝑡))^𝑇𝑒^−2𝛼ℎ2𝑈𝑥𝑡−ℎ₁((𝑡−𝑥((𝑡−𝑥𝑡−ℎ))1−𝛽𝑡−ℎ))−𝑥𝑡−ℎ₂^𝑇𝑒^−2𝛼ℎ2𝑈𝑥(𝑡−ℎ(𝑡))−𝑥𝑡−ℎ₂=𝑒^2𝛼𝑡𝜁^𝑇(𝑡)1−𝛽ℳ₁+𝛽ℳ₂𝜁(𝑡), (3.20) where ℳ₁ and ℳ₂ are defined as in (3.1), respectively, and 𝜁(𝑡)=[𝑥(𝑡)𝑥(𝑡−ℎ(𝑡))𝑥(𝑡−ℎ₁)𝑥(𝑡−ℎ₂)𝑓(𝜎(𝑡))𝑥̇(𝑡)𝑥̇(𝑡−𝜂(𝑡))𝑝(𝑡)]. By (1−𝛽)ℳ₁+𝛽ℳ₂<0 holds if and only if ℳ₁<0 and ℳ₂<0. For showing the convergence rate, we have 𝑉̇(𝑥(𝑡))≤0, and then 𝑉(𝑥(𝑡))≤𝑉(𝑥(0)). However, 𝑉₁(𝑥(0))=𝑒^2𝛼𝑡𝑥^𝑇(0)𝑃𝑥(0)≤𝜆_max(𝑃)𝜙²,𝑉₂(𝑥(0=))_0−ℎ1𝑒^2𝛼𝑠𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠,≤𝜆_max(𝑄)⁰_−ℎ1𝑒^2𝛼𝑠𝑑𝑠𝑀₁²=𝜆_max(𝑄)1−𝑒^−2𝛼ℎ2𝑀2𝛼¹²,𝑉₃(𝑥(0=))_0−ℎ2𝑒^2𝛼𝑠𝑥^𝑇(𝑠)𝑄𝑥(𝑠)𝑑𝑠,≤𝜆_max(𝑄)⁰_−ℎ2𝑒^2𝛼𝑠𝑑𝑠𝑀₁²=𝜆_max(𝑄)1−𝑒^−2𝛼ℎ2𝑀2𝛼¹²,𝑉₄(𝑥(0))=ℎ₁⁰_−ℎ1⁰_𝑠𝑒^2𝛼𝑠𝑥̇^𝑇(𝜏)𝑅𝑥̇(𝜏)𝑑𝜏𝑑𝑠=ℎ₁⁰_−ℎ1𝑒^2𝛼𝑠𝑥^𝑇(0)𝑅𝑥(0)−𝑥^𝑇(𝑠)𝑅𝑥(𝑠)𝑑𝑠≤ℎ₂𝜆_𝑀(𝑅)⁰_−ℎ1𝑒^2𝛼𝑠𝑑𝑠𝜙²−ℎ₂𝜆_𝑀(𝑅)⁰_−ℎ1𝑒^2𝛼𝑠𝑑𝑠𝑀₁²=ℎ₂𝜆_𝑀(𝑅)1−𝑒^−2𝛼ℎ2𝜙2𝛼²−ℎ₂𝜆_𝑀(𝑅)1−𝑒^−2𝛼ℎ2𝑀2𝛼¹²,𝑉₅(𝑥(0))=ℎ₂⁰_−ℎ2⁰_𝑠𝑒^2𝛼𝑠𝑥̇^𝑇(𝜏)𝑅𝑥̇(𝜏)𝑑𝜏𝑑𝑠≤ℎ₂𝜆_𝑀(𝑅)1−𝑒^−2𝛼ℎ2𝜙2𝛼²−ℎ₂𝜆_𝑀(𝑅)1−𝑒^−2𝛼ℎ2𝑀2𝛼¹²,𝑉₆(𝑥(0))=ℎ₂^−ℎ1_−ℎ2⁰_𝑠𝑒^2𝛼𝑠𝑥̇^𝑇(𝜏)𝑈𝑥̇(𝜏)𝑑𝜏𝑑𝑠≤ℎ₂𝜆_𝑀(𝑈)1−𝑒^−2𝛼ℎ2𝜙2𝛼²−ℎ₂𝜆_𝑀(𝑈)1−𝑒^−2𝛼ℎ2𝑀2𝛼¹²,𝑉₇(𝑥(0=))_0𝜂(0)𝑒^2𝛼𝑠𝑥̇^𝑇(𝑠)𝐹𝑥̇(𝑠)𝑑𝑠,≤𝜆_𝑀(𝐹)⁰_−𝜂𝑒^2𝛼𝑠𝑑𝑠𝑀₂²≤𝜆_𝑀(𝐹)1−𝑒^−2𝛼𝜂𝑀2𝛼²²,𝑉₈(𝑥(𝑡))=2^𝑚_𝑖=1𝜆_𝑖𝑒^{2𝛼𝑡𝜎𝑖}₀𝑓_𝑖𝜎_𝑖𝑑𝜎_𝑖≤2^𝑚_𝑖=1𝜆_𝑖𝑒^{2𝛼𝑡𝜎𝑖}₀𝑘𝜎_𝑖(𝑡)𝑑𝜎_𝑖≤2^𝑚_𝑖=1𝑒^2𝛼𝑡𝜆_𝑖𝑘𝜎²_𝑖(𝑡)≤2^𝑚_𝑖=1𝑒^2𝛼𝑡𝜆_𝑖𝑘𝑥^𝑇(𝑡)𝐻𝐻^𝑇𝑥(𝑡),𝑉₈(𝑥(0))≤2𝜆_𝑀𝐻𝑍𝐾𝐻^𝑇𝜙², (3.21) and 𝑉(𝑥(𝑡))≥𝑒^2𝛼𝑡𝑥^𝑇(𝑡)𝑃𝑥(𝑡)≥𝑒^2𝛼𝑡𝜆_𝑚(𝑃)‖𝑥(𝑡)‖². Then, from Definition 2.1, we conclude that the equilibrium point is globally exponentially stable. This completes the proof.

Remark 3.3. For system (2.1) with uncertainty located in the sector conditions (2.4), if there exists 𝐽=diag(𝑗₁,𝑗₂,…,𝑗_𝑚), it follows that 𝑟_𝑖𝑓_𝑖𝜎_𝑖(𝑡)ℎ^𝑇_𝑖𝑥(𝑡)≥0,𝑖=1,2,…,𝑚, (3.22) which is equivalent to 𝑥^𝑇(𝑡)𝐻𝐽𝑓(𝜎(𝑡))≥0. (3.23)
By replacing (3.18) with (3.23) and letting 𝛼=0, we obtain the following robust stability criterion.

Corollary 3.4. Under Assumption 3.1, the system (2.1) satisfying the sector condition (2.4) is asymptotically stable if there exist symmetric positive definite matrices 𝑃, 𝑄, 𝑅, 𝑈, 𝐹, 𝐿; symmetric positive semidefinite matrices 𝑍=diag(𝑧₁,𝑧₂,…,𝑧_𝑚) and 𝐽=diag(𝑗₁,𝑗₂,…,𝑗_𝑚); scalars 𝜖₁>0 and 𝜖₂>0; matrices 𝑁_𝑖; 𝑖=1,2,3 of appropriate dimension such that the following LMI holds: ℳ₁=ℳ−0𝐼−𝐼000000^𝑇×𝑒^−2𝛼ℎ2𝑈ℳ0−𝐼𝐼000000<0,₂=ℳ−0−𝐼0𝐼00000^𝑇×𝑒^−2𝛼ℎ2𝑈𝜙0𝐼0−𝐼00000<0,ℳ=¹¹𝜙₁₂𝜙₁₃𝜙₁₄𝜛₁₅𝜙₁₆𝜙₁₇𝜙₁₈𝜙₁₉∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅𝜙₂₆𝜙₂₇𝜙₂₈𝜙₂₉∗∗𝜙₃₃000000∗∗∗𝜙₄₄00000∗∗∗∗𝜛₅₅𝜙₅₆𝜙₅₇𝜙₅₈𝜙₅₉∗∗∗∗∗𝜙₆₆𝜙₆₇𝜙₆₈0∗∗∗∗∗∗𝜙₇₇00∗∗∗∗∗∗∗𝜙₈₈0∗∗∗∗∗∗∗∗𝜙₉₉, (3.24) where 𝜛₁₅=𝐴^𝑇𝐻𝑍^𝑇+𝑃𝐵+𝐻𝐽,𝜛₅₅=𝑍𝐻^𝑇𝐵+𝐵^𝑇𝐻𝑍^𝑇. (3.25)

Remark 3.5. The following stability criteria are presented for finite and infinite sector conditions, for systems without uncertainties.

Corollary 3.6. Under Assumption 3.1, given 𝛼>0. The system (2.1) without uncertainties satisfying the sector condition (2.3) is 𝛼-exponentially stabilizable if there exist symmetric positive definite matrices 𝑃, 𝑄, 𝑅, 𝑈, 𝐹, 𝐿; symmetric positive semidefinite matrices 𝑍=diag(𝑧₁,𝑧₂,…,𝑧_𝑚) and 𝐽=diag(𝑗₁,𝑗₂,…,𝑗_𝑚); matrices 𝑁_𝑖; 𝑖=1,2,3 of appropriate dimension such that the following LMI holds: ℳ₁=ℳ−0𝐼−𝐼0000^𝑇×𝑒^−2𝛼ℎ2𝑈ℳ0−𝐼𝐼0000<0,₂=ℳ−0−𝐼0𝐼000^𝑇×𝑒^−2𝛼ℎ2𝑈𝜙0𝐼0−𝐼000<0,ℳ=¹¹𝜙₁₂𝜙₁₃𝜙₁₄𝜙₁₅𝜙₁₆𝜙₁₇∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅𝜙₂₆𝜙₂₇∗∗𝜙₃₃0000∗∗∗𝜙₄₄000∗∗∗∗𝜙₅₅𝜙₅₆𝜙₅₇∗∗∗∗∗𝜙₆₆𝜙₆₇∗∗∗∗∗∗𝜙₇₇. (3.26)

Corollary 3.7. Under Assumption 3.1, the system (2.1) without uncertainties satisfying the sector condition (2.4) is asymptotically stable if there exist symmetric positive definite matrices 𝑃, 𝑄, 𝑅, 𝑈, 𝐹, 𝐿; symmetric positive semidefinite matrices 𝑍=diag(𝑧₁,𝑧₂,…,𝑧_𝑚) and 𝐽=diag(𝑗₁,𝑗₂,…,𝑗_𝑚); matrices 𝑁_𝑖; 𝑖=1,2,3 of appropriate dimension such that the following LMI holds: ℳ₁=ℳ−0𝐼−𝐼0000^𝑇×𝑒^−2𝛼ℎ2𝑈ℳ0−𝐼𝐼0000<0,₂=ℳ−0−𝐼0𝐼000^𝑇×𝑒^−2𝛼ℎ2𝑈𝜙0𝐼0−𝐼000<0,ℳ=¹¹𝜙₁₂𝜙₁₃𝜙₁₄𝜛₁₅𝜙₁₆𝜙₁₇∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅𝜙₂₆𝜙₂₇∗∗𝜙₃₃0000∗∗∗𝜙₄₄000∗∗∗∗𝜛₅₅𝜙₅₆𝜙₅₇∗∗∗∗∗𝜙₆₆𝜙₆₇∗∗∗∗∗∗𝜙₇₇. (3.27)

Remark 3.8. In this paper, the restriction that the state delay is differentiable is not required which allows the state delay to be fast time-varying. Meanwhile, this restriction is required in some existing result, see [8, 9, 16, 17].

Remark 3.9. It is worth pointing out that we can extend this method to more complex dynamical network models, such as neutral-type neural networks [18, 19] or BAM neutral-type neural networks [20, 21].

4. Numerical Examples

In this section, we provide numerical examples to show the effectiveness of our theoretical results.

Example 4.1. Consider the following nominal Lur’e system with time-varying delays which is studied in [8, 9, 17]: 𝑥̇(𝑡)−𝐶𝑥̇𝑡−𝜂(𝑡)=𝐴𝑥(𝑡)+𝐴₁𝑥((𝑡(𝜎(𝑡,𝜎(𝑡)𝑡−ℎ))+𝐵𝑓))=𝐻^𝑇𝑥(𝑡)=ℎ₁ℎ₂^𝑇𝑥(𝑡),∀≥0, (4.1) where 𝐴=−20.50−1,𝐴₁=,.10.40.4−1,𝐵=−0.5−0.75𝐶=0.20.10.10.2,𝐻=0.20.6 (4.2) Tables 1 and 2 give comparison of maximum allowable value of ℎ₂ for (4.1) obtained in Corollary 3.6 with nonlinearity satisfying (2.3), where 𝑘=100 and Corollary 3.7 with nonlinearity satisfying (2.4), respectively. We see that, when ℎ₁=0, the maximum allowable bounds for ℎ₂ obtained from Corollaries 3.6 and 3.7 are much better than those obtained in [8, 9, 17]. The results obtained in [8, 9] may not be used for the case when ℎ₁≠0. Moreover, the differentiability of the time delay ℎ(𝑡) is not required in Corollaries 3.6 and 3.7.

Table 1 

Upper delay bound ℎ₂ with ℎ₁=0.

Table 2 

Upper delay bound ℎ₂ with ℎ₁=0.

We let ℎ(𝑡)=0.7|cos𝑡|, 𝜂(𝑡)=1, 𝜙(𝑡)=[−0.1cos𝑡,cos𝑡], for all 𝑡∈[−1,0], and 𝑓(𝑥(𝑡))=𝛿|𝑥(𝑡)|, |𝛿|≤0.5. Figure 1 shows the trajectories of solutions 𝑥₁(𝑡) and 𝑥₂(𝑡) of the nominal Lur’e system with time-varying delays (4.1).

Figure 1 

The trajectories of 𝑥1(𝑡) and 𝑥2(𝑡) of the the nominal Lur’e system with time-varying delays (4.1).

Example 4.2. Consider the following uncertain Lur’e system with interval time-varying delay with the following parameters: 𝑥̇(𝑡)−𝐶𝑥̇𝑡−𝜂(𝑡)=(𝐴+Δ𝐴(𝑡))𝑥(𝑡)+𝐴₁+Δ𝐴₁(𝑡)𝑥((𝑡+((𝑡𝑓(𝜎(𝑡,𝑡−ℎ))𝐵+Δ𝐵)))) (4.3) where 𝐴=−200−0.9,𝐴₁=,−10−1−1,𝐵=−0.2−0.3,𝐶=0.2000.2𝐻=0.40.8,𝐷=𝐸₁=𝐸₃=0.1000.1,𝐸₂=0.0.1 (4.4)

Solution. From the conditions (3.1) of Theorem 3.2, we let 𝛼=0.4, ℎ₁=0.1, ℎ₂=0.4, 𝑡=0.8, 𝜂=0.1, and 𝐾=0.5. By using the LMI Tool-box in MATLAB, we obtain ,,,𝑃=28.6713−0.2237−0.22372.4026,𝑄=25.11760.76160.76161.0772,𝑈=23.67983.00453.00453.1148𝐹=7.0939−0.3657−0.36570.8018,𝑁1=−3.17360.4426−1.46011.2008,𝐿=16.55540016.5554𝑁3=−5.5006−0.4441−0.9892−14.4101,𝑁2=−1.64100.2843,𝑅=12.09060.23130.23132.2354𝑍=0.9717,𝐽=4.7851,𝑒₁=10.2112,𝑒₂=18.1870. (4.5) Thus, the system (2.1), is 0.4-exponentially stabilizable. Given 𝛼>0, we will give the values of the maximum allowable upper bounds of the uncertain Lur’e system with interval time-varying delay (4.3) for difference 𝜂_𝑑 of the delay for different decay rates 0.1≤𝛼≤0.4. From Theorem 3.2, we obtain the maximum allowable upper bound of the time-varying delay ℎ₂, as shown in Table 3.
We let ℎ(𝑡)=0.1+0.65|sin𝑡|, 𝜂(𝑡)=0.5+0.4|sin𝑡|, 𝜙(𝑡)=[−cos𝑡,cos𝑡], for all 𝑡∈[−0.9,0], and 𝑓(𝑥(𝑡))=𝛿|𝑥(𝑡)|, |𝛿|≤0.5. Figure 2 shows the trajectories of solutions 𝑥₁(𝑡) and 𝑥₂(𝑡) of the uncertain Lur’e system with interval time-varying delay (4.3).

Table 3 

Maximum allowable upper bounds ℎ₂ of the uncertain Lur'e system with interval time-varying delay (4.3) for different values of the 𝜂_𝑑 and decay rates.

Figure 2 

The trajectories of 𝑥1(𝑡) and 𝑥2(𝑡) of the uncertain Lur’e system with interval time-varying delay (4.3).

5. Conclusions

In this paper, we have investigated the delay-dependent robust stability criteria for uncertain neutral and Lur’e dynamical systems with sector-bounded nonlinearity. Based on Lyapunov-krasovskii theory, new delay-dependent sufficient conditions for robust stability have been derived in terms of LMIs. The interval time-varying delay function is not required to be differentiable which allows time-delay function to be a fast time-varying function. The global exponential stability for uncertain neutral and Lur’e dynamical systems with some conditions are investigated. Numerical examples are given to illustrate the effectiveness of the theoretic results which show that our results are much less conservative than some existing results in the literature.

Acknowledgments

The financial support from the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0355/2552) to W. Weera and P. Niamsup is acknowledged. The first author is also supported by the Graduate School, Chiang Mai University. The second author is also supported by the the Center of Excellence in Mathematics, CHE, Thailand. The authors also wish to thank the National Research University Project under Thailand’s Office of the Higher Education Commission for the financial support.