A Novel Approach to 𝐻_∞ Control Design for Linear Neutral Time-Delay Systems

Abstract

This paper is concerned with the problem of 𝐻_∞ control of linear neutral systems with time-varying delay. Firstly, by applying a novel Lyapunov-Krasovskii functional which is constructed with the idea of delay partitioning approach, appropriate free-weighting matrices, an improved delay-dependent bounded real lemma (BRL) for neutral systems is established. By using the obtained BRL, a delay-dependent sufficient condition for the existence of a state-feedback controller, which ensures asymptotic stability and a prescribed 𝐻_∞ performance level of the corresponding closed-loop system, is formulated in terms of linear matrix inequalities. Some numerical examples are given to illustrate the effectiveness of the proposed design method.

1. Introduction

A neutral time-delay system contains delays both in its states and in its derivatives of states, which occurs in various dynamic systems, such as economical systems, biological systems, metallurgical processing systems, nuclear reactor, power systems, and long-transmission lines in pneumatic and hydraulic systems [1–10]. It has been recognized that time delays can degrade system performance and even result in instability [3–8]. Therefore, researchers have paid considerable attention to the problems of analysis and synthesis for time-delay systems in the last decades (e.g., [1–16]).

In practical applications, however, it is desirable to design a controller such that the closed-loop system is not only stable but also possesses an adequate level of performance [6, 7, 10, 11]. One approach to cope with this problem is the so-called 𝐻_∞ control approach. The main objective of the 𝐻_∞ control is to obtain a controller such that the resulted closed system allows a maximum delay size for a fixed 𝐻_∞ performance bound or achieves a minimum 𝐻_∞ performance bound for a fixed delay size [10–12]. The conservatism in the 𝐻_∞ control is hence measured by the allowable delay size or performance bound obtained. Recently, many results on 𝐻_∞ control of neutral systems have appeared in the literature; see [4, 10–12, 16, 17] and the references therein.

In general, the existing literature for time-delay systems can be roughly divided into two types: delay-independent results [8–10] and delay-dependent ones [16–27]. The former is irrelevant to the delay size while the latter includes the information of delay size. Obviously, it has been recognized that delay-dependent results are generally less conservative than delay-independent ones, particularly when the delay size is time varying [20–24, 28, 29]. In order to further reduce the conservatism, some improved delay-dependent stability conditions are derived by introducing free-weighting matrices in [19]. In fact, Wu et al. and He et al. [20, 28] have proposed some effective methods for dealing with time-delay systems, which employ free-weighting matrices to express the relationships between the terms in the Leibinz-Newton formula. The method therein reduces the conservativeness of methods involving a fixed model transformation. Additional studies can be found in [8, 16, 20, 25, 28, 30] and references cited therein.

On the other hand, some other efforts on improving the delay-dependent conditions were made through introducing new Lyapunov-Krasovskii functional. To mention a few, new classes of Lyapunov functional and augmented Lyapunov functional were introduced to study the delay-dependent stability for systems with time-varying delay in [28], which is shown to possess less conservatism than the existing ones. In [21], based on a novel fuzzy Lyapunov-Krasovskii functional which is constructed using a delay partitioning method, a delay-dependent criterion is developed for the stability analysis of fuzzy time-varying state delay systems. In [24], a less conservative delay-dependent robust 𝐻_∞ control is proposed for uncertain linear systems with a state-delay based on a new Lyapunov-Krasovskii functional. A new criterion of asymptotic stability is derived in [26] by introducing a novel Lyapunov functional with the idea of partitioning the lower bound of the time-varying delay. Recently, there is enormous growth of interest in using the delay partitioning technique to deal with time-delay systems; see, for example [21–23, 26, 31–33]. The basic idea of this approach is to evenly partition time delay into several components. By constructing a Lyapunov-Krasovskii functional (LKF) with every delay component, one can obtain a less conservative stability condition as discussed by F. Gouaisbaut and Peaucelle [22] and Wu et al. [21, 29]. More results can be found in articles [17, 21–27, 29–33] and the references therein.

In this context, motivated by Wu et al. [21] and Zhang and Li [23], we will study the delay-dependent 𝐻_∞ control problem of a class of neutral time-delay systems based on a delay partitioning technique. The remainder of the paper is organized as follows. Section 2 gives problem formulation and a necessary lemma. In Section 3, dividing the delay interval into multiple segments, using the Lyapunov functional technology combined with matrix inequality technology, a new delay-dependent bounded real lemma is proposed. Based on the BRL, a condition for the existence of a state-feedback 𝐻_∞ controller is introduced in terms of linear matrix inequalities. Numerical examples are given in Section 4, followed by the conclusions, which are presented in Section 5.

Notation. Throughout this paper, “𝑇” stands for matrix transposition. “𝐼” denotes the identity matrix of appropriate dimensions. “𝑃>0” means that 𝑃 is positive definite. “∗” represents the elements below the main diagonal of a symmetric matrix.

2. Problem Formulations

Consider a class of linear time-varying discrete neutral system: 𝑥̇(𝑡)−𝐶_𝑑𝑥̇(𝑡−𝜏)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥((𝑡(𝑡)𝑡−𝑑))+𝐵𝑤+𝐵_𝑢𝑢(𝑡)𝑧(𝑡)(𝑡),𝑡>0,=𝐶𝑥+𝐶_𝑢𝑢(𝑡)(𝑡),𝑥(𝑡)(𝑡)[],+𝐷𝑤=𝜙,𝑡∈−𝑟,0 (1) where 𝑥(𝑡) is the state vector; the matrices 𝐴, 𝐴_𝑑, 𝐵, 𝐶, 𝐶_𝑑, and  𝐷 are known constant matrices of appropriate dimensions, and the eigenvalue of the matrix 𝐶_𝑑, 𝜌(𝐶_𝑑) satisfies 𝜌(𝐶_𝑑)<1. 𝑤(𝑡) is the disturbance input that belongs to 𝐿₂[0,∞). 𝑧(𝑡) is the controlled output, and 𝑢(𝑡) is the controlled input. 𝜙(𝑡)(𝑡∈[−𝑟,0]) is the system’s initial function which is continuous differentiable on [−𝑟,0]. The scalar 𝜏 is a positive constant time delay. Time delay 𝑑(𝑡) is a continuously differentiable function, satisfying the following conditions: (𝑡)𝑑̇(𝑡)0≤𝑑≤ℎ,≤𝜇, (2) where ℎ,  and 𝜇 are known positive real constants, and it is assumed that 𝑟=max{ℎ,𝜏}. In this paper, we are interested in designing a memoryless state-feedback controller 𝑢(𝑡)=𝐾𝑥(𝑡), (3) where 𝐾 is a constant matrix, such that for a given scalar 𝛾, the following requirements are satisfied:(I)the corresponding closed-loop system is asymptotically stable when 𝑤(𝑡)=0;(II)under zero initial condition (i.e., 𝑥(𝑡)=0(𝑡∈[−𝑟,0])), the corresponding closed-loop system satisfies ‖𝑧‖₂≤𝛾‖𝑤‖₂,∀𝑤∈𝐿₂[0,∞), (4)where 𝛾>0 is a prescribed scalar.

In obtaining the main results of this paper, the following lemma plays an important role.

Lemma 1 (Schur complement). For the symmetrical matrix 𝐿=^𝐿11𝐿12_{𝐿𝑇12𝐿22}, the followings are equivalent: (1)(2)𝐿𝐿<0,₁₁<0,𝐿₂₂−𝐿^𝑇₁₂𝐿⁻¹₁₁𝐿₁₂(3)𝐿<0,₂₂<0,𝐿₁₁−𝐿₁₂𝐿⁻¹₂₂𝐿^𝑇₁₂<0. (5)

3. Main Results

In this section, we discuss the problem of 𝐻_∞ performance and state-feedback 𝐻_∞ controller design of system (1).

3.1. 𝐻_∞ Performance Analysis

In the following theorem, we present a new version of delay-dependent bounded real lemma for neutral system (1) with 𝑢(𝑡)≡0; that is, we consider the following system: 𝑥̇(𝑡)−𝐶_𝑑𝑥̇(𝑡−𝜏)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥((𝑡(𝑡),𝑧(𝑡)(𝑡)(𝑡).𝑡−𝑑))+𝐵𝑤=𝐶𝑥+𝐷𝑤 (6)

Theorem 2. For given positive scalars ℎ>0 and 𝜇>0, the neutral system (6) with delay restrictions (2) is asymptotically stable and satisfies ‖𝑧‖₂≤𝛾‖𝑤‖₂ for any nonzero 𝑤(𝑡)∈𝐿₂[0,∞) under the zero initial condition if there exist matrices 𝑃=𝑃^𝑇>0, 𝑄_𝑖=𝑄^T_𝑖≥0(𝑖=1,…,4), 𝑅=𝑅^𝑇>0,𝑍=𝑍^𝑇>0, 𝑋𝑋=¹¹𝑋₁₂𝑋₁₃𝑋₁₄𝑋₁₅𝑋₁₆∗𝑋₂₂𝑋₂₃𝑋₂₄𝑋₂₅𝑋₂₆∗∗𝑋₃₃𝑋₃₄𝑋₃₅𝑋₃₆∗∗∗𝑋₄₄𝑋₄₅𝑋₄₆∗∗∗∗𝑋₅₅𝑋₅₆∗∗∗∗∗𝑋₆₆≥0, (7) and free-weighting matrices 𝑁_𝑖(𝑖=1,…,6) with appropriate dimensions, such that the following LMIs hold: ΦΦ=¹¹Φ₁₂Φ₁₃Φ₁₄Φ₁₅Φ₁₆𝐴^𝑇𝐻0𝐶^𝑇∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅Φ₂₆000∗∗Φ₃₃Φ₃₄Φ₃₅Φ₃₆𝐴^𝑇_𝑑𝐻00∗∗∗Φ₄₄Φ₄₅Φ₄₆000∗∗∗∗Φ₅₅Φ₅₆000∗∗∗∗∗Φ₆₆𝐶^𝑇_𝑑𝐻00∗∗∗∗∗∗−𝐻00∗∗∗∗∗∗∗−𝛾²𝐼𝐷^𝑇∗∗∗∗∗∗∗∗−𝐼<0,𝑋Ψ=¹¹𝑋₁₂𝑋₁₃𝑋₁₄𝑋₁₅𝑋₁₆𝑁₁∗𝑋₂₂𝑋₂₃𝑋₂₄𝑋₂₅𝑋₂₆𝑁₂∗∗𝑋₃₃𝑋₃₄𝑋₃₅𝑋₃₆𝑁₃∗∗∗𝑋₄₄𝑋₄₅𝑋₄₆𝑁₄∗∗∗∗𝑋₅₅𝑋₅₆𝑁₅∗∗∗∗∗𝑋₆₆𝑁₆∗∗∗∗∗∗𝑍≥0, (8)(9) where Φ₁₁=𝑃𝐴+𝐴^𝑇𝑃+𝑄₁+𝑁₁+𝑁^𝑇₁+ℎ𝑋₁₁,Φ₁₂=𝑁^T₂+ℎ𝑋₁₂,Φ₁₃=𝑃𝐴_𝑑−𝑁₁+𝑁^T₃+ℎ𝑋₁₃,Φ₁₄=𝑁^T₄+ℎ𝑋₁₄,Φ₁₅=𝑁^T₅+ℎ𝑋₁₅,Φ₁₆=𝑃𝐶_𝑑+𝑁^T₆+ℎ𝑋₁₆,Φ₂₂𝜇=−1−2𝑄₁+𝑄₂+ℎ𝑋₂₂,Φ₂₃=−𝑁₂+ℎ𝑋₂₃,Φ₂₄=ℎ𝑋₂₄,Φ₂₅=ℎ𝑋₂₅,Φ₂₆=ℎ𝑋₂₆,Φ₃₃=−1−𝜇𝑄₂+𝑄₃−𝑁₃−𝑁^𝑇₃+ℎ𝑋₃₃,Φ₃₄=−𝑁^𝑇₄+ℎ𝑋₃₄,Φ₃₅=−𝑁^𝑇₅+ℎ𝑋₃₅Φ₃₆=−𝑁^𝑇₆+ℎ𝑋₃₆,Φ₄₄𝜇=−1−2𝑄₃+𝑄₄+ℎ𝑋₄₄,Φ₄₅=ℎ𝑋₄₅,Φ₄₆=ℎ𝑋₄₆,Φ₅₅=−𝑄₄+ℎ𝑋₅₅,Φ₅₆=ℎ𝑋₅₆,Φ₆₆=−𝑅+ℎ𝑋₆₆,𝐻=𝑅+ℎ𝑍. (10)

Proof. Under the condition of the theorem, we first show the asymptotic stability of system (6). To this end, we consider system (6) with 𝑤(𝑡)=0, that is, 𝑥̇(𝑡)−𝐶_𝑑𝑥̇(𝑡−𝜏)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−𝑑(𝑡)). (11)
Inspired by the works of [23], we divided the delay interval [0,ℎ] into [0,𝑑(𝑡)/2], [𝑑(𝑡)/2,𝑑(𝑡)], [𝑑(𝑡),(𝑑(𝑡)+ℎ)/2], and [(𝑑(𝑡)+ℎ)/2,ℎ] for system (6). Corresponding to such a division, the following Lyapunov-Krasovskii functional is chosen for this system: 𝑉𝑡,𝑥_𝑡=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑡_{𝑡−𝑑(𝑡)/2}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+𝑑𝑠_{(𝑡)𝑡−𝑑(𝑡)/2𝑡−𝑑}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)+𝑑𝑠_{(𝑡)𝑡−𝑑(𝑡)𝑡−(𝑑+ℎ)/2}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠)+𝑑𝑠_{𝑡−(𝑑(𝑡)+ℎ)/2𝑡−ℎ}𝑥^𝑇(𝑠)𝑄₄𝑥(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏𝑥̇^𝑇(𝑠)𝑅𝑥̇(𝑠)+𝑑𝑠_{0−ℎ𝑡𝑡+𝜃}𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠𝑑𝜃, (12) where 𝑃=𝑃^𝑇>0, 𝑄_𝑖=𝑄^T_𝑖≥0(𝑖=1,…,4), 𝑅=𝑅^𝑇>0, and 𝑍=𝑍^𝑇>0 are matrices to be determined. By using the Leibniz-Newton formula, one has 𝑥(𝑡−𝑑(𝑡))=𝑥(𝑡)−^𝑡_{(𝑡)𝑡−𝑑}𝑥̇(𝑠)𝑑𝑠. (13) Since (13) can be rewritten as 𝑥(𝑡)−𝑥(𝑡−𝑑(𝑡))−∫^𝑡_{𝑡−𝑑(𝑡)}𝑥̇(𝑠)𝑑𝑠=0. Due to this relation, one can introduce some appropriate dimensional matrices 𝑁_𝑖(𝑖=1,…,6), such that 2𝑥^𝑇(𝑡)𝑁₁+𝑥^𝑇𝑑(𝑡)𝑡−2𝑁₂+𝑥^𝑇(𝑡−𝑑(𝑡))𝑁₃+𝑥^𝑇𝑑(𝑡)𝑡−+ℎ2𝑁₄+𝑥^𝑇(𝑡−ℎ)𝑁₅+𝑥̇^𝑇(𝑡−𝜏)𝑁₆×𝑥(𝑡)−𝑥(𝑡−𝑑(𝑡))−^𝑡_{(𝑡)𝑡−𝑑}𝑥̇(𝑠)𝑑𝑠=0. (14) Moreover, it follows from (2) that for any appropriate dimensional matrix 𝑋≥0, ℎ𝜂^𝑇(𝑡)𝑋𝜂(𝑡)−^𝑡_{(𝑡)𝑡−𝑑}𝜂^𝑇(𝑡)𝑋𝜂(𝑡)𝑑𝑠≥0, (15) where𝜂(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇𝑑(𝑡)𝑡−2𝑥^𝑇(𝑡−𝑑(𝑡))𝑥^𝑇𝑑(𝑡)𝑡−+ℎ2𝑥^𝑇(𝑡−ℎ)𝑥̇^𝑇(𝑡−𝜏)^𝑇. (16)
According to (14) and (15), and taking the time derivative of the Lyapunov-Krasovskii functional candidate (12) gives that 𝑉̇𝑡,𝑥_𝑡=2𝑥^𝑇(𝑡)𝑃𝑥̇(𝑡)+𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)−𝑑̇(𝑡)1−2𝑥^𝑇𝑑(𝑡)𝑡−2𝑄₁𝑥𝑑(𝑡)𝑡−2+𝑑̇(𝑡)1−2𝑥^𝑇𝑑(𝑡)𝑡−2𝑄₂𝑥𝑑(𝑡)𝑡−2−1−𝑑̇(𝑡)𝑥^𝑇(𝑡−𝑑(𝑡))𝑄₂𝑥((𝑡+𝑑̇(𝑡)𝑡−𝑑))1−2𝑥^𝑇𝑑(𝑡)𝑡−2𝑄₂𝑥𝑑(𝑡)𝑡−2−1−𝑑̇(𝑡)𝑥^𝑇(𝑡−𝑑(𝑡))𝑄₂𝑥((𝑡+𝑑̇(𝑡)𝑥𝑡−𝑑))1−^𝑇(𝑡−𝑑(𝑡))𝑄₃𝑥((𝑡−𝑑̇(𝑡)𝑡−𝑑))1−2𝑥^𝑇𝑑(𝑡)𝑡−+ℎ2𝑄₃𝑑(𝑡)×𝑥𝑡−+ℎ2+𝑑̇(𝑡)1−2𝑥^𝑇𝑑(𝑡)𝑡−+ℎ2𝑄₄𝑑(𝑡)×𝑥𝑡−+ℎ2−𝑥^𝑇(𝑡−ℎ)𝑄₄𝑥()+𝑥̇𝑡−ℎ_𝑇(𝑡)𝑅𝑥̇(𝑡)−𝑥̇^𝑇()𝑅𝑥̇()𝑥̇𝑡−𝜏𝑡−𝜏+ℎ^𝑇(𝑡)𝑍𝑥̇(𝑡)−^𝑡_𝑡−ℎ𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑑𝑠≤2𝑥^𝑇(𝑡)𝑃𝐴𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−𝑑(𝑡))+𝐶_𝑑𝑥̇(𝑡−𝜏)+𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)−𝜇1−2𝑥^𝑇𝑑(𝑡)𝑡−2𝑄₁𝑥𝑑(𝑡)𝑡−2+𝑑̇(𝑡)1−2𝑥^𝑇𝑑(𝑡)𝑡−2𝑄₂𝑥𝑑(𝑡)𝑡−2−1−𝜇𝑥^𝑇(𝑡−𝑑(𝑡))𝑄₂𝑥((𝑡+𝑑̇(𝑡)𝑥𝑡−𝑑))1−^𝑇(𝑡−𝑑(𝑡))𝑄₃𝑥((𝑡−𝜇𝑡−𝑑))1−2𝑥^𝑇𝑑(𝑡)𝑡−+ℎ2𝑄₃𝑥𝑑(𝑡)𝑡−+ℎ2+𝑑̇(𝑡)1−2𝑥^𝑇𝑑(𝑡)𝑡−+ℎ2𝑄₄𝑑(𝑡)×𝑥𝑡−+ℎ2−𝑥^𝑇(𝑡−ℎ)𝑄₄𝑥()+𝑥̇𝑡−ℎ_𝑇(𝑡)𝑅𝑥̇(𝑡)−𝑥̇^𝑇()𝑅𝑥̇()𝑥̇𝑡−𝜏𝑡−𝜏+ℎ^𝑇(𝑡)𝑍𝑥̇(𝑡)−^𝑡_{(𝑡)𝑡−𝑑}𝑥̇^𝑇(𝑠)𝑍𝑥̇(𝑠)𝑥𝑑𝑠+2^𝑇(𝑡)𝑁₁+𝑥^𝑇𝑑(𝑡)𝑡−2𝑁₂+𝑥^𝑇(𝑡−𝑑(𝑡))𝑁₃+𝑥^𝑇𝑑(𝑡)𝑡−+ℎ2𝑁₄+𝑥^𝑇(𝑡−ℎ)𝑁₅+𝑥̇^𝑇(𝑡−𝜏)𝑁₆×𝑥(𝑡)−𝑥(𝑡−𝑑(𝑡))−^𝑡_{(𝑡)𝑡−𝑑}𝑥̇(𝑠)𝑑𝑠+ℎ𝜂^𝑇(𝑡)𝑋𝜂(𝑡)−^𝑡_{(𝑡)𝑡−𝑑}𝜂^𝑇(𝑡)𝑋𝜂(𝑡)𝑑𝑠=𝜂^𝑇(𝑡)Ξ𝜂(𝑡)−^𝑡_{(𝑡)𝑡−𝑑}𝜉^𝑇()()−𝑑̇(𝑡)𝑡,𝑠Ψ𝜉𝑡,𝑠𝑑𝑠2𝑥^𝑇𝑑(𝑡)𝑡−2𝑄₂𝑥𝑑(𝑡)𝑡−2−𝑑̇(𝑡)𝑥^𝑇(𝑡−𝑑(𝑡))𝑄₃𝑥((𝑡−𝑑̇(𝑡)𝑡−𝑑))2𝑥^𝑇𝑑(𝑡)𝑡−+ℎ2𝑄₄𝑥𝑑(𝑡)𝑡−+ℎ2, (17)where𝜉(𝑡)=𝜂^𝑇(𝑡)𝑥̇^𝑇(𝑠)^𝑇,ΦΞ=¹¹+𝐴^𝑇𝐻𝐴Φ₁₂Φ₁₃+𝐴^𝑇𝐻𝐴_𝑑Φ₁₄Φ₁₅Φ₁₆+𝐴^𝑇𝐻𝐶_𝑑∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅Φ₂₆∗∗Φ₃₃+𝐴^𝑇_𝑑𝐻𝐴_𝑑Φ₃₄Φ₃₅Φ₃₆+𝐴^𝑇_𝑑𝐻𝐶_𝑑∗∗∗Φ₄₄Φ₄₅Φ₄₆∗∗∗∗Φ₅₅Φ₅₆∗∗∗∗∗Φ₆₆+𝐶^𝑇_𝑑𝐻𝐶_𝑑 (18)and Ψ is denoted in (9). The last three items of (17) are not more than zero since 𝑄_𝑖≥0(𝑖=2,…,4). Therefore, if Ξ<0 and Ψ≥0, there exists a positive scalar 𝜀 such that 𝑉̇(𝑡,𝑥_𝑡)≤−𝜀‖𝑥(𝑡)‖², which guarantees system (6) is asymptotically stable. By Lemma 1, we can conclude that if the matrix inequality (8) is feasible then the inequality Ξ<0 is feasible. This implies that system (6) is asymptotically stable if LMIs (8) and (9) are feasible.
Next, we shall establish the 𝐻_∞ performance of system (6) under zero initial condition. To this end, we introduce 𝐽_𝛾=^𝑡₀𝑧^𝑇(𝜃)𝑧(𝜃)−𝛾²𝑤^𝑇(𝜃)𝑤(𝜃)𝑑𝜃, (19) where 𝑡>0.
By combining (17) with those results analyzed above, now it is interesting to note that 𝑉̇𝑡,𝑥_𝑡≤𝜂^𝑇(𝑡)Ξ𝜂(𝑡)−^𝑡_{(𝑡)𝑡−𝑑}𝜉^𝑇(𝑡,𝑠)Ψ𝜉(𝑡,𝑠)𝑑𝑠. (20)
Considering zero initial condition, it is easy to see that for any nonzero 𝑤∈𝐿₂[0,∞) and 𝑡>0, the following expression holds. 𝐽_𝛾=^𝑡₀𝑧^𝑇(𝜃)𝑧(𝜃)−𝛾²𝑤^𝑇(𝜃)𝑤(𝜃)+𝑉̇(𝜃)(𝑡)≤𝑑𝜃−𝑉_𝑡0𝑧^𝑇(𝜃)𝑧(𝜃)−𝛾²𝑤^𝑇(𝜃)𝑤(𝜃)+𝑉̇(𝜃)=𝑑𝜃_𝑡0𝜂^𝑇(𝜃)Ξ^′𝜂(𝜃)−𝑑𝜃_{𝑡0𝜃(𝜃)𝜃−𝑑}𝜉^𝑇(𝜃,𝑠)Ψ𝜉(𝜃,𝑠)𝑑𝑠𝑑𝜃, (21) where𝜂^𝑇(𝑡)=𝜂^𝑇(𝑡)𝑤^𝑇(𝑡)Ξ^′=Φ₁₁+𝐴^𝑇𝐻𝐴+𝐶^𝑇𝐶Φ₁₂Φ₁₃+𝐴^𝑇𝐻𝐴_𝑑Φ₁₄Φ₁₅Φ₁₆+𝐴^𝑇𝐻𝐶_𝑑𝐶^𝑇𝐷∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅Φ₂₆0∗∗Φ₃₃+𝐴^𝑇_𝑑𝐻𝐴_𝑑Φ₃₄Φ₃₅Φ₃₆+𝐴^𝑇_𝑑𝐻𝐶_𝑑0∗∗∗Φ₄₄Φ₄₅Φ₄₆0∗∗∗∗Φ₅₅Φ₅₆0∗∗∗∗∗Φ₆₆+𝐶^𝑇_𝑑𝐻𝐶_𝑑0∗∗∗∗∗∗𝐷^𝑇𝐷−𝛾²𝐼<0. (22)
By carrying out some algebraic manipulations, the aforementioned matrix inequality (22) can be rewritten as follows: Φ₁₁Φ₁₂Φ₁₃Φ₁₄Φ₁₅Φ₁₆𝐴^𝑇𝐻0∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅Φ₂₆00∗∗Φ₃₃Φ₃₄Φ₃₅Φ₃₆𝐴^𝑇_𝑑𝐻0∗∗∗Φ₄₄Φ₄₅Φ₄₆00∗∗∗∗Φ₅₅Φ₅₆00∗∗∗∗∗Φ₆₆𝐶^𝑇_𝑑𝐻0∗∗∗∗∗∗−𝐻0∗∗∗∗∗∗∗−𝛾²𝐼+𝐶^𝑇000000𝐷^𝑇×𝐶000000𝐷<0. (23)
By Schur complement (Lemma 1) and some matrices primary manipulations, it is easy to see that the above-mentioned matrix inequality (23) is equivalent to (8). Combining (8) and (9), we have 𝜂^𝑇(𝑡)Ξ^′𝜂(𝑡)≤0 for any 𝜂(𝑡). Therefore, the following expression holds for any 𝑡>0: 𝐽_𝛾=^𝑡₀𝑧^𝑇(𝜃)𝑧(𝜃)−𝛾²𝑤^𝑇(𝜃)𝑤(𝜃)𝑑𝜃≤0. (24)
By letting 𝑡→∞, we lead to ^∞₀𝑧^𝑇(𝜃)𝑧(𝜃)𝑑𝜃≤𝛾^2∞₀𝑤^𝑇(𝜃)𝑤(𝜃)𝑑𝜃. (25) And hence, (4) is satisfied for any nonzero 𝑤(𝑡)∈𝐿₂[0,∞).
The proof is thus completed.

Remark 3. Theorem 2 presents an improved bounded real lemma for linear neutral system with time-varying delay by defining a novel Lyapunov-Krasovskii functional. The merit of the proposed BRL lies in its reduced conservatism, which is based on a time-delay fractioning approach.

Remark 4. About the delay partitioning technique, the classical approach is to represent the time delay as two parts: constant part and time-varying part, and then a Lyapunov-Krasovskii functional is introduced by applying the idea of delay partitioning to the constant part. However, in this brief, we partition the whole time-varying delay interval into multiparts and a novel Lyapunov-Krasovskii functional (LKF) is constructed with every delay component. Then, we obtain a time-varying LKF since it is dependent on the time-varying delay, which constitutes the major difference from most existing results in the literature.

3.2. 𝐻_∞ Control of Neutral Time-Delay Systems

Now, we are in a position to state the 𝐻_∞ control result based on the BRL derived in the previous section. A sufficient condition under which there exists a memoryless state-feedback 𝐻_∞ controller for the neutral system (1) is given in Theorem 5.

Theorem 5. For given positive scalars 𝜇 and 𝜆, there exists a state-feedback controller in the form of (3) such that the resulting closed-loop system satisfies the requirements (I) and (II) if there exist matrices 𝑃>0, 𝑄_𝑖>0(𝑖=1,…,4), 𝑅>0, 𝑍>0, 𝑋𝑋=¹¹𝑋₁₂𝑋₁₃𝑋₁₄𝑋₁₅𝑋₁₆∗𝑋₂₂𝑋₂₃𝑋₂₄𝑋₂₅𝑋₂₆∗∗𝑋₃₃𝑋₃₄𝑋₃₅𝑋₃₆∗∗∗𝑋₄₄𝑋₄₅𝑋₄₆∗∗∗∗𝑋₅₅𝑋₅₆∗∗∗∗∗𝑋₆₆≥0, (26) free-weighting matrices 𝑁_𝑖(𝑖=1,…,6) of appropriate dimensions, and a scalar 𝛾>0, such that the following matrix inequalities hold: Φ₁₁Φ₁₂Φ₁₃Φ₁₄Φ₁₅Φ₁₆𝜆𝑃𝐵^𝑇+𝜆𝑌^𝑇𝐵^𝑇_𝑢0𝑃𝐶^𝑇+𝑌^𝑇𝐶^𝑇_𝑢∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅Φ₂₆Φ000∗∗₃₃Φ₃₄Φ₃₅Φ₃₆𝜆𝑃𝐴^𝑇_𝑑Φ00∗∗∗₄₄Φ₄₅Φ₄₆Φ000∗∗∗∗₅₅Φ₅₆Φ000∗∗∗∗∗₆₆𝜆𝑃𝐶^𝑇_𝑑00∗∗∗∗∗∗−𝜆𝑃00∗∗∗∗∗∗∗−𝛾²𝐼𝐷^𝑇∗∗∗∗∗∗∗∗−𝐼<0,𝑋Ψ=¹¹𝑋₁₂𝑋₁₃𝑋₁₄𝑋₁₅𝑋₁₆𝑁₁∗𝑋₂₂𝑋₂₃𝑋₂₄𝑋₂₅𝑋₂₆𝑁₂∗∗𝑋₃₃𝑋₃₄𝑋₃₅𝑋₃₆𝑁₃∗∗∗𝑋₄₄𝑋₄₅𝑋₄₆𝑁₄∗∗∗∗𝑋₅₅𝑋₅₆𝑁₅∗∗∗∗∗𝑋₆₆𝑁₆∗∗∗∗∗∗𝑍≥0, (27)(28)where Φ₁₁=𝐵_𝑢𝑌+𝑌^𝑇𝐵^𝑇_𝑢+𝑃𝐵^𝑇+𝐵𝑃+𝑄₁+𝑁₁+𝑁^𝑇₁+ℎ𝑋₁₁,Φ₁₂=𝑁^𝑇₂+ℎ𝑋₁₂,Φ₁₃=𝐴_𝑑𝑃−𝑁₁+𝑁^T₃+ℎ𝑋₁₃,Φ₁₄=𝑁^T₄+ℎ𝑋₁₄,Φ₁₅=𝑁^T₅+ℎ𝑋₁₅Φ₁₆=𝐶_𝑑𝑃+𝑁^T₆+ℎ𝑋₁₆,Φ₂₂𝜇=−1−2𝑄₁+𝑄₂+ℎ𝑋₂₂,Φ₂₃=−𝑁₂+ℎ𝑋₂₃,Φ₂₄=ℎ𝑋₂₄,Φ₂₅=ℎ𝑋₂₅,Φ₂₆=ℎ𝑋₂₆,Φ₃₃=−1−𝜇𝑄₂+𝑄₃−𝑁₃−𝑁^𝑇₃+ℎ𝑋₃₃,Φ₃₄=−𝑁₄+ℎ𝑋₃₄,Φ₃₅=−𝑁₅+ℎ𝑋₃₅,Φ₃₆=−𝑁₆+ℎ𝑋₃₆,Φ₄₄=𝜇2−1𝑄₃+𝑄₄+ℎ𝑋₄₄,Φ₄₅=ℎ𝑋₄₅,Φ₄₆=ℎ𝑋₄₆,Φ₅₅=−𝑄₄+ℎ𝑋₅₅,Φ₅₆=ℎ𝑋₅₆,Φ₆₆=−𝑅+ℎ𝑋₆₆. (29) In this case, an 𝐻_∞ state-feedback controller can be chosen as 𝑢(𝑡)=𝑌𝑃⁻¹𝑥(𝑡). (30)

Proof. Assume that 𝑢(𝑡)=𝐾𝑥(𝑡); 𝐵_𝑢𝑢(𝑡) and 𝐶_𝑢𝑢(𝑡) in (1) are replaced by 𝐵_𝑢𝐾𝑥(𝑡) and 𝐶_𝑢𝐾𝑥(𝑡), respectively. Define 𝐴_𝑘≐𝐴+𝐵_𝑢𝐾 and 𝐶_𝑘≐𝐶+𝐶_𝑢𝐾. Taking this into account, the condition in Theorem 2 is replaced by Φ^′₁₁Φ₁₂Φ₁₃Φ₁₄Φ₁₅Φ₁₆𝐴^𝑇_𝑘𝐻0𝐶^𝑇_𝑘∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅Φ₂₆000∗∗Φ₃₃Φ₃₄Φ₃₅Φ₃₆𝐴^𝑇_𝑑𝐻00∗∗∗Φ₄₄Φ₄₅Φ₄₆000∗∗∗∗Φ₅₅Φ₅₆000∗∗∗∗∗Φ₆₆𝐶^𝑇_𝑑𝐻00∗∗∗∗∗∗−𝐻00∗∗∗∗∗∗∗−𝛾²𝐼𝐷^𝑇∗∗∗∗∗∗∗∗−𝐼<0, (31) where Φ^′₁₁=𝑃𝐴_𝑘+𝐴^𝑇_𝑘𝑃+𝑄₁+𝑁₁+𝑁^𝑇₁+ℎ𝑋₁₁. (32)
Pre- and postmultiplying (31) by diag {𝑃^−𝑇,𝑃^−𝑇,𝑃^−𝑇,𝑃^−𝑇,𝑃^−𝑇,𝑃^−𝑇,𝑃^−𝑇,𝐼,𝐼} and diag {𝑃⁻¹,𝑃⁻¹,𝑃⁻¹,𝑃⁻¹,𝑃⁻¹,𝑃⁻¹,𝑃⁻¹,𝐼,𝐼}, respectively, and introducing an additional constraint 𝐻=𝜆𝑃, then one can obtain an equivalent expression of (31) as follows:Φ^′₁₁Φ₁₂Φ₁₃Φ₁₄Φ₁₅Φ₁₆𝜆𝑃^−𝑇𝐵^𝑇+𝜆𝑃^−𝑇𝐾^𝑇𝐵^𝑇_𝑢0𝑃^−𝑇𝐶^𝑇+𝑃^−𝑇𝐾^𝑇𝐶^𝑇_𝑢∗Φ₂₂Φ₂₃Φ₂₄Φ₂₅Φ₂₆000∗∗Φ₃₃Φ₃₄Φ₃₅Φ₃₆𝜆𝑃^−𝑇𝐴^𝑇_𝑑00∗∗∗Φ₄₄Φ₄₅Φ₄₆000∗∗∗∗Φ₅₅Φ₅₆000∗∗∗∗∗Φ₆₆𝜆𝑃^−𝑇𝐶^𝑇_𝑑00∗∗∗∗∗∗−𝜆𝑃^−𝑇00∗∗∗∗∗∗∗−𝛾²𝐼𝐷^𝑇∗∗∗∗∗∗∗∗−𝐼<0, (33)where Φ^′₁₁=𝐵_𝑢𝐾𝑃⁻¹+𝑃^−𝑇𝐾^𝑇𝐵^𝑇_𝑢+𝑃^−𝑇𝐵^𝑇+𝐵𝑃⁻¹+𝑄₁+𝑁₁+𝑁^𝑇₁+ℎ𝑋₁₁,Φ₁₂=𝑁^𝑇₂+ℎ𝑋₁₂,Φ₁₃=𝐴_𝑑𝑃⁻¹−𝑁₁+𝑁^𝑇₃+ℎ𝑋₁₃,Φ₁₄=𝑁^𝑇₄+ℎ𝑋₁₄,Φ₁₅=𝑁^𝑇₅+ℎ𝑋₁₅,Φ₁₆=𝐶_𝑑𝑃⁻¹+𝑁^𝑇₆+ℎ𝑋₁₆,Φ₂₂𝜇=−1−2𝑄₁+𝑄₂+ℎ𝑋₂₂,Φ₂₃=−𝑁₂+ℎ𝑋₂₃,Φ₂₄=ℎ𝑋₂₄,Φ₂₅=ℎ𝑋₂₅,Φ₂₆=ℎ𝑋₂₆,Φ₃₃=−1−𝜇𝑄₂+𝑄₃−𝑁₃−𝑁^𝑇₃+ℎ𝑋₃₃,Φ₃₄=−𝑁₄+ℎ𝑋₃₄,Φ₃₅=−𝑁₅+ℎ𝑋₃₅,Φ₃₆=−𝑁₆+ℎ𝑋₃₆,Φ₄₄𝜇=−1−2𝑄₃+𝑄₄+ℎ𝑋₄₄,Φ₄₅=ℎ𝑋₄₅,Φ₄₆=ℎ𝑋₄₆,Φ₅₅=−𝑄₄+ℎ𝑋₅₅,Φ₅₆=ℎ𝑋₅₆,Φ₆₆=−𝑅+ℎ𝑋₆₆,𝑄_𝑖=𝑃^−𝑇𝑄_𝑖𝑃⁻¹(𝑖=1,…,4),𝑁_𝑖=𝑃^−𝑇𝑁_𝑖𝑃⁻¹(𝑖=1,…,6),𝑅=𝑃^−𝑇𝑅𝑃⁻¹,𝑋_𝑖𝑗=𝑃^−𝑇𝑋_𝑖𝑗𝑃⁻¹𝑖=1,…,6,𝑗=1,…,6,𝑍=𝑃^−𝑇𝑍𝑃⁻¹. (34)
Introducing change of those above-mentioned variables such that 𝑃=𝑃^−𝑇,𝑌=𝐾𝑃,𝑄_𝑖=𝑄_𝑖(),𝑁𝑖=1,…,4_𝑖=𝑁_𝑖(),𝑋𝑖=1,…,6_𝑖𝑗=𝑋_𝑖𝑗𝑖=1,…,6,𝑗=1,…,6,𝑅=𝑅,𝑍=𝑍, (35) thus we can obtain (27).
The proof is thus completed.

Remark 6. It should be pointed out that Theorem 5 gives a sufficient condition for the existence of a sate feedback compensation controller with the 𝐻_∞ performance bound 𝛾 in the form of (3) for system (1), which guarantees the closed-loop system to be asymptotically stable.

Remark 7. Since conditions (27) and (28) in Theorem 5 are in the LMI forms, for a given scalar 𝛾>0, the solutions can be easily obtained using LMI Toolbox. The problem is then how to find the optimal values of 𝛾. A feasible optimizing approach is given in Corollary 8 as follows, which can be completed using the Matlab command mincx.

Corollary 8. A suboptimal 𝐻_∞ controller in the form of (3) for the neutral time-delay system (1) can be found by solving the following optimization problem: min𝛾s.t.(27),(28). (36)

4. Numerical Examples and Discussions

This section presents some examples to illustrate the effectiveness of the methods described above.

Example 9. Consider the neutral system 𝑥̇(𝑡)−𝐶_𝑑𝑥̇(𝑡−𝜏)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥((𝑡(𝑡),𝑧(𝑡)(𝑡)(𝑡)𝑡−𝑑))+𝐵𝑤=𝐶𝑥+𝐷𝑤 (37) with 1,𝐴𝐴=−1.700−0.9,𝐵=0.5_𝑑=−10−1−1,𝐶_𝑑=,0.1000.1𝐶=10,𝐷=0.1. (38)
By Theorem 2, the 𝐻_∞ index 𝛾 is listed in Table 1 for various values of 𝜇 and ℎ. It is clear that our results presented in this paper are feasible.

Table 1 

𝐻_∞ index 𝛾 for various values of 𝜇 and ℎ.

Example 10. Consider the neutral system 𝑥̇(𝑡)−𝐶_𝑑𝑥̇(𝑡−𝜏)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥((𝑡(𝑡)𝑡−𝑑))+𝐵𝑤+𝐵_𝑢𝑢(𝑡),𝑧(𝑡)=𝐶𝑥(𝑡)+𝐶_𝑢𝑢(𝑡)+𝐷𝑤(𝑡), (39) with 𝐴=−1−0.20−1,𝐴_𝑑=,𝐶1001.2_𝑑=0.2000.1,𝐵_𝑢=1111,𝐵=,𝐶=00.5,𝐶_𝑢=0.5,𝐷=0.2. (40)
For 𝜇=0.5, ℎ=0.8, 𝑥(𝑡)=[1−1]^𝑇, a minimum of 𝛾=0.8 with a corresponding gain 𝐾=[0.9646−12.1649] was obtained with Theorem 5, which implies that the proposed method is effective and feasible.

5. Conclusions

In this contribution, the 𝐻_∞ performance for linear neutral system with time-varying delay is discussed and a new bounded real lemma is presented by introducing a novel Lyapunov-Krasovskii functional. Based on the BRL, an approach to design memoryless state-feedback 𝐻_∞ controller using LMI technique for linear neutral system with time-varying delay is proposed, which can be solved readily by using existing LMI optimization techniques. The numerical example simulation results demonstrate that the method is feasible and effective. Therefore, how to further reduce the conservatism constitutes is an important problem for future investigation.

Acknowledgments

The work is supported by Development Program for Outstanding Young Teachers in Harbin Institute of Technology (HITQNJS.2009.007), National Natural Science Foundation of China (NSFC 61273094) and the Project for Distinguished Young Scholars of the Basic Research Plan in Shenzhen City under Contract No. JCJ201110001. The authors are very thankful to the reviewers for their valuable suggestions and comments.