Asymptotic Stability Analysis and Optimality Algorithm for Uncertain Neutral Systems with Saturation

Abstract

The certain and uncertain neutral systems with time-delay and saturating actuator are considered in this paper. In order to analyse and optimize the system, auxiliary functions are presented based on additive decomposition approach and the relationship among them is discussed. As the novel stability criterion, two sufficient conditions are obtained for asymptotic stability of the neutral systems. Furthermore, the paper gives the stability analysis algorithm and optimality algorithm to optimize the result. Finally, from the two-stage dissolution tank of solid caustic soda in a chemical plant, three numerical examples are implemented to show the effectiveness of the proposed method.

1. Introduction

Delay is often inevitable in various practical systems; examples include population ecology [1], steam or water pipes, heat exchanges [2], and many others [3–5]. In the control engineering language, these delays can be categorized as state delay, input or output delay (retarded systems), delay in the state derivative (neutral systems), and so forth. Guaranteeing the stability of systems with delay is one core design objective both in theory and in practice. Particularly, in terms of neutral systems, the focus has mainly been on systems with identical delays in neutral and discrete terms [6–10]. Results also exist that depend only on the size of the discrete delays but not on the size of the neutral delays [11–13].

Besides delays, the saturated controller is apt to cause instability as well. In the presence of actuator saturation, the problem of estimating asymptotic stability regions for linear systems subject to it has been studied by many researches in the past years in [14]. Generally speaking, the existing methods for estimating the stability regions for linear systems with saturating actuators are based on the concept of Lyapunov level set. LMI optimization-based approaches were proposed to estimate the stability regions by using quadratic Lyapunov functions and the Lur’s-type Lyapunov functions in [15–19].

For the studies in response to both issues of delay and saturation, the sufficient conditions for systems with delay and saturated actuator are obtained in [18, 20–22]; Lyapunov-Krasovskii functional is employed to investigate the delay-dependent robust stabilization for uncertain neutral systems with saturated actuators in [20]; a controller is constructed in terms of linear matrix inequalities using descriptor model transformation in [18], just to name a few. However, this paper wants to provide a new method to find the system stability region and give the optimality algorithm to obtain the largest region in this method. Besides, it gives the application in the chemical process of the plant.

In this paper, a novel Lyapunov functional is proposed based on the delay-dividing approach, which leads to less conservative stability conditions for linear systems with time-delay and saturated actuator. This is done by introducing auxiliary functions based on the additive decomposition approach [23]. We also propose an algorithm to obtain the optimal auxiliary function. Finally, we design a two-stage dissolution tank of the chemical process by modeling it as a neutral delay system with actuator saturation and demonstrate the effectiveness of the proposed method.

Notations. ⋆ denotes the symmetric part to be a symmetric block matrix, ℝ^𝑛 denotes the 𝑛 dimensional Euclidean space, and ℝ^𝑚×𝑛 is the set of all real 𝑚×𝑛 matrices. 𝐼 is the identity matrix with proper dimensions. 𝐶₀ is the set of all continuous functions from [−𝜏,0] to ℝ^𝑛 where 𝜏 is a constant representing the neutral time-delay. 𝐴^𝑇 is the transpose of matrix 𝐴. |𝐴|=[|𝑎_𝑖𝑗|], with 𝐴=[𝑎_𝑖𝑗]. For real symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (respectively, 𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (respectively, positive definite). 𝜆_max(min)(𝐴) is the eigenvalue of matrix 𝐴 with maximum (minimum) real part. ‖𝑣‖ is the Euclidean norm of vector 𝑣, ‖𝑣‖=(𝑣^𝑇𝑣)^1/2, while ‖𝐴‖ is spectral norm of matrix 𝐴, ‖𝐴‖=[𝜆_max(𝐴^𝑇𝐴)]^1/2. 𝜌 represents the domain of attraction. diag{⋯} denotes a block-diagonal matrix decided by the corresponding elements in the brace and finally 𝐵(𝜌)={𝑦∈𝑅^𝑛∶‖𝑦‖≤𝜌}.

2. Problem Statement and Preliminaries

The following neutral system with time-delay and actuator saturation is considered: 𝑥̇(𝑡)−𝐷𝑥̇(𝑡−𝜏)=𝐴𝑥(𝑡)+𝐵𝑥(𝑡−ℎ)+𝐶Sat(𝑢(𝑡)), (1) where 𝑥(𝑡)∈ℝ^𝑛 is the system state and 𝑢(𝑡) is the control input. ℎ>0 is the constant discrete time delay and 𝜏>0 is the constant neutral time-delay. 𝐴, 𝐵, 𝐶, and 𝐷 are known real constant parameter matrices of appropriate dimensions with ‖𝐷‖<1. Sat(⋅) is used to denote the standard saturation function defined for 𝑢=[𝑢₁,𝑢₂,…,𝑢_𝑚]: Sat(𝑢)=Sat𝑢₁,Sat𝑢₂,…,Sat𝑢_𝑚^𝑇,Sat𝑢_𝑖=𝑢⁺_𝑖if𝑢_𝑖>𝑢⁺_𝑖𝑢_𝑖otherwise−𝑢⁺_𝑖if𝑢_𝑖<−𝑢⁺_𝑖. (2)

The following linear state feedback is to be designed: 𝑢(𝑡)=−𝐾𝑥(𝑡), (3) where the linear state feedback gain 𝐾=[𝐾^𝑇₁,𝐾^𝑇₂,…,𝐾^𝑇_𝑚]^𝑇; 𝐾_𝑖∈ℝ^𝑛 is an 𝑛-dimensional row vector.

Here we have slightly abused the notation by using Sat to denote both a scalar valued and a vector valued function. We have also assumed a unity saturation level for the saturation function without loss of generality.

Define 𝐷𝑧(𝑢(𝑡))=𝑢(𝑡)−Sat(𝑢(𝑡)) where 𝐷𝑧(𝑢(𝑡))=[𝑑𝑧(𝑢₁),𝑑𝑧(𝑢₂),…,𝑑𝑧(𝑢_𝑚)]^𝑇 and 𝑑𝑧𝑢_𝑖=𝑢_𝑖−𝑢⁺_𝑖if𝑢_𝑖>𝑢⁺_𝑖𝑢0otherwise_𝑖+𝑢⁺_𝑖if𝑢_𝑖<−𝑢⁺_𝑖. (4)

The saturated system can now be written as follows: 𝑥̇(𝑡)−𝐷𝑥̇(𝑡−𝜏)=𝐴_𝑐𝑥(𝑡)+𝐵𝑥(𝑡−ℎ)+𝐶𝐷𝑧(𝐾𝑥(𝑡)), (5) where 𝐴_𝑐=𝐴−𝐶𝐾.

The neutral system (5) then leads to the following by model transformation: 𝑑[ℒ(𝑥(𝑡=𝐴𝑑𝑡))]_𝑐𝑥(𝑡)+𝐶𝐷𝑧(𝐾𝑥), (6) where 𝐴_𝑐=𝐴_𝑐+𝐵=𝐴+𝐵−𝐶𝐾 and ℒ(𝑥(𝑡))=𝑥(𝑡)−𝐷𝑥(𝑡−𝜏)+𝐵∫^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠.

The following definitions and lemmas are required before proceeding with the main contributions presented in the next section.

Definition 1 (see [24]). The operator ℒ∶𝐶₀→ℝ^𝑛 is said to be stable if the zero solution of the homogeneous difference equation ℒ(𝑥(𝑡𝑥))=0,𝑡≥0,₀=𝜓∈𝜑∈𝐶₀∶ℒ𝜑=0 (7) is uniformly asymptotically stable. The stability of operator ℒ is necessary for the stability of neutral system (1) with (3), which is always satisfied when ‖ℎ|𝐵|+𝐷‖<1.

Lemma 2 (see [24]). For any matrix 𝐵,𝐷∈ℝ^𝑚×𝑛, if  ‖ℎ|𝐵|+𝐷‖<1, then the operator ℒ∶𝐶₀→ℝ^𝑛 with ℒ(𝑥(𝑡))=𝑥(𝑡)+𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠−𝐷𝑥(𝑡−𝜏) (8) is stable.

Lemma 3 (see [25]). Let 𝑈∈ℝ^𝑛×𝑛 and  𝑉∈ℝ^𝑛×𝑛, and let 𝑥∈ℝ^𝑛. Then we have 𝑥^𝑇𝑈𝑉𝑥≤‖𝑈𝑉‖𝑥^𝑇𝑥≤‖𝑈‖‖𝑉‖𝑥^𝑇𝑥,2𝑥^𝑇𝑈𝑉𝑥≤𝑥^𝑇1𝛾𝑈𝑈^𝑇+𝛾𝑉^𝑇𝑉𝑥,∀𝛾>0. (9)

Definition 4 (see [23]). Auxiliary functions 𝑊₁(𝜌₁) and 𝑊₂(𝜌₂) are described in the following 𝑊₁𝜌₁=^𝑚_𝑖=1𝑢1−^+𝑖𝐾_𝑖𝜌₁²𝐾^𝑇_𝑖𝐾_𝑖if𝐾_𝑖𝜌₁>𝑢⁺_𝑖,𝑊0otherwise,₂𝜌₂=^𝑚_𝑖=1𝜌₂4𝑢⁺_𝑖²𝐾^𝑇_𝑖𝐾_𝑖²if𝐾_𝑖𝜌₂>𝑢⁺_𝑖,0otherwise. (10)

3. Main Results

In this section, we firstly construct the auxiliary functions 𝑊₃(𝜌₃),𝑊₄(𝜌₄),…,𝑊_𝑛(𝜌_𝑛) using the geometric method and with them present a new delay-dependent stabilization criterion. Then the relationship between those auxiliary functions is explored which helps to obtain the optimal 𝑊_𝑖(𝜌_𝑖).

Consider the polynomial function 𝑦=𝑎𝑥^𝑛 which is tangent with saturated function 𝑦=𝑥−𝑢⁺_𝑖 in positive axis. 𝑎 can be firstly determined by the geometric relation and auxiliary polynomial functions can be obtained.

Let 𝑦=𝑎𝑥^𝑛,𝑦=𝑥−𝑢⁺_𝑖. (11)

The slope of polynomial function 𝑦=𝑎𝑥^𝑛 equals to 1 at the tangent point. Thus we have 𝑦̇=𝑎𝑛𝑥^𝑛−1=1. (12)

It can be determined by solving the resulting equations simultaneously and we obtain 𝑎=𝑛−1𝑢⁺_𝑖^𝑛−11𝑛^𝑛. (13)

Noting in the polynomial function, we replace 𝑥 with 𝑢_𝑖 here so that they have the same independent variable with 𝑑𝑧(𝑢_𝑖). Consider 𝑦=𝑛−1𝑢⁺_𝑖^𝑛−11𝑛^𝑛𝑢^𝑛_𝑖. (14)

Thus we compare the nonlinear function 𝑑𝑧(𝑢_𝑖) with the polynomial function above. For simplicity, we consider the cases 𝑛=2 and 𝑛=3. As shown in Figure 1, the graphics of quadratic and cubic polynomial functions are above the graphic of |𝑑𝑧(𝑢_𝑖)|.

Figure 1 

Two polynomial functions are 𝑦=(1/4𝑢+𝑖)𝑢2𝑖 and 𝑦=(4/27)(1/𝑢+𝑖)2𝑢3𝑖 (when 𝑢+𝑖=2) and the piecewise linear function is 𝑑𝑧(𝑢𝑖).

Furthermore, according to the nature of the polynomial function when 𝑛=4,5,…, the graphic of function 𝑦=((𝑛−1)/𝑢⁺_𝑖)^𝑛−1(1/𝑛^𝑛)𝑢^𝑛_𝑖 is also above the graphic of |𝑑𝑧(𝑢_𝑖)|. So we have the following inequality: 𝑑𝑧𝑢_𝑖≤𝑛−1𝑢⁺_𝑖^𝑛−11𝑛^𝑛𝑢^𝑛_𝑖, (15) where 𝑛=2,3,4,….

Lemma 5. When 𝑛=2,3,4,…, the following inequality holds: 𝐷𝑧^𝑇(𝐾𝑥(𝑡))𝐷𝑧(𝐾𝑥(𝑡))≤𝑥^𝑇𝑊_𝑛𝜌_𝑛𝑥, (16) where 𝑊_𝑛𝜌_𝑛=^𝑚_𝑖=1(𝑛−1)𝜌_𝑛𝑛^{𝑛/(𝑛−1)}𝑢⁺_𝑖^2𝑛−2𝐾^𝑇_𝑖𝐾_𝑖^𝑛𝑖𝑓𝐾_𝑖𝜌_𝑛>𝑢⁺_𝑖0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (17)

Proof. From (15) we obtain that 𝐷𝑧^𝑇(𝐾𝑥)𝐷𝑧(𝐾𝑥)=^𝑚_𝑖=1𝑑𝑧²𝑢_𝑖≤^𝑚_𝑖=1𝑛−1𝑢⁺_𝑖^2𝑛−21𝑛^2𝑛𝑢_𝑖^2𝑛≤^𝑚_𝑖=1𝑛−1𝑢⁺_𝑖^2𝑛−2×1𝑛^2𝑛𝑥^𝑇𝐾^𝑇_𝑖𝐾_𝑖𝑥⋯𝑥^𝑇𝐾^𝑇_𝑖𝐾_𝑖𝑥≤^𝑚_𝑖=1𝑛−1𝑢⁺_𝑖^2𝑛−2×1𝑛^2𝑛𝑥𝑥^𝑇𝑛−1𝑥^𝑇𝐾^𝑇_𝑖𝐾_𝑖⋯𝐾^𝑇_𝑖𝐾_𝑖𝑥≤^𝑚_𝑖=1𝑛−1𝑢⁺_𝑖^2𝑛−21𝑛^2𝑛𝜌_𝑛^2𝑛−2𝑥^𝑇𝐾^𝑇_𝑖𝐾_𝑖^𝑛𝑥≤𝑥^𝑇𝑊_𝑛𝜌_𝑛𝑥. (18) This completes the proof.

3.1. Asymptotic Stability for Certain Neutral System

Theorem 6. The neutral system with time-delay and actuator saturation as described in (1) and (3) is asymptotic stability if ‖ℎ|𝐵|+𝐷‖<1 and there exist scalars 𝛾_𝑖>0, 𝑖=1,2,3, 𝑃>0, 𝑃>0, 𝑄=[𝑄_𝑖𝑗]_2×2, 𝑄_𝑖𝑗>0, 𝑅=[𝑅_𝑖𝑗]_2×2, 𝑅_𝑖𝑗>0, 𝑇_𝑖>0(𝑖=1,2,3,4,5,6), and 𝑇_𝑖>0(𝑖=3,4) such that the following symmetric linear matrix inequality holds: ΦΦ=¹¹Φ₁₂0Φ₁₄Φ₁₅Φ₁₆0⋆Φ₂₂Φ₂₃0000⋆⋆Φ₃₃0000⋆⋆⋆Φ₄₄Φ₄₅00⋆⋆⋆⋆Φ₅₅00⋆⋆⋆⋆⋆Φ₆₆0⋆⋆⋆⋆⋆⋆Φ₇₇<0, (19) where 𝑊_𝑛𝜌=^𝑚_𝑖=1(𝑛−1)𝜌_𝑛𝑛^{𝑛/(𝑛−1)}𝑢⁺_𝑖^2𝑛−2𝐾^𝑇_𝑖𝐾_𝑖^𝑛𝑖𝑓𝐾_𝑖𝜌_𝑛>𝑢⁺_𝑖,Φ0𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,₁₁=𝐴^𝑇_𝑐𝑃+𝑃𝐴_𝑐+1𝛾₁𝑃+𝛾₁+𝛾₂+𝛾₃𝐶^𝑇𝐶𝑊_𝑖𝜌_𝑖+𝑄₁₁+𝑅₁₁+𝑇₁+𝑇₂+ℎ²2𝑇₃+𝑇₅+𝜏²2𝑇₄+𝑇₆,Φ₂₂=𝑄₂₂−𝑄₁₁,Φ₃₃=−𝑄₂₂−𝑇₁,Φ₄₄=𝑅₂₂−𝑅₁₁,Φ₅₅=1𝛾₂𝐷^𝑇𝑃𝐷−𝑅₂₂−𝑇₂,Φ₆₆=1𝛾₃𝐵^𝑇𝑃𝐵−𝑇₃,Φ₇₇=−𝑇₄,Φ₁₂=𝑄₁₂,Φ₁₄=𝑅₁₂,Φ₁₅=−𝐴^𝑇_𝑐Φ𝑃𝐷,₁₆=𝐴^𝑇_𝑐Φ𝑃𝐵,₂₃=−𝑄₁₂,Φ₄₅=−𝑅₁₂. (20)

Proof. Define a legitimate Lyapunov functional candidate as follows: 𝑉(𝑥(𝑡))=𝑉₁(𝑥(𝑡))+𝑉₂(𝑥(𝑡))+𝑉₃(𝑥(𝑡))+𝑉₄(𝑥(𝑡))+𝑉₅(𝑥(𝑡))+𝑉₆(𝑥(𝑡)), (21) where 𝑉₁(𝑥(𝑡))=ℒ^𝑇(𝑥(𝑡(𝑥(𝑡,𝑉))𝑃ℒ))₂(𝑥(𝑡))=^𝑡_{()𝑡−ℎ/2}𝑥(𝑠)𝑥ℎ𝑠−2^𝑇𝑄₁₁𝑄₁₂⋆𝑄₂₂×𝑥(𝑠)𝑥ℎ𝑠−2+𝑑𝑠_{𝑡()𝑡−𝜏/2}𝑥(𝑠)𝑥𝜏𝑠−2^𝑇×𝑅₁₁𝑅₁₂⋆𝑅₂₂𝑥(𝑠)𝑥𝜏𝑠−2𝑉𝑑𝑠,₃(𝑥(𝑡))=^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑇₁𝑥(𝑠)𝑑𝑠+^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑇₂𝑥(𝑠)𝑉𝑑𝑠,₄(𝑥(𝑡))=^𝑡_𝑡−ℎ^𝑡_𝑠𝑥^𝑇𝑠₁𝑑𝑠₁𝑇₃^𝑡_𝑠𝑥𝑠₂𝑑𝑠₂𝑉𝑑𝑠,₅(𝑥(𝑡))=^𝑡_𝑡−𝜏^𝑡_𝑠𝑥^𝑇𝑠₁𝑑𝑠₁𝑇₄^𝑡_𝑠𝑥^𝑇𝑠₂𝑑𝑠₂𝑉𝑑𝑠,₆(𝑥(𝑡=1))2^𝑡_𝑡−ℎ(𝑠−𝑡+ℎ)²𝑥^𝑇(𝑠)𝑇₅𝑥(𝑠)+1𝑑𝑠2^𝑡_𝑡−𝜏(𝑠−𝑡+𝜏)²𝑥^𝑇(𝑠)𝑇₆𝑥(𝑠)𝑑𝑠, (22)(23) where 𝑃>0, 𝑄=[𝑄_𝑖𝑗]_2×2, 𝑄_𝑖𝑗>0, 𝑅=[𝑅_𝑖𝑗]_2×2, 𝑅_𝑖𝑗>0, and 𝑇_𝑖>0, 𝑖=1,2,3,4,5,6.
Then 𝑉̇(𝑥(𝑡))=𝑉̇₁(𝑥(𝑡))+𝑉̇₂(𝑥(𝑡))+𝑉̇₃(𝑥(𝑡+𝑉̇))₄(𝑥(𝑡))+𝑉̇₅(𝑥(𝑡))+𝑉̇₆(𝑥(𝑡)). (24) By (19)-(22), we obtain that 𝑉̇₁(𝑥(𝑡))=2𝑥^𝑇(𝑡)−𝑥^𝑇(𝑡−𝜏)𝐷^𝑇+^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇×𝑃𝐴_𝑐𝑥(𝑡)+𝐶𝐷𝑧(𝐾𝑥)=2𝑥^𝑇𝑃𝐴_𝑐𝑥+2𝑥^𝑇𝑃𝐶𝐷𝑧(𝐾𝑥)−2𝑥^𝑇𝐴^𝑇_𝑐𝑃𝐷𝑥(𝑡−𝜏)−2𝑥^𝑇(𝑡−𝜏)𝐷^𝑇𝑃𝐶𝐷𝑧(𝐾𝑥)+2𝑥^𝑇(𝑡)𝐴^𝑇_𝑐𝑃𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠+2^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇𝑃𝐶𝐷𝑧(𝐾𝑥)≤𝑥^𝑇𝐴^𝑇_𝑐𝑃+𝑃𝐴_𝑐+1𝛾₁𝑃+𝛾₁+𝛾₂+𝛾₃𝐶^𝑇𝐶𝑊_𝑖𝜌_𝑖𝑥+1𝛾₂𝑥^𝑇(𝑡−𝜏)𝐷^𝑇()+1𝑃𝐷𝑥𝑡−𝜏𝛾₃^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇𝑃𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠−2𝑥^𝑇(𝑡)𝐴^𝑇_𝑐𝑃𝐷𝑥(𝑡−𝜏)+2𝑥^𝑇(𝑡)𝐴^𝑇_𝑐𝑃𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠, (25) where 𝑃>0, since 𝑃=𝑃². Consider 𝑉̇₂(𝑥(𝑡=𝑥(𝑡)𝑥ℎ))𝑡−2^𝑇𝑄₁₁𝑄₁₂⋆𝑄₂₂𝑥(𝑡)𝑥ℎ𝑡−2−𝑥ℎ𝑡−2𝑥(𝑡−ℎ)^𝑇𝑄₁₁𝑄₁₂⋆𝑄₂₂𝑥ℎ𝑡−2𝑥()+𝑥(𝑡)𝑥𝜏𝑡−ℎ𝑡−2^𝑇𝑅₁₁𝑅₁₂⋆𝑅₂₂𝑥(𝑡)𝑥𝜏𝑡−2−𝑥𝜏𝑡−2𝑥(𝑡−𝜏)^𝑇𝑅₁₁𝑅₁₂⋆𝑅₂₂𝑥𝜏𝑡−2𝑥(),𝑉̇𝑡−𝜏₃(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑇₁𝑥(𝑡)−𝑥^𝑇(𝑡−ℎ)𝑇₁𝑥(𝑡−ℎ)+𝑥^𝑇(𝑡)𝑇₂𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)𝑇₂𝑥(),𝑉̇𝑡−𝜏₄(𝑥(𝑡))=−^𝑡_𝑡−ℎ𝑥^𝑇𝑠₁𝑑𝑠₁𝑇₃^𝑡_𝑡−ℎ𝑥𝑠₂𝑑𝑠₂+^𝑡_𝑡−ℎ𝑥^𝑇(𝑡)𝑇₃^𝑡_𝑠𝑥𝑠₂𝑑𝑠₂+𝑑𝑠_{𝑡𝑡−ℎ𝑡𝑠}𝑥𝑠₂𝑑𝑠₂𝑇₃𝑥(𝑡)𝑑𝑠=−^𝑡_𝑡−ℎ𝑥^𝑇𝑠₁𝑑𝑠₁𝑇₃^𝑡_𝑡−ℎ𝑥𝑠₂𝑑𝑠₂+^𝑡_𝑡−ℎ(𝑠−𝑡+ℎ)𝑥^𝑇(𝑡)𝑇₃𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−ℎ}(𝑠−𝑡+ℎ)𝑥^𝑇(𝑠)𝑇₃𝑥(𝑡)𝑑𝑠=−^𝑡_𝑡−ℎ𝑥^𝑇𝑠₁𝑑𝑠₁𝑇₃^𝑡_𝑡−ℎ𝑥𝑠₂𝑑𝑠₂−^𝑡_𝑡−ℎ(𝑠−𝑡+ℎ)𝑇₃𝑥(𝑡)−𝑇₅𝑥(𝑠)^𝑇×𝑇₅⁻¹𝑇₃𝑥(𝑡)−𝑇₅𝑥(𝑠+ℎ)𝑑𝑠₂2𝑥^𝑇(𝑡)𝑇₃𝑇₅⁻¹𝑇₃𝑥(𝑡)+^𝑡_𝑡−ℎ(𝑠−𝑡+ℎ)𝑥^𝑇(𝑠)𝑇₅𝑥(𝑠)𝑑𝑠≤−^𝑡_𝑡−ℎ𝑥^𝑇𝑠₁𝑑𝑠₁𝑇₃^𝑡_𝑡−ℎ𝑥𝑠₂𝑑𝑠₂+ℎ²2𝑥^𝑇(𝑡)𝑇₃𝑥(𝑡)+^𝑡_𝑡−ℎ(𝑠−𝑡+ℎ)𝑥^𝑇(𝑠)𝑇₅𝑥(𝑠)𝑑𝑠, (26) where 𝑇₃>0, since 𝑇₃=𝑇₃𝑇₅⁻¹𝑇₃.
Similarly we have 𝑉̇₅(𝑥(𝑡))≤−^𝑡_𝑡−𝜏𝑥^𝑇𝑠₁𝑑𝑠₁𝑇₄^𝑡_𝑡−𝜏𝑥𝑠₂𝑑𝑠₂+𝜏²2𝑥^𝑇(𝑡)𝑇₄𝑥(𝑡)+^𝑡_𝑡−𝜏(𝑠−𝑡+𝜏)𝑥^𝑇(𝑠)𝑇₆𝑥(𝑠)𝑑𝑠, (27) where 𝑇₄>0, since 𝑇₄=𝑇₄𝑇₆⁻¹𝑇₄. Consider 𝑉̇₆(𝑥(𝑡=1))2ℎ²𝑥^𝑇(𝑡)𝑇₅𝑥(𝑡)−^𝑡_𝑡−ℎ(𝑠−𝑡+ℎ)𝑥^𝑇(𝑠)𝑇₅𝑥(𝑠)+1𝑑𝑠2𝜏²𝑥^𝑇(𝑡)𝑇₆𝑥(𝑡)−^𝑡_𝑡−𝜏(𝑠−𝑡+𝜏)𝑥^𝑇(𝑠)𝑇₆𝑥(𝑠)𝑑𝑠. (28) Substituting these into (24), the time-derivative of 𝑉 has new upper bound as follows: 𝑉̇(𝑥(𝑡))≤𝜉^𝑇(𝑡)Φ𝜉(𝑡), (29)
where𝜉^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇ℎ𝑡−2𝑥^𝑇(𝑡−ℎ)𝑥^𝑇𝜏𝑡−2𝑥^𝑇(𝑡−𝜏)^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑑𝑠; (30)Φ is defined as stated in (19).
If linear matrix inequality (19) is feasible, then we can get 𝑉̇(𝑥(𝑡))<0, for all 𝑥∈𝐵(𝜌). Therefore, if constant scalar ℎ, constant parameter matrices 𝐵,𝐷 such that ‖ℎ|𝐵|+𝐷‖<1 and there exist 𝑃>0, 𝑃>0, 𝑄=[𝑄_𝑖𝑗]_2×2, 𝑄_𝑖𝑗>0, 𝑅=[𝑅_𝑖𝑗]_2×2, 𝑅_𝑖𝑗>0, 𝑇_𝑖>0(𝑖=1,2,3,4,5,6), and 𝑇_𝑖>0(𝑖=3,4) satisfying (19) for real scalars 𝛾_𝑖>0(𝑖=1,2,3), from Hale and Verduyn Lunel [24], we can draw the neutral system which can be described by (1) and (3) is asymptotic stability. This completes the proof.

Remark 7. 𝑊_𝑖(𝜌_𝑖)(𝑖=1,2,…,𝑛) are created by the parameter 𝜌_𝑖 which is a measure tool for domain of attraction. With these functions, we obtain the novel stability criterion. However, in Theorem 6 we need to look for the largest value of 𝜌_𝑖 with the optimal 𝑊_𝑖(𝜌_𝑖). These can be seen in Section 3.3 below.

Remark 8. Theorem 6 gives a delay-dependent stability criterion for neutral system with (1) and (3) using a delay-dividing approach. The delay differential conditions in other works, such as in [26], are usually more strict. These facts mean that our result is less conservative than some previous approaches.

The delay-dependent stability criterion for system (1) with 𝜏≡ℎ is presented in the following corollary.

Corollary 9. The neutral systems (1) and (3) with 𝜏≡ℎ are asymptotic stability if ‖ℎ|𝐵|+𝐷‖<1 and there exist 𝑃>0, 𝑃>0, 𝑄=[𝑄_𝑖𝑗]_2×2, 𝑄_𝑖𝑗>0, 𝑇_𝑖>0(𝑖=1,2,3), and 𝑇₂>0 such that the following symmetric linear matrix inequality holds for real constant scalars 𝛾_𝑖>0, 𝑖=1,2,3: Φ=Φ₁₁Φ₁₂Φ₁₃Φ₁₄⋆Φ₂₂Φ₂₃0⋆⋆Φ₃₃0⋆⋆⋆Φ₄₄<0, (31) where 𝑊_𝑖(𝜌_𝑖), 𝑖=1,2,3,…,𝑛, are defined as before. Consider Φ₁₁=𝐴^𝑇_𝑐𝑃+𝑃𝐴_𝑐+1𝛾₁𝑃+𝛾₁+𝛾₂+𝛾₃𝐶^𝑇𝐶𝑊_𝑖𝜌_𝑖+𝑄₁₁+𝑇₁+ℎ²2𝑇₂+𝑇₃,Φ₂₂=𝑄₂₂−𝑄₁₁,Φ₃₃=1𝛾₂𝐷^𝑇𝑃𝐷−𝑄₂₂−𝑇₁,Φ₄₄=1𝛾₃𝐵^𝑇𝑃𝐵−𝑇₂,Φ₁₂=𝑄₁₂,Φ₁₃=−𝐴^𝑇_𝑐𝑃𝐷,Φ₁₄=𝐴_𝑐^𝑇𝑃𝐵,Φ₂₃=−𝑄₁₂. (32)

Proof. Define a legitimate Lyapunov functional candidate as 𝑉(𝑥(𝑡))=𝑉₁(𝑥(𝑡))+𝑉₂(𝑥(𝑡))+𝑉₃(𝑥(𝑡))+𝑉₄(𝑥(𝑡))+𝑉₅(𝑥(𝑡)), (33) where 𝑉₁(𝑥(𝑡))=ℒ^𝑇(𝑥(𝑡(𝑥(𝑡,𝑉))𝑃ℒ))₂(𝑥(𝑡=))_{𝑡()𝑡−ℎ/2}𝑥(𝑠)𝑥ℎ𝑠−2^𝑇𝑄₁₁𝑄₁₂⋆𝑄₂₂𝑥(𝑠)𝑥ℎ𝑠−2𝑉𝑑𝑠,₃(𝑥(𝑡))=^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑇₁𝑥(𝑠)𝑉𝑑𝑠,₄(𝑥(𝑡))=^𝑡_𝑡−ℎ^𝑡_𝑠𝑥^𝑇𝑠₁𝑑𝑠₁𝑇₂^𝑡_𝑠𝑥𝑠₂𝑑𝑠₂𝑉𝑑𝑠,₅(𝑥(𝑡=1))2^𝑡_𝑡−ℎ(𝑠−𝑡+ℎ)²𝑥^𝑇(𝑠)𝑇₃𝑥(𝑠)𝑑𝑠, (34) where 𝑃>0, 𝑄=[𝑄_𝑖𝑗]_2×2, 𝑄_𝑖𝑗>0, and 𝑇_𝑖>0, 𝑖=1,2,3.
According to (34) we obtain 𝑉̇₁(𝑥(𝑡))≤𝑥^𝑇𝐴^𝑇_𝑐𝑃+𝑃𝐴_𝑐+1𝛾₁𝑃+𝛾₁+𝛾₂+𝛾₃𝐶^𝑇𝐶𝑊_𝑖𝜌_𝑖𝑥+1𝛾₂𝑥^𝑇(𝑡−ℎ)𝐷^𝑇()+1𝑃𝐷𝑥𝑡−ℎ𝛾₃^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇𝑃𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠−2𝑥^𝑇(𝑡)𝐴^𝑇_𝑐𝑃𝐷𝑥(𝑡−ℎ)+2𝑥^𝑇(𝑡)𝐴^𝑇_𝑐𝑃𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠, (35) where 𝑃>0, since 𝑃=𝑃². Consider 𝑉̇₂(𝑥(𝑡=𝑥(𝑡)𝑥ℎ))𝑡−2^𝑇𝑄₁₁𝑄₁₂⋆𝑄₂₂𝑥(𝑡)𝑥ℎ𝑡−2−𝑥ℎ𝑡−2𝑥(𝑡−ℎ)^𝑇𝑄₁₁𝑄₁₂⋆𝑄₂₂𝑥ℎ𝑡−2𝑥(),𝑉̇𝑡−ℎ₃(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑇₁𝑥(𝑡)−𝑥^𝑇(𝑡−ℎ)𝑇₁𝑥(),𝑉̇𝑡−ℎ₄(𝑥(𝑡))≤−^𝑡_𝑡−ℎ𝑥^𝑇𝑠₁𝑑𝑠₁𝑇₂^𝑡_𝑡−ℎ𝑥𝑠₂𝑑𝑠₂+ℎ²2𝑥^𝑇(𝑡)𝑇₂𝑥(𝑡)+^𝑡_𝑡−ℎ(𝑠−𝑡+ℎ)𝑥^𝑇(𝑠)𝑇₃𝑥(𝑠)𝑑𝑠, (36) where 𝑇₂>0, since 𝑇₂=𝑇₂𝑇₃⁻¹𝑇₂. Consider 𝑉̇₅(𝑥(𝑡))=12ℎ²𝑥^𝑇(𝑡)𝑇₃𝑥(𝑡)−^𝑡_𝑡−ℎ(𝑠−𝑡+ℎ)𝑥^𝑇(𝑠)𝑇₃𝑥(𝑠)𝑑𝑠. (37)
Then, the time-derivative of 𝑉 has new upper bound as follows: 𝑉̇(𝑥(𝑡))=𝑉̇₁(𝑥(𝑡))+𝑉̇₂(𝑥(𝑡))+𝑉̇₃(𝑥(𝑡+𝑉̇))₄(𝑥(𝑡))+𝑉̇₅(𝑥(𝑡))≤𝜉^𝑇(𝑡)Φ𝜉(𝑡), (38) where Φ is defined as stated in (31) and 𝜉^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇ℎ𝑡−2𝑥^𝑇(𝑡−ℎ)^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠. (39)
The corollary can then be proved following [24].

3.2. Asymptotic Stability for Uncertain Neutral System

Consider the following uncertain neutral system with time-delay and actuator saturation: 𝑥̇(𝑡)𝑥̇()=((𝑡𝑥(𝑡)()+((𝑡(𝑢(𝑡,−𝐷𝑡−𝜏𝐴+Δ𝐴))+𝐵𝑥𝑡−ℎ𝐶+Δ𝐶))Sat)) (40) where Δ𝐴(𝑡) and Δ𝐶(𝑡) stand for the uncertainties. For simplicity, the constant parameter matrices 𝐴, 𝐵, 𝐶, and 𝐷 are square matrices. The spectral norm bound of the unknown uncertainties is ‖Δ𝐴(𝑡)‖≤𝛼,‖Δ𝐶(𝑡)‖≤𝛽,∀𝑡≥0. (41)

Using the nonlinear function 𝐷𝑧(⋅), rewrite the uncertain neutral system as follows: 𝑥̇(𝑡)−𝐷𝑥̇(𝑡−𝜏)=𝐴_𝑐(𝑡)(𝑡)𝐾𝑥(𝑡)()+((𝑡((𝑡,+Δ𝐴−Δ𝐶+𝐵𝑥𝑡−ℎ𝐶+Δ𝐶))𝐷𝑧𝐾𝑥)) (42) where 𝐴_𝑐 is defined as the same with certain neutral system with time-delay and actuator saturation. In particular, when ‖Δ𝐴(𝑡)‖=0 and ‖Δ𝐶(𝑡)‖=0, the uncertain neutral system becomes the certain case.

Similarly, we employ the operator ℒ∶𝐶₀→ℝ^𝑛 with ℒ(𝑥(𝑡))=𝑥(𝑡)+𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠−𝐷𝑥(𝑡−𝜏). (43)

The following transformed system is then obtained: 𝑑ℒ(𝑥(𝑡=𝑑𝑡))𝐴_𝑐𝑥(𝑡)+(𝐶+Δ𝐶(𝑡))𝐷𝑧(𝐾𝑥(𝑡)), (44) where 𝐴_𝑐=𝐴_𝑐+Δ𝐴(𝑡)−Δ𝐶(𝑡)𝐾=𝐴+𝐵−𝐶𝐾+Δ𝐴(𝑡)−Δ𝐶(𝑡)𝐾.

Theorem 10. The uncertain neutral system in (40) with feedback control (3) is asymptotic stability if ‖ℎ|𝐵|+𝐷‖<1 and there exist scalars 𝛾_𝑖>0, 𝑖=1,2,3,4,5,6, 𝑃>0, 𝑃>0, 𝑄=[𝑄_𝑖𝑗]_2×2, 𝑄_𝑖𝑗>0, 𝑅=[𝑅_𝑖𝑗]_2×2, 𝑅_𝑖𝑗>0, 𝑇_𝑖>0(𝑖=1,2,3,4,5,6), and 𝑇_𝑖>0(𝑖=3,4) such that the following symmetric linear matrix inequality holds: ΞΞ=¹¹Ξ₁₂0Ξ₁₄Ξ₁₅Ξ₁₆0⋆Ξ₂₂Ξ₂₃0000⋆⋆Ξ₃₃0000⋆⋆⋆Ξ₄₄Ξ₄₅00⋆⋆⋆⋆Ξ₅₅00⋆⋆⋆⋆⋆Ξ₆₆0⋆⋆⋆⋆⋆⋆Ξ₇₇<0, (45) where Ξ₁₁=𝐴^𝑇_𝑐𝑃+𝑃𝐴_𝑐+1𝛾₁1𝑃+𝛾₂𝑃+𝛾₁+𝛾₃+𝛾₅𝛼²+2𝛼𝛽‖𝐾‖+𝛽²‖𝐾‖²𝐼+𝛾₂+𝛾₄+𝛾₆𝛽²+2𝛽‖𝐶‖+‖𝐶‖²𝑊_𝑖𝜌_𝑖+𝑄₁₁+𝑅₁₁+𝑇₁+𝑇₂+ℎ²2𝑇₃+𝑇₅+𝜏²2𝑇₄+𝑇₆,Ξ₂₂=𝑄₂₂−𝑄₁₁,Ξ₃₃=−𝑄₂₂−𝑇₁,Ξ₄₄=𝑅₂₂−𝑅₁₁,Ξ₅₅=1𝛾₃+1𝛾₄𝐷^𝑇𝑃𝐷−𝑅₂₂−𝑇₂,Ξ₆₆=1𝛾₅+1𝛾₆𝐵^𝑇𝑃𝐵−𝑇₃,Ξ₇₇=−𝑇₄,Ξ₁₂=𝑄₁₂,Ξ₁₄=𝑅₁₂,Ξ₁₅=−𝐴^𝑇_𝑐Ξ𝑃𝐷,₁₆=𝐴^𝑇_𝑐Ξ𝑃𝐵,₂₃=−𝑄₁₂,Ξ₄₅=−𝑅₁₂. (46)

Proof. Define the legitimate Lyapunov functional candidate as 𝑉(𝑥(𝑡))=𝑉₁(𝑥(𝑡))+𝑉₂(𝑥(𝑡))+𝑉₃(𝑥(𝑡))+𝑉₄(𝑥(𝑡))+𝑉₅(𝑥(𝑡))+𝑉₆(𝑥(𝑡)), (47) where 𝑉₁(𝑥(𝑡)), 𝑉₂(𝑥(𝑡)), 𝑉₃(𝑥(𝑡)), 𝑉₄(𝑥(𝑡)), 𝑉₅(𝑥(𝑡)), and 𝑉₆(𝑥(𝑡)) are the same as in Theorem 6.
The time-derivative of 𝑉(𝑥(𝑡)) along the trajectories of closed system (44) is given by the following: 𝑉̇(𝑥(𝑡))=𝑉̇₁(𝑥(𝑡))+𝑉̇₂(𝑥(𝑡))+𝑉̇₃(𝑥(𝑡+𝑉̇))₄(𝑥(𝑡))+𝑉̇₅(𝑥(𝑡))+𝑉̇₆(𝑥(𝑡)). (48) Then 𝑉̇₁(𝑥(𝑡))=2𝑥^𝑇(𝑡)−𝑥^𝑇(𝑡−𝜏)𝐷^𝑇+^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇×𝑃𝐴_𝑐𝑥(𝑡)+(𝐶+Δ𝐶)𝐷𝑧(𝐾𝑥)=2𝑥^𝑇𝑃𝐴_𝑐𝑥+2𝑥^𝑇𝑃(𝐶+Δ𝐶)𝐷𝑧(𝐾𝑥)−2𝑥^𝑇𝐴_𝑐^𝑇𝑃𝐷𝑥(𝑡−𝜏)−2𝑥^𝑇(𝑡−𝜏)𝐷^𝑇×𝑃(𝐶+Δ𝐶)𝐷𝑧(𝐾𝑥)+2𝑥^𝑇(𝑡)𝐴_𝑐^𝑇𝑃𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠+2^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇𝑃(𝐶+Δ𝐶)𝐷𝑧(𝐾𝑥)=2𝑥^𝑇𝑃𝐴_𝑐𝑥+2𝑥^𝑇𝑃(Δ𝐴−Δ𝐶𝐾)𝑥+2𝑥^𝑇𝑃(𝐶+Δ𝐶)𝐷𝑧(𝐾𝑥)−2𝑥^𝑇(𝑡−𝜏)𝐷^𝑇𝑃𝐴_𝑐𝑥(𝑡)−2𝑥^𝑇(𝑡−𝜏)𝐷^𝑇𝑃(Δ𝐴−Δ𝐶𝐾)𝑥−2𝑥^𝑇(𝑡−𝜏)𝐷^𝑇𝑃(𝐶+Δ𝐶)𝐷𝑧(𝐾𝑥)+2^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇𝑃𝐴_𝑐𝑥+2^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇𝑃(Δ𝐴−Δ𝐶𝐾)𝑥+2^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇𝑃(𝐶+Δ𝐶)𝐷𝑧(𝐾𝑥)≤𝑥^𝑇𝐴^𝑇_𝑐𝑃+𝑃𝐴_𝑐𝑥−2𝑥^𝑇(𝑡)𝐴_𝑐^𝑇𝑃𝐷𝑥(𝑡−𝜏)+2𝑥^𝑇(𝑡)𝐴^𝑇_𝑐𝑃𝐵^𝑡_𝑡−ℎ𝑥(𝑠)+1𝑑𝑠𝛾₁𝑥^𝑇𝑃𝑥+𝛾₁𝑥^𝑇(Δ𝐴−Δ𝐶𝐾)^𝑇()𝑥+1Δ𝐴−Δ𝐶𝐾𝛾₂𝑥^𝑇𝑃𝑥+𝛾₂𝐷𝑧^𝑇(𝐾𝑥)(𝐶+Δ𝐶)^𝑇×()()+1𝐶+Δ𝐶𝐷𝑧𝐾𝑥𝛾₃𝑥^𝑇(𝑡−𝜏)𝐷^𝑇𝑃𝐷𝑥(𝑡−𝜏)+𝛾₃𝑥^𝑇(Δ𝐴−Δ𝐶𝐾)^𝑇()𝑥+1Δ𝐴−Δ𝐶𝐾𝛾₄𝑥^𝑇(𝑡−𝜏)𝐷^𝑇𝑃𝐷𝑥(𝑡−𝜏)+𝛾₄𝐷𝑧^𝑇(𝐾𝑥)(𝐶+Δ𝐶)^𝑇()()+1𝐶+Δ𝐶𝐷𝑧𝐾𝑥𝛾₅^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇𝑃𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠+𝛾₅𝑥^𝑇(Δ𝐴−Δ𝐶𝐾)^𝑇()𝑥+1Δ𝐴−Δ𝐶𝐾𝛾₆^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠𝐵^𝑇𝑃𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠+𝛾₆𝐷𝑧^𝑇(𝐾𝑥)(𝐶+Δ𝐶)^𝑇(𝐶+Δ𝐶)𝐷𝑧(𝐾𝑥)≤𝑥^𝑇𝐴_𝑐^𝑇𝑃+𝑃𝐴_𝑐+1𝛾₁𝑃+𝛾₁𝛼²+2𝛼𝛽‖𝐾‖+𝛽²‖𝐾‖²𝐼+1𝛾₂𝑃+𝛾₂𝛽²+2𝛽‖𝐶‖+‖𝐶‖²𝑊_𝑖𝜌+𝛾₃𝛼²+2𝛼𝛽‖𝐾‖+𝛽²‖𝐾‖²𝐼+𝛾₄𝛽²+2𝛽‖𝐶‖+‖𝐶‖²𝑊_𝑖𝜌_𝑖+𝛾₅𝛼²+2𝛼𝛽‖𝐾‖+𝛽²‖𝐾‖²𝐼+𝛾₆𝛽²+2𝛽‖𝐶‖+‖𝐶‖²𝑊_𝑖𝜌_𝑖𝑥+𝑥^𝑇()1𝑡−𝜏𝛾₃𝐷^𝑇1𝑃𝐷+𝛾₄𝐷^𝑇𝑥()+𝑃𝐷𝑡−𝜏_{𝑡𝑡−ℎ}𝑥^𝑇(𝑠)1𝑑𝑠𝛾₅𝐵^𝑇1𝑃𝐵+𝛾₆𝐵^𝑇×𝑃𝐵_{𝑡𝑡−ℎ}𝑥(𝑠)𝑑𝑠−2𝑥^𝑇(𝑡)𝐴^𝑇_𝑐𝑃𝐷𝑥(𝑡−𝜏)+2𝑥^𝑇(𝑡)𝐴^𝑇_𝑐𝑃𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠, (49) where 𝑃>0, since 𝑃=𝑃².
𝑉̇₂(𝑥(𝑡)), 𝑉̇₃(𝑥(𝑡)), 𝑉̇₄(𝑥(𝑡)), 𝑉̇₅(𝑥(𝑡)), and 𝑉̇₆(𝑥(𝑡)) are obtained similarly as in Theorem 6. Substituting these into (48), the time-derivative of 𝑉 has new upper bound as follows: 𝑉̇(𝑥(𝑡))≤𝜉^𝑇(𝑡)Ξ𝜉(𝑡), (50) where𝜉^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇ℎ𝑡−2𝑥^𝑇(𝑡−ℎ)𝑥^𝑇𝜏𝑡−2𝑥^𝑇(𝑡−𝜏)^𝑡_𝑡−ℎ𝑥^𝑇(𝑠)𝑑𝑠^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑑𝑠; (51)Ξ is defined as stated in (45).
If linear matrix inequality (45) is feasible, then 𝑉̇(𝑥(𝑡))<0, for all 𝑥∈𝐵(𝜌). The theorem can then be proved following [24].

Theorem 10 provides new asymptotic stability conditions for the uncertain neutral systems in (40) and (3). The following corollary is presented as a special case of the theorem.

Corollary 11. The uncertain neutral system in (40) and (3) with 𝜏≡ℎ is asymptotic stability if ‖ℎ|𝐵|+𝐷‖<1 and there exist 𝑃>0, 𝑃>0, 𝑄=[𝑄_𝑖𝑗]_2×2, 𝑄_𝑖𝑗>0, 𝑇_𝑖>0(𝑖=1,3), and 𝑇₂>0 such that the following symmetric linear matrix inequality holds for real constant scalars 𝛾_𝑖>0, 𝑖=1,2,3,4,5,6: Ξ=Ξ₁₁Ξ₁₂Ξ₁₃Ξ₁₄⋆Ξ₂₂Ξ₂₃0⋆⋆Ξ₃₃0⋆⋆⋆Ξ₄₄<0, (52) where 𝑊_𝑖(𝜌), 𝑖=1,2,3,…,𝑛 are defined as before and Ξ₁₁=𝐴^𝑇_𝑐𝑃+𝑃𝐴_𝑐+1𝛾₁1𝑃+𝛾₂𝑃+𝛾₁+𝛾₃+𝛾₅𝛼²+2𝛼𝛽‖𝐾‖+𝛽²‖𝐾‖²𝐼+𝛾₂+𝛾₄+𝛾₆𝛽²+2𝛽‖𝐶‖+‖𝐶‖²𝑊_𝑖𝜌_𝑖+𝑄₁₁+𝑇₁+ℎ²2𝑇₂+𝑇₃,Ξ₂₂=𝑄₂₂−𝑄₁₁,Ξ₃₃=1𝛾₃+1𝛾₄𝐷^𝑇𝑃𝐷−𝑄₂₂−𝑇₁,Ξ₄₄=1𝛾₅+1𝛾₆𝐵^𝑇𝑃𝐵−𝑇₂,Ξ₁₂=𝑄₁₂,Ξ₁₃=−𝐴^𝑇_𝑐𝑃𝐷,Ξ₁₄=𝐴^𝑇_𝑐𝑃𝐵,Ξ₂₃=−𝑄₁₂. (53)

Proof. Choose a legitimate Lyapunov functional candidate as 𝑉(𝑥(𝑡))=𝑉₁(𝑥(𝑡))+𝑉₂(𝑥(𝑡))+𝑉₃(𝑥(𝑡))+𝑉₄(𝑥(𝑡))+𝑉₅(𝑥(𝑡)) (54) which are the same as (34).
𝑉̇₁(𝑥(𝑡)) can be evaluated similarly as in Theorem 10 and Corollary 9. The proof can be readily obtained.

3.3. The Algorithm with 𝑊₂(𝜌₂) and the Algorithm to Solve the Optimal 𝑊_𝑖(𝜌_𝑖)

From Definition 4, it is seen that 𝑊₁(𝜌₁) are different from 𝑊₂(𝜌₂), 𝑊₃(𝜌₃), and 𝑊₄(𝜌₄),…,𝑊_𝑛(𝜌_𝑛). We compare between 𝑊₂(𝜌₂), 𝑊₃(𝜌₃), and 𝑊₄(𝜌₄),…,𝑊_𝑛(𝜌_𝑛) and intend to reduce the conservativeness of the result. To that end, we should obtain 𝜌₂ in the first place. In what follows we present the stability analysis algorithm with 𝑊₂(𝜌₂) to solve 𝜌₂.

Step 1. Give 𝛼,𝛽,𝐾.

Step 2. Set positive values 𝛾_𝑖(𝑖=1,2,3,4,5,6).

Step 3. Initialize 𝜌₂.

Step 4. With 𝑊₂(𝜌₂), solve linear matrix inequality (45) by Matlab LMI Toolbox.

Step 5. If the solution satisfies the stability condition, go to Step 6; otherwise, reduce 𝜌₂ and go to Step 3.

Step 6. Increase 𝜌₂ and go to Step 3.

Step 7. End.

Remark 12. The above algorithm is stated with respect to uncertain systems (40) and (3). Other cases can be dealt with similarly.

After defining 𝜌₂, we replace 𝑊₂(𝜌₂) by 𝑊_𝑖(𝜌_𝑖) to reduce the conservativeness. We analyse these functions 𝑊₂(𝜌₂),𝑊₃(𝜌₃),𝑊₄(𝜌₄),…,𝑊_𝑛(𝜌_𝑛) and find the optimal 𝑊_𝑖(𝜌_𝑖) to obtain the maximum 𝜌_𝑖 among 𝜌₂,𝜌₃,…,𝜌_𝑛.

Theorem 13. Given 𝑊₂(𝜌₂),𝑊₃(𝜌₃),𝑊₄(𝜌₄),…,𝑊_𝑛(𝜌_𝑛), if there exists 𝑗≤𝑛, such that 𝜌_𝑗−1≥𝜌_𝑘 and ‖𝑊_𝑗(𝜌_𝑗−1)‖≤‖𝑊_𝑘(𝜌_𝑘)‖, for all 𝑘=2,3,…,𝑗−1, then we have 𝐾_𝑖𝜌_𝑗−1𝑢⁺_𝑖≤𝑓_𝑘𝑗,∀𝑘=2,3,…,𝑗−1, (55) where 𝜌_𝑗 is the domain attraction obtained by 𝑊_𝑗(𝜌_𝑗) and 𝑓_𝑘𝑗=(𝑘−1)^{(𝑘−1)/(𝑗−𝑘)}𝑘^{𝑘/(𝑗−𝑘)}×𝑗^{𝑗/(𝑗−𝑘)}𝑗−1^{(𝑗−1)/(𝑗−𝑘)}. (56)

Proof. Recall the definition of 𝑊_𝑗(𝜌_𝑗), we know it can be expressed in the following equality: 𝑊_𝑗𝜌_𝑗=^𝑚_𝑖=1𝑗−1𝜌_𝑗𝑗^{𝑗/(𝑗−1)}𝑢⁺_𝑖^2𝑗−2𝐾^𝑇_𝑖𝐾_𝑖^𝑗. (57) If we have 𝜌_𝑗−1≥𝜌_𝑘, for all 𝑘=2,3,…,𝑗−1, then from the above equality we obtain 𝑊_𝑘𝜌_𝑘≤𝑊_𝑘𝜌_𝑗−1,∀𝑘=2,3,…,𝑗−1. (58) With regard to 𝑊_𝑗(𝜌_𝑗−1), we have 𝑊_𝑗𝜌_𝑗−1=^𝑚_𝑖=1𝑗−1𝜌_𝑗−1𝑗^{𝑗/(𝑗−1)}𝑢⁺_𝑖^2𝑗−2𝐾^𝑇_𝑖𝐾_𝑖^𝑘𝐾^𝑇_𝑖𝐾_𝑖^𝑗−𝑘=^𝑚_𝑖=1(𝑘−1)𝜌_𝑗−1𝑘^{𝑘/(𝑘−1)}𝑢⁺_𝑖^2𝑘−2𝐾^𝑇_𝑖𝐾_𝑖^𝑘𝐾^𝑇_𝑖𝐾_𝑖^𝑗−𝑘×𝜌_𝑗−1𝑢⁺_𝑖^{2(𝑗−𝑘)}𝑘^2𝑘(𝑘−1)^2𝑘−2𝑗−1^2𝑗−2𝑗^2𝑗=𝐾_𝑖^{2(𝑗−𝑘)}𝜌_𝑗−1𝑢⁺_𝑖^{2(𝑗−𝑘)}×𝑘^2𝑘(𝑘−1)^2𝑘−2𝑗−1^2𝑗−2𝑗^2𝑗𝑊_𝑘𝜌_𝑗−1. (59) If we have ‖𝑊_𝑗(𝜌_𝑗−1)‖≤‖𝑊_𝑘(𝜌_𝑘)‖, then we can obtain 𝑊_𝑗𝜌_𝑗−1=𝐾_𝑖^{2(𝑗−𝑘)}𝜌_𝑗−1𝑢⁺_𝑖^{2(𝑗−𝑘)}𝑘^2𝑘(𝑘−1)^2𝑘−2×𝑗−1^2𝑗−2𝑗^2𝑗𝑊_𝑘𝜌_𝑗−1≤𝑊_𝑘𝜌_𝑘. (60) By (58), we have 𝐾_𝑖𝜌_𝑗−1𝑢⁺_𝑖^{2(𝑗−𝑘)}𝑘^2𝑘(𝑘−1)^2𝑘−2𝑗−1^2𝑗−2𝑗^2𝑗≤1, (61) where 𝑗=3,4,5,…, and 𝑘=2,3,…,𝑗−1.
Then we easily obtain 𝐾_𝑖𝜌_𝑗−1𝑢⁺_𝑖≤𝑓_𝑘𝑗=(𝑘−1)^{(𝑘−1)/(𝑗−𝑘)}𝑘^{𝑘/(𝑗−𝑘)}×𝑗^{𝑗/(𝑗−𝑘)}𝑗−1^{(𝑗−1)/(𝑗−𝑘)} (62) which completes the proof.

Remark 14. Theorem 13 provides us a criterion to find the optimal 𝑊_𝑗(𝜌_𝑗). That is, if condition (55) cannot be satisfied, the procedure to find the optimal one should be ended. For example, when 𝜌₄ is obtained, we compute ‖𝐾_𝑖‖𝜌₄/𝑢⁺_𝑖 and the following inequalities need to be satisfied: 𝐾_𝑖𝜌₄𝑢⁺_𝑖≤𝑓₂(5),𝐾_𝑖𝜌₄𝑢⁺_𝑖≤𝑓₃(5),𝐾_𝑖𝜌₄𝑢⁺_𝑖≤𝑓₄(5). (63) Otherwise, the algorithm terminates with 𝜌₄.

The following optimality algorithm is used to obtain the optimal 𝑊_𝑛(𝜌_𝑛).

Step 1. Set positive value 𝛿 (small value, for example, 10⁻⁵).

Step 2. Solve 𝜌₂ based on the above stability analysis algorithm with 𝑊₂(𝜌₂).

Step 3. Compute ‖𝐾_𝑖‖𝜌₂/𝑢⁺_𝑖, if (‖𝐾_𝑖‖𝜌₂/𝑢⁺_𝑖)−𝑓_𝑘(𝑗)≤𝑓₂(3), then go to Step 4; otherwise, go to Step 7.

Step 4. Set 𝜌_𝑗+1=𝜌_𝑗, repeat the stability analysis algorithm with 𝑊_𝑗+1(𝜌_𝑗) then obtain 𝜌_𝑗+1 iteratively.

Step 5. Compute ‖𝐾_𝑖‖𝜌_𝑗+1/𝑢⁺_𝑖, if 𝐾_𝑖𝜌_𝑗+1𝑢⁺_𝑖≤𝑓_𝑘,𝐾𝑗+2^𝑖𝜌_𝑗+1𝑢⁺_𝑖−𝑓_𝑘𝑗+2>𝛿,∀𝑘=2,3,…,𝑗+1 (64) then go to Step 6; otherwise go to Step 7.

Step 6. Set 𝑗=𝑗+1 and go to Step 4.

Step 7. End.

4. Application and Numerical Examples

The two-stage system has been applied to many factories worldwide. For the interest of efficiency, a lot of equipment such as converter, cylinder, and ejector is designed in the two-stage form. The natural, social, and economic systems also exhibit the two-stage mode. A typical example is the two-stage ditch design which is a conservation tool supported by the conservancy in Indiana. The advantages of a two-stage ditch against the typical agricultural ditch include both improved drainage function and ecological function. The two-stage design improves ditch stability by reducing water flow and the need for maintenance, saving both labor and money.

Consider Figure 2 which shows two-stage dissolution tank in the chemical process. The solute in the hopper is transported by the conveyor belt and falls into DT1 within a certain time. The solute is part of the solution in DT1; then within a certain time the solution in DT1 and the undissolved solute flow into DT2 to continue to dissolve and dilute.

Figure 2 

Two-stage dissolution tank.

Under normal circumstances, the dilute fluid flow of DT1 and the velocity of the conveyor belt are constant. The dilute fluid flow of DT2 is controllable. In practical system design, the pipeline of the DT2 dilute fluid is thin and the regulating action of DT2 is tiny, which can thus be represented by the saturating form Sat(𝑢(𝑡)). Let 𝜆₁(𝑡) and 𝜆₂(𝑡) be the solution concentration of DT1 and DT2 and 𝜆₁₀ and 𝜆₂₀ the solution concentration of DT1 and DT2 when the system reaches an equilibrium, respectively. 𝑥₁(𝑡)=𝜆₁(𝑡)−𝜆₁₀ and 𝑥₂(𝑡)=𝜆₂(𝑡)−𝜆₂₀ are the considered system states and the dilute fluid flow into DT2 𝑢(𝑡) is the control input. The latter is of the saturating form Sat(𝑢(𝑡)) because the dilute fluid flow into DT2 𝑢(𝑡) is a fine adjustment role. The chemical plant puts the solute into DT1 with constant speed; that is, the velocity of the conveyor belt is constant in a period of time interval. In addition, the velocity of the dilute fluid flow into DT1 is also constant in a period of time interval.

4.1. No Uncertainty Case

Example 1. The model for the above described system is 𝑥̇(𝑡)−𝑥̇()=𝑥(𝑡)+𝑥()01(𝑢(𝑡,0.2000.2𝑡−𝜏11.50.3−20−100𝑡−ℎSat)) (65) where the saturation limit is 𝑢⁺=15. The discrete time-delay ℎ=0.2 and neutral time-delay 𝜏=0.4.
It is seen that ‖ℎ|𝐵|+𝐷‖=0.2+0−1000.2000.2=0.1047<1 (66) and the operator ℒ∶𝐶₀→ℝ^𝑛 with ℒ(𝑥(𝑡))=𝑥(𝑡)+𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠−𝐷𝑥(𝑡−𝜏) (67) is stable.
The following linear state feedback control is used and the same as [18]: 𝑢=−𝐾𝑥=−0.13250.0153𝑥 (68)
To begin with the stability analysis algorithm, we have 𝜌₂=161.73, and compute ‖𝐾‖𝜌₂/𝑢⁺=1.4383. It is seen that ‖𝐾‖𝜌1<²𝑢⁺≤𝑓₂(3)=2716≐1.6875. (69)
According to the optimality algorithm, let initial data 𝜌₃=𝜌₂. Repeating the stability algorithm with 𝑊₃(𝜌₃), we obtain 𝜌₃=163.46. Similarly, we compute ‖𝐾‖𝜌₃/𝑢⁺ and (‖𝐾‖𝜌₃/𝑢⁺)=1.4537.
It is seen that ‖𝐾‖𝜌₃𝑢⁺<𝑓₂(4)‖𝐾‖𝜌≐1.5396,but³𝑢⁺>𝑓₃(4)≐1.4047 (70) which terminates the algorithm. In this example, we obtain that the optimal auxiliary function is 𝑊₃(𝜌₃). By Theorem 13, we must verify ‖𝑊₃(𝜌₃)‖≤‖𝑊₂(𝜌₂)‖, which is true as ‖𝑊₃(𝜌₃)‖=0.0017 and ‖𝑊₂(𝜌₂)‖=0.0023. So we infer that 𝑊₃(𝜌₃) is the optimal auxiliary function among 𝑊₂(𝜌₂),𝑊₃(𝜌₃),𝑊₄(𝜌₄),…,𝑊_𝑛(𝜌_𝑛) and 𝜌₃ is the largest one among 𝜌₂,𝜌₃,𝜌₄,…,𝜌_𝑛.
The comparison of these results can be seen in Figure 3. Notice that 𝜌 is not the domain of attraction.
We obtain the largest parameter 𝜌_𝑖 with optimal 𝑊_𝑖(𝜌_𝑖) to enlarge the stability region. Thus the chemical plant can make the conveyor belt and dilute fluid flow of DT1 speed up accordingly. In this way, this chemical plant can improve the production efficiency and reduce the cost of production.

Figure 3 

Attraction domain expanding schematic diagram: the largest parameter 𝜌3 with optimal 𝑊3(𝜌3) is in red pentagram, with 𝑊2(𝜌2) in blue, and another one with 𝑊4(𝜌4) in red.

However, in some circumstances, we cannot enlarge the domain of attraction by replacing 𝑊₂(𝜌₂) with 𝑊_𝑖(𝜌_𝑖), 𝑖=3,4,…,𝑛. For example, let us see the following example.

Example 2. Consider the following linear neutral system with time-delay and actuator saturation: 𝑥̇(𝑡)−𝑥̇()=𝑥(𝑡)+𝑥()+(𝑢(𝑡,0.2000.2𝑡−𝜏0.1−0.10.1−3.00.5−100.5𝑡−ℎ0.7000.1Sat)) (71) where the saturation limit values are 𝑢⁺₁=10 and 𝑢⁺₂=4. The discrete time-delay ℎ=0.2 and neutral time-delay 𝜏=0.1.
Compute ‖ℎ|𝐵|+𝐷‖ and we also know that the operator ℒ∶𝐶₀→ℝ^𝑛 with ℒ(𝑥(𝑡))=𝑥(𝑡)+𝐵^𝑡_𝑡−ℎ𝑥(𝑠)𝑑𝑠−𝐷𝑥(𝑡−𝜏) (72) is stable. Then we obtain the linear state feedback control without considering the saturation using optimal control law in [27]. Consider 𝑢=−𝐾𝑥=−0.84560.06270.01421.3754𝑥. (73)
To begin with the stability analysis algorithm, we have 𝜌₂=17.84, and compute 𝐾₁𝜌₂𝑢⁺₁𝐾=1.5127,²𝜌₂𝑢⁺₂=6.1780. (74) It is seen that 𝐾1<¹𝜌₂𝑢⁺₁≤𝑓₂(3)=27𝐾16≐1.6875,but²𝜌₂𝑢⁺₂>𝑓₂(3) (75) which terminates the algorithm. So we infer that 𝑊₂(𝜌₂) is the optimal auxiliary function among 𝑊₂(𝜌₂),𝑊₃(𝜌₃),𝑊₄(𝜌₄),…,𝑊_𝑛(𝜌_𝑛). 𝜌₂ has been the largest one among 𝜌₂,𝜌₃,𝜌₄,…,𝜌_𝑛 and we cannot maximize the value of 𝜌_𝑖.
The comparison of these results can be seen in Figure 4.

Figure 4 

Attraction domain expanding schematic diagram: the largest parameter 𝜌2 with optimal 𝑊2(𝜌2) is in red and another one with 𝑊3(𝜌3) in blue.

4.2. Uncertainty Case

In this subsection, we consider the existence of uncertainty due to outside interference. The numerical example demonstrates the effectiveness of the proposed algorithm.

Example 3. Consider system (65) with norm bounded uncertainties: ‖Δ𝐴(𝑡)‖≤0.1,‖Δ𝐵(𝑡)‖≤0.1. (76)
It is known that this uncertainty decreases the initial state bound that guarantees the asymptotic stability of the system (65). To compare with the above case, the feedback control (68) is applied and we use (45) in Theorem 10.
Using the stability analysis algorithm and the optimal algorithm, we see that 𝑊₃(𝜌₃) is the optimal auxiliary function among 𝑊₂(𝜌₂),𝑊₃(𝜌₃),𝑊₄(𝜌₄),…,𝑊_𝑛(𝜌_𝑛). 𝜌₃=43.57 is the largest one among 𝜌₂,𝜌₃,𝜌₄,…,𝜌_𝑛. This is shown in Figure 5.

Figure 5 

Attraction domain expanding schematic diagram: the largest parameter 𝜌3 with optimal 𝑊3(𝜌3) is in red pentagram, with 𝑊2(𝜌2) in blue, and another one with 𝑊4(𝜌4) in red.

5. Conclusions

By model transformation and Lyapunov method, delay-dependent criteria for a class of uncertain neutral delay systems with saturation are derived in terms of the spectral radius and LMIs. Both conditions for asymptotic stability and algorithms are presented. These theoretical findings are successfully verified via a two-stage dissolution tank in the chemical plant. Potentially, the result is of great significance to the chemical production process.

Conflict of Interests

It has no any conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the Associate Editor and the anonymous reviewers for their constructive comments and suggestions to improve the quality of the paper. This work was supported in part by the Fundamental Research Funds for the Central Universities.