#### 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, denotes the dimensional Euclidean space, and is the set of all real matrices. is the identity matrix with proper dimensions. is the set of all continuous functions from to where is a constant representing the neutral time-delay. is the transpose of matrix . , with . For real symmetric matrices and , the notation (respectively, ) means that the matrix is positive semidefinite (respectively, positive definite). is the eigenvalue of matrix with maximum (minimum) real part. is the Euclidean norm of vector , , while is spectral norm of matrix , . represents the domain of attraction. denotes a block-diagonal matrix decided by the corresponding elements in the brace and finally .

#### 2. Problem Statement and Preliminaries

The following neutral system with time-delay and actuator saturation is considered: where is the system state and is the control input. is the constant discrete time delay and is the constant neutral time-delay. , , , and are known real constant parameter matrices of appropriate dimensions with . is used to denote the standard saturation function defined for :

The following linear state feedback is to be designed: where the linear state feedback gain ; is an -dimensional row vector.

Here we have slightly abused the notation by using 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 where and

The saturated system can now be written as follows: where .

The neutral system (5) then leads to the following by model transformation: where and .

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

*Definition 1 (see [24]). *The operator is said to be stable if the zero solution of the homogeneous difference equation
is uniformly asymptotically stable. The stability of operator is necessary for the stability of neutral system (1) with (3), which is always satisfied when .

Lemma 2 (see [24]). *For any matrix , if , then the operator with
**
is stable.*

Lemma 3 (see [25]). *Let and , and let . Then we have
*

*Definition 4 (see [23]). *Auxiliary functions and are described in the following

#### 3. Main Results

In this section, we firstly construct the auxiliary functions 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 .

Consider the polynomial function which is tangent with saturated function in positive axis. can be firstly determined by the geometric relation and auxiliary polynomial functions can be obtained.

Let

The slope of polynomial function equals to 1 at the tangent point. Thus we have

It can be determined by solving the resulting equations simultaneously and we obtain

Noting in the polynomial function, we replace with here so that they have the same independent variable with . Consider

Thus we compare the nonlinear function with the polynomial function above. For simplicity, we consider the cases and . As shown in Figure 1, the graphics of quadratic and cubic polynomial functions are above the graphic of .

Furthermore, according to the nature of the polynomial function when , the graphic of function is also above the graphic of . So we have the following inequality: where .

Lemma 5. *When , the following inequality holds:
**
where
*

*Proof. *From (15) we obtain that
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 and there exist scalars , , , , , , , , , and such that the following symmetric linear matrix inequality holds:
**
where
*

*Proof. *Define a legitimate Lyapunov functional candidate as follows:
where
where , , , , , and , .

Then
By (19)-(22), we obtain that
where , since . Consider
where , since .

Similarly we have
where , since . Consider
Substituting these into (24), the time-derivative of has new upper bound as follows:

where is defined as stated in (19).

If linear matrix inequality (19) is feasible, then we can get , for all . Therefore, if constant scalar , constant parameter matrices such that and there exist , , , , , , , and satisfying (19) for real scalars , 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. * are created by the parameter 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 with the optimal . 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 is presented in the following corollary.

Corollary 9. *The neutral systems (1) and (3) with are asymptotic stability if and there exist , , , , , and such that the following symmetric linear matrix inequality holds for real constant scalars , :
**
where , , are defined as before. Consider
*

*Proof. *Define a legitimate Lyapunov functional candidate as
where
where , , , and , .

According to (34) we obtain
where , since . Consider
where , since . Consider

Then, the time-derivative of has new upper bound as follows:
where is defined as stated in (31) and

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: where and stand for the uncertainties. For simplicity, the constant parameter matrices , , , and are square matrices. The spectral norm bound of the unknown uncertainties is

Using the nonlinear function , rewrite the uncertain neutral system as follows: where is defined as the same with certain neutral system with time-delay and actuator saturation. In particular, when and , the uncertain neutral system becomes the certain case.

Similarly, we employ the operator with

The following transformed system is then obtained: where .

Theorem 10. *The uncertain neutral system in (40) with feedback control (3) is asymptotic stability if and there exist scalars , , , , , , , , , and such that the following symmetric linear matrix inequality holds:
**
where
*

*Proof. *Define the legitimate Lyapunov functional candidate as
where , , , , , and are the same as in Theorem 6.

The time-derivative of along the trajectories of closed system (44) is given by the following:
Then
where , since .

, , , , and are obtained similarly as in Theorem 6. Substituting these into (48), the time-derivative of has new upper bound as follows:
where is defined as stated in (45).

If linear matrix inequality (45) is feasible, then , for all . 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 is asymptotic stability if and there exist , , , , , and such that the following symmetric linear matrix inequality holds for real constant scalars , :
**
where , are defined as before and
*

*Proof. *Choose a legitimate Lyapunov functional candidate as
which are the same as (34).

can be evaluated similarly as in Theorem 10 and Corollary 9. The proof can be readily obtained.

##### 3.3. The Algorithm with and the Algorithm to Solve the Optimal

From Definition 4, it is seen that are different from , , and . We compare between , , and and intend to reduce the conservativeness of the result. To that end, we should obtain in the first place. In what follows we present the stability analysis algorithm with to solve .

*Step 1. *Give .

*Step 2. *Set positive values .

*Step 3. *Initialize .

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

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

*Step 6. *Increase 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 , we replace by to reduce the conservativeness. We analyse these functions and find the optimal to obtain the maximum among .

Theorem 13. *Given , if there exists , such that and , for all , then we have
**
where is the domain attraction obtained by and
*

*Proof. *Recall the definition of , we know it can be expressed in the following equality:
If we have , for all , then from the above equality we obtain
With regard to , we have
If we have , then we can obtain
By (58), we have
where , and .

Then we easily obtain
which completes the proof.

*Remark 14. *Theorem 13 provides us a criterion to find the optimal . That is, if condition (55) cannot be satisfied, the procedure to find the optimal one should be ended. For example, when is obtained, we compute and the following inequalities need to be satisfied:
Otherwise, the algorithm terminates with .

The following optimality algorithm is used to obtain the optimal .

*Step 1. *Set positive value (small value, for example, ).

*Step 2. *Solve based on the above stability analysis algorithm with .

*Step 3. *Compute , if , then go to Step 4; otherwise, go to Step 7.

*Step 4. *Set =, repeat the stability analysis algorithm with then obtain iteratively.

*Step 5. *Compute , if