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ISRN Applied Mathematics
Volume 2014 (2014), Article ID 805798, 14 pages
Asymptotic Stability Analysis and Optimality Algorithm for Uncertain Neutral Systems with Saturation
Department of Auto, School of Information Science and Technology, University of Science and Technology of China, Anhui 230027, China
Received 29 December 2013; Accepted 18 February 2014; Published 27 March 2014
Academic Editors: C.-Y. Lu and X. Meng
Copyright © 2014 Xinghua Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
Delay is often inevitable in various practical systems; examples include population ecology , steam or water pipes, heat exchanges , 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 . 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 ; a controller is constructed in terms of linear matrix inequalities using descriptor model transformation in , 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 . 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 ). 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 ). For any matrix , if , then the operator with is stable.
Lemma 3 (see ). Let and , and let . Then we have
Definition 4 (see ). 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.
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 , , , , , 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 , 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 , 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 , , , 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 .
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 .
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
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 6. Increase and go to Step 3.
Step 7. End.
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 4. Set =, repeat the stability analysis algorithm with then obtain iteratively.
Step 6. Set 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.
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 . Let and be the solution concentration of DT1 and DT2 and and the solution concentration of DT1 and DT2 when the system reaches an equilibrium, respectively. and are the considered system states and the dilute fluid flow into DT2 is the control input. The latter is of the saturating form because the dilute fluid flow into DT2 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
where the saturation limit is . The discrete time-delay and neutral time-delay .
It is seen that and the operator with is stable.
The following linear state feedback control is used and the same as :
To begin with the stability analysis algorithm, we have , and compute . It is seen that
According to the optimality algorithm, let initial data . Repeating the stability algorithm with , we obtain . Similarly, we compute