Abstract
This paper is concerned with finite-time extended dissipative analysis and nonfragile control for a class of uncertain switched neutral systems with time delay, and the controller is assumed to have either additive or multiplicative form. By employing the average dwell-time and linear matrix inequality technique, sufficient conditions for finite-time boundedness of the switched neutral system are provided. Then finite-time extended dissipative performance for the switched neutral system is addressed, where we can solve , , Passivity, and ()-dissipativity performance in a unified framework based on the concept of extended dissipative. Furthermore, nonfragile state feedback controllers are proposed to guarantee that the closed-loop system is finite-time bounded with extended dissipative performance. Finally, numerical examples are given to demonstrate the effectiveness of the proposed method.
1. Introduction
Switched system is an important class of hybrid systems, which consists of a family of subsystems and a logical rule that orchestrates switching between them. For its practical importance, switched systems have received considerable attention in the last decades [1–5]. Meanwhile, time delay exists widely in many practical systems and may cause undesirable system performance or even instability [6–9]. Switched system with time delay is a main issue in recent years. As a special time delay system, switched neutral systems have received much attention [10–14]. For example, the problem of stability analysis and control for several switched neutral systems were considered in [10] and [13], respectively.
Up to now, most researches for switched neutral systems focus on Lyapunov asymptotic stability, which is defined over an infinite time interval. However, in practice, the transient performance of a system is also of great significance. In many practical applications such as missile systems and robot control systems, the main concern is the system behavior over a finite-time interval. Therefore, finite-time analysis of switched systems is worth researching. Recently, some related research results were published in the literatures [15–20]. More specifically, finite-time control of switched systems was addressed in [16], and finite-time stabilization and boundedness of switched linear system were investigated in [19].
On the other hand, the controller coefficients are generally exact values when designing a desired controller. However, in practice, uncertainty cannot be avoided in controller design, and it may be caused by many reasons, such as numerical round-off errors and actuator degradation. The existence of uncertainty motivates the study of nonfragile control. Over decades, much attention has been devoted to the issue of controller fragility and related remedies [21–24]. To name a few, the problem of passivity-based nonfragile control for Markovian jump systems with aperiodic sampling is studied in [22], and nonfragile control for linear systems with multiplicative controller gain variations is investigated in [24], respectively. More recently, an effective tool named extended dissipative was firstly proposed by Zhang et al. in [25] to deal with the problem of robust control. By adjusting weighting matrices, the extended dissipative covers some well-known performance indices such as performance, performance, Passivity performance, and ()-dissipativity performance. This concept has been successfully applied to the stability analysis for several neural networks [26–30]. Could this concept be applied to switched systems To the best of our knowledge, the topic of nonfragile finite-time extended dissipative control for a class of uncertain switched neutral systems has not been investigated yet, which motivates our study.
This paper is organized as follows. In Section 2, preliminaries and problem statement are formulated and some necessary lemmas are given. In Section 3, by employing the average dwell-time and linear matrix inequality approach, some sufficient conditions of finite-time boundness and finite-time extended dissipative performance for switched neutral systems are established. Furthermore, existence and the design method of the nonfragile state feedback controllers are proposed. All of the results are in terms of a set of linear matrix inequalities which can be easily resolved using the LMIs toolbox. In Section 4, numerical examples are given to show the effectiveness of the proposed approach. The main contributions of this paper include the following. We firstly apply the concept of extended dissipative to the nonfragile finite-time analysis and control to the uncertain switched neutral systems. More general switched systems are considered in our paper, including the time-varying delay and distributed delay, neutral parameters, and additive and multiplicative form controller.
Notation. The notations used in this paper are standard. denotes the -dimensional Euclidean space and represents the transpose of the matrix . The notation (≥0) is used to denote a symmetric positive definite (positive-semidefinite) matrix. represents the Euclidean norm of the matrix ; and denote the minimum and maximum eigenvalue of matrix , respectively. is the identity matrix with appropriate dimension. stands for a block-diagonal matrix. The asterisk in a matrix is used to denote a term that is induced by symmetry.
2. Preliminaries and Problem Statement
Consider the following switched neutral system with time-varying delay:where is the state vector, is the control input, is the exogenous disturbance which belongs to , and is the output. The switching signal is a piecewise continuous function, where is the number of subsystems and means that the th subsystem is activated. is the initial condition, and , , and denote the time-varying delay and satisfyFurthermore, , and for each , , , , and are uncertain real-valued matrices with appropriate dimensions. We assume that the uncertainties are norm-bounded and of the formwhere , , , , , , , and are known real-valued constant matrices with appropriate dimensions and is unknown and possibly time-varying matrix with Lebesgue measurable elements satisfying
In this paper, we consider the following nonfragile state feedback controller: , where , is the controller gain and is a perturbed matrix with the following forms.
Case 1. has an additive uncertainty which is assumed to bewhere and are known real constant matrices with appropriate dimensions and the time-varying matrix satisfies .
Case 2. has a multiplicative uncertaintywhere is the element of and and are real uncertain parameters which satisfy
Assumption 1. For a given time constant , the external disturbance satisfies
Assumption 2. For a given time constant , the state vector is time-varying and satisfies the constraint where is a fixed sufficient large constant number.
Assumption 3 (see [25]). Matrices , , , and satisfy the following conditions:(1);(2)
Assumption 4. For , we have where denotes the switching number of over and denotes a positive number.
Definition 5 (see [25]). For given matrices , , , and satisfying Assumption 3, system (1) is said to be extended dissipative if the following inequality holds for any and all :where
Remark 6. The concept of extended dissipative introduced in Definition 5 contains a few of well-known performance indices as special cases by setting the weighting matrices:(1) performance: , , , and ;(2) performance: , , , and ;(3)Passivity performance: , , , and ;(4)()-dissipativity performance: , , , and .
Definition 7 (see [17]). Given three positive constants , , and with , a positive definite matrix and a switching signal , assume that , , and switched neutral system (1) is said to be finite-time bounded with respect to , if, , Furthermore, if the condition above holds with , , the system is said to be finite-time stable.
Definition 8 (see [17]). For any , let denote the switching number of over . Ifholds for and an integer , then is called an average dwell-time. Without loss of generality, in this paper we choose .
Lemma 9 (see [29]). Let and be real matrices of appropriate dimensions and satisfy .
Lemma 10 (see [31]). For any positive definite symmetric matrix , scalar , and a vector function , the following integral inequality is satisfied:
Lemma 11 (see [32]). Let and be real matrices of appropriate dimensions. Then, for any scalar , one has when satisfies .
Lemma 12 (see [33]). Let be real positive definite symmetric matrices and let and be appropriate dimensional real matrices. Then, one has
3. Main Results
3.1. Finite-Time Boundedness Analysis
Consider the following unforced switched neutral system without uncertainties:In this section, the problem of finite-time boundedness analysis of the switched neutral system is proposed, by using the average dwell-time approach, sufficient conditions are derived by solving some linear matrix inequalities, and the results are shown as follows.
Theorem 13. For given positive scalars , , , , and , if there exist positive definite symmetric matrices , , , , and and matrices , , , , , and with appropriate dimensions, thenhold, where meanwhile, the average dwell-time satisfiesWe define where satisfiesThen, switched system (17) is finite-time bounded with respect to
Proof. Choose the piecewise Lyapunov-Krasovskii functional candidate aswherein which is a given scalar and , , , , and are positive definite matrices to be determined. Taking the derivative of with respect to along the trajectory of system (17) yields Using the Leibniz-Newton formula, we have Let and it obviously holds that By Lemma 10, it is easy to obtainThus we have whereConsidering (19) and (20), we can obtain thatIntegrating (36), it can be obtained from (26) and (36) that, ,From Definition 8, we can deduce that , and then we can obtainOn the other handFrom (38)-(39), we can obtainWhen , it is obvious that by (22).
When , by virtue of (22), we have that From (23), we can obtainSubstituting (42) into (40) yields The proof is completed.
Based on Theorem 13, when we set , the following corollary is proposed to solve the finite-time stable problem.
Corollary 14. Consider system (17) with . For given positive scalars , , , , and , if there exist positive definite symmetric matrices , , , , and and matrices , , , , and with appropriate dimensions, thenhold, where meanwhile, the average dwell-time satisfieswhere satisfying (26) and , , , , , , , , , , and are defined just the same as (25). Then the switched system (17) is finite-time stable with respect to .
Proof. The proof is similar to that of Theorem 13, and it is omitted here.
3.2. Finite-Time Extended Dissipative Analysis
In this section, the finite-time extended dissipative analysis is considered in the following theorem.
Theorem 15. For given positive scalars , , , , , and , if there exist positive definite symmetric matrices , , , , and and matrices , , , , and with appropriate dimensions, thenhold, where meanwhile, the average dwell-time satisfieswhereThen, system (17) is finite-time bounded with extended dissipative performance with respect to .
Proof. Choose the same Lyapunov-Krasovskii function as in Theorem 13, similar to the proof of Theorem 13, and we obtain where by virtue of (49)-(50) we can obtain that follows the proof line of (37); it is easy to obtain the following inequality: under zero initial condition , it can be calculated that and it is equivalent to by Assumption 4, we have so we obtain considering inequality when , one obtains when , by Assumption 3 we have , , and , and then we obtain thus, for , we have it follows from (47) that so we get Thus the proof of extended dissipative is completed.
Next, we proof finite-time boundedness. Following the proof above, we can deduce that When , we can obtain so we get by and by Lemma 9, we have According to (106), we can obtain From (52), we can conclude that . Thus the proof is completed.
3.3. Nonfragile Finite-Time Extended Dissipative Control
Consider system (1), under the controller , the corresponding closed-loop system is given by
Firstly, for the additive gain variation model satisfying the form of Case , that is, , we have the following theorem.
Theorem 16. For given positive scalars , , , , , ,, and , if there exist positive definite symmetric matrices , , , , and and matrices , , , , , , and with appropriate dimensions, thenhold, where the matrices are defined as follows: meanwhile, the average dwell-time satisfieswhereThen the switched linear neutral system is finite-time bounded with extended dissipative performance. Furthermore, the nonfragile controller can be chosen by
Proof. Replacing , , , and in (50) with , , , and and by Schur complement, we obtain where can be rewritten as where with by using Lemma 11, there exists a scalar , such that where Here we consider the norm-bounded uncertainties, and we set , where with By Lemma 11, there exists a scalar , such thatThen pre- and postmultiplying (78) by , we havewhere Based on above discussion, from , by Schur complement, we can conclude that . Similar to the proof of Theorem 15, we can obtain where is given in (77). The following proof is similar to that of Theorem 15; it is omitted here.
Furthermore, for the multiplicative gain variation model with the form in (5) of Case , we have the following theorem.
Theorem 17. For given positive scalars , , , , , , , , and , if there exist positive definite symmetric matrices , , , , and and matrices , , , , and , then where the matrices are defined as follows: the average dwell-time satisfiesThe controller gains can be given by . Then the switched linear neutral system is finite-time bounded with extended dissipative performance under the nonfragile controller .
Proof. Replacing , , , and in (50) with , , , and and by Schur complement, we obtainwhere (108) can be rewritten aswhere with considering (5), (110) can be rewritten as where based on (6) and Lemma 12, for some , () and , it can be verified that On the other hand, by Schur complement, is equal to . Then holds. It can be proven that Hence, we have Here we consider the norm-bounded uncertainties, and we set where with By Lemma 11, there exists a scalar , such that Then pre- and postmultiplying (102) by , we have where Based on above discussion, from , by Schur complement, we can conclude that Similar to the proof of Theorem 15, we can obtain where is given in (101). The following proof is similar to that of Theorem 15; it is omitted here.
Remark 18. The concept of extended dissipative could be employed to lots of other systems, for example, the T-S fuzzy systems [34–37], which shows the effectiveness of the powerful tool.
4. Numerical Example
In this section, we present an example to illustrate the effectiveness of the controller design method.
Example 1. Consider system (1) with two subsystems with parameters as follows:
Case 1. When satisfies additive form (4), we set
Case 2. When satisfies multiplicative form (5), we choose Furthermore, just as the discussion in Remark 6, we choose the values for the extended dissipative parameters in Table 1.
Then, solve the LMIs from (47) to (50) in Theorem 15, and we can get the results of optimized variables of four performances in Table 2.
Furthermore, solve the LMIs presented in Theorems 16 and 17, and we can obtain the controller gain for the additive form controller uncertainty and the multiplicative form controller uncertainty in Tables 3 and 4, respectively.
5. Conclusion
In this paper, we have investigated the problem of finite-time extended dissipative analysis and nonfragile control of switched neutral system with unknown time-varying disturbance. The average dwell-time approach is utilized for finite-time boundedness and extended dissipative performance analysis; controllers are designed to guarantee that the system is finite-time bounded and satisfies the extended dissipative performance. Based on extended dissipative performance, we can solve , , Passivity, and ()-dissipativity performance in a unified framework. All the results are given in terms of linear matrix inequalities (LMIs), and numerical examples are provided to show the effectiveness of the proposed method. In our future research, the nonfragile control and extended dissipative performance will be extended to more complex systems, such as Markovian jump delayed systems, sliding control systems, and T-S fuzzy systems, which deserve further study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by Natural Science Foundation of China (nos. 61573177, 61773191, and 61573008); the Natural Science Foundation of Shandong Province for Outstanding Young Talents in Provincial Universities under Grant R2016JL025; Special Fund Plan for Local Science and Technology Development Led by Central Authority.