Complexity
Volume 2017 (2017), Article ID 6581308, 22 pages
https://doi.org/10.1155/2017/6581308

Research Article

Nonfragile Finite-Time Extended Dissipative Control for a Class of Uncertain Switched Neutral Systems

Hui Gao,¹ Jianwei Xia,¹ Guangming Zhuang,¹ Zhen Wang,² and Qun Sun³

¹School of Mathematics Science, Liaocheng University, Liaocheng 252000, China
²College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China
³School of Mechanical and Automotive Engineering, Liaocheng University, Liaocheng 252000, China

Correspondence should be addressed to Jianwei Xia; njustxjw@126.com

Received 5 April 2017; Accepted 16 September 2017; Published 14 November 2017

Academic Editor: Sigurdur F. Hafstein

Copyright © 2017 Hui Gao et al. 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.

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 satisfy

Furthermore, , and for each , , , , and are uncertain real-valued matrices with appropriate dimensions. We assume that the uncertainties are norm-bounded and of the form

where , , , , , , , 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 be

where and are known real constant matrices with appropriate dimensions and the time-varying matrix satisfies .

Case 2. has a multiplicative uncertainty

where 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 . If

holds 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, then

hold, where

meanwhile, the average dwell-time satisfies

We define

where satisfies

Then, switched system (17) is finite-time bounded with respect to

Proof. Choose the piecewise Lyapunov-Krasovskii functional candidate as

where

in 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 obtain

Thus we have

where

Considering (19) and (20), we can obtain that

Integrating (36), it can be obtained from (26) and (36) that, ,

From Definition 8, we can deduce that , and then we can obtain

On the other hand

From (38)-(39), we can obtain

When , it is obvious that by (22).
When , by virtue of (22), we have that

From (23), we can obtain

Substituting (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, then

hold, where

meanwhile, the average dwell-time satisfies

where 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, then

hold, where

meanwhile, the average dwell-time satisfies

where

Then, 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, then

hold, where

the matrices are defined as follows:

meanwhile, the average dwell-time satisfies

where

Then 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 that

Then pre- and postmultiplying (78) 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 (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 satisfies

The 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 obtain

where

(108) can be rewritten as

where

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.

Table 1: Matrices for each case.

Analysis

performance

performance

Passivity

Dissipativity

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.

Table 2: Optimized variable for each case.

performance
performance
Passivity
Dissipativity

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.

Table 3: Controller gain of the additive form controller uncertainty for each case.

Subsystem

performance

performance

Passivity

Dissipativity

Table 4: Controller gain of the multiplicative form controller uncertainty for each case.

Subsystem

performance

performance

Passivity

Dissipativity

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.

References

S. Wang, T. Shi, L. Zhang, A. Jasra, and M. Zeng, “Extended finite-time control for uncertain switched linear neutral systems with time-varying delays,” Neurocomputing, vol. 152, pp. 377–387, 2015.

X. Zhao, Y. Yin, and X. Zheng, “State-dependent switching control of switched positive fractional-order systems,” ISA Transactions, pp. 103–108, 2016.

X. Zhao, P. Shi, and L. Zhang, “Asynchronously switched control of a class of slowly switched linear systems,” Systems & Control Letters, vol. 61, no. 12, pp. 1151–1156, 2012.

L. Zhang, S. Zhuang, and R. D. Braatz, “Switched model predictive control of switched linear systems: feasibility, stability and robustness,” Automatica, vol. 67, pp. 8–21, 2016.

X. Zhao, X. Liu, S. Yin, and H. Li, “Improved results on stability of continuous-time switched positive linear systems,” Automatica, vol. 50, no. 2, pp. 614–621, 2014.

J. Xia, J. H. Park, and H. Zeng, “Improved delay-dependent robust stability analysis for neutral-type uncertain neural networks with Markovian jumping parameters and time-varying delays,” Neurocomputing, vol. 149, pp. 1198–1205, 2015.

J. Xia, J. H. Park, H. Zeng, and H. Shen, “Delay-difference-dependent robust exponential stability for uncertain stochastic neural networks with multiple delays,” Neurocomputing, vol. 140, pp. 210–218, 2014.

Z. Wang, X. Huang, and J. Zhou, “A numerical method for delayed fractional-order differential equations: based on G-L definition,” Applied Mathematics & Information Sciences, vol. 7, no. 2L, pp. 525–529, 2013.

Z. Wang, X. Huang, and G. Shi, “Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1531–1539, 2011.

Y. Zhang, X. Liu, H. Zhu, and S. Zhong, “Stability analysis and control synthesis for a class of switched neutral systems,” Applied Mathematics and Computation, vol. 190, no. 2, pp. 1258–1266, 2007.

D. Zhang and L. Yu, “Exponential stability analysis for neutral switched systems with interval time-varying mixed delays and nonlinear perturbations,” Nonlinear Analysis: Hybrid Systems, vol. 6, no. 2, pp. 775–786, 2012.

C.-H. Lien, K.-W. Yu, Y.-J. Chung, Y.-F. Lin, L.-Y. Chung, and J.-D. Chen, “Exponential stability analysis for uncertain switched neutral systems with interval-time-varying state delay,” Nonlinear Analysis: Hybrid Systems, vol. 3, no. 3, pp. 334–342, 2009.

Y. Zhang, H. Zhu, X. Liu, and S. Zhong, “Reliable control for a class of switched neutral systems,” Complex System and Applications: Modeling, Control and Simulations, vol. 14, supplement 2, pp. 1724–1729, 2007.

Y.-E. Wang, J. Zhao, and B. Jiang, “Stabilization of a class of switched linear neutral systems under asynchronous switching,” Institute of Electrical and Electronics Engineers Transactions on Automatic Control, vol. 58, no. 8, pp. 2114–2119, 2013.

Z. Zhang, Z. Zhang, and H. Zhang, “Finite-time stability analysis and stabilization for uncertain continuous-time system with time-varying delay,” Journal of The Franklin Institute, vol. 352, no. 3, pp. 1296–1317, 2015.

H. Liu and X. Zhao, “Finite-time control of switched systems with mode-dependent average dwell time,” Journal of The Franklin Institute, vol. 351, no. 3, pp. 1301–1315, 2014.

X. Lin, H. Du, and S. Li, “Finite-time boundedness and -gain analysis for switched delay systems with norm-bounded disturbance,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5982–5993, 2011.

S. He and F. Liu, “Stochastic finite-time boundedness of Markovian jumping neural network with uncertain transition probabilities,” Applied Mathematical Modelling, vol. 35, no. 6, pp. 2631–2638, 2011.

H. Liu, Y. Shen, and X. Zhao, “Finite-time stabilization and boundedness of switched linear system under state-dependent switching,” Journal of The Franklin Institute, vol. 350, no. 3, pp. 541–555, 2013.

Z. Xiang, Y.-N. Sun, and M. S. Mahmoud, “Robust finite-time control for a class of uncertain switched neutral systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1766–1778, 2012.

K. Mathiyalagan, J. H. Park, H. Y. Jung, and R. Sakthivel, “Non-fragile observer-based control for discrete-time systems using passivity theory,” Circuits, Systems and Signal Processing, vol. 34, no. 8, pp. 2499–2516, 2015.

Y.-Q. Wu, H. Su, R. Lu, Z.-G. Wu, and Z. Shu, “Passivity-based non-fragile control for Markovian jump systems with aperiodic sampling,” Systems & Control Letters, vol. 84, pp. 35–43, 2015.

D. Yue and J. Lam, “Non-fragile guaranteed cost control for uncertain descriptor systems with time-varying state and input delays,” Optimal Control Applications and Methods, vol. 26, no. 2, pp. 85–105, 2005.

G.-H. Yang and J. L. Wang, “Non-fragile control for linear systems with multiplicative controller gain variations,” Automatica, vol. 37, no. 5, pp. 727–737, 2001.

B. Zhang, W. X. Zheng, and S. Xu, “Filtering of Markovian jump delay systems based on a new performance index,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 5, pp. 1250–1263, 2013.

J. Xiao, Y. Li, S. Zhong, and F. Xu, “Extended dissipative state estimation for memristive neural networks with time-varying delay,” ISA Transactions, vol. 64, pp. 113–128, 2016.

H. Yang, L. Shu, S. Zhong, and X. Wang, “Extended dissipative exponential synchronization of complex dynamical systems with coupling delay and sampled-data control,” Journal of The Franklin Institute, vol. 353, no. 8, pp. 1829–1847, 2016.

H. Wei, R. Li, C. Chen, and Z. Tu, “Extended dissipative analysis for memristive neural networks with two additive time-varying delay components,” Neurocomputing, vol. 216, pp. 429–438, 2016.

H. Shen, Y. Zhu, L. Zhang, and J. H. Park, “Extended dissipative state estimation for Markov jump neural networks with unreliable links,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 2, pp. 346–358, 2017.

T. H. Lee, M.-J. Park, J. H. Park, O.-M. Kwon, and S.-M. Lee, “Extended dissipative analysis for neural networks with time-varying delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 25, no. 10, pp. 1936–1941, 2014.

S. Lakshmanan, J. H. Park, H. Y. Jung, O. M. Kwon, and R. Rakkiyappan, “A delay partitioning approach to delay-dependent stability analysis for neutral type neural networks with discrete and distributed delays,” Neurocomputing, vol. 111, pp. 81–89, 2013.

L. Xie, “Output feedback control of systems with parameter uncertainty,” International Journal of Control, vol. 63, no. 4, pp. 741–750, 1996.

Y. Y. Wang, L. Xie, and C. E. de Souza, “Robust control of a class of uncertain nonlinear systems,” Systems & Control Letters, vol. 19, no. 2, pp. 139–149, 1992.

J. Cheng, J. H. Park, Y. Liu, Z. Liu, and L. Tang, “Finite-time fuzzy control of nonlinear Markovian jump delayed systems with partly uncertain transition descriptions,” Fuzzy Sets and Systems, vol. 314, pp. 99–115, 2017.

J. Cheng, J. H. Park, L. Zhang, and Y. Zhu, “An Asynchronous Operation Approach to Event-triggered Control for Fuzzy Markovian Jump Systems with General Switching Policies,” IEEE Transactions on Fuzzy Systems, pp. 1–1.

B. Wang, J. Cheng, A. Al-Barakati, and H. M. Fardoun, “A mismatched membership function approach to sampled-data stabilization for T-S fuzzy systems with time-varying delayed signals,” Signal Processing, vol. 140, pp. 161–170, 2017.

B. Wang, J. Cheng, and J. Zhan, “A sojourn probability approach to fuzzy-model-based reliable control for switched systems with mode-dependent time-varying delays,” Nonlinear Analysis: Hybrid Systems, vol. 26, pp. 239–253, 2017.