Journal of Control Science and Engineering
Volume 2019 (2019), Article ID 5454179, 8 pages
https://doi.org/10.1155/2019/5454179

Research Article

Improved Absolute Stability for a Class of Nonlinear Switched Delay Systems via Mode-Dependent Average Dwell Time

Juan Liu1,2 and Qingdao Huang¹

¹Jilin University, Changchun 130012, China
²Shenyang University, Shenyang 110044, China

Correspondence should be addressed to Juan Liu; liujuan11@mails.jlu.edu.cn

Received 31 October 2018; Revised 24 December 2018; Accepted 21 January 2019; Published 11 February 2019

Academic Editor: Ai-Guo Wu

Copyright © 2019 Juan Liu and Qingdao Huang. 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.

In this paper, the absolute stability for a class of switched delay systems with time-varying uncertainties is analyzed. By constructing an appropriate Lyapunov-Krasovskii functional, a new and less conservative criterion is proposed based on the MDADT method. Besides, the idea of –segmentation is also utilized to make it more flexible to solve the LMIs. Finally, a numerical example is given to show the feasibility and advantages of the method proposed in this paper.

1. Introduction

Switched system is an important class of hybrid system which can be described by a series of continuous or discrete subsystems with a rule organizing the switching among them. As well known, many kinds of practical systems, such as power systems and chemical procedure control systems, may be properly modeled by the switched systems [1]. The main problem of switched systems is the stability analysis. In the last two decades, there have been increases in interests in the stability analysis for switched systems; see, for example, [2–5] and the references cited therein.

Apart from that, in many physical systems, such as chemical process and electric network, time-delay phenomenon is also unavoidable. Time-delays may lead to instability or even poor performance such as chaos [6]. Recently, the stability of systems with delays has aroused extensive research of scholars (see [7–10] and the references cited therein). For systems with time delay, the most important technique is to construct appropriate Lyapunov-Krasovskii functional.

Consequently, the stability of switched systems with time delay is worth considering. In fact, switched delay systems have a wide range of practical backgrounds in network control [11], chemical reactors [12], drilling system [13], and so on. To cope with the stability and design problem of switched delay systems, a good deal of methods have been proposed in terms of state-dependent switching and time-dependent switching. For instance, in [14], a delay-dependent stability criterion is derived under a state-driven law for switched delay systems. With regard to time-dependent switching, the dwell time (DT) method [15] and the average dwell time (ADT) method [16], as well as the mode-dependent average dwell time (MDADT) method [17], are put forward successively. In particular, for impulsive dynamical networks, in order to describe the frequency of occurrence of impulses, in 2010, by referring to the concept of ADT, Jianquan, Daniel W.C. Ho, and Jinde Cao [18] presented the concept of average impulsive interval. Recently, more and more work has been done on some special switched systems. For instance, for a class of impulsive stochastic differential systems, the stability properties are investigated under Markovian switching [19]. For switched cellular neural networks, by the MDADT approach, the exponential stability is considered in two cases where all subsystems are stable or unstable subsystems exist [20]. For a class of genetic regulatory networks (GRNs) with mixed delays, adaptive feedback control scheme is used to obtain the stability criteria [21].

In addition, since it was raised in 1940s, Lur’e systems have been deeply concerned. The main concern of Lur’e systems is absolute stability which covers systems without delays and those with delays [22–24]. To list a few, [25, 26] survey the absolute stability of Lur’e systems with constant delays and time-varying delays, respectively. From then on, more and more results have appeared to get less conservative stability criteria. Free-matrix-weighing method [27] and improved Lyapunov-Krasovskii functional method [28] are adopted in succession to reduce the conservativeness of previous results.

Integrated with the features of switched systems and Lur’e systems, switched Lur’e systems have also been paid much attention. A number of systems in control communities can be modeled by switched Lur’e systems, such as Hopfield neural network [29] and variable structure system [30]. However, as far as we know, up to now, absolute stability of switched Lur’e systems has not been completely considered.

Recently, [31] studies the absolute stability of switched delay Lur’e systems. By the method of ADT and Lyapunov functional, the efficient delay-dependent condition is obtained in the form of LMIs. However, the system considered in [31] is linear; meanwhile, the Lyapunov function chosen in [31] is too simple and restrictive, as only quadratic term and the integration term of state are included. Therefore, the result obtained in [31] is more conservative, which is one motivation of present work. Also, when a system with delay is considered, one chief objective is to get maximum allowed delay upper bounds, and the bigger, the better. Considering the complexity of switched systems, it is challenging to design a switching law for switched delay systems to get bigger maximum allowed delay upper bounds and better performance, which is the other driving force of this paper. Besides, sometimes, in previous work, such as [32], it is difficult to get the same positive matrix which satisfies the corresponding LMIs.

In this paper, the absolute exponential stability of switched Lur’e systems with delays is investigated. The main contribution of this paper is as follows. Employing the integration of state derivative, an appropriate Lyapunov-Krasovskii functional is constructed to get new stability criteria which are less conservative. The switching law is designed based on MDADT method to reduce the conservativeness. The criteria proposed in this paper are more flexible than the previous results [31], involving those criteria obtained in [25, 31]. On top of that, compared with those in [32], the method derived in this paper can also be used to get less conservative results for general switched delay systems.

The rest of this paper is organized as follows. The problem is stated in Section 2. Section 3 shows the main results and a simulation example is given in Section 4. In the end of the paper, conclusions are presented.

Notations. Throughout this paper, denotes the real number set; denotes the nonnegative integer set; is the Euclid space with dimension ; is the real matrices set with dimension . For the two given symmetric matrices and , we use to denote that is a positive definite (positive semidefinite) matrix. “” denotes the transpose of matrix ; I denotes the identity matrix with appropriate dimension. For any matrix and two symmetric matrices , , we use to denote the symmetric part of the symmetric matrix.

2. Problem Statement

Consider the following system

where , , are the state vector, input vector, and output vector, respectively; is system delay; is a continuous vector-valued function and , , , , are all constant matrices; : is the switching signal; is a nonlinear vector-valued function which is continuous about and globally Lipchitz at with . Besides also satisfies

where is a symmetric positive definite matrix with constant matrices . The nonlinear function which satisfies (2) is said to belong to . Initially, we will give some definitions and lemmas needed in this paper.

Definition 1 (see [22]). System described by (1) and (2) is called to be absolutely exponentially stable in if the zero solutions of system (1) are all globally exponentially stable under some switching signal .

Definition 2 (see [15]). For a switching signal and any , let be the switching numbers, that is, the subsystem is activated over the interval , and denote the total running time of the subsystem over . We say that has a mode-dependent average dwell time (MDADT) if there exist positive numbers (chatter bounds) and such that

Lemma 3 (see [25]). For any symmetric positive definite matrix , scalar , and function , if the integrations related are well defined, the following holds.

Lemma 4 (see [26]). For given matrices , , with appropriate dimensions,

holds for all satisfying if and only if there exists such that

3. Main Results

In the previous work [31], the Lyapunov function used is of the following form.

Inspired by the method in [33] and letting , we choose the following Lyapunov-Krasovskii functional:

where , , and , , are positive definite matrices to be determined.

As in [31], the case when the nonlinear function will be firstly considered and we have the following result.

Theorem 5. For given , , , system (1) is absolutely exponentially stable in sector with MDADT if there exist a real number and symmetric matrices , , , , such that the following LMIs hold

where

Proof. Differentiating function (8) about time along the solutions of system (1), we will get the following.

By use of Lemma 3, we can get

and from (1), it is true that

where

From (1) and considering , the following holds.

According to (12)-(15), for a scalar , it yields that

where

with

By Schur complement, it yields that is equivalent to inequality (9). Then it can be concluded from [17] that system (1) is absolutely exponentially stable.
When , the following corollary can be derived from Theorem 5.

Corollary 6. For given , , , system (1) is absolutely exponentially stable in sector with MDADT if there exist a real number and symmetric matrices , , , such that the following LMIs hold

where

Remark 7. In fact, Corollary 6 is the same as Proposition 3 in [25] when ; also it is Theorem 1 in [31] with . So Proposition 3 in [25] and Theorem 1 in [31] are included in Theorem 5 in this paper.

Remark 8. In fact, if we set , , the Lyapunov-Krasovskii functional (8) will reduce to corresponding Lyapunov function (7) ([31, 32]), so (7) can be seen as a special case of (8) which considers both the integration of state and the integration of state’s derivative. As is well known, when a system is considered, the character of the state’s derivative also plays an important role in the stability analysis. Besides, sometimes, it is difficult to get the same in [31]. Fortunately, in our result, is allowed to be different. Therefore, the LMIs obtained in this paper is more flexible and general than those obtained by ADT method in [31]. Meanwhile, the method can also be used for the stability analysis of general switched delay system to derive less conservative conditions than those in [32] (,).
Similarly, when , we have the following result.

Corollary 9. For given , , , system (1) is absolutely exponentially stable in sector with MDADT if there exist a real number and symmetric matrices , , , , such that the following LMIs hold

where

and , , , are defined in Theorem 5.

Proof. Using the transform in [34] and Theorem 5, it is easy to get this Corollary.
Except the absolute stability of system (1), we will also consider the more general system as follows:

with

where , , are constant matrices with appropriate dimensions and for any , satisfying

For system (23), Corollary 10 can be easily derived from Corollary 9.

Corollary 10. For given , , , system (23) is absolutely exponentially stable in sector with MDADT if there exist real numbers , and symmetric matrices , , , , such that the following LMIs hold

where

and , , are defined in Theorem 5 and , are defined in Corollary 9.

Proof. Replacing and in (21) with and , making use of Schur complement, Lemma 4, and Corollary 9, it is easy to get Corollary 10.

Remark 11. The results of this paper are based on LMI technology. In further research, when nonconvex matrix inequality conditions (nonlinear coupling) are encountered, the Chang-Yang decoupling method ([35]) can be used for nonlinear coupling.

4. Numerical Example

Consider uncertain system (23) with the following parameters:

for , , , , , , , , . Using the Matlab software, the LMI in Corollary 10 is solvable, and we can get the following.

The simulation is shown in Figures 1 and 2.

Figure 1: Switching signal.

Figure 2: The response curve of the states in example.

Table 1 lists the maximum allowed time-delay bounds obtained by Corollary 10 for various in comparison with those obtained by Corollary 2 in [31]. Clearly, the results in this paper are better than those in [31]. Besides, by setting , Corollary 10 reduces to Theorem 2 in [28].

Table 1: Comparison with the result in [31].

Corollary 2 [31]
2.2845
2.2352
2.0243

Corollary 10, n=2
3.0079
2.6738
2.1654

Corollary 10, n=3
3.1085
2.7602
2.4731

5. Conclusions

In this article, we considered the stability of switched Lur’e systems. In order to derive less conservative criteria, we introduce an appropriate Lyapunov-Krasovskii functional by the method of delay length segmentation ([33]). Besides, the MDADT method is also used to gain more applicable results. In further research, nonlinear switched Lur’e systems with time-varying delays will be considered; in particular, we will consider if the concept “average impulsive interval” [18] can be used in our further study.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (11171131) and Jilin Provincial Natural Science Foundation of China (201115043).

References

D. Liberzon and A. S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Systems Magazine, vol. 19, no. 5, pp. 59–70, 1999.

L. Liu, Q. Zhou, H. Liang, and L. Wang, “Stability and stabilization of nonlinear switched systems under average dwell time,” Applied Mathematics and Computation, vol. 298, pp. 77–94, 2017.

J. Zheng, J.-G. Dong, and L. Xie, “Stability of discrete-time positive switched linear systems with stable and marginally stable subsystems,” Automatica, vol. 91, pp. 294–300, 2018.

X. Liu, Q. Zhao, and S. Zhong, “Stability analysis of a class of switched nonlinear systems with delays: a trajectory-based comparison method,” Automatica, vol. 91, pp. 36–42, 2018.

M. Parada Contzen, “Stability of switched linear systems with possible Zeno behavior: a polytopic approach,” European Journal of Control, vol. 40, pp. 40–47, 2018.

A. Ucar, “A prototype model for chaos studies,” International Journal of Engineering Science, vol. 40, no. 3, pp. 251–258, 2002.

I. V. Alexandrova and A. P. Zhabko, “A new LKF approach to stability analysis of linear systems with uncertain delays,” Automatica, vol. 91, pp. 173–178, 2018.

Z.-Y. Li, J. Lam, and Y. Wang, “Stability analysis of linear stochastic neutral-type time-delay systems with two delays,” Automatica, vol. 91, pp. 179–189, 2018.

J. Chen, C. Lin, B. Chen, and Q.-G. Wang, “Improved stability criterion and output feedback control for discrete time-delay systems,” Applied Mathematical Modelling, vol. 52, pp. 82–93, 2017.

B. Zhou and W. Luo, “Improved Razumikhin and Krasovskii stability criteria for time-varying stochastic time-delay systems,” Automatica, vol. 89, pp. 382–391, 2018.

D. Xie, X. Chen, L. Lv, and N. Xu, “Asymptotical stabilisability of networked control systems: time-delay switched system approach,” IET Control Theory & Applications, vol. 2, no. 9, pp. 743–751, 2008.

Z. Duan, Z. Xiang, and H. R. Karimi, “Delay-dependent exponential stabilization of positive 2D switched state-delayed systems in the Roesser model,” Information Sciences, vol. 272, pp. 173–184, 2014.

B. Saldivar, S. Mondie, J. J. Loiseau, and V. Rasvan, “Exponential stability analysis of the drilling system described by a switched neutral type delay qeuation with nonlinear perturbations,” in proceedings of the 50th IEEE conference on decision and control and european control conference (CDC-ECC), pp. 4164–4169, orlando, FL, USA, 2011.

J.-S. Chiou, C.-J. Wang, and C.-M. Cheng, “On delay-dependent stabilization analysis for the switched time-delay systems with the state-driven switching strategy,” Journal of The Franklin Institute, vol. 348, no. 2, pp. 261–276, 2011.

P. Colaneri, “Dwell time analysis of deterministic and stochastic switched systems,” European Journal of Control, vol. 15, no. 3-4, pp. 228–248, 2009.

A. S. Morse, “Supervisory control of families of linear set-point controllers - Part I. Exact matching,” IEEE Transactions on Automatic Control, vol. 41, no. 10, pp. 1413–1431, 1996.

X. Zhao, L. Zhang, P. Shi, and M. Liu, “Stability and stabilization of switched linear systems with mode-dependent average dwell time,” IEEE Transactions on Automatic Control, vol. 57, no. 7, pp. 1809–1815, 2012.

J. Lu, D. W. C. Ho, and J. Cao, “A unified synchronization criterion for impulsive dynamical networks,” Automatica, vol. 46, no. 7, pp. 1215–1221, 2010.

F. Yao, J. Cao, P. Cheng, and L. Qiu, “Generalized average dwell time approach to stability and input-to-state stability of hybrid impulsive stochastic differential systems,” Nonlinear Analysis: Hybrid Systems, vol. 22, pp. 147–160, 2016.

C. Huang, J. Cao, and J. Cao, “Stability analysis of switched cellular neural networks: A mode-dependent average dwell time approach,” Neural Networks, vol. 82, pp. 84–99, 2016.

J. Hu, J. Liang, and J. Cao, “Stabilization of genetic regulatory networks with mixed time-delays: an adaptive control approach,” IMA Journal of Mathematical Control and Information, vol. 32, no. 2, pp. 343–358, 2015.

L. Guo, H. Pan, and X. Nian, “Adaptive pinning control of cluster synchronization in complex networks with Lurie-type nonlinear dynamics,” Neurocomputing, vol. 182, pp. 294–303, 2016.

B. Liu and X.-C. Jia, “New absolute stability criteria for uncertain Lure systems with time-varying delays,” Journal of The Franklin Institute, vol. 355, no. 9, pp. 4015–4031, 2018.

F. Long, C.-K. Zhang, Y. He, L. Jiang, Q.-G. Wang, and M. Wu, “Stability analysis of Lur'e systems with additive delay components via a relaxed matrix inequality,” Applied Mathematics and Computation, vol. 328, pp. 224–242, 2018.

Q.-L. Han, “Absolute stability of time-delay systems with sector-bounded nonlinearity,” Automatica, vol. 41, no. 12, pp. 2171–2176, 2005.

Q.-L. Han and D. Yue, “Absolute stability of Lur'e systems with time-varying delay,” IET Control Theory & Applications, vol. 1, no. 3, pp. 854–859, 2007.

M. Wu, Z. Y. Feng, Y. He, and J. H. She, “Improved delay-dependent absolute stability and robust stability for a class of nonlinear systems with a time-varying delay,” International Journal of Robust and Nonlinear Control, vol. 20, no. 6, pp. 694–702, 2010.

M. Wu, Z. Y. Feng, and Y. He, “Improved delay-dependent absolute stability of Lure systems with time-delay,” International Journal of Control, Automation, and Systems, vol. 7, pp. 1009–1014, 2009.

E. Kaszkurewiez and A. Bhaya, “On a class of globally stable neural circuits,” IEEE Transactions on Circuits System, vol. 41, no. 2, pp. 171–174, 1994.

E. Kaszkurewicz and A. Bhaya, “Robust stability and diagonal Lyapunov functions,” SIAM Journal on Matrix Analysis and Applications, pp. 508–250, 1993.

J. Liu and Q. Huang, “Absolute exponential stability for a class of switched delay systems,” in Proceedings of the 29th Chinese Control and Decision Conference, CCDC 2017, pp. 5585–5591, May 2017.

X.-M. Sun and J. Zhao, “Robust exponential stability of linear switched delay systems: an average dwell time method,” Dynamics of continunos, Discrete and Impulsive Systems-Series B, 2007.

F. Gouaisbaut and D. Peaucelle, “Delay-dependent stability analysis of linear time delay systems,” in Proceedings of the sixth ifac workshop on time delay systems, L’Aquila, Italy, 2006.

H. K. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, USA, 1996.

X. Chang and G. Yang, “New results on output feedback control for linear discrete-time systems,” IEEE Transactions on Automatic Control, vol. 59, no. 5, pp. 1355–1359, 2014.