Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 852303, 10 pages
http://dx.doi.org/10.1155/2015/852303

Research Article

Robust Control for Switched Nonlinear System with Multiple Delays

Juan Liu1,2 and Qingdao Huang¹

¹College of Mathematics, Jilin University, Changchun 130012, China
²Shenyang University, Shenyang 110044, China

Received 20 January 2015; Revised 13 April 2015; Accepted 16 April 2015

Academic Editor: Carlo Cosentino

Copyright © 2015 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.

This paper focuses on the problem of robust control for a class of switched nonlinear systems with multiple time delays which are allowed to appear not only in state but also in input. We will design a controller and a switching law under which the system is “robust stable” and has performance -gain . Besides, multiple Lyapunov-Razumikhin functions which are relatively less conservative are introduced to investigate the stability of the candidate switched system. In detail, firstly, for uncertain nonlinear switched system with time-varying delays, sufficient condition is given by inequalities which are equivalent to LMIs, so it is easy to be solved; then, as a special case, nonlinear switched system with constant time delay is also considered. Finally, by providing numerical examples, the feasibility of the proposed approach is demonstrated.

1. Introduction

Switched systems have constituted a very active field of current scientific research; many real world processes and systems can be modeled as switched systems, such as chemical processes and computer controlled systems. Besides, switched systems are extensively applied in many domains, including mechanical systems, automation, aircraft and air traffic control, and many other fields. And stability analysis and control, as the most important topics for the study of switched systems, have been studied widely [1–11]. Up to now, some methods are given to solve the stability analysis for switched system. For example, by constructing a common Lyapunov function, stability under arbitrary switching can be obtained. However, most switched systems still may be stable under certain switching laws, though they do not own a common Lyapunov function. The key idea to select such switching laws is the dwell time [12, 13]. Lately, control theory, which is another effective tool to solve robust stabilization problems, has been well built [14].

On the other hand, time-delay phenomenon is very common in many kinds of engineering systems, for instance, long-distance transportation systems, hydraulic pressure systems, network control systems, and so on. It is regularly a source of instability and often causes undesirable performance and even makes the system out of control. So time-delay systems have also drawn more and more attention [15–23]. Based on the above two aspects, the problem of stability analysis and controller synthesis for switched system with time delay (switched delay system) has aroused growing interest. And switched delay system, as a new brand, can be found in many applications and there are quite a lot of related results [24–34]. However, in the above papers, systems with only one delay are considered. And system with multiple delays, which is widely used in practice, has gained more and more attention [35–38]. For switched systems with multiple time delays, the problem of robust stability is considered in [39]. In [39], delays are constant and appear only in state. In fact, in engineering control design for the actual system, time-varying delays are universal, and due to various reasons delays can also appear in control input. On this occasion, the system will become more complicated and then the response of state may also be affected. In [40], the problem of robust control for nonswitched system with input delays is considered. Motivated by this, it is meaningful to investigate the following problem: if the system in [40] is switched, how is its stability and also how do we design a controller and a switching law to make the system stable? As yet, as far as we know, it has not been fully investigated despite its potential in practical applications, and the purpose of this paper is therefore to shorten such a gap by providing a rather general framework.

The contribution of this paper lies in the following aspects. First, we address the problem of control for a more general class of switched systems with multiple time-varying delays, in which delays not only appear in the state but also appear in the input. Second, by choosing a new set of multiple Lyapunov-Razumikhin functionals (which is relatively less conservative than the CLF approach), we derive sufficient conditions of the existence of state-feedback controller for a class of switching signals with average dwell time scheme; besides, the specific expression of controller is also given in the paper. Third, in this paper, the inequality we derived is easy to compute for it can be rewritten as LMIs.

The remainder of this paper is organized as follows. Firstly, problem formulation and preliminaries are stated in Section 2. In the following section, the main results are addressed. And a numerical example is given in Section 4. Section 5 draws the conclusion.

Notations. Throughout this paper, the notation means that is a positive definite (positive-semidefinite, negative-semidefinite, and negative definite) matrix . and denote the maximum and minimum eigenvalues of , respectively; the superscript “” stands for the transpose; denotes the Euclidean norm; represents the space of square integrable vector functions over ; is an identity matrix with appropriate dimension, and if not stated, matrices are assumed to have compatible dimensions. denotes a symmetric matrix.

2. Problem Formulation and Preliminaries

Consider a class of uncertain switched delay systems of the form

where , , and are the state vector, input, and output, respectively. which is assumed to belong to is external disturbance input; is the switching signal; , , , , and are given constant matrices with appropriate dimensions; () are the time-varying delays which satisfy the following assumption.

Assumption 1. Consider , , , for all .

Remark 2. With regard to the switching signal , there exists a switching sequence , which means that when , the th subsystem is activated.

Remark 3. In general, the nonlinear uncertainties , , and which refer to the nonlinear internal parameter uncertainties with respect to the current state and delayed states are assumed to satisfy the following inequalities.

Assumption 4. Consider

where , , and are known constant matrices.

Remark 5. Compared with the switched system in [39] (where the control input is not contained), in this paper, delays are time varying and control input is also contained with delays, so system (1) is more general and more practical for we can choose a controller to make an unstable system stable.

In this paper, we are interested in designing a state-feedback controller which is described by and a switching law such that system (1) is “robust stable.” Firstly, the relevant definitions are given below.

Definition 6. System (1) is said to be exponentially stable under controller and switching law , if, with , the solution of system (1) satisfies

where and .

Definition 7 (see [11]). For any , let represent the switching number of on . If holds for , , then is called average dwell time.

Remark 8. The concept of average dwell time, which was an effective tool for the stability analysis of switched systems, was put forward for continuous switched systems firstly by Hespanha and Morse (see [11]). As commonly used in the literature, we choose in this paper.

Definition 9 (see [41]). For , , consider system (1) and design a switching law and controller , if it holds that

(1)system (1) is exponentially stable when ;(2)under zero initial conditions system (1) satisfies

Then system (1) is said to have exponential performance (weighted -gain).

Now the following lemma which will be used to draw the main results in this paper is presented.

Lemma 10 (see [40]). For any function vector , symmetric matrix , and constant matrix , it holds that

3. Main Results

In this paper, we focus on the switched delay system with time-varying delays and constant delays, respectively. And we will tackle the robust control problem for switched delay system (1). When , the “th” subsystem of system (1) is activated. At this moment it can be seen as a “nonswitched” system corresponding to system (1). Based on this, we will first consider the nonswitched system:

Lemma 11. Consider the nonswitched system (6). For a given positive definite matrix and constants , , , and , , define

with

the linear feedback controller is designed by , , such that the inequality

has a positive definite solution , where

Then, along the trajectory of the system, when ,

For general ,

where .

Proof. The following proof is motivated by the method in [40].
For the function defined by (7),

Using Assumptions 1 and 4 and Lemma 10, we get the following inequalities:

Substituting inequalities (14) into (13), it holds that

Considering (8), inequality (15) can be expressed as

Let in (16); we can get

Hence

When , it yields that

Then, integrating inequalities (18) and (19), respectively, we can get inequalities (11) and (12) in Lemma 11. This completes the proof.

When the delays in the nonswitched system (6) are constant, we assume that the following conditions hold:

where , , and are positive real constants.

Lemma 12. For given constants , , , , , , and , such that

define

where is a positive symmetric definite matrix of the following inequality:

Then, along the trajectory of system (6), when ,

and, for general ,

where .

Proof. The proof is similar to that of Lemma 11.
For and ,

Using a similar step of the proof in Lemma 11, we can obtain the following equality:

If we assume

inequality (27) becomes

Considering that , it holds that

Hence from inequalities (29) and (30), we have

Then, when ,

Similar to Lemma 11, we can get (24) and (25).

Remark 13. Assumption (28), that is, , , , in the above proof is commonly used in papers (see [42]).

Now, we will design a controller and a switching law , under which switched system (1) has exponential performance . Firstly, for system (1) with time-varying delays, the following theorem is given.

Theorem 14. For a given matrix and constants , , and , , , if there exist matrices , , and a linear state-feedback controller , , such that

has a positive definite solution , where

then system (1) is exponentially stable and has weighted -gain for any switching signal with average dwell time satisfying , where satisfies , , , , , , where

Proof. Define a set of Lyapunov-Razumikhin function candidates as follows:

Then when and , it follows from Lemma 11 that

and from (36) the following inequality holds at switching instant :

Letting , it follows from inequalities (37) and (38) that

Let

Then

Combining inequalities (39) and (41), we obtain

which leads to the exponential stability of the switched system (1) with .
Now we will show the weighted -gain of system (1). Firstly for any , using Lemma 10, we have

Then considering both inequalities (38) and (43), it holds that

This gives

Multiplying both sides in (45) by , it yields

Observing that and , we have ; consequently

Integrating both sides of inequality (47) from to , we obtain

This ends the proof.

Remark 15. Inequality (33) in Theorem 14 is equivalent to LMIs as follows:

where . So it can be calculated easily.

When the delays in system (1) are constant, we can get the following corollary.

Corollary 16. For given constants , , , , , and , , if there exists a linear state feedback controller such that

where is a positive symmetric definite matrix of the following inequality:

then switched system (1) with constant delays is exponentially stable and has weighted -gain for any switching signal with average dwell time satisfying , where satisfies , .

Proof. Define the Lyapunov-Razumikhin function candidate in Theorem 14 as follows:

the proof is similar to that of Theorem 14 by Lemma 12.

Remark 17. Inequality (51) is easy to be solved for it can be rewritten as a LMI:

where .

Remark 18. In [40], conditions are given to guarantee that the nonswitched system with input delays is asymptotically stable. Compared with [40], we get the conditions when the switched system is exponentially stable.

4. Examples

To illustrate the main results, we consider the following examples.

Example 1. Consider the time-varying delay switched system

where and , , , ; thus , , , , and, , , , . So we have , , , , , , , . And , , , , , , , , , , , , , , , , , , , , , with , , . Then . Let , and , ; solving (33) using Matlab software, we can get and ; besides , , , , . And we can get satisfying the condition of Theorem 14.
Therefore according to Theorem 14, system (54) is exponentially stable and has -gain for any switching signal with average dwell time satisfying . The simulation result is depicted in Figure 1.

Figure 1: The state response of the closed-loop system (54) in Example 1.

Example 2. Consider the constant time-delay switched system with , , , , , , , , , , , , , , , , , , , , with , , . Then let , , and , and let , , , , ; solving (51) by the Matlab software, we can get and and , .
It is obvious that , , and we can get satisfying , . Therefore according to Corollary 16, this system is exponentially stable and has -gain for any switching signal with average dwell time satisfying . The simulation result is depicted in Figure 2.

Figure 2: The state response of the closed-loop system (54) in Example 2.

Remark 19. In the figures, horizontal axis stands for time and vertical axis stands for the states and . The red and blue lines denote the state response of and , respectively. We can see from the figures that the two systems are both exponentially stable under given controllers and switching law and this shows that the method proposed in this paper is feasible.

Remark 20. It is easy to verify that both subsystems are unstable. And using the method given in [40], controllers are only given to make the systems asymptotically stable. However, compared with [40], in this paper, we can see that, under given controllers, the switched systems in the examples are exponentially stable from the simulation results.

5. Summary

In this paper, we considered the problem of robust control for a class of uncertain nonlinear switched systems with state and input delays. The controller and switching law are designed and sufficient conditions have been presented for uncertain nonlinear switched delay systems under which the system is robust stable and has performance . Firstly, we considered the system with time-varying delays; then, as a special case, system with constant delays was also considered. The feasibility of the developed results has been proved using numerical examples.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (11171131) and Jilin Provincial Natural Science Foundation of China (201115043). The authors are very much thankful to the Associate Editor and anonymous reviewers for their careful reading, constructive comments, and fruitful suggestions to improve the quality of this paper.

References

B. A. Francis, A Course in H∞ Control Theory, vol. 88 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 1987.

K. S. Narendra and J. Balakrishnan, “A common Lyapunov function for stable LTI systems with commuting A-matrices,” IEEE Transactions on Automatic Control, vol. 39, no. 12, pp. 2469–2471, 1994.

D. Liberzon, J. P. Hespanha, and A. S. Morse, “Stability of switched systems: a Lie-algebraic condition,” Systems & Control Letters, vol. 37, no. 3, pp. 117–122, 1999.

D. Cheng, L. Guo, and J. Huang, “On quadratic Lyapunov functions,” IEEE Transactions on Automatic Control, vol. 48, no. 5, pp. 885–890, 2003.

D. Angeli and D. Liberzon, “A note on uniform global asymptotically stability of nonlinear switched systems in triangular form,” in Proceedings of the International Symposium on Mathematical Theory of Networks and Systems (MTNS '00), Perpignan, France, 2000.

J. P. Hespanha and A. S. Morse, “Switching between stabilizing controllers,” Automatica, vol. 38, no. 11, pp. 1905–1917, 2002.

P. Peleties and R. DeCarlo, “Asymptotic stability of m-switched systems using Lyapunov-like functions,” in Proceedings of the 1991 American Control Conference, pp. 1679–1684, San Diego, Calif, USA, June 1991.

M. S. Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,” IEEE Transactions on Automatic Control, vol. 43, no. 4, pp. 475–482, 1998.

J. P. Hespanha and A. S. Morse, “Stability of switched systems with average dwell-time,” in Proceedings of the 38th IEEE Conference on Decision and Control (CDC '99), pp. 2655–2660, Phoenix, Ariz, USA, December 1999.

B. Niu and J. Zhao, “Robust stabilization and tracking control for a class of switched nonlinear systems,” Asian Journal of Control, vol. 15, no. 5, pp. 1496–1502, 2013.

D. Liberzon, Switching in Systems and Control, Birkhäuser, Boston, Mass, USA, 2003.

J. P. Hespanha and A. S. Morse, “Stability of switched systems with average dwell-time,” in Proceedings of the 38th IEEE Conference on Decision and Control, pp. 2655–2660, IEEE, Phoenix, Ariz, USA, December 1999.

X.-M. Sun, “State feedback control for discrete delay systems with controller failures based on average dwell-time method,” IET Control Theory and Applications, vol. 2, no. 2, pp. 126–132, 2008.

A. Isidori, Nonlinear Control Systems, Springer, Berlin, Germany, 3rd edition, 1995.

H. Ghadiri, M. R. Jahed-Motlagh, and M. B. Yazdi, “Robust output observer-based guaranteed cost control of a class of uncertain switched neutral systems with interval time-varying mixed delays,” International Journal of Control, Automation, and Systems, vol. 12, no. 6, pp. 1167–1179, 2014.

E. Fridman and U. Shaked, “An improved stabilization method for linear time-delay systems,” IEEE Transactions on Automatic Control, vol. 47, no. 11, pp. 1931–1937, 2002.

H. J. Gao and C. H. Wang, “Delay-dependent robust and filtering for a class of uncertain nonlinear time-delay systems,” IEEE Transactions on Automatic Control, vol. 48, no. 9, pp. 1661–1666, 2003.

Y. Zhang, F. Yang, and Q.-L. Han, “H ∞ control of LPV systems with randomly multi-step sensor delays,” International Journal of Control, Automation and Systems, vol. 12, no. 6, pp. 1207–1215, 2014.

Q.-L. Han and K. Gu, “On robust stability of time-delay systems with norm-bounded uncertainty,” IEEE Transactions on Automatic Control, vol. 46, no. 9, pp. 1426–1431, 2001.

Y. Lai and G. Lu, “Over networks filtering for discrete singular jump systems with interval time-varying delay,” International Journal of Control, Automation and Systems, vol. 12, no. 6, pp. 1187–1196, 2014.

T. Guo and G. Liu, “Adaptive fuzzy control for unknown nonlinear time-delay systems with virtual control functions,” International Journal of Control, Automation and Systems, vol. 9, no. 6, pp. 1227–1234, 2011.

W. Michiels, V. Van Assche, and S.-I. Niculescu, “Stabilization of time-delay systems with a controlled time-varying delay and applications,” IEEE Transactions on Automatic Control, vol. 50, no. 4, pp. 493–504, 2005.

Z. Wang, B. Huang, and H. Unbehauen, “Robust reliable control for a class of uncertain nonlinear state-delayed systems,” Automatica, vol. 35, no. 5, pp. 955–963, 1999.

R. H. Wang, J. C. Xing, C. Zhou, and P. Wang, “Robust finite-time fault-tolerant control of switched systems with asynchronous switching,” Control and Intelligent Systems, vol. 42, no. 3, pp. 254–267, 2014.

G. Zhai, Y. Sun, X. Chen, and A. N. Michel, “Stability and L ₂-gain analysis for switched symmetric systems with time delay,” in Proceedings of the American Control Conference, pp. 2682–2687, Denver, Colo, USA, June 2003.

X.-M. Sun, J. Zhao, and D. J. Hill, “Stability and -gain analysis for switched delay systems: a delay-dependent method,” Automatica, vol. 42, no. 10, pp. 1769–1774, 2006.

E. Tian, W. K. Wong, and D. Yue, “Robust control for switched systems with input delays: a sojourn-probability-dependent method,” Information Sciences, vol. 283, pp. 22–35, 2014.

C.-H. Lien, K.-W. Yu, L.-C. Wu, L.-Y. Chung, and J.-D. Chen, “Robust switching control and switching signal design for uncertain discrete switched systems with interval time-varying delay,” Journal of the Franklin Institute, vol. 351, no. 1, pp. 565–578, 2014.

Z. Li, H. Gao, and H. R. Karimi, “Stability analysis and controller synthesis of discrete-time switched systems with time delay,” Systems & Control Letters, vol. 66, pp. 85–93, 2014.

M. Kermania and A. Saklya, “Stability analysis for a class of switched nonlinear time-delay systems,” Systems Science & Control Engineering, vol. 2, no. 1, pp. 80–89, 2014.

L. Bao, S. Fei, and L. Yu, “Exponential stability of linear distributed parameter switched systems with time-delay,” Journal of Systems Science & Complexity, vol. 27, no. 2, pp. 263–275, 2014.

G.-X. Zhong and G.-H. Yang, “Robust control and fault detection for continuous-time switched systems subject to a dwell time constraint,” International Journal of Robust and Nonlinear Control, 2015.

X. Xie, H. Xu, and R. Zhang, “Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control,” Abstract and Applied Analysis, vol. 2014, Article ID 126836, 8 pages, 2014.

Y. Chen and W. X. Zheng, “Stability analysis and control for switched stochastic delayed systems,” International Journal of Robust and Nonlinear Control, 2015.

F. Liao, J. Wu, and M. Tomizuka, “An improved delay-dependent stability criterion for linear uncertain systems with multiple time-varying delays,” International Journal of Control, vol. 87, no. 4, pp. 861–873, 2014.

Y. He, M. Wu, and J.-H. She, “Delay-dependent stability criteria for linear systems with multiple time delays,” Proceedings of the IEE Control Theory & Applications, vol. 153, no. 4, pp. 447–452, 2006.

J. H. Park and S. Won, “Asymptotic stability of neutral systems with multiple delays,” Journal of Optimization Theory and Applications, vol. 103, no. 1, pp. 183–200, 1999.

K. Ramakrishnan and G. Ray, “Delay-dependent stability criteria for linear systems with multiple time-varying delays,” in Proceedings of the IEEE Region 10 Conference (TENCON '09), pp. 1–6, Singapore, November 2009.

X. Zhang, “Analysis of robust stability for switched systems with multiple time-delays,” Journal of Control Theory and Applications, vol. 9, no. 2, pp. 283–288, 2011.

W. Li, Y. Dong, and X. Wang, “Robust control of uncertain nonlinear systems with state and input time-varying delays,” in Proceedings of the Chinese Control and Decision Conference (CCDC '10), pp. 317–321, May 2010.

G. S. Zhai, B. Hu, K. Yasuda, and A. N. Michel, “Disturbance attenuation properties of time-controlled switched systems,” Journal of the Franklin Institute. Engineering and Applied Mathematics, vol. 338, no. 7, pp. 765–779, 2001.

B. G. Xu and Y. Q. Liu, “An improved Razumikhin-type theorem and its applications,” IEEE Transactions on Automatic Control, vol. 39, no. 4, pp. 839–841, 1994.